laeken/0000755000176200001440000000000014127345133011514 5ustar liggesuserslaeken/NAMESPACE0000644000176200001440000000312614127307134012734 0ustar liggesusers# Generated by roxygen2: do not edit by hand S3method(bootVar,indicator) S3method(calibVars,data.frame) S3method(calibVars,default) S3method(calibVars,matrix) S3method(plot,paretoTail) S3method(print,arpr) S3method(print,indicator) S3method(print,minAMSE) S3method(print,paretoScale) S3method(print,paretoTail) S3method(print,rmpg) S3method(replaceTail,paretoTail) S3method(reweightOut,paretoTail) S3method(shrinkOut,paretoTail) S3method(subset,arpr) S3method(subset,indicator) S3method(subset,rmpg) export(arpr) export(arpt) export(bootVar) export(calibVars) export(calibWeights) export(eqInc) export(eqSS) export(fitPareto) export(gini) export(gpg) export(incMean) export(incMedian) export(incQuintile) export(is.arpr) export(is.gini) export(is.gpg) export(is.indicator) export(is.prop) export(is.qsr) export(is.rmpg) export(meanExcessPlot) export(minAMSE) export(paretoQPlot) export(paretoScale) export(paretoTail) export(prop) export(qsr) export(replaceOut) export(replaceTail) export(reweightOut) export(rmpg) export(shrinkOut) export(thetaHill) export(thetaISE) export(thetaLS) export(thetaMoment) export(thetaPDC) export(thetaQQ) export(thetaTM) export(thetaWML) export(variance) export(weightedMean) export(weightedMedian) export(weightedQuantile) importFrom(MASS,ginv) importFrom(boot,boot) importFrom(boot,boot.ci) importFrom(graphics,abline) importFrom(graphics,identify) importFrom(graphics,par) importFrom(graphics,plot) importFrom(stats,aggregate) importFrom(stats,optimize) importFrom(stats,qexp) importFrom(stats,quantile) importFrom(stats,runif) importFrom(stats,uniroot) importFrom(stats,weighted.mean) laeken/data/0000755000176200001440000000000014127307305012424 5ustar liggesuserslaeken/data/datalist0000644000176200001440000000001314127307305014146 0ustar liggesuserseusilc ses laeken/data/ses.RData0000644000176200001440000323057414127307310014145 0ustar liggesusers7zXZi"6!X9])TW"nRʟ#M ^fvH'jFV/Lw*hNR9!\򽊪қ H c؏'Gtg^ub|Jz߀'lq%mBwXB0 6Y]k+(CppzmLC#VC[]!+⿩ldr‹.5\~~$Bxَ8ChC@ҕ!KLpM\Iuq.7>6Fb$'^:-~˺:/<\fֆbd;@^ cO+c$;pGՂ4Pn}Jz:@9Q s b=j{1VM E]s1Py(VItZO'Yצfp nGe307J6쫣(!tv1U|=#-1pꎗ;Iߡ#2(oUe˪J) }s , H[o1  zg~])805v^3WE%À}Qg N%;U '6O[2SL XC P=5 })cd%FUiz+H=8:`DHװ >_bimuX;ʉn{jaU'ƶJ%طn$k\ [$Y'H4٥e|Va5S̻%5y-^LoMzWK B[](Cm-E 'yPɯRK4j' @,(?-!A~0tdyۼؐhThCDbߋgܗo)R͂ wд-@jGPš*>*KQ"%<"CtSG'F}`Ugp죅IYF#YmA:ƥ35+7,!$uf+֛1LWٔ8j2qilSgk,ْ@}#5S53;IpzC:ÏP}br=˻$tP철Ho.?ַjNI2u̶?lJW#s=nD S![J\CF~zp.,g?8%qS :_5^b5EJK7]a3.,(< ȉhn5c3< I_^^"ȚZV{h3l`&}Ԭȋ~TRI${h׉:5#I2!% 436nx狯}C[9rtɶvmeXf@# *7evgr!fJAơʰldHhK4WXq*ՒhYn녘Y3JL*ƜGJ E!7d2ؕ"o{J}|8ZM~} I鍊^vWFO,Ghe Ngjq{ }JO <Y2`s7u8hn)` RcFH|lle.ӃUZ!erG#BAFo^i^ijaI!$م9}փT܎xD2)s-ٵtZ/erPƊ1{UR>paP㤨~0Ij'^ʓcu-_C'CcAOSպ``9FMB),fG|3dm~k/I*TCK>M+Om?wʷ䇒(Ծ#"꣐htd2l5y𩸧qI n-qd n W54ō`\*SջLۣ(=q,T¾{ڪVJfO9mi]6c,7IIax:0 A_iMkޒ}9)n=!`iŇNc޼78F=D& yfxXbl@-͙&j)hFGBh%^ɶ 8rmxk( BlK~U>7vJ<XDY>[j0?@zkG csp>Gl}f0aL('c(g_Nr&U{Kn~ҮCizN8Ȅqg2yX~BHQԾ=bItFҎțkã"E~~ˀ*uGZ{7G{^R~33c\Xsm$)i>}JCbV Lf(:r.C-PX|e GF2`N_SNʴwƫc}c4oUG휇&(yI$C\N6ޔ?uTpin7H%2)*)溹OEsVZN7t}cd SX5&\PlA.o^F ECѾU\;r|Yz$!|Gp4wy2GJIKG K.t**- K\cbE%yjcR\}9!?CUӟx*2>#ij =/oUu`ڏ r^P]/GJRtb#V;mD3*4uZf`)Ox%vZOA>>O/ЊtU{T99+ޭj~ƃam+z(.5exȩ)4ӏ",$_NjP3t[ͽ"C8oQyh?|g sXŶ7c,t` ܃sF'[-a5s?̗PNa2ak1&Jѩ!( *epcb455HO Ê:ytFaZjfO%x#͑E3∢WbWX$Z'P }aįxd &̭tcG@K@Q'5)9opY*v8WQU|rG[o)\в!9!=$AL:}%C^ Ld$Ѕlvf (xu 9Fxaw=Zovҗ9!$*&ͪ>GIB0qI 崉׀lP,jƖiucIJ?KVL V@Ph x9wfLZw1~Ƃen,~9"2Vp㢵ib@l07E|&j&oz跇:1ww0#7X?ߏYLى,hJ椩Kܒhh6;_f$+u4'SI5{:jRpjiQ a1/lM˽ysRD`{8Wњ#{kD߃<pK͍UOd (zHc]t it$@t@zIvĝ!mpE;|a鸋%30{1Γ5}klsUl{z 5}XЗQ#.1w1[ fb*V0u M2XG]絥9jzϙj1#Wldg!a#{@ !\Xܣ@,ŗ29:( u^ [-pX'# `Y;H9Y_%]!L,HOEn;BJwܝua*_ }h3c噁}P~I{TDNFR&sx?F,!9#JkzYޣG@{0T˴AEnmg0' mѩ08pe.Ǯsہv6Ƶ9*;CC2>)akD÷2u ~J  SoBq&-*lD<]OSA3i9zWLKO-vfE% w^~Rဇ~p£ۿOd$\Ivw9SѠxNEhLi]%f3O?PPvR=Fr NayM zpT{5l^)02,s64.dQVJ5 ,Vê'EB26xU(j1*L1{-Q+d8q$oBl,Ӛk ed֙JDa〗A/1ݠ=!ިW(9%y^W7>QaxB:]WAPMP? 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License: GPL (>= 2) Authors@R: c(person("Andreas", "Alfons", email = "alfons@ese.eur.nl", role = c("aut", "cre")), person("Josef", "Holzer", role = "aut"), person("Matthias", "Templ", role = "aut"), person("Alexander", "Haider", role = "ctb")) Author: Andreas Alfons [aut, cre], Josef Holzer [aut], Matthias Templ [aut], Alexander Haider [ctb] Maintainer: Andreas Alfons Encoding: UTF-8 RoxygenNote: 7.1.2 NeedsCompilation: no Packaged: 2021-10-06 11:55:49 UTC; andreas Repository: CRAN Date/Publication: 2021-10-06 16:10:02 UTC laeken/build/0000755000176200001440000000000014127307300012605 5ustar liggesuserslaeken/build/vignette.rds0000644000176200001440000000073714127307300015153 0ustar liggesusersN0C`}ЄT7ؑn>$vbK~v(jD5С]ٳ\.3?Rk u8#@ŘKU:Z1=ec )Kn18m,+xiVl2G6bC0CV}H?.#1+M@Hv9Jfl s/(pP սTc \ssN}$BVhԇ{x)쏽pCiLT_kfհbM1wv*rWbגP)L΅7YQ.,fs$zѨ)[4C_C:\Xg2rzz_EA{˕ܭFIm Z`k $vz?d_5x`ZWAFv~Ѱn+%PQv98O ٟg*d\laeken/build/partial.rdb0000644000176200001440000033410014127307235014742 0ustar liggesusers^ɖ&`6H1`7յ)YJ9S¦g{s3gzc+_z\_J!I9"R)"bMZ#ZrēNka+]<~šQ0ldd.ϻFale-VJȚ%bK/d5H GWߗMWb›_vm\Mbo3+TBBwφսn2V,l@˭}plb/h\ J %}2/S3y rK̷o֖͔~,i nut6w\#wt;Yvŭ7щDZ[#B]z],/oLW7f>;3MdGr3t^7>^HvN@˧lJϓ _J?T :_;TkG-~ nCs;wR4ndo~ZQ(%*0Vaeo ކ|[pH8'ʝ(F_/> aDUer P~ U{ 3y{.4T߷;!KeL֣zԝ]rB C<1|i6Vnhvm:[d+BxmT2KB ^|5yz!I})Յ7OC ģ-Gi]ڴjCyTLs̐0v0hec ss98c 9'2XvIUBUfu]C3k *:؆m֧45NA>Lᘛr_ɗaK?tiV]J#)qF 0vϤ$ސ/n^VL(eΖۗ1DaZ?z6(΋cpT*%L4ZrB[`[vͰ6HY ۳Hv-;ti~Z':l@M|LGX'Yr[Xۆmf؎)Y64w¦[)X6wܮFeMh9Fޠ6"k\$sAS C$(LJ0h*ˈ1 H];dP7/a@N.Լ"&v5̖n!ot?r˓.Z-+ȯNM3烚)3Nv㦢UY|y,'NAjnM@T怏!?nǑi*:-D+ |2>FcRёhB~Z[M'l\z10ƹtwq(=\!]Rύ'$Mr DGGhMY? 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Y|Q{rF'I%q%M9J @yi _ alGjvA}P:YD8Yݣ=D.pYiNt.D! nR7 |YB,o`#62QIGR .IO4߬!c+qY" )q Est6.@9{]Xj +GiS-_\ _,3:BSYY / nf]&l܃D(Nd!K/mK!'upPl_DO@>!a ,|"ug II>*xcd! 9ͫH(p|lp 1X.B^ ^)~#JI ui` +<`^VQk 3|sfU)Gz ;<%p置JGo>`7n:Sq,'[Srj1lAM)gU04%d c>)=mH3-Ij}+|J拺S o[Ɵ[~WiY*/*4}OVZjQ-xsuM=K>ϰ,!3$fEUTeyBiM֞<G m[=^Jz*+~G}bK=]1 cܜ4r_lN75J+UƛCx%{37s?"Ϣ7A+SɣjU3vZnfE!dVdߴ2}R͸p?҄ B{azko3^-ڸlp $ Ae"Lnh9[Žݟ'O/CꀉY}*sޭ>Exsw3.Snx'; dʷDm=_}-dwU9VBEqxpGRSӷ䜴t0_*GhNBmU$Sb0lUaӼ-މ-:4f ب1 Eh*i6{,4vތݨY(Y= d+$_OA ĩ> _hM,F= T!?:9}S\f g9nfdNì;vqܩqaş?-t֋E4>= options(prompt="R> ") @ \paragraph{Abstract} This vignette illustrates the application of variance estimation procedures to indicators on social exclusion and poverty using the \proglang{R} package \pkg{laeken}. To be more precise, it describes a general framework for estimating variance and confidence intervals of indicators under complex sampling designs. Currently, the package is focused on bootstrap approaches. While the naive bootstrap does not modify the weights of the bootstrap samples, a calibrated version allows to calibrate each bootstrap sample on auxiliary information before deriving the bootstrap replicate estimate. % ------------ % introduction % ------------ \section{Introduction} When point estimates of indicators are computed from samples, it is important to also obtain variance estimates and confidence intervals in order to account for variability due to sampling. Other sources of variability such as data editing or imputation may need to be considered as well, but this is not further discussed in this paper. While this vignette targets the topic of variance and confidence interval estimation for the indicators on social exclusion and poverty according to \citet{EU-SILC04, EU-SILC09}, the aim is not to describe and evaluate the different approaches that have been proposed to date. Instead, the aim is to present the functionality for the statistical environment \proglang{R} \citep{RDev} implemented in the add-on package \pkg{laeken} \citep{laeken}. It should be noted that the basic design of the package, as well as standard point estimation of the indicators on social exclusion and poverty, is discussed in detail in vignette \code{laeken-standard} \citep{templ11a}. In addition, vignette \code{laeken-pareto} \citep{alfons11a} presents more sophisticated methods for point estimation of the indicators, which are less influenced by outliers. Those documents can be viewed from within \proglang{R} with the following commands: <>= vignette("laeken-standard") vignette("laeken-pareto") @ Morover, a general introduction to package \pkg{laeken} is published as \citet{alfons13b}. The data basis for the estimation of the indicators on social exclusion and poverty is the \emph{European Union Statistics on Income and Living Conditions} (EU-SILC), which is an annual panel survey conducted in EU member states and other European countries. Package \pkg{laeken} provides the synthetic example data \code{eusilc} consisting of $14\,827$ observations from $6\,000$ households. Furthermore, the data were generated from Austrian EU-SILC survey data from 2006 using the data simulation methodology proposed by \citet{alfons11c} and implemented in the \proglang{R} package \pkg{simPopulation} \citep{simPopulation}. The data set \code{eusilc} is used in the code examples throughout the paper. % ----- <<>>= library("laeken") data("eusilc") @ The rest of the paper is organized as follows. Section~\ref{sec:variance} presents the general wrapper function for estimating variance and confidence intervals of indicators in package \pkg{laeken}. The naive and calibrated bootstrap approaches are discussed in Sections~\ref{sec:naive} and~\ref{sec:calib}, respectively. Section~\ref{sec:concl} concludes. % --------------- % general wrapper % --------------- \section{General wrapper function for variance estimation} \label{sec:variance} The function \code{variance()} provides a flexible framework for estimating the variance and confidence intervals of indicators such as the \emph{at-risk-of-poverty rate}, the \emph{Gini coefficient}, the \emph{quintile share ratio} and the \emph{relative median at-risk-of-poverty gap}. For a mathematical description and details on the implementation of these indicators in the \proglang{R} package \pkg{laeken}, the reader is referred to vignette \code{laeken-standard} \citep{templ11a}. In any case, \code{variance()} acts as a general wrapper function for computing variance and confidence interval estimates of indicators on social exclusion and poverty with package \pkg{laeken}. The arguments of function \code{variance()} are shown in the following: <<>>= args(variance) @ All these arguments are fully described in the \proglang{R} help page of function \code{variance()}. The most important arguments are: \begin{description} \item[inc:] the income vector. \item[weights:] an optional vector of sample weights. \item[breakdown:] an optional vector giving different domains in which variances and confidence intervals should be computed. \item[design:] an optional vector or factor giving different strata for stratified sampling designs. \item[data:] an optional \code{data.frame}. If supplied, each of the above arguments should be specified as a character string or an integer or logical vector specifying the corresponding column. \item[indicator:] an object inheriting from the class \code{"indicator"} that contains the point estimates of the indicator, such as \code{"arpr"} for the at-risk-of-poverty rate, \code{"qsr"} for the quintile share ratio, \code{"rmpg"} for the relative median at-risk-of-poverty gap, or \code{"gini"} for the Gini coefficient. \item[type:] a character string specifying the type of variance estimation to be used. Currently, only \code{"bootstrap"} is implemented for variance estimation based on bootstrap resampling. \end{description} In the following sections, two bootstrap methods for estimating the variance and confidence intervals of point estimates for complex survey data are described. Furthermore, their application using the function \code{variance()} from package \pkg{laeken} is demonstrated. % --------------- % naive bootstrap % --------------- \section{Naive bootstrap} \label{sec:naive} Let $\boldsymbol{X} := (\boldsymbol{x}_{1}, \ldots, \boldsymbol{x}_{n})'$ denote a survey sample with $n$ observations and $p$ variables. Then the \emph{naive bootstrap algorithm} for estimating the variance and confidence interval of an indicator can be summarized as follows: \begin{enumerate} \item Draw $R$ independent bootstrap samples $\boldsymbol{X}_{1}^{*}, \ldots, \boldsymbol{X}_{R}^{*}$ from $\boldsymbol{X}$. \item Compute the bootstrap replicate estimates $\hat{\theta}_{r}^{*} := \hat{\theta}(\boldsymbol{X}_{r}^{*})$ for each bootstrap sample $\boldsymbol{X}_{r}^{*}$, $r = 1, \ldots, R$, where $\hat{\theta}$ denotes an estimator for a certain indicator of interest. Of course the sample weights always need to be considered for the computation of the bootstrap replicate estimates. \item Estimate the variance $V(\hat{\theta})$ by the variance of the $R$ bootstrap replicate estimates: \begin{equation} \hat{V}(\hat{\theta}) := \frac{1}{R-1} \sum_{r=1}^{R} \left( \hat{\theta}_{r}^{*} - \frac{1}{R} \sum_{s=1}^{R} \hat{\theta}_{s}^{*} \right)^{2}. \end{equation} \item Estimate the confidence interval at confidence level $1 - \alpha$ by one of the following methods \citep[for details, see][]{davison97}: \begin{description} \item[Percentile method:] $\left[ \hat{\theta}_{((R+1) \frac{\alpha}{2})}^{*}, \hat{\theta}_{((R+1)(1-\frac{\alpha}{2}))}^{*} \right]$, as suggested by \cite{efron93}. \item[Normal approximation:] $\hat{\theta} \pm z_{1-\frac{\alpha}{2}} \cdot \hat{V}(\hat{\theta})^{1/2}$ with $z_{1-\frac{\alpha}{2}} = \Phi^{-1}(1 - \frac{\alpha}{2})$. \item[Basic bootstrap method:] $\left[ 2\hat{\theta} - \hat{\theta}_{((R+1)(1-\frac{\alpha}{2}))}^{*}, 2\hat{\theta} - \hat{\theta}_{((R+1)\frac{\alpha}{2})}^{*} \right]$. \end{description} For the percentile and the basic bootstrap method, $\hat{\theta}_{(1)}^{*} \leq \ldots \leq \hat{\theta}_{(R)}^{*}$ denote the order statistics of the bootstrap replicate estimates. \end{enumerate} In the following example, the variance and confidence interval of the at-risk-of-poverty rate are estimated with the naive bootstrap procedure. The output of function \code{variance()} is an object of the same class as the point estimate supplied as the \code{indicator} argument, but with additional components for the variance and confidence interval. In addition to the point estimate, the income and the sample weights need to be supplied. Furthermore, a stratified sampling design can be considered by specifying the \code{design} argument, in which case observations are resampled separately within the strata. To ensure reproducibility of the results, the seed of the random number generator is set. <<>>= a <- arpr("eqIncome", weights = "rb050", data = eusilc) variance("eqIncome", weights = "rb050", design = "db040", data = eusilc, indicator = a, bootType = "naive", seed = 123) @ One of the most convenient features of package \pkg{laeken} is that indicators can be evaluated for different subdomains using a single command. This also holds for variance estimation. Using the \code{breakdown} argument, the example below produces variance and confidence interval estimates for each NUTS2 region in addition to the overall values. <<>>= b <- arpr("eqIncome", weights = "rb050", breakdown = "db040", data = eusilc) variance("eqIncome", weights = "rb050", breakdown = "db040", design = "db040", data = eusilc, indicator = b, bootType = "naive", seed = 123) @ It should be noted that the workhorse function \code{bootVar()} is called internally by \code{variance()} for bootstrap variance and confidence interval estimation. The function \code{bootVar()} could also be called directly by the user in exactly the same manner. Moreover, variance and confidence interval estimation for any other indicator implemented in package \pkg{laeken} is straightforward---the application using function \code{variance()} or \code{bootVar()} remains the same. % -------------------- % calibrated bootstrap % -------------------- \section{Calibrated bootstrap} \label{sec:calib} \cite{rao88} showed that the naive bootstrap is biased when used in the complex survey context. They propose to increase the variance estimate in the $h$-th stratum by a factor of $\frac{n_{h} - 1}{n_{h}}$ (if the bootstrap sample is of the same size). In addition, they describe extensions to sampling without replacement, unequal probability sampling, and two-stage cluster sampling with equal probabilities and without replacement. \cite{deville92} and \cite{deville93} provide a general description on how to calibrate sample weights to account for known population totals. The naive bootstrap does not include the recalibration of bootstrap samples in order to fit known population totals and therefore is, strictly formulated, not suitable for many practical applications. However, even though a bias might be introduced, the naive bootstrap works well in many situations and is faster to compute than the calibrated version. Hence it is a popular method often used in practice. In real-world data, the inclusion probabilities for observations in the population are in general not all equal, resulting in different \emph{design weights} for the observations in the sample. Furthermore, the initial design weights are in practice often adjusted by calibration, e.g., to account for non-response or so that certain known population totals can be precisely estimated from the survey sample. To give a simplified example, if the population sizes in different regions are known, the sample weights may be calibrated so that the Horvitz-Thompson estimates \citep{horvitz52} of the population sizes equal the known true values. However, when bootstrap samples are drawn from survey data, resampling observations has the effect that such known population totals can no longer be precisely estimated. As a remedy, the sample weights of each bootstrap sample should be calibrated. The calibrated version of the bootstrap thus results in more precise variance and confidence interval estimation, but comes with higher computational costs than the naive approach. In any case, the \emph{calibrated bootstrap algorithm} is obtained by adding the following step between Steps~1 and~2 of the naive bootstrap algorithm from Section~\ref{sec:naive}: \begin{itemize} \item[1b.] Calibrate the sample weights for each bootstrap sample $\boldsymbol{X}_{r}^{*}$, $r = 1, \ldots, R$. Generalized raking procedures are thereby used for calibration: either a multiplicative method known as \emph{raking}, an additive method or a logit method \citep[see][]{deville92, deville93}. \end{itemize} The function call to \code{variance()} for the calibrated bootstrap is very similar to its counterpart for the naive bootstrap. A matrix of auxiliary calibration variables needs to be supplied via the argument \code{X}. In addition, the argument \code{totals} can be used to supply the corresponding population totals. If the \code{totals} argument is omitted, as in the following example, the population totals are computed from the sample weights of the original sample. This follows the assumption that those weights are already calibrated on the supplied auxiliary variables. % ----- <<>>= variance("eqIncome", weights = "rb050", design = "db040", data = eusilc, indicator = a, X = calibVars(eusilc$db040), seed = 123) @ % ----- Note that the function \code{calibVars()} transforms a factor into a matrix of binary variables, as required by the calibration function \code{calibWeights()}, which is called internally. While the default is to use raking for calibration, other methods can be specified via the \code{method} argument. % ----------- % conclusions % ----------- \section{Conclusions} \label{sec:concl} Both bootstrap procedures for variance and confidence interval estimation of indicators on social exclusion and poverty currently implemented in the \proglang{R} package \pkg{laeken} have their strengths. While the naive bootstrap is faster to compute, the calibrated bootstrap in general leads to more precise results. The implementation of other procedures such as linearization techniques \citep{kovacevic97, deville99, hulliger06, osier09} or the delete-a-group jackknife \citep{kott01} is future work. Furthermore, \citet{alfons09} demonstrated how the variance of indicators computed from data with imputed values may be underestimated in bootstrap procedures, depending on the indicator itself and the imputation procedure used. They proposed to use the method described in \cite{little02}, which consists of drawing bootstrap samples from the original data with missing values, and to impute the missing data for each bootstrap sample before computing the corresponding bootstrap replicate estimate. Of course, this results in an additional increase of the computation time. The implementation of this procedure in package \pkg{laeken} is future work. It should also be noted that multiple imputation is a further possibility to consider the additional uncertainty from imputation when estimating the variance of an indicator \citep[see][]{little02}. % --------------- % acknowledgments % --------------- \section*{Acknowledgments} This work was partly funded by the European Union (represented by the European Commission) within the 7$^{\mathrm{th}}$ framework programme for research (Theme~8, Socio-Economic Sciences and Humanities, Project AMELI (Advanced Methodology for European Laeken Indicators), Grant Agreement No. 217322). Visit \url{http://ameli.surveystatistics.net} for more information on the project. % ------------ % bibliography % ------------ \bibliographystyle{plainnat} \bibliography{laeken} \end{document} laeken/vignettes/laeken-pareto.Rnw0000644000176200001440000011610214127275361016751 0ustar liggesusers\documentclass[a4paper,10pt]{scrartcl} \usepackage[OT1]{fontenc} \usepackage{Sweave} %% additional packages \usepackage{natbib} \bibpunct{(}{)}{,}{a}{}{,} \usepackage{amsmath, amssymb} \usepackage{hyperref} \hypersetup{colorlinks, citecolor=blue, linkcolor=blue, urlcolor=blue} \usepackage[top=30mm, bottom=30mm, left=30mm, right=30mm]{geometry} %% additional commands \newcommand{\code}[1]{\texttt{#1}} \newcommand{\pkg}[1]{\mbox{\textbf{#1}}} \newcommand{\proglang}[1]{\mbox{\textsf{#1}}} %%\VignetteIndexEntry{Robust Pareto Tail Modeling for the Estimation of Indicators on Social Exclusion using the R Package laeken} %%\VignetteDepends{laeken} %%\VignetteKeywords{social exclusion, indicators, robust estimation, Pareto distribution} %%\VignettePackage{laeken} \begin{document} \title{Robust Pareto Tail Modeling for the Estimation of Indicators on Social Exclusion using the \proglang{R} Package \pkg{laeken}} %\author{ % Andreas Alfons\footnote{Vienna University of Technology, % \href{mailto:alfons@statistik.tuwien.ac.at}{alfons@statistik.tuwien.ac.at}}, % Matthias Templ\footnote{Vienna University of Technology \& Statistics Austria, % \href{mailto:templ@tuwien.ac.at}{templ@tuwien.ac.at}}, % Peter Filzmoser\footnote{Vienna University of Technology, % \href{mailto:p.filzmoser@tuwien.ac.at}{p.filzmoser@tuwien.ac.at}}, % Josef Holzer\footnote{Landesstatistik Steiermark, % \href{mailto:josef.holzer@stmk.gv.at}{josef.holzer@stmk.gv.at}} %} \author{ Andreas Alfons$^{1}$, Matthias Templ$^{2}$, Peter Filzmoser$^{3}$, Josef Holzer$^{4}$ } \date{} \maketitle \setlength{\footnotesep}{11pt} \footnotetext[1]{ \begin{tabular}[t]{l} Erasmus School of Economics, Erasmus University Rotterdam\\ E-mail: \href{mailto:alfons@ese.eur.nl}{alfons@ese.eur.nl} \end{tabular} } \footnotetext[2]{ \begin{tabular}[t]{l} Zurich University of Applied Sciences\\ E-mail: \href{mailto:matthias.templ@zhaw.ch}{matthias.templ@zhaw.ch} \end{tabular} } \footnotetext[3]{ \begin{tabular}[t]{l} Vienna University of Technology\\ E-mail: \href{mailto:p.filzmoser@tuwien.ac.at}{p.filzmoser@tuwien.ac.at} \end{tabular} } \footnotetext[4]{ \begin{tabular}[t]{l} Landesstatistik Steiermark\\ E-mail: \href{mailto:josef.holzer@stmk.gv.at}{josef.holzer@stmk.gv.at} \end{tabular} } % change R prompt <>= options(prompt="R> ") @ %% specify folder and name for Sweave graphics %\SweaveOpts{prefix.string=figures-pareto/fig} \paragraph{Abstract} In this vignette, robust semiparametric estimation of social exclusion indicators using the \proglang{R} package \pkg{laeken} is discussed. Special emphasis is thereby given to income inequality indicators, as the standard estimates for these indicators are highly influenced by outliers in the upper tail of the income distribution. This influence can be reduced by modeling the upper tail with a Pareto distribution in a robust manner. While the focus of the paper is to demonstrate the functionality of \pkg{laeken} beyond the standard estimation techniques, a brief mathematical description of the implemented procedures is given as well. % ------------ % introduction % ------------ \section{Introduction} From a robustness point of view, the standard estimators for some of the social exclusion indicators defined by \citet{EU-SILC04, EU-SILC09} are problematic. In particular the income inequality indicators \emph{quintile share ratio} (QSR) and \emph{Gini coefficient} suffer from a lack of robustness. Consider, e.g., the QSR, which is estimated as the ratio of estimated totals or means (see Section~\ref{sec:QSR} for an exact definition). It is well known that the classical estimates for totals or means have a breakdown point of 0, meaning that even a single outlier can distort the results to an arbitrary extent. In fact, the influence of a single observation in the upper tail of the income distribution on the estimation of the QSR is linear and therefore unbounded. For practical purposes, the standard QSR estimator thus cannot be recommended in many situations \citep[cf.][]{hulliger09a}. It is also important to note that the behavior of the Gini coefficient is similar to the behavior of the QSR. The data basis for the estimation of the social exclusion indicators according to \citet{EU-SILC04, EU-SILC09} is the \emph{European Union Statistics on Income and Living Conditions} (EU-SILC), which is an annual panel survey conducted in EU member states and other European countries. On the one hand, EU-SILC data typically contain a considerable amount of \emph{representative} outliers in the upper tail of the income distribution, i.e., correct observations that behave differently from the main part of the data, but that are not unique in the population and hence need to be considered for computing estimates of the indicators. On the other hand, EU-SILC data frequently contain some even more extreme \emph{nonrepresentative} outliers, i.e., observations that are either incorrect or can be considered unique in the population. Consequently, such nonrepresentative outliers need to be excluded from the estimation process or downweighted. As a remedy, the upper tail of the income distribution may be modeled with a \emph{Pareto distribution} in order to recalibrate the sample weights or use fitted income values for observations in the upper tail when estimating the indicators (see Section~\ref{sec:fit}). %This is highly applicable because the upper tail of the income distribution in %EU-SILC data virtually always contains a considerable amount of representative %outliers. Nevertheless, classical estimators for the parameters of the Pareto distribution are highly influenced by the nonrepresentative outliers themselves. Using robust methods reduces the influence on fitting the Pareto distribution to the representative outliers and therefore on the estimation of the indicators. Rather than evaluating these methods, the paper concentrates on showing how they can be applied in the statistical environment \proglang{R} \citep{RDev} with the add-on package \pkg{laeken} \citep{laeken}. The basic design of the package, as well as standard estimation of the social exclusion indicators is discussed in detail in vignette \code{laeken-standard} \citep{templ11a}. Furthermore, the general framework for variance estimation is illustrated in vignette \code{laeken-variance} \citep{templ11b}. Those documents can be viewed from within \proglang{R} with the following commands: <>= vignette("laeken-standard") vignette("laeken-variance") @ Morover, a general introduction to package \pkg{laeken} is published as \citet{alfons13b}. Throughout the paper, the example data from package \pkg{laeken} is used. The data set is called \code{eusilc} and consists of $14\,827$ observations from $6\,000$ households. In addition, it was synthetically generated from Austrian EU-SILC survey data from 2006 using the data simulation methodology proposed by \citet{alfons11c} and implemented in the \proglang{R} package \pkg{simPopulation} \citep{simPopulation}. More information on the example data can be found in vignette \code{laeken-standard} or in the corresponding \proglang{R} help page. <<>>= library("laeken") data("eusilc") @ The rest of the paper is organized as follows. Section~\ref{sec:laeken} gives a mathematical description of the Eurostat definitions of the social exclusion indicators QSR and Gini coefficient. In Section~\ref{sec:Pareto}, the Pareto distribution is briefly discussed. Section~\ref{sec:threshold} discusses a rule of thumb for estimating the threshold for the upper tail of the distribution, and illustrates graphical methods for exploring the data in order to find the threshold. Classical and robust estimators for the shape parameter of the Pareto distribution are described in Section~\ref{sec:shape}. How to use Pareto tail modeling to estimate the social exclusion indicators is then shown in Section~\ref{sec:fit}. Finally, Section~\ref{sec:concl} concludes. % ------------------- % selected indicators % ------------------- \section{Social exclusion indicators} \label{sec:laeken} This paper is focused on the inequality indicators \emph{quintile share ratio} (QSR) and \emph{Gini coefficient}, which are both highly influenced by outliers in the upper tail of the distribution. Note that for the estimation of the social exclusion indicators, each person in a household is assigned the same \emph{eqivalized disposable income}. See vignette \code{laeken-standard} \citep{templ11a} for the computation of the equivalized disposable income with the \proglang{R} package \pkg{laeken}. For the following definitions, let $\boldsymbol{x} := (x_{1}, \ldots, x_{n})'$ be the equivalized disposable income with $x_{1} \leq \ldots \leq x_{n}$ and let $\boldsymbol{w} := (w_{i}, \ldots, w_{n})'$ be the corresponding personal sample weights, where $n$ denotes the number of observations. \subsection{Quintile share ratio (QSR)} \label{sec:QSR} The income \emph{quintile share ratio} (QSR) is defined as the ratio of the sum of the equivalized disposable income received by the 20\% of the population with the highest equivalized disposable income to that received by the 20\% of the population with the lowest equivalized disposable income \citep{EU-SILC04, EU-SILC09}. For the estimation of the quintile share ratio from a sample, let $\hat{q}_{0.2}$ and $\hat{q}_{0.8}$ denote the weighted 20\% and 80\% quantiles, respectively. With $0 \leq p \leq 1$, these weighted quantiles are given by \begin{equation} \label{eq:wq} \hat{q}_{p} = \hat{q}_{p} (\boldsymbol{x}, \boldsymbol{w}) := \begin{cases} \frac{1}{2} (x_{j} + x_{j+1}), & \quad \text{if } \sum_{i=1}^{j} w_{i} = p \sum_{i=1}^{n} w_{i}, \\ x_{j+1}, & \quad \text{if } \sum_{i=1}^{j} w_{i} < p \sum_{i=1}^{n} w_{i} < \sum_{i=1}^{j+1} w_{i}. \end{cases} \end{equation} %See also vignette \code{laeken-standard} \citep{templ11a} for the computation %of these quantiles with package \pkg{laeken}. Using index sets \mbox{$I_{\leq \hat{q}_{0.2}} := \{ i \in \{ 1, \ldots, n \} : x_{i} \leq \hat{q}_{0.2} \}$} and \mbox{$I_{> \hat{q}_{0.8}} := \{ i \in \{ 1, \ldots, n \} : x_{i} > \hat{q}_{0.8} \}$}, the quintile share ratio is estimated by \begin{equation} \widehat{QSR} := \frac{\sum_{i \in I_{> \hat{q}_{0.8}}} w_{i} x_{i}}{\sum_{i \in I_{\leq \hat{q}_{0.2}}} w_{i} x_{i}}. \end{equation} With package \pkg{laeken}, the quintile share ratio can be estimated using the function \code{qsr()}. Sample weights can thereby be supplied via the \code{weights} argument. <<>>= qsr("eqIncome", weights = "rb050", data = eusilc) @ \subsection{Gini coefficient} \label{sec:Gini} The \emph{Gini coefficient} is defined as the relationship of cumulative shares of the population arranged according to the level of equivalized disposable income, to the cumulative share of the equivalized total disposable income received by them \citep{EU-SILC04, EU-SILC09}. For the estimation of the Gini coefficient from a sample, the sample weights need to be taken into account. In mathematical terms, the Gini coefficient is estimated by \begin{equation} \widehat{Gini} := 100 \left[ \frac{2 \sum_{i=1}^{n} \left( w_{i} x_{i} \sum_{j=1}^{i} w_{j} \right) - \sum_{i=1}^{n} w_{i}^{\phantom{i}2} x_{i}}{\left( \sum_{i=1}^{n} w_{i} \right) \sum_{i=1}^{n} \left(w_{i} x_{i} \right)} - 1 \right]. \end{equation} The function \code{gini()} is available in \pkg{laeken} to estimate the Gini coefficient. As before, sample weights can be specified with the \code{weights} argument. <<>>= gini("eqIncome", weights = "rb050", data = eusilc) @ % ------------------- % Pareto distribution % ------------------- \section{The Pareto distribution} \label{sec:Pareto} The \emph{Pareto distribution} is well studied in the literature and is defined in terms of its cumulative distribution function \begin{equation} \label{eq:CDF} F_{\theta}(x) = 1 - \left( \frac{x}{x_{0}} \right) ^{-\theta}, \qquad x \geq x_{0}, \end{equation} where $x_{0} > 0$ is the scale parameter and $\theta > 0$ is the shape parameter \citep{kleiber03}. Furthermore, its density function is given by \begin{equation} f_{\theta}(x) = \frac{\theta x_{0}^{\theta}}{x^{\theta + 1}}, \qquad x \geq x_{0}. \end{equation} Figure~\ref{fig:Pareto} visualizes the Pareto probability density function with scale parameter $x_{0} = 1$ and different values of the shape parameter $\theta$. Clearly, the Pareto distribution is a highly right-skewed distribution with a heavy tail. It is therefore reasonable to assume that a random variable following a Pareto distribution contains extreme values. The effect of changing the shape parameter $\theta$ is visible in the probability mass at the scale parameter $x_{0}$: the higher $\theta$, the higher the probability mass at $x_{0}$. <>= x <- seq(1, 6, length.out=1000) dpareto <- function(x, x0 = 1, theta = 1) theta*x0^theta / x^(theta+1) y1 <- dpareto(x, theta=1) y2 <- dpareto(x, theta=2) y3 <- dpareto(x, theta=3) @ \begin{figure} \begin{center} <>= par(mar = c(4, 4, 0.5, 0.5) + 0.1) plot(x, y3, type = "l", lty = 3, ylab = "f(x)", xlim = c(0.75, 6), panel.first = { abline(h = 0, col = grey(0.75)) abline(v = 1, col = grey(0.75)) }) lines(x, y2, lty = 2) lines(x, y1, lty = 1) leg <- expression(paste(theta, " = 1"), paste(theta, " = 2"), paste(theta, " = 3")) legend("topright", legend = leg, lty = 1:3) @ \caption{Pareto probability density functions with parameters $x_{0} = 1$ and $\theta = 1, 2, 3$.} \label{fig:Pareto} \end{center} \end{figure} In Pareto tail modeling, the cumulative distribution function on the whole range of $x$ is modeled as \begin{equation} \label{eq:tail} F(x) = \left\{ \begin{array}{ll} G(x), & \quad \text{if } x \leq x_{0}, \\ G(x_{0}) + (1 - G(x_{0})) F_{\theta}(x), & \quad \text{if } x > x_{0}, \end{array} \right. \end{equation} where $G$ is an unknown distribution function \citep{dupuis06}. Let $n$ be the number of observations and let $\boldsymbol{x} = (x_{1}, \ldots, x_{n})'$ denote the observed values with $x_{1} \leq \ldots \leq x_{n}$. In addition, let $k$ be the number of observations to be used for tail modeling. In this scenario, the threshold $x_{0}$ is estimated by % Let $k$ be the number of observations to be used for tail modeling and let % $x_{(1)} \leq \ldots \leq x_{(n)}$, denote the sorted observations. In this % scenario, the threshold $x_{0}$ is estimated by \begin{equation} \hat{x}_{0} := x_{n-k}. \end{equation} If an estimate $\hat{x}_{0}$ for the scale parameter of the Pareto distribution has been obtained, $k$ is given by the number of observations larger than $\hat{x}_{0}$. Thus estimating $x_{0}$ and $k$ directly corresponds with each other. In the remainder of this package vignette, the equivalized disposable income of the EU-SILC example data is of main interest. Consequently, the Pareto distribution will be modeled at the household level rather than the individual level. Moreover, the focus of this vignette is on robust estimation of the social exclusion indicators. Hence the equivalized disposable income of the household with the largest income is replaced by a large outlier. <<>>= hID <- eusilc$db030[which.max(eusilc$eqIncome)] eusilc[eusilc$db030 == hID, "eqIncome"] <- 10000000 @ Since the aim is to model a Pareto distribution at the household level, the following command creates a data set that contains only the equivalized disposable income and the sample weights on the household level. This data set will be used in Sections~\ref{sec:threshold} and~\ref{sec:shape} to estimate the parameters of the Pareto distribution. <<>>= eusilcH <- eusilc[!duplicated(eusilc$db030), c("eqIncome", "db090")] @ % --------- % threshold % --------- \section{Finding the threshold} \label{sec:threshold} The aim of the methods presented in this sections is to find the threshold $x_{0}$ for modeling the Pareto distribution. Several methods for the estimation of the threshold $x_{0}$ or the number of observations $k$ in the tail have been proposed in the literature, but those proposals typically do not consider sample weights. \citet{beirlant96a, beirlant96b} developed a procedure that analytically determines the optimal choice of $k$ for the Hill estimator of the shape parameter \citep[see also Section~\ref{sec:Hill} of this paper]{hill75} by minimizing the asymptotic mean squared error (AMSE). In package \pkg{laeken}, this approach is implemented in the function \code{minAMSE()}. However, the procedure is designed for the non-robust Hill estimator and is therefore not further discussed in this paper. Furthermore, \citet{danielsson01} proposed a bootstrap method to find the optimal $k$ for the Hill estimator with respect to the AMSE, which has less analytical requirements than the approach by \citet{beirlant96a, beirlant96b}. Please note that this method is not robust either and that it is currently not available in package \pkg{laeken}. A robust prediction error criterion for choosing the number of observations $k$ in the tail and estimating the shape parameter $\theta$ was developed by \citet{dupuis06}. Nevertheless, our implementation of this robust criterion was unstable and is therefore not included in \pkg{laeken}. In any case, \citet{holzer09} concludes that graphical methods for finding the threshold outperform those analytical approaches in the case of EU-SILC data. While this section is thus focused graphical methods, a simple rule of thumb designed specifically for the equivalized disposable income in EU-SILC data is described in the following as well. \subsection{Van Kerm's rule of thumb} \label{sec:vanKerm} \citet{vankerm07} presented a formula that is more of a rule of thumb for the threshold of the equivalized disposable income in EU-SILC data. Is is given by \begin{equation} \hat{x}_{0} := \min(\max(2.5\bar{x}, q_{0.98}), q_{0.97}), \end{equation} where $\bar{x}$ is the weighted mean, and $q_{0.98}$ and $q_{0.97}$ are weighted quantiles as defined in Equation~(\ref{eq:wq}). In package \pkg{laeken}, the function \code{paretoScale()} provides functionality for computing the threshold with van Kerm's rule of thumb. The argument \code{w} is available to supply sample weights. %In the example below, the household IDs are supplied via the argument %\code{groups} to estimate the threshold on the houshold level rather than the %personal level. %<<>>= %paretoScale(eusilc$eqIncome, eusilc$db090, groups = eusilc$db030) %@ <<>>= ts <- paretoScale(eusilcH$eqIncome, w = eusilcH$db090) ts @ It should be noted that the function returns an object of class \code{"paretoScale"}, which consists of a component \code{x0} for the threshold (scale parameter) and a component \code{k} for the number of observations in the tail of the distribution, i.e., that are larger than the threshold. \subsection{Pareto quantile plot} The \emph{Pareto quantile plot} is a graphical method for inspecting the parameters of a Pareto distribution. For the case without sample weights, it is described in detail in \citet{beirlant96a}. If the Pareto model holds, there exists a linear relationship between the lograrithms of the observed values and the quantiles of the standard exponential distribution, since the logarithm of a Pareto distributed random variable follows an exponential distribution. Hence the logarithms of the observed values, $\log (x_{i})$, $i = 1, \ldots, n$, are plotted against the theoretical quantiles. In the case without sample weights, the theoretical quantiles of the standard exponential distribution are given by \begin{equation} \label{eq:quantiles} -\log \left( 1 - \frac{i}{n+1} \right), \qquad i = 1, \ldots, n, \end{equation} i.e., by dividing the range into $n + 1$ equally sized subsets and using the resulting $n$ inner gridpoints as probabilities for the quantiles. If the data contain sample weights, the range of the exponential distribution needs to be divided according to the weights of the $n$ observations. The Pareto quantile plot is thus generalized by using the theoretical quantiles \begin{equation} -\log \left( 1 - \frac{\sum_{j=1}^{i} w_{j}}{\sum_{j=1}^{n} w_{j}} \frac{n}{n+1} \right), \qquad i = 1, \ldots, n, \end{equation} where the correction factor $\frac{n}{n+1}$ ensures that the quantiles reduce to (\ref{eq:quantiles}) if all sample weights are equal. If the tail of the data follows a Pareto distribution, those observations form almost a straight line. The leftmost point of a fitted line can thus be used as an estimate of the threshold $x_{0}$, the scale parameter. All values starting from the point after the threshold may be modeled by a Pareto distribution, but this point cannot be determined exactly. Furthermore, the slope of the fitted line is in turn an estimate of $\frac{1}{\theta}$, the reciprocal of the shape parameter. Figure~\ref{fig:ParetoQuantile} displays the Pareto quantile plot for the example data \code{eusilc} on the household level with the largest observation replaced by an outlier. The plot is generated using the function \code{paretoQPlot()}, which allows to supply sample weights via the argument \code{w}. In addition, the threshold can be selected interactively by clicking on a data point. Information on the selected threshold is then printed on the \proglang{R} console. When the interactive selection is terminated, which is typically done by a secondary mouse click, the selected threshold is returned as an object of class \code{"paretoScale"}. Another advantage of the Pareto quantile plot is also illustrated in Figure~\ref{fig:ParetoQuantile}. Nonrepresentative outliers such as the large income introduced into the example data in Section~\ref{sec:Pareto}, i.e., extreme observations in the upper tail that deviate from the Pareto model, are clearly visible. \begin{figure} \begin{center} \setkeys{Gin}{width=.75\textwidth} <>= paretoQPlot(eusilcH$eqIncome, w = eusilcH$db090) @ \caption{Pareto Quantile plot for the example data \code{eusilc} on the household level with the largest observation replaced by an outlier.} \label{fig:ParetoQuantile} \end{center} \end{figure} \subsection{Mean excess plot} The \emph{mean excess plot} is another graphical method for inspecting the threshold for Pareto tail modeling, but it does not provide information on the shape parameter. It is based on the excess function \begin{equation} \label{eq:excess} e(x_{0}) := \mathbb{E}(x - x_{0}|x > x_{0}), \qquad x_{0} \geq 0. \end{equation} A detailed description for the case without sample weights can be found in \citet{borkovec00}. For the following definition of the mean excess plot, keep in mind that the observations are sorted such that $x_{1} \leq \ldots \leq x_{n}$. For each observation $x_{i}$, $i = 1, \ldots, \lfloor n-\sqrt{n} \rfloor$, the empirical excess function $e_{n}$ is computed. In the case without sample weights, the expectation in Equation~(\ref{eq:excess}) is replaced by the arithmetic mean, and the empirical excess function is given by \begin{equation} e_{n}(x_{i}) := \frac{1}{n-i} \sum_{j=i+1}^{n} (x_{j} - x_{i}), \qquad i = 1, \ldots, \lfloor n-\sqrt{n} \rfloor. \end{equation} The values of the empirical excess function $e_{n}(x_{i})$ are then plotted against the corresponding $x_{i}$, $i = 1, \ldots, \lfloor n-\sqrt{n} \rfloor$. If sample weights are available in the data, the mean excess plot is simply generalized by using the weighted mean for the empirical excess function: \begin{equation} e_{n}(x_{i}) := \frac{1}{\sum_{j=i+1}^{n} w_{j}} \sum_{j=i+1}^{n} w_{j} (x_{j} - x_{i}), \qquad i = 1, \ldots, \lfloor n-\sqrt{n} \rfloor. \end{equation} If the tail of the data follows a Pareto distribution, those observations show a positive linear trend. The leftmost point of a fitted line can thus be used as an estimate of the threshold $x_{0}$, the scale parameter. As for the Pareto quantile plot, a disadvantage of the mean excess plot is that the threshold cannot be determined exactly. \begin{figure} \begin{center} \setkeys{Gin}{width=.75\textwidth} <>= meanExcessPlot(eusilcH$eqIncome, w = eusilcH$db090) @ \caption{Mean excess plot for the example data \code{eusilc} on the household level with the largest observation replaced by an outlier.} \label{fig:meanExcess} \end{center} \end{figure} Figure~\ref{fig:meanExcess} shows the mean excess plot for the example data \code{eusilc} on the household level with the largest observation replaced by an outlier. The function \code{meanExcessPlot()} is thereby used to produce the plot. Sample weights can be supplied via the argument \code{w}. Interactive selection of the threshold works just like for the Pareto quantile plot. Again, the selected threshold is returned as an object of class \code{"paretoScale"}. % --------------- % shape parameter % --------------- \section{Estimation of the shape parameter} \label{sec:shape} This section is focused on methods for estimating the shape parameter $\theta$ once the threshold $x_0$ is fixed. It should be noted that none of the original proposals takes sample weights into account. Most estimators presented in the following were therefore adjusted for the case of sample weights. \subsection{Hill estimator} \label{sec:Hill} The maximum likelihood estimator for the shape parameter of the Pareto distribution was introduced by \citet{hill75} and is referred to as the \emph{Hill} estimator. If the data do not contain sample weights, it is given by \begin{equation} \label{eq:Hill} \hat{\theta}_{\mathrm{Hill}} = \frac{k}{\sum_{i = 1}^{k} \log x_{n-k+i} - k \log x_{n-k}}. \end{equation} In the case of sample weights, the \emph{weighted Hill} (wHill) estimator is given by generalizing Equation~(\ref{eq:Hill}) to \begin{equation} \label{eq:wHill} \hat{\theta}_{\mathrm{wHill}} = \frac{\sum_{i = 1}^{k} w_{n-k+i}}{\sum_{i = 1}^{k} w_{n-k+i} \left( \log x_{n-k+i} - \log x_{n-k} \right)} . \end{equation} Package \pkg{laeken} provides the function \code{thetaHill()} to compute the Hill estimator. It requires to specify either the number of observations in the tail via the argument \code{k}, or the threshold via the argument \code{x0}. Furthermore, the argument \code{w} can be used to supply sample weights. In the following example, the shape parameter is estimated using the largest observations (first command) and the threshold (second command) as computed with van Kerm's rule of thumb in Section~\ref{sec:vanKerm}. <<>>= thetaHill(eusilcH$eqIncome, k = ts$k, w = eusilcH$db090) thetaHill(eusilcH$eqIncome, x0 = ts$x0, w = eusilcH$db090) @ \subsection{Weighted maximum likelihood estimator} The \emph{weighted maximum likelihood} (WML) estimator \citep{dupuis02, dupuis06} falls into the class of M-estimators and is given by the solution $\hat{\theta}$ of \begin{equation} \sum_{i = 1}^{k} \mathrm{\Psi}(x_{n-k+i}, \theta) = 0 \end{equation} with \begin{equation} \mathrm{\Psi}(x, \theta) := u(x, \theta) \frac{\partial}{\partial \theta} \log f(x, \theta) = u(x, \theta) \left( \frac{1}{\theta} - \log \frac{x}{x_{0}} \right), \end{equation} where $u(x, \theta)$ is a weight function with values in $[0,1]$. In the implementation in package \pkg{laeken}, a Huber type weight function is used by default, as proposed by \citet{dupuis06}. Let the logarithms of the relative excesses be denoted by \begin{equation} z_{i} := \log \left( \frac{x_{n-k+i}}{x_{n-k}} \right), \qquad i = 1, \ldots, k. \end{equation} In the Pareto model, these can be predicted by \begin{equation} \hat{z}_{i} := -\frac{1}{\theta} \log \left( \frac{k+1-i}{k+1} \right), \qquad i = 1, \ldots, k. \end{equation} The variance of $z_{i}$ is given by \begin{equation} \sigma_{i}^{\phantom{i}2} := \sum_{j = 1}^{i} \frac{1}{\theta^{2} (k-i+j)^{2}}, \qquad i = 1, \ldots, k. \end{equation} Using the standardized residuals \begin{equation} r_{i} := \frac{z_{i} - \hat{z}_{i}}{\sigma_{i}}, \end{equation} the Huber type weight function with tuning constant $c$ is defined as \begin{equation} u(x_{n-k+i}, \theta) := \left\{ \begin{array}{cl} 1, & \quad \text{if } |r_{i}| \leq c, \\ \frac{c}{|r_{i}|}, & \quad \text{if } |r_{i}| > c. \end{array} \right. \end{equation} For this choice of weight function, the bias of $\hat{\theta}$ is approximated by \begin{equation} \hat{B}(\hat{\theta}) = - \frac{\sum_{i=1}^{k} \left( u_{i} \frac{\partial}{\partial \theta} \log f_{i} \right) \vert_{\hat{\theta}} \left( F_{\hat{\theta}}(x_{n-k+i}) - F_{\hat{\theta}}(x_{n-k+i-1}) \right)}{\sum_{i=1}^{k} \left( \frac{\partial}{\partial \theta} u_{i} \frac{\partial}{\partial \theta} \log f_{i} + u_{i} \frac{\partial^{2}}{\partial \theta^{2}} \log f_{i} \right) \vert_{\hat{\theta}} \left( F_{\hat{\theta}}(x_{n-k+i}) - F_{\hat{\theta}}(x_{n-k+i-1}) \right)}, \end{equation} where $u_{i} := u(x_{n-k+i}, \theta)$ and $f_{i} := f(x_{n-k+i}, \theta)$. This term is used to obtain a bias-corrected estimator \begin{equation} \tilde{\theta} := \hat{\theta} - \hat{B}(\hat{\theta}). \end{equation} For details and proofs of the above statements, as well as for information on a probability-based weight function $u(x, \theta)$, the reader is referred to \citet{dupuis02} and \citet{dupuis06}. However, note the WML estimator does not consider sample weights. An adjustment of the estimator to take sample weights into account is currently not available due to its complexity. For sampling designs that lead to equal sample weights, the WML estimator may still be useful, though. The function \code{thetaWML()} is available in \pkg{laeken} to compute the WML estimator. Again, either the argument \code{k} or \code{x0} needs to be used to specify the number of observations in the tail or the threshold. Since the sample weights in the example data are not equal, the following example is only included to demonstrate the use of the function. <<>>= thetaWML(eusilcH$eqIncome, k = ts$k) thetaWML(eusilcH$eqIncome, x0 = ts$x0) @ \subsection{Integrated squared error estimator} For the \emph{integrated squared error} (ISE) estimator \citep{vandewalle07}, the Pareto distribution is modeled in terms of the relative excesses \begin{equation} y_{i} := \frac{x_{n-k+i}}{x_{n-k}}, \qquad i = 1, \ldots, k. \end{equation} The density function of the Pareto distribution for the relative excesses is approximated by \begin{equation} f_{\theta}(y) = \theta y^{-(1+\theta)}. \end{equation} The ISE estimator is then given by minimizing the integrated squared error criterion \citep{terrell90}: \begin{equation} \hat{\theta} = \arg \min_{\theta} \left[ \int f_{\theta}^{2}(y) dy - 2 \mathbb{E}(f_{\theta}(Y)) \right] . \end{equation} If there are no sample weights in the data, the mean is used as an unbiased estimator of $\mathbb{E}(f_{\theta}(Y))$ in order to obtain the ISE estimate \begin{equation} \label{eq:ISE} \hat{\theta}_{\mathrm{ISE}} = \arg \min_{\theta} \left[ \int f_{\theta}^{2}(y) dy - \frac{2}{k} \sum_{i=1}^{k} f_{\theta}(y_{i}) \right] . \end{equation} See \citet{vandewalle07} for more information on the ISE estimator for the case without sample weights. If sample weights are available in the data, the mean in Equation~(\ref{eq:ISE}) is simply replaced by a weighted mean to obtain the \emph{weighted integrated squared error} (wISE) estimator: \begin{equation} \label{eq:wISE} \hat{\theta}_{\mathrm{wISE}} = \arg \min_{\theta} \left[ \int f_{\theta}^{2}(y) dy - \frac{2}{\sum_{i=1}^{k} w_{n-k+i}} \sum_{i=1}^{k} w_{n-k+i} f_{\theta}(y_{i}) \right] . \end{equation} With package \pkg{laeken}, the ISE estimator can be computed using the function \code{thetaISE()}. The arguments \code{k} and \code{x0} are available to specify either the number of observations in the tail or the threshold, and sample weights can be supplied via the argument \code{w}. <<>>= thetaISE(eusilcH$eqIncome, k = ts$k, w = eusilcH$db090) thetaISE(eusilcH$eqIncome, x0 = ts$x0, w = eusilcH$db090) @ \subsection{Partial density component estimator} For the \emph{partial density component} (PDC) estimator \cite{vandewalle07} minimizes the integrated squared error criterion using an incomplete density mixture model $u f_{\theta}$. If the data do not contain sample weights, the PDC estimator in is thus given by \begin{equation} \label{eq:PDC} \hat{\theta}_{\mathrm{PDC}} = \arg \min_{\theta} \left[ u^{2} \int f_{\theta}^{2}(y) dy - \frac{2 u}{k} \sum_{i = 1}^{k} f_{\theta}(y_{i}) \right]. \end{equation} The parameter $u$ can be interpreted as a measure of the uncontaminated part of the sample and is estimated by \begin{equation} \label{eq:u} \hat{u} = \frac{\frac{1}{k} \sum_{i = 1}^{k} f_{\hat{\theta}}(y_{i})}{\int f_{\hat{\theta}}^{2}(y) dy}. \end{equation} See \cite{vandewalle07} and references therein for more information on the PDC estimator for the case without sample weights. Taking sample weights into account, the \emph{weighted partial density component} (wPDC) estimator is obtained by generalizing Equations~(\ref{eq:PDC}) and~(\ref{eq:u}) to \begin{align} \label{eq:wPDC} \hat{\theta}_{\mathrm{wPDC}} =& \arg \min_{\theta} \left[ u^{2} \int f_{\theta}^{2}(y) dy - \frac{2u}{\sum_{i=1}^{k} w_{n-k+i}} \sum_{i = 1}^{k} w_{n-k+i} f_{\theta}(y_{i}) \right] , \\ \hat{u} =& \frac{\frac{1}{\sum_{i=1}^{k} w_{n-k+i}} \sum_{i = 1}^{k} w_{n-k+i} f_{\hat{\theta}}(y_{i})}{\int f_{\hat{\theta}}^{2}(y) dy} . \end{align} The function \code{thetaPDC()} is implemented in package \pkg{laeken} to compute the PDC estimator. As for the other estimators, it is necessary to specify either the number of observations in the tail via the argument \code{k}, or the threshold via the argument \code{x0}. Sample weights can be supplied using the argument \code{w}. <<>>= thetaPDC(eusilcH$eqIncome, k = ts$k, w = eusilcH$db090) thetaPDC(eusilcH$eqIncome, x0 = ts$x0, w = eusilcH$db090) @ % ---------------------------- % estimation of the indicators % ---------------------------- \section{Estimation of the indicators using Pareto tail modeling} \label{sec:fit} Three approaches based on Pareto tail modeling for reducing the influence of outliers on the social exclusion indicators are implemented in the \proglang{R} package \pkg{laeken}: \begin{description} \item[Calibration for nonrepresentative outliers (CN):] Values larger than a certain quantile of the fitted distribution are declared as nonrepresentative outliers. Since these are considered to be unique to the population data, the sample weights of the corresponding observations are set to $1$ and the weights of the remaining observations are adjusted accordingly by calibration. \item[Replacement of nonrepresentative outliers (RN):] Values larger than a certain quantile of the fitted distribution are declared as nonrepresentative outliers. Only these nonrepresentative outliers are replaced by values drawn from the fitted distribution, thereby preserving the order of the original values. \item[Replacement of the tail (RT):] All values above the threshold are replaced by values drawn from the fitted distribution. The order of the original values is preserved. \end{description} An evaluation of the RT approach by means of a simulation study can be found in \citet{alfons10b}. Keep in mind that the largest observation in the example data \code{eusilc} was replaced by a large outlier in Section~\ref{sec:Pareto}. With the following command, the Gini coefficient is estimated according to the Eurostat definition to show that even a single outlier can completely distort the results for the standard estimation (see Section~\ref{sec:Gini} for the original value). <<>>= gini("eqIncome", weights = "rb050", data = eusilc) @ For Pareto tail modeling, the function \code{paretoTail()} is implemented in \pkg{laeken}. It returns an object of class \code{"paretoTail"}, which contains all the necessary information for further analysis using the three approaches described above. Note that the household IDs are supplied via the argument \code{groups} such that the Pareto distribution is fitted on the household level rather than the individual level. In addition, the PDC is used by default to estimate the shape parameter. Other estimators can be specified via the \code{method} argument. <<>>= fit <- paretoTail(eusilc$eqIncome, k = ts$k, w = eusilc$db090, groups = eusilc$db030) @ The function \code{reweightOut()} is available for semiparametric estimation with the CN approach. It returns a vector of the recalibrated weights. In this example, regional information is used as auxiliary variables for calibration. The function \code{calibVars()} thereby transforms a factor into a matrix of binary variables, as required by the calibration function \code{calibWeights()}, which is called internally. These recalibrated weights are then simply used to estimate the Gini coefficient with function \code{gini()}. <<>>= w <- reweightOut(fit, calibVars(eusilc$db040)) gini(eusilc$eqIncome, w) @ For the RN approach, the function \code{replaceOut()} is implemented. Since values are drawn from the fitted distribution to replace the observations flagged as outliers, the seed of the random number generator is set first for reproducibility of the results. The returned vector of incomes is then supplied to \code{gini()} to estimate the Gini coefficient. <<>>= set.seed(1234) eqIncome <- replaceOut(fit) gini(eqIncome, weights = eusilc$rb050) @ Similarly, the function \code{replaceTail()} is available for the RT approach. Again, the seed of the random number generator is set beforehand. <<>>= set.seed(1234) eqIncome <- replaceTail(fit) gini(eqIncome, weights = eusilc$rb050) @ It should be noted that \code{replaceTail()} can also be used for the RN approach by setting the argument \code{all} to \code{FALSE}. In fact, \code{replaceOut(x, ...)} is a simple wrapper for \code{replaceTail(x, all = FALSE, ...)}. In any case, the estimates for the semiparametric approaches based on Pareto tail modeling are very close to the original value before the outlier has been introduced (see Section~\ref{sec:Gini}), whereas the standard estimation is corrupted by the outlier. Furthermore, the estimation of other indicators such as the quintile share ratio (see Section~\ref{sec:QSR}) using the semiparametric approaches is straightforward and hence not shown here. % ----------- % conclusions % ----------- \section{Conclusions} \label{sec:concl} This vignette shows the functionality of package \pkg{laeken} for robust semiparametric estimation of social exclusion indicators based on Pareto tail modeling. Most notably, it demonstrates that the functions are easy to use and that the implementation follows an object-oriented design. While the focus of the paper lies on the use of the package, a mathematical description of the methods is given as well. Furthermore, it is shown that the standard estimation of the inequality indicators can be corrupted by a single outlier, thus underlining the need for robust alternatives. Three approaches for robust semiparametric estimation based on Pareto tail modeling are thereby implemented such that the corresponding functions share a common interface for ease of use. % --------------- % acknowledgments % --------------- \section*{Acknowledgments} This work was partly funded by the European Union (represented by the European Commission) within the 7$^{\mathrm{th}}$ framework programme for research (Theme~8, Socio-Economic Sciences and Humanities, Project AMELI (Advanced Methodology for European Laeken Indicators), Grant Agreement No. 217322). Visit \url{http://ameli.surveystatistics.net} for more information on the project. % ------------ % bibliography % ------------ \bibliographystyle{plainnat} \bibliography{laeken} \end{document} laeken/vignettes/laeken-standard.Rnw0000644000176200001440000010667114127275434017272 0ustar liggesusers\documentclass[a4paper,10pt]{scrartcl} \usepackage[OT1]{fontenc} \usepackage{Sweave} %% additional packages \usepackage{natbib} \bibpunct{(}{)}{,}{a}{}{,} \usepackage{amsmath, amssymb} \usepackage{hyperref} \hypersetup{colorlinks, citecolor=blue, linkcolor=blue, urlcolor=blue} \usepackage[top=30mm, bottom=30mm, left=30mm, right=30mm]{geometry} \usepackage{enumerate} \usepackage{engord} %% additional commands \newcommand{\code}[1]{\texttt{#1}} \newcommand{\pkg}[1]{\mbox{\textbf{#1}}} \newcommand{\proglang}[1]{\mbox{\textsf{#1}}} %%\VignetteIndexEntry{Standard Methods for Point Estimation of Indicators on Social Exclusion and Poverty using the R Package laeken} %%\VignetteDepends{laeken} %%\VignetteKeywords{social exclusion, poverty, indicators, point estimation} %%\VignettePackage{laeken} \begin{document} \title{Standard Methods for Point Estimation of Indicators on Social Exclusion and Poverty using the \proglang{R} Package \pkg{laeken}} \author{Matthias Templ$^{1}$, Andreas Alfons$^{2}$} \date{} \maketitle \setlength{\footnotesep}{11pt} \footnotetext[1]{ \begin{tabular}[t]{l} Zurich University of Applied Sciences\\ E-mail: \href{mailto:matthias.templ@zhaw.ch}{matthias.templ@zhaw.ch} \end{tabular} } \footnotetext[2]{ \begin{tabular}[t]{l} Erasmus School of Economics, Erasmus University Rotterdam\\ E-mail: \href{mailto:alfons@ese.eur.nl}{alfons@ese.eur.nl} \end{tabular} } % change R prompt <>= options(prompt="R> ") @ \paragraph{Abstract} This vignette demonstrates the use of the \proglang{R} package \pkg{laeken} for standard point estimation of indicators on social exclusion and poverty according to the definitions by Eurostat. The package contains synthetically generated data for the European Union Statistics on Income and Living Conditions (EU-SILC), which is used in the code examples throughout the paper. Furthermore, the basic object-oriented design of the package is discussed. Even though the paper is focused on showing the functionality of package \pkg{laeken}, it also provides a brief mathematical description of the implemented indicators. % ------------ % introduction % ------------ \section{Introduction} The \emph{European Union Statistics on Income and Living Conditions} (EU-SILC) is a panel survey conducted in EU member states and other European countries, and serves as basis for measuring risk-of-poverty and social cohesion in Europe. %and for evaluating the Lisbon~2010 strategy and for monitoring the %Europe~2020 goals of the European Union. A short overview of the $11$ most important indicators on social exclusion and poverty according to \cite{EU-SILC04} %and \cite{EU-SILC09} is given in the following. \paragraph{Primary indicators} \begin{enumerate} \item At-risk-of-poverty rate (after social transfers) \begin{enumerate}[a.] \item At-risk-of-poverty rate by age and gender \item At-risk-of-poverty rate by most frequent activity status and gender \item At-risk-of-poverty rate by household type \item At-risk-of-poverty rate by accommodation tenure status \item At-risk-of-poverty rate by work intensity of the household \item At-risk-of-poverty threshold (illustrative values) \end{enumerate} \item Inequality of income distribution: S80/S20 income quintile share ratio \item At-persistent-risk-of-poverty rate by age and gender ($60\%$ median) \item Relative median at-risk-of-poverty gap, by age and gender \newcounter{enumi_last} \setcounter{enumi_last}{\value{enumi}} \end{enumerate} \paragraph{Secondary indicators} \begin{enumerate} \setcounter{enumi}{\value{enumi_last}} \item Dispersion around the at-risk-of-poverty threshold \item At-risk-of-poverty rate anchored at a moment in time \item At-risk-of-poverty rate before social transfers by age and gender \item Inequality of income distribution: Gini coefficient \item At-persistent-risk-of-poverty rate, by age and gender ($50\%$ median) \setcounter{enumi_last}{\value{enumi}} \end{enumerate} \paragraph{Other indicators} \begin{enumerate} \setcounter{enumi}{\value{enumi_last}} \item Mean equivalized disposable income \item The gender pay gap \end{enumerate} \paragraph{} Note that especially the Gini coefficient is very well studied due to its importance in many fields of research. The add-on package \pkg{laeken} \citep{laeken} aims is to bring functionality for the estimation of indicators on social exclusion and poverty to the statistical environment \proglang{R} \citep{RDev}. In the examples in this vignette, standard estimates for the most important indicators are computed according to the Eurostat definitions \citep{EU-SILC04, EU-SILC09}. More sophisticated methods that are less influenced by outliers are described in vignette \code{laeken-pareto} \citep{alfons11a}, while the basic framework for variance estimation is discussed in vignette \code{laeken-variance} \citep{templ11b}. Those documents can be viewed from within \proglang{R} with the following commands: <>= vignette("laeken-pareto") vignette("laeken-variance") @ Morover, a general introduction to package \pkg{laeken} is published as \citet{alfons13b}. The example data set of package \pkg{laeken}, which is called \code{eusilc} and consists of $14\,827$ observations from $6\,000$ households, is used throughout the paper. It was synthetically generated from Austrian EU-SILC survey data from 2006 using the data simulation methodology proposed by \citet{alfons11c} and implemented in the \proglang{R} package \pkg{simPopulation} \citep{simPopulation}. The first three observations of the synthetic data set \code{eusilc} are printed below. <<>>= library("laeken") data("eusilc") head(eusilc, 3) @ Only a few of the large number of variables in the original survey are included in the example data set. The variable names are rather cryptic codes, but these are the standardized names used by the statistical agencies. Furthermore, the variables \code{hsize} (household size), \code{age}, \code{eqSS} (equivalized household size) and \code{eqIncome} (equivalized disposable income) are not included in the standardized format of EU-SILC data, but have been derived from other variables for convenience. Moreover, some very sparse income components were not included in the the generation of this synthetic data set. Thus the equivalized household income is computed from the available income components. For the remainder of the paper, the variable \code{eqIncome} (equivalized disposable income) is of main interest. Other variables are in some cases used to break down the data in order to evaluate the indicators on the resulting subsets. It is important to note that EU-SILC data are in practice conducted through complex sampling designs with different inclusion probabilities for the observations in the population, which results in different weights for the observations in the sample. Furthermore, calibration is typically performed for non-response adjustment of these initial design weights. Therefore, the sample weights have to be considered for all estimates, otherwise biased results are obtained. The rest of the paper is organized as follows. Section \ref{sec:design} briefly illustrates the basic object-oriented design of the package. The calculation of the equivalized household size and the equivalized disposable income is then described in Section \ref{sec:income}. Afterwards, Section~\ref{sec:w} introduces the Eurostat definitions of the weighted median and weighted quantiles, which are required for the estimation of some of the indicators. In Section~\ref{sec:ind}, a mathematical description of the most important indicators on social exclusion and poverty is given and their estimation with package \pkg{laeken} is demonstrated. Section~\ref{sec:sub} discusses a useful subsetting method, and Section~\ref{sec:concl} concludes. % ------------ % basic design % ------------ \section{Basic design of the package} \label{sec:design} The implementation of the package follows an object-oriented design using \proglang{S3} classes \citep{chambers92}. Its aim is to provide functionality for point and variance estimation of Laeken indicators with a single command, even for different years and domains. Currently, the following indicators are available in the \proglang{R} package \pkg{laeken}: \begin{itemize} \item \emph{At-risk-of-poverty rate}: function \code{arpr()} \item \emph{Quintile share ratio}: function \code{qsr()} \item \emph{Relative median at-risk-of-poverty gap}: function \code{rmpg()} \item \emph{Dispersion around the at-risk-of-poverty threshold}: also function \code{arpr()} \item \emph{Gini coefficient}: function \code{gini()} \end{itemize} Note that the implementation strictly follows the Eurostat definitions \citep{EU-SILC04,EU-SILC09}. %In addition, robust estimators are also implemented. Here, the focus is on %Pareto tail modeling. \subsection{Class structure} In this section, the class structure of package \pkg{laeken} is briefly discussed. Section~\ref{sec:indicator} describes the basic class \code{"indicator"}, while the different subclasses for the specific indicators are listed in Section~\ref{sec:classes}. \subsubsection{Class \code{"indicator"}} \label{sec:indicator} The basic class \code{"indicator"} acts as the superclass for all classes in the package corresponding to specific indicators. It consists of the following components: % \begin{description} \item[\code{value}:] A numeric vector containing the point estimate(s). \item[\code{valueByStratum}:] A \code{data.frame} containing the point estimates by domain. \item[\code{varMethod}:] A character string specifying the type of variance estimation used. \item[\code{var}:] A numeric vector containing the variance estimate(s). \item[\code{varByStratum}:] A \code{data.frame} containing the variance estimates by domain. \item[\code{ci}:] A numeric vector or matrix containing the confidence interval(s). \item[\code{ciByStratum}:] A \code{data.frame} containing the confidence intervals by domain. \item[\code{alpha}:] The confidence level is given by $1 - $\code{alpha}. \item[\code{years}:] A numeric vector containing the different years of the survey. \item[\code{strata}:] A character vector containing the different strata of the breakdown. % \item[\code{seed}:] The seed of the random number generator before the computations. \end{description} These list components are inherited by each indicator in the package. One of the most important features of \pkg{laeken} is that indicators can be evaluated for different years and domains. The latter of which can be regions (e.g., NUTS2), but also any other breakdown given by a categorical variable (see the examples in Section~\ref{sec:ind}). In any case, the advantage of the object-oriented implementation is the possibility of sharing code among the indicators. To give an example, the following methods for the basic class \code{"indicator"} are implemented in the package: <<>>= methods(class="indicator") @ The \code{print()} and \code{subset()} methods are called by their respective generic functions if an object inheriting from class \code{"indicator"} is supplied. While the \code{print()} method defines the output of objects inheriting from class \code{"indicator"} shown on the \proglang{R} console, the \code{subset()} method allows to extract subsets of an object inheriting from class \code{"indicator"} and is discussed in detail in Section~\ref{sec:sub}. Furthermore, the function \code{is.indicator()} is available to test whether an object is of class \code{"indicator"}. \subsubsection{Additional classes} \label{sec:classes} For the specific indicators on social exclusion and poverty, the following classes are implemented in package \pkg{laeken}: % \begin{itemize} \item Class \code{"arpr"} with the following additional components: \begin{description} \item[\code{p}:] The percentage of the weighted median used for the at-risk-of-poverty threshold. \item[\code{threshold}:] The at-risk-of-poverty threshold(s). \end{description} \item Class \code{"qsr"} with no additional components. \item Class \code{"rmpg"} with the following additional components: \begin{description} \item[\code{threshold}:] The at-risk-of-poverty threshold(s). \end{description} \item Class \code{"gini"} with no additional components. \end{itemize} % All these classes are subclasses of the basic class \code{"indicator"} and therefore inherit all its components and methods. In addition, functions to test whether an object is a member of one of these subclasses are implemented. Similarly to \code{is.indicator()}, these are called \code{is.foo()}, where \code{foo} is the name of the respective class (e.g., \code{is.arpr()}). % ----------------------------- % equivalized disposable income % ----------------------------- \section{Calculation of the equivalized disposable income} \label{sec:income} For each person, the equivalized disposable income is defined as the total household disposable income divided by the equivalized household size. It follows that each person in the same household receives the same equivalized disposable income. The total disposable income of a household is calculated by adding together the personal income received by all of the household members plus the income received at the household level. The equivalized household size is defined according to the modified OECD scale, which gives a weight of 1.0 to the first adult, 0.5 to other household members aged 14 or over, and 0.3 to household members aged less than 14 \citep{EU-SILC04, EU-SILC09}. In practice, the equivalized disposable income needs to be computed from the income components included in EU-SILC for the estimation of the indicators on social exclusion and poverty. Therefore, this section outlines how to perform this step with package \pkg{laeken}, even though the variable \code{eqIncome} containing the equivalized disposable income is already available in the example data set \code{eusilc}. Note that not all variables that are required for an exact computation of the equivalized income are included in the synthetic example data. However, the functions of the package can be applied in exactly the same manner to real EU-SILC data. First, the equivalized household size according to the modified OECD scale needs to be computed. This can be done with the function \code{eqSS()}, which requires the household ID and the age of the individuals as arguments. In the example data, household~ID and age are stored in the variables \code{db030} and \code{age}, respectively. It should be noted that the variable \code{age} is not in the standardized format of EU-SILC data and needs to be calculated from the data beforehand. Nevertheless, these computations are very simple and are therefore not shown here \citep[for details, see][]{EU-SILC09}. The following two lines of code calculate the equivalized household size, add it to the data set, and print the first eight observations of the variables involved. <<>>= eusilc$eqSS <- eqSS("db030", "age", data=eusilc) head(eusilc[,c("db030", "age", "eqSS")], 8) @ Then the equivalized disposable income can be computed with the function \code{eqInc()}. It requires the following information to be supplied: the household~ID, the household income components to be added and subtracted, respectively, the personal income components to be added and subtracted, respectively, as well as the equivalized household size. With the following commands, the equivalized disposable income is calculated and added to the data set, after which the first eight observations of the important variables in this context are printed. <<>>= hplus <- c("hy040n", "hy050n", "hy070n", "hy080n", "hy090n", "hy110n") hminus <- c("hy130n", "hy145n") pplus <- c("py010n", "py050n", "py090n", "py100n", "py110n", "py120n", "py130n", "py140n") eusilc$eqIncome <- eqInc("db030", hplus, hminus, pplus, character(), "eqSS", data=eusilc) head(eusilc[,c("db030", "eqSS", "eqIncome")], 8) @ % Note that the net income is considered in this example, therefore no personal income component needs to be subtracted \citep[see][]{EU-SILC04, EU-SILC09}. This is reflected in the call to \code{eqInc()} by the use of an empty character vector \code{character()} for the corresponding argument. % ------------------ % weighted quantiles % ------------------ \section{Weighted median and quantile estimation} \label{sec:w} Some of the indicators on social exclusion and poverty require the estimation of the median income or other quantiles of the income distribution. Hence functions that strictly follow the definitions according to \citet{EU-SILC04, EU-SILC09} are implemented in package \pkg{laeken}. They are used internally for the estimation of the respective indicators, but can also be called by the user directly. In the analysis of income distributions, the median income is typically of higher interest than the arithmetic mean. This is because income distributions commonly are strongly right-skewed with a heavy tail of \emph{representative outliers} (correctly measured units that are not unique to the population) and \emph{nonrepresentative outliers} (either measurement errors or correct observations that can be considered unique in the population). Therefore, the center of the distribution is more reliably estimated by a weighted median than by a weighted mean, as the latter is highly influenced by extreme values. In mathematical terms, quantiles are defined as $q_{p} := F^{-1}(p)$, where $F$ is the distribution function on the population level and $0 \leq p \leq 1$. The median as an important special case is given by $p = 0.5$. For the following definitions, let $n$ be the number of observations in the sample, let $\boldsymbol{x} := (x_{1}, \ldots, x_{n})'$ denote the equivalized disposable income with \mbox{$x_{1} \leq \ldots \leq x_{n}$}, and let $\boldsymbol{w} := (w_{i}, \ldots, w_{n})'$ be the corresponding personal sample weights. Weighted quantiles for the estimation of the population values according to \citet{EU-SILC04, EU-SILC09} are then given by \begin{equation} \label{eq:wq} \hat{q}_{p} = \hat{q}_{p} (\boldsymbol{x}, \boldsymbol{w}) := \begin{cases} \frac{1}{2} (x_{j} + x_{j+1}), & \quad \text{if } \sum_{i=1}^{j} w_{i} = p \sum_{i=1}^{n} w_{i}, \\ x_{j+1}, & \quad \text{if } \sum_{i=1}^{j} w_{i} < p \sum_{i=1}^{n} w_{i} < \sum_{i=1}^{j+1} w_{i}. \end{cases} \end{equation} This definition of weighted quantiles is available in \pkg{laeken} through the function \code{weightedQuantile()}. The following command computes the weighed 20\% quantile, the weighted median, and the weighted 80\% quantile. In the context of social exclusion indicators, these are of most importance. % ----- <>= weightedQuantile(eusilc$eqIncome, eusilc$rb050, probs = c(0.2, 0.5, 0.8)) @ % ----- For the important special case of the weighted median, the function \code{weightedMedian()} is available for convenience. % ----- <<>>= weightedMedian(eusilc$eqIncome, eusilc$rb050) @ In addition, the functions \code{incMedian()} and \code{incQuintile()} are more tailored towards application in the case of indicators on social exclusion and poverty and provide a similar interface as the functions for the indicators (see Section~\ref{sec:ind}). In particular, they allow to supply an additional variable to be used as tie-breakers for sorting, and to compute the weighted median and income quintiles, respectively, for several years of the survey. With the following lines of code, the median income as well as the \engordnumber{1} and \engordnumber{4} income quintile (i.e., the weighted 20\% and 80\% quantiles) are estimated. <<>>= incMedian("eqIncome", weights = "rb050", data = eusilc) incQuintile("eqIncome", weights = "rb050", k = c(1, 4), data = eusilc) @ % ------------------- % selected indicators % ------------------- \section{Indicators on social exclusion and poverty} \label{sec:ind} In this section, the most important indicators on social exclusion and poverty are described in detail. Furthermore, the functionality of package \pkg{laeken} to estimate these indicators is demonstrated. It should be noted that all functions for the implemented indicators provide a very similar interface. Most importantly, it is possible to compute estimates for several years of the survey and different subdomains with a single command. Furthermore, the functions allow to supply an additional variable to be used as tie-breakers for sorting. However, not all of the implemented functionality is shown in this vignette. For a complete description of the functions and their arguments, the reader is referred to the corresponding \proglang{R} help pages. In addition, only point estimation of the indicators on social exclusion and poverty is illustrated here, statistical significance of these estimates is not discussed. The functionality for variance estimation of the indicators is described in the package vignette \code{laeken-variance} \citep{templ11b}. For the following definitions of the estimators according to \citet{EU-SILC04, EU-SILC09}, let $\boldsymbol{x} := (x_{1}, \ldots, x_{n})'$ be the equivalized disposable income with $x_{1} \leq \ldots \leq x_{n}$ and let $\boldsymbol{w} := (w_{i}, \ldots, w_{n})'$ be the corresponding personal sample weights, where $n$ denotes the number of observations. Furthermore, define the following index sets for a certain threshold $t$: \begin{align} I_{< t} &:= \{ i \in \{ 1, \ldots, n \} : x_{i} < t \},\label{eq:01-Ilt}\\ I_{\leq t} &:= \{ i \in \{ 1, \ldots, n \} : x_{i} \leq t \},\label{eq:01-Ileqt}\\ I_{> t} &:= \{ i \in \{ 1, \ldots, n \} : x_{i} > t\}\label{eq:01-Igt}. \end{align} \subsection{At-risk-at-poverty rate} \label{sec:ARPR} In order to define the \emph{at-risk-of-poverty rate} (ARPR), the \emph{at-risk-of-poverty threshold} (ARPT) needs to be introduced first, which is set at $60\%$ of the national median equivalized disposable income. Then the at-risk-at-poverty rate is defined as the proportion of persons with an equivalized disposable income below the at-risk-at-poverty threshold \citep{EU-SILC04, EU-SILC09}. In a more mathematical notation, the at-risk-at-poverty rate is defined as \begin{equation} \label{eq:ARPR} ARPR := P(x < 0.6 \cdot q_{0.5}) \cdot 100,% = F(0.6 \cdot q_{0.5}) \cdot 100, \end{equation} where $q_{0.5} := F^{-1}(0.5)$ denotes the population median (50\% quantile) and $F$ is the distribution function of the equivalized income on the population level. For the estimation of the at-risk-at-poverty rate from a sample, the sample weights need to be taken into account. %Let $n$ be the number of observations in the sample, let $\boldsymbol{x} := %(x_{1}, \ldots, x_{n})'$ denote the equivalized disposable income with %\mbox{$x_{1} \leq \ldots \leq x_{n}$}, and let $\boldsymbol{w} := (w_{i}, %\ldots, w_{n})'$ be the corresponding personal sample weights. Then the %at-risk-at-poverty threshold is estimated by First, the at-risk-at-poverty threshold is estimated by \begin{equation} \label{eq:ARPT} \widehat{ARPT} = 0.6 \cdot \hat{q}_{0.5}, \end{equation} where $\hat{q}_{0.5}$ is the weighted median as defined in Equation~(\ref{eq:wq}). %Furthermore, define an index set of observations with an equivalized disposable %income below the estimated at-risk-at-poverty threshold as %\begin{equation} %I_{< \widehat{ARPT}} := \{ i \in \{ 1, \ldots, n \} : x_{i} < \widehat{ARPT} \}. %\end{equation} %With these definitions, the at-risk-at-poverty rate can be estimated by Then the at-risk-at-poverty rate can be estimated by \begin{equation} \widehat{ARPR} := \frac{\sum_{i \in I_{< \widehat{ARPT}}} w_{i}}{\sum_{i=1}^{n} w_{i}} \cdot 100, \end{equation} where $I_{< \widehat{ARPT}}$ is an index set of persons with an equivalized disposable income below the estimated at-risk-of-poverty threshold as defined in Equation~(\ref{eq:01-Ilt}). In package \pkg{laeken}, the functions \code{arpt()} and \code{arpr()} are implemented for the estimation of the at-risk-of-poverty threshold and the at-risk-of-poverty rate. Whenever sample weights are available in the data, they should be supplied as the \code{weights} argument. Even though \code{arpt()} is called internally by \code{arpr()}, it can also be called by the user directly. <<>>= arpt("eqIncome", weights = "rb050", data = eusilc) arpr("eqIncome", weights = "rb050", data = eusilc) @ It is also possible to use these functions for the estimation of the indicator \emph{dispersion around the at-risk-of-poverty threshold}, which is defined as the proportion of persons with an equivalized disposable income below $40\%$, $50\%$ and $70\%$ of the national weighted median equivalized disposable income. The proportion of the median equivalized income to be used can thereby be adjusted via the argument \code{p}. <<>>= arpr("eqIncome", weights = "rb050", p = 0.4, data = eusilc) arpr("eqIncome", weights = "rb050", p = 0.5, data = eusilc) arpr("eqIncome", weights = "rb050", p = 0.7, data = eusilc) @ In order to compute estimates for different subdomains, a breakdown variable simply needs to be supplied as the \code{breakdown} argument. Note that in this case the same overall at-risk-of-poverty threshold is used for all subdomains \citep[see][]{EU-SILC04, EU-SILC09}. The following command computes the overall estimate, as well as estimates for all NUTS2 regions. <<>>= arpr("eqIncome", weights = "rb050", breakdown = "db040", data = eusilc) @ However, any kind of breakdown can be supplied, e.g., the breakdowns defined by \citet{EU-SILC04, EU-SILC09}. With the following lines of code, a breakdown variable with all possible combinations of age categories and gender is defined and added to the data set, before it is used to compute estimates for the corresponding domains. <<>>= ageCat <- cut(eusilc$age, c(-1, 16, 25, 50, 65, Inf), right=FALSE) eusilc$breakdown <- paste(ageCat, eusilc$rb090, sep=":") arpr("eqIncome", weights = "rb050", breakdown = "breakdown", data = eusilc) @ Clearly, the results are even more heterogeneous than for the breakdown into NUTS2 regions. %The results are even more different when considering household size %(\code{hsize}) and citizenship (\code{pb220a}) as the domain level for %estimation. %<<>>= %eusilc$breakdown <- paste(eusilc$hsize, eusilc$pb220a, sep=":") %arpr("eqIncome", weights = "rb050", breakdown = "breakdown", data = eusilc) %@ \subsection{Quintile share ratio} The income \emph{quintile share ratio} (QSR) is defined as the ratio of the sum of the equivalized disposable income received by the 20\% of the population with the highest equivalized disposable income to that received by the 20\% of the population with the lowest equivalized disposable income \citep{EU-SILC04, EU-SILC09}. For the estimation of the quintile share ratio from a sample, let $\hat{q}_{0.2}$ and $\hat{q}_{0.8}$ denote the weighted 20\% and 80\% quantiles, respectively, as defined in Equation~(\ref{eq:wq}). Using index sets $I_{\leq \hat{q}_{0.2}}$ and $I_{> \hat{q}_{0.8}}$ as defined in Equations~(\ref{eq:01-Ileqt}) and~(\ref{eq:01-Igt}), respectively, the quintile share ratio is estimated by \begin{equation} \widehat{QSR} := \frac{\sum_{i \in I_{> \hat{q}_{0.8}}} w_{i} x_{i}}{\sum_{i \in I_{\leq \hat{q}_{0.2}}} w_{i} x_{i}}. \end{equation} With package \pkg{laeken}, the quintile share ratio can be estimated using the function \code{qsr()}. As for the at-risk-of-poverty rate, sample weights can be supplied via the \code{weights} argument. <<>>= qsr("eqIncome", weights = "rb050", data = eusilc) @ Computing estimates for different subdomains is again possible by specifying the \code{breakdown} argument. In the following example, estimates for each NUTS2 region are computed in addition to the overall estimate. <<>>= qsr("eqIncome", weights = "rb050", breakdown = "db040", data = eusilc) @ Nevertheless, it should be noted that the quintile share ratio is highly influenced by outliers \citep[see][]{hulliger09a, alfons10b}. Since the upper tail of income distributions virtually always contains nonrepresentative outliers, robust estimators of the quintile share ratio should preferably be used. Thus robust semi-parametric methods based on Pareto tail modeling are implemented in package \pkg{laeken} as well. Their application is discussed in vignette \code{laeken-pareto} \citep{alfons11a}. \subsection{Relative median at-risk-of-poverty gap (by age and gender)} The \emph{relative median at-risk-of-poverty gap} (RMPG) is defined as the difference between the median equivalized disposable income of persons below the at-risk-of-poverty threshold and the at-risk of poverty threshold itself, expressed as a percentage of the at-risk-of-poverty threshold \citep{EU-SILC04, EU-SILC09}. %Let $wmed_{(poor)}$ the weighted median of the people who having an income %below $ARPR$ defined in Equation~\ref{eq:ARPR}. Then the relative median %at-risk-of-poverty gap is estimated by %\begin{displaymath} %RMPG = \frac{ARPR - wmed_{(poor)}}{ARPR} \cdot 100 %\end{displaymath} For the estimation of the relative median at-risk-of-poverty gap from a sample, let $\widehat{ARPT}$ be the estimated at-risk-of-poverty threshold according to Equation~(\ref{eq:ARPT}), and let $I_{< \widehat{ARPT}}$ be an index set of persons with an equivalized disposable income below the estimated at-risk-of-poverty threshold as defined in Equation~(\ref{eq:01-Ilt}). Using this index set, define $\boldsymbol{x}_{< \widehat{ARPT}} := (x_{i})_{i \in I_{< \widehat{ARPT}}}$ and $\boldsymbol{w}_{< \widehat{ARPT}} := (w_{i})_{i \in I_{< \widehat{ARPT}}}$. Furthermore, let $\hat{q}_{0.5} (\boldsymbol{x}_{< \widehat{ARPT}}, \boldsymbol{w}_{< \widehat{ARPT}})$ be the corresponding weighted median according to the definition in Equation~(\ref{eq:wq}). Then the relative median at-risk-of-poverty gap is estimated by \begin{equation} \widehat{RMPG} = \frac{\widehat{ARPT} - \hat{q}_{0.5} (\boldsymbol{x}_{< \widehat{ARPT}}, \boldsymbol{w}_{< \widehat{ARPT}})}{\widehat{ARPT}} \cdot 100. \end{equation} In package \pkg{laeken}, the function \code{rmpg()} is implemented for the estimation of the relative median at-risk-of-poverty gap. If available in the data, sample weights should be supplied as the \code{weights} argument. Note that the function \code{arpt()} for the estimation of the at-risk-of-poverty threshold is called internally (cf. function \code{arpr()} for the at-risk-of-poverty rate in Section~\ref{sec:ARPR}). <<>>= rmpg("eqIncome", weights = "rb050", data = eusilc) @ Estimates for different subdomains can be computed by making use of the \code{breakdown} argument. With the following command, the overall estimate and estimates for all NUTS2 regions are computed. <<>>= rmpg("eqIncome", weights = "rb050", breakdown = "db040", data = eusilc) @ For the relative median at-risk-of-poverty gap, the breakdown by age and gender is of particular interest. In the following example, a breakdown variable with all possible combinations of age categories and gender is defined and added to the data set. Afterwards, estimates for the corresponding domains are computed. <<>>= ageCat <- cut(eusilc$age, c(-1, 16, 25, 50, 65, Inf), right=FALSE) eusilc$breakdown <- paste(ageCat, eusilc$rb090, sep=":") rmpg("eqIncome", weights = "rb050", breakdown = "breakdown", data = eusilc) @ \subsection{Gini coefficient} The \emph{Gini coefficient} is defined as the relationship of cumulative shares of the population arranged according to the level of equivalized disposable income, to the cumulative share of the equivalized total disposable income received by them \citep{EU-SILC04, EU-SILC09}. For the estimation of the Gini coefficient from a sample, the sample weights need to be taken into account. In mathematical terms, the Gini coefficient is estimated by \begin{equation} \widehat{Gini} := 100 \left[ \frac{2 \sum_{i=1}^{n} \left( w_{i} x_{i} \sum_{j=1}^{i} w_{j} \right) - \sum_{i=1}^{n} w_{i}^{\phantom{i}2} x_{i}}{\left( \sum_{i=1}^{n} w_{i} \right) \sum_{i=1}^{n} \left(w_{i} x_{i} \right)} - 1 \right]. \end{equation} The function \code{gini()} is available in \pkg{laeken} to estimate the Gini coefficient. As for the other indicators, sample weights can be specified with the \code{weights} argument. <<>>= gini("eqIncome", weights = "rb050", data = eusilc) @ Using the \code{breakdown} argument in the following command, estimates for the NUTS2 regions are computed in addition to the overall estimate. <<>>= gini("eqIncome", weights = "rb050", breakdown = "db040", data = eusilc) @ Since outliers have a strong influence on the Gini coefficient, robust estimators are preferred to the standard estimation described above \citep[see][]{alfons10b}. Vignette \code{laeken-pareto} \citep{alfons11a} describes how to apply the robust semi-parametric methods implemented in package \pkg{laeken}. % ------------------ % extracting subsets % ------------------ \section{Extracting information using the \code{subset()} method} \label{sec:sub} If estimates of an indicator have been computed for several subdomains, it may sometimes be desired to extract the results for some domains of particular interest. In package \pkg{laeken}, this is implemented by taking advantage of the object-oriented design of the package. Each of the functions for the indicators described in Section~\ref{sec:ind} returns an object belonging to a class of the same name as the respective function, e.g., function \code{arpr()} returns an object of class \code{"arpr"}. All these classes thereby inherit from the basic class \code{"indicator"} (see Section~\ref{sec:design}). <<>>= a <- arpr("eqIncome", weights = "rb050", breakdown = "db040", data = eusilc) print(a) is.arpr(a) is.indicator(a) class(a) @ To extract a subset of results from such an object, a \code{subset()} method for the class \code{"indicator"} is implemented in \pkg{laeken}. The method \code{subset.indicator()} is hidden from the user and is called internally by the generic function \code{subset()} whenever an object of class \code{"indicator"} is supplied. In the following example, the estimates of the at-risk-of-poverty rate for the regions Lower Austria and Vienna are extracted from the object computed above. <<>>= subset(a, strata = c("Lower Austria", "Vienna")) @ % ----------- % conclusions % ----------- \section{Conclusions} \label{sec:concl} This vignette demonstrates the use of package \pkg{laeken} for point estimation of the European Union indicators on social exclusion and poverty. Since the description of the indicators in \citet{EU-SILC04, EU-SILC09} is weak from a mathematical point of view, a more precise notation is given in this paper. Currently, the most important indicators are implemented in \pkg{laeken}. Their estimation is made easy with the package, as it is even possible to compute estimates for several years and different subdomains with a single command. Concerning the inequality indicators quintile share ratio and Gini coefficient, it is clearly visible from their definitions that the standard estimators are highly influenced by outliers \citep[see also][]{hulliger09a, alfons10b}. Therefore, robust semi-parametric methods are implemented in \pkg{laeken} as well. These are described in vignette \code{laeken-pareto} \citep{alfons11a}, while variance and confidence interval estimation for the indicators on social exclusion and poverty with package \pkg{laeken} is treated in vignette \code{laeken-variance} \citep{templ11b}. % --------------- % acknowledgments % --------------- \section*{Acknowledgments} This work was partly funded by the European Union (represented by the European Commission) within the 7$^{\mathrm{th}}$ framework programme for research (Theme~8, Socio-Economic Sciences and Humanities, Project AMELI (Advanced Methodology for European Laeken Indicators), Grant Agreement No. 217322). Visit \url{http://ameli.surveystatistics.net} for more information on the project. % ------------ % bibliography % ------------ \bibliographystyle{plainnat} \bibliography{laeken} \end{document} laeken/vignettes/laeken-intro.Rnw0000644000176200001440000016232714127277276016633 0ustar liggesusers\documentclass[article,nojss]{jss} % \documentclass[article,shortnames]{jss} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% declarations for jss.cls %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% almost as usual \author{Andreas Alfons\\ Erasmus University Rotterdam \And Matthias Templ\\Zurich University of Applied Sciences} \title{Estimation of Social Exclusion Indicators from Complex Surveys: The \proglang{R} Package \pkg{laeken}} %% for pretty printing and a nice hypersummary also set: \Plainauthor{Andreas Alfons, Matthias Templ} %% comma-separated \Plaintitle{Estimation of Social Exclusion Indicators from Complex Surveys: The R Package laeken} %% without formatting \Shorttitle{\pkg{laeken}: Estimation of Social Exclusion Indicators} %% a short title (if necessary) %% an abstract and keywords \Abstract{ This package vignette is an up-to-date version of \citet{alfons13b}, published in the \emph{Journal of Statistical Software}. Units sampled from finite populations typically come with different inclusion probabilities. Together with additional preprocessing steps of the raw data, this yields unequal sampling weights of the observations. Whenever indicators are estimated from such complex samples, the corresponding sampling weights have to be taken into account. In addition, many indicators suffer from a strong influence of outliers, which are a common problem in real-world data. The \proglang{R} package \pkg{laeken} is an object-oriented toolkit for the estimation of indicators from complex survey samples via standard or robust methods. In particular the most widely used social exclusion and poverty indicators are implemented in the package. A general calibrated bootstrap method to estimate the variance of indicators for common survey designs is included as well. Furthermore, the package contains synthetically generated close-to-reality data for the European Union Statistics on Income and Living Conditions and the Structure of Earnings Survey, which are used in the code examples throughout the paper. Even though the paper is focused on showing the functionality of package \pkg{laeken}, it also provides a brief mathematical description of the implemented indicator methodology. } \Keywords{indicators, robust estimation, sample weights, survey methodology, \proglang{R}} \Plainkeywords{indicators, robust estimation, sample weights, survey methodology, R} %% without formatting %% at least one keyword must be supplied %% publication information %% NOTE: Typically, this can be left commented and will be filled out by the technical editor %% \Volume{50} %% \Issue{9} %% \Month{June} %% \Year{2012} %% \Submitdate{2012-06-04} %% \Acceptdate{2012-06-04} %% The address of (at least) one author should be given %% in the following format: \Address{ Andreas Alfons \\ Erasmus School of Economics \\ Erasmus University Rotterdam \\ Burgemeester Oudlaan 50 \\ 3062PA Rotterdam, Netherlands \\ E-mail: \email{alfons@ese.eur.nl} \\ URL: \url{https://personal.eur.nl/alfons/} \bigskip Matthias Templ \\ Zurich University of Applied Sciences \\ Rosenstra\ss e 3 \\ 8400 Winterthur, Switzerland \\ E-mail: \email{matthias.templ@zhaw.ch} \\ URL: \url{https://data-analysis.at/} } %% It is also possible to add a telephone and fax number %% before the e-mail in the following format: %% Telephone: +43/512/507-7103 %% Fax: +43/512/507-2851 %% for those who use Sweave please include the following line (with % symbols): %% need no \usepackage{Sweave.sty} %%\VignetteIndexEntry{Estimation of Social Exclusion Indicators From Complex Surveys: The R Package laeken} %%\VignetteDepends{laeken} %%\VignetteKeywords{indicators, robust estimation, sample weights, survey methodology, R} %%\VignettePackage{laeken} %% end of declarations %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% additional packages \usepackage{amsfonts} \usepackage{amsmath} \usepackage{amssymb} \usepackage{engord} \usepackage{enumerate} \usepackage{soul} \begin{document} % \SweaveOpts{concordance=TRUE} %% include your article here, just as usual %% Note that you should use the \pkg{}, \proglang{} and \code{} commands. %% load package "laeken" <>= options(prompt = "R> ", continue = "+ ", width = 72, useFancyQuotes = FALSE) library("laeken") @ %% some references have to many authors to list them in the text \shortcites{AMELI-D7.1} % ------------ % Introduction % ------------ \section{Introduction} Estimation of indicators is one of the main tasks in survey statistics. They are usually estimated from complex surveys with many thousands of observations, conducted in a harmonized manner over many countries. Indicators are designed to reflect major developments in society, for example with respect to poverty, social cohesion or gender inequality, in order to quantify and monitor progress towards policy objectives. Moreover, by implementing a monitoring system across countries via a harmonized set of indicators, different policies can be compared based on quantitative information regarding their impact on society. Thus statistical indicators are an important source of information on which policy makers can base their decisions. Nevertheless, for policy decisions to be effective, the underlying quantitative information from the indicators needs to be reliable. Not only should the variability of the indicators be kept in mind, but also the impact of data collection and preprocessing needs to be considered. Indicators are typically based on complex surveys, in which units are drawn from finite populations, most often with unequal inclusion probabilities. Hence the observations in the sample represent different numbers of units in the population, giving them unequal sample weights. In addition, those initial weights are often modified by preprocessing steps such as calibration for nonresponse. Therefore, sample weights always need to be taken into account in the estimation of indicators from survey samples, otherwise the estimates may be biased. The focus of this paper is on socioeconomic indicators on poverty, social cohesion and gender differences. In economic data, extreme outliers are a common problem. Such outliers can have a disproportionally large influence on the estimates of indicators and may completely distort them. If indicators are corrupted by outliers, wrong conclusions could be drawn by policy makers. Robust estimators that give reliable estimates even in the presence of extreme outliers are therefore necessary. We introduce the add-on package \pkg{laeken} \citep{laeken} for the open source statistical computing environment \proglang{R} \citep{RDev}. It provides functionality for standard and robust estimation of indicators on social exclusion and poverty from complex survey samples. The aim of the paper is to present the most important functionality of the package. A more complete overview of the available functionality is given in additional package vignettes on specialized topics. A list of the available vignettes can be viewed from within \proglang{R} with the following command: <>= vignette(package="laeken") @ Even though official statistical agencies usually rely on commercial software, \proglang{R} has gained some traction in the survey statistics community over the years. Various add-on packages for survey methodology are now available. For instance, an extensive collection of methods for the analysis of survey samples is implemented in package \pkg{survey} \citep{lumley04, survey}. The accompanying book by \citet{lumley10} also serves as an excellent introduction to survey statistics with \proglang{R}. Other examples for more specialized functionality are package \pkg{sampling} \citep{sampling} for finite population sampling, and package \pkg{EVER} \citep{EVER} for variance estimation based on efficient resampling. For the common problem of nonresponse, package \pkg{VIM} \citep{VIM} allows to explore the structure of missing data via visualization techniques \citep[see][]{templ12}, and to impute the missing values via advanced imputation methods \citep[e.g.,][]{templ11}. Even a general framework for simulation studies in survey statistics is available through package \pkg{simFrame} \citep{alfons10c, simFrame}. Package \pkg{laeken} provides functionality for the estimation of indicators that is not available in any of the packages listed above, including a novel approach for robust estimation of indicators. While packages \pkg{survey} and \pkg{EVER} require the generation of certain objects describing the survey design prior to analysis, the methods in \pkg{laeken} can be directly applied to the data. This allows \pkg{laeken} to be used more efficiently in simulations, for instance with the \pkg{simFrame} framework. Furthermore, \pkg{laeken} can easily be used on samples drawn with the \pkg{sampling} package or preprocessed with the \pkg{VIM} package. The rest of the paper is organized as follows. Section~\ref{sec:data} introduces the data sets that are used in the examples throughout the paper. In Section~\ref{sec:indicators}, the most widely used indicators on social exclusion and poverty are briefly described. The basic design of the package and its core functionality are then presented in Section~\ref{sec:design}. More advanced topics such as robust estimation and variance estimation via bootstrap techniques are discussed in Sections~\ref{sec:rob} and~\ref{sec:var}, respectively. The final Section~\ref{sec:conclusions} concludes. % --------- % Data sets % --------- \section{Data sets} \label{sec:data} Package \pkg{laeken} contains example data sets for two well-known surveys: the \emph{European Union Statistics on Income and Living Conditions} (EU-SILC) and the \emph{Structure of Earnings Survey} (SES). Since original data from those surveys are confidential, the example data sets are simulated using the methodology described in \citet{alfons11c} and implemented in the \proglang{R} package \pkg{simPopulation} \citep{simPopulation}. Such close-to-reality data sets provide nearly the same multivariate structure as the confidential original data sets and allow researchers to test and compare methods. However, for policy making purposes and economic interpretation, estimations need to be based on the original data. In any case, the simulated data sets are used in the code examples throughout the paper. \subsection{European Union Statistics on Income and Living Conditions} \label{sec:eusilc} EU-SILC is an annual household survey conducted in EU member states and other European countries. Samples consist of about 450 variables containing information on demographics, income and living conditions \citep[see][]{EU-SILC}. Most notably, EU-SILC serves as data basis for measuring risk-of-poverty and social cohesion in Europe. A subset of the indicators computed from EU-SILC is presented in Section~\ref{sec:laeken}. The EU-SILC example data set in \pkg{laeken} is called \code{eusilc} and contains $14\,827$ observations from $6\,000$ households on the 28 most important variables. The data are synthetically generated from preprocessed Austrian EU-SILC data from 2006 provided by Statistics Austria. A description of all the variables is given in the \proglang{R} help page of the data set. To give an overview of what the data look like, the first three observations of the first ten variables of \code{eusilc} are printed below. <<>>= data("eusilc") head(eusilc[, 1:10], 3) @ For this paper, the variable \code{eqIncome} (equivalized disposable income) is of main interest. Other variables are in some cases used to break down the data into different demographics in order to estimate the indicators on those subsets. \subsection{Structure of Earnings Survey} \label{sec:ses} The Structure of Earnings Survey (SES) \citep{SES} is an enterprise survey that aims at providing harmonized data on earnings for almost all European countries. SES data not only contain information on the enterprise level, but also on the individual employment level from a large sample of employees. The most important indicator on the basis of SES data is the gender pay gap, which is described in Section~\ref{sec:GPG}. The SES example data set in \pkg{laeken} is called \code{ses} and contains information on 27 variables and 15\,691 employees from 500 places of work. It is a subset of synthetic data that are simulated from preprocessed Austrian SES 2006 data provided by Statistics Austria. The first three observations of the first seven variables are shown below. <>= data("ses") head(ses[, 1:7], 3) @ In this paper, the SES data is used to illustrate the estimation of the gender pay gap. Hence the most important variables for our purposes are \code{earningsHour}, \code{sex} and \code{education}. For a description of all the variables in the data set, the reader is referred to its \proglang{R} help page. % ---------- % Indicators % ---------- \section{Indicators} \label{sec:indicators} This section gives a brief description of the most widely used indicators on poverty, social cohesion and gender differences. Unless otherwise stated, the presented definitions strictly follow \citet{EU-SILC04, EU-SILC09}. While quick examples for their computation are provided in this section, a detailed discussion on the respective functions is given later on in Section~\ref{sec:design}. % ------------------ % weighted quantiles % ------------------ \subsection{Weighted median and quantile estimation} \label{sec:w} Nearly all of the indicators considered in the paper require the estimation of the median income or other quantiles of the income distribution. Note that in the analysis of income distributions, the median income is of higher interest than the arithmetic mean, since income distributions typically are strongly right-skewed. In mathematical terms, quantiles are defined as $q_{p} := F^{-1}(p)$, where $F$ is the distribution function on the population level and $0 \leq p \leq 1$. The median as an important special case is given by $p = 0.5$. For the following definitions, let $n$ be the number of observations in the sample, let $\boldsymbol{x} := (x_{1}, \ldots, x_{n})^{\top}$ denote the income with \mbox{$x_{1} \leq \ldots \leq x_{n}$}, and let $\boldsymbol{w} := (w_{i}, \ldots, w_{n})^{\top}$ be the corresponding sample weights. Weighted quantiles for the estimation of the population values are then given by \begin{equation} \label{eq:wq} \hat{q}_{p} = \hat{q}_{p} (\boldsymbol{x}, \boldsymbol{w}) := \begin{cases} \frac{1}{2} (x_{j} + x_{j+1}), & \quad \text{if } \sum_{i=1}^{j} w_{i} = p \sum_{i=1}^{n} w_{i}, \\ x_{j+1}, & \quad \text{if } \sum_{i=1}^{j} w_{i} < p \sum_{i=1}^{n} w_{i} < \sum_{i=1}^{j+1} w_{i}. \end{cases} \end{equation} % ------------------- % selected indicators % ------------------- \subsection{Indicators on social exclusion and poverty} \label{sec:laeken} The indicators described in this section are estimated from EU-SILC data based on household income rather than personal income. For each person, this \emph{equivalized disposable income} is defined as the total household disposable income divided by the equivalized household size. It follows that each person in the same household receives the same equivalized disposable income. The total disposable income of a household is thereby calculated by adding together the personal income received by all of the household members plus the income received at the household level. The equivalized household size is defined according to the modified OECD scale, which gives a weight of 1.0 to the first adult, 0.5 to other household members aged 14 or over, and 0.3 to household members aged less than 14. For the definitions of the following indicators, let $\boldsymbol{x} := (x_{1}, \ldots, x_{n})^{\top}$ be the equivalized disposable income with $x_{1} \leq \ldots \leq x_{n}$ and let $\boldsymbol{w} := (w_{i}, \ldots, w_{n})^{\top}$ be the corresponding sample weights, where $n$ denotes the number of observations. Furthermore, define the following index sets for a certain threshold $t$: \begin{align} I_{< t} &:= \{ i \in \{1, \ldots, n\} : x_{i} < t \},\label{eq:01-Ilt}\\ I_{\leq t} &:= \{ i \in \{ 1,\ldots, n\} : x_{i} \leq t \},\label{eq:01-Ileqt}\\ I_{> t} &:= \{ i \in \{1, \ldots, n\} : x_{i} > t\}\label{eq:01-Igt}. \end{align} \subsubsection{At-risk-at-poverty rate} % \label{sec:ARPR} In order to define the \emph{at-risk-of-poverty rate} (ARPR), the \emph{at-risk-of-poverty threshold} (ARPT) needs to be introduced first, which is set at $60\%$ of the national median equivalized disposable income. Then the at-risk-at-poverty rate is defined as the proportion of persons with an equivalized disposable income below the at-risk-at-poverty threshold. In a more mathematical notation, the at-risk-at-poverty rate is defined as \begin{equation} \label{eq:ARPR} ARPR := P(x < 0.6 \cdot q_{0.5}) \cdot 100,% = F(0.6 \cdot q_{0.5}) \cdot 100, \end{equation} where $q_{0.5} := F^{-1}(0.5)$ denotes the population median (50\% quantile) and $F$ is the distribution function of the equivalized income on the population level. For the estimation of the at-risk-at-poverty rate from a sample, first the at-risk-at-poverty threshold is estimated by \begin{equation} \label{eq:ARPT} \widehat{ARPT} = 0.6 \cdot \hat{q}_{0.5}, \end{equation} where $\hat{q}_{0.5}$ is the weighted median as defined in Equation~\ref{eq:wq}. Then the at-risk-at-poverty rate can be estimated by \begin{equation} \widehat{ARPR} := \frac{\sum_{i \in I_{< \widehat{ARPT}}} w_{i}}{\sum_{i=1}^{n} w_{i}} \cdot 100, \end{equation} where $I_{< \widehat{ARPT}}$ is an index set of persons with an equivalized disposable income below the estimated at-risk-of-poverty threshold as defined in Equation~\ref{eq:01-Ilt}. In package \pkg{laeken}, the function \code{arpr()} is implemented to estimate the at-risk-at-poverty rate. <<>>= arpr("eqIncome", weights = "rb050", data = eusilc) @ Note that the at-risk-of-poverty threshold is computed internally by \code{arpr()}. If necessary, it can also be computed by the user through function \code{arpt()}. % <<>>= % arpt("eqIncome", weights = "rb050", data = eusilc) % @ In addition, a highly related indicator is the \emph{dispersion around the at-risk-of-poverty threshold}, which is defined as the proportion of persons with an equivalized disposable income below $40\%$, $50\%$ and $70\%$ of the national weighted median equivalized disposable income. For the estimation of this indicator with function \code{arpr()}, the proportion of the median equivalized income to be used can easily be adjusted via the argument \code{p}. <<>>= arpr("eqIncome", weights = "rb050", p = c(0.4, 0.5, 0.7), data = eusilc) @ \subsubsection{Quintile share ratio} The income \emph{quintile share ratio} (QSR) is defined as the ratio of the sum of the equivalized disposable income received by the 20\% of the population with the highest equivalized disposable income to that received by the 20\% of the population with the lowest equivalized disposable income. For a given sample, let $\hat{q}_{0.2}$ and $\hat{q}_{0.8}$ denote the weighted 20\% and 80\% quantiles, respectively, as defined in Equation~\ref{eq:wq}. Using index sets $I_{\leq \hat{q}_{0.2}}$ and $I_{> \hat{q}_{0.8}}$ as defined in Equations~\ref{eq:01-Ileqt} and~\ref{eq:01-Igt}, respectively, the quintile share ratio is estimated by \begin{equation} \widehat{QSR} := \frac{\sum_{i \in I_{> \hat{q}_{0.8}}} w_{i} x_{i}}{\sum_{i \in I_{\leq \hat{q}_{0.2}}} w_{i} x_{i}}. \end{equation} To estimate the quintile share ratio, the function \code{qsr()} is available. <<>>= qsr("eqIncome", weights = "rb050", data = eusilc) @ \subsubsection{Relative median at-risk-of-poverty gap} The \emph{relative median at-risk-of-poverty gap} (RMPG) is given by the difference between the median equivalized disposable income of persons below the at-risk-of-poverty threshold and the at-risk of poverty threshold itself, expressed as a percentage of the at-risk-of-poverty threshold. For the estimation of the relative median at-risk-of-poverty gap from a sample, let $\widehat{ARPT}$ be the estimated at-risk-of-poverty threshold according to Equation~\ref{eq:ARPT}, and let $I_{< \widehat{ARPT}}$ be an index set of persons with an equivalized disposable income below the estimated at-risk-of-poverty threshold as defined in Equation~\ref{eq:01-Ilt}. Using this index set, define $\boldsymbol{x}_{< \widehat{ARPT}} := (x_{i})_{i \in I_{< \widehat{ARPT}}}$ and $\boldsymbol{w}_{< \widehat{ARPT}} := (w_{i})_{i \in I_{< \widehat{ARPT}}}$. Furthermore, let $\hat{q}_{0.5} (\boldsymbol{x}_{< \widehat{ARPT}}, \boldsymbol{w}_{< \widehat{ARPT}})$ be the corresponding weighted median according to the definition in Equation~\ref{eq:wq}. Then the relative median at-risk-of-poverty gap is estimated by \begin{equation} \widehat{RMPG} = \frac{\widehat{ARPT} - \hat{q}_{0.5} (\boldsymbol{x}_{< \widehat{ARPT}}, \boldsymbol{w}_{< \widehat{ARPT}})}{\widehat{ARPT}} \cdot 100. \end{equation} The relative median at-risk-of-poverty gap is implemented in the function \code{rmpg()}. <<>>= rmpg("eqIncome", weights = "rb050", data = eusilc) @ \subsubsection{Gini coefficient} The \emph{Gini coefficient} is defined as the relationship of cumulative shares of the population arranged according to the level of equivalized disposable income, to the cumulative share of the equivalized total disposable income received by them. Mathematically speaking, the Gini coefficient is estimated from a sample by \begin{equation} \widehat{Gini} := 100 \left[ \frac{2 \sum_{i=1}^{n} \left( w_{i} x_{i} \sum_{j=1}^{i} w_{j} \right) - \sum_{i=1}^{n} w_{i}^{\phantom{i}2} x_{i}}{\left( \sum_{i=1}^{n} w_{i} \right) \sum_{i=1}^{n} \left(w_{i} x_{i} \right)} - 1 \right]. \end{equation} For estimating the Gini coefficient, the function \code{gini()} can be used. <<>>= gini("eqIncome", weights = "rb050", data = eusilc) @ % -------------- % gender pay gap % -------------- \newpage \subsection{The gender pay gap} \label{sec:GPG} Probably the most important indicator derived from the SES data is the \textit{gender pay gap} (GPG). The calculation of the gender pay gap is based on each person's hourly earnings, which are given by the gross monthly earnings from employment divided by the number of hours usually worked per week in employment during $4.33$ weeks. The gender pay gap in unadjusted form is then defined as the difference between average gross earnings of male paid employees and of female paid employees divided by the earnings of male paid employees \citep{EU-SILC04}. Further discussion on the gender pay gap in Europe can be found in, e.g., \citet{beblot03}. For the following definitions, let $\boldsymbol{x} := (x_{1}, \ldots, x_{n})^{\top}$ be the hourly earnings with \mbox{$x_{1} \leq \ldots \leq x_{n}$}, where $n$ is the number of observations. As in the previous sections, $\boldsymbol{w} := (w_{i}, \ldots, w_{n})^{\top}$ denotes the corresponding sample weights. Then define the index set \begin{align*} I_{M} := \{ i \in \{ 1, \ldots, n\} : & \ \text{worked as least 1 hour per week} \ \wedge \\ & \ (16 \leq \text{age} \leq 65) \wedge \, \text{person is male} \}, \end{align*} and define $I_{F}$ analogously as the index set which differs from $I_{M}$ in the fact that it includes females instead of males. With these index sets, the gender pay gap in unadjusted form is estimated by \begin{equation} \label{eq:GPGmean} GPG_{(mean)} = \left( \frac{\sum_{i \in I_{M}} w_i x_i}{\sum_{i \in I_{M}} w_i} - \frac{\sum_{i \in I_{F}} w_i x_i}{\sum_{i \in I_{F} w_i}} \right) \Bigg/ \ \frac{\sum_{i \in I_{M}} w_i x_i}{\sum_{i \in I_{M}} w_i}. \end{equation} The function \code{gpg()} is implemented in \pkg{laeken} to estimate the gender pay gap. <>= gpg("earningsHour", gender = "sex", weigths = "weights", data = ses) @ While \citet{EU-SILC04} proposes the weighted mean as a measure for the average in the definition of the gender pay gap, the U.S. Census Bureau uses the weighted median %as a robust alternative to better reflect the average in skewed earnings distributions \citep[see, e.g.,][]{Weinberg07}. In this case, the estimate of the gender pay gap in unadjusted form changes to \begin{equation} GPG_{(med)} = \frac{\hat{q}_{0.5}(\boldsymbol{x}_{I_{M}}) - \hat{q}_{0.5}(\boldsymbol{x}_{I_{F}})} {\hat{q}_{0.5}(\boldsymbol{x}_{I_{M}})}, \end{equation} where $\boldsymbol{x}_{I_{M}} := (x_{i})_{i \in I_{M}}$ and $\boldsymbol{x}_{I_{F}} := (x_{i})_{i \in I_{F}}$. It should be noted that even though Eurostat proposes to estimate the gender pay gap via weighted means, Statistics Austria for example uses the variant based on weighted medians as well. In function \code{gpg()}, using the weighted median rather than the weighted mean can be specified via the \code{method} argument. <>= gpg("earningsHour", gender = "sex", weigths = "weights", data = ses, method = "median") @ % ------------ % basic design % ------------ \section{Basic design and core functionality} \label{sec:design} This section discusses the basic design of package \pkg{laeken} and its core functions for the estimation of indicators. \subsection{Indicators and class structure} \label{sec:class} Small examples for computing the social exclusion and poverty indicators with package \pkg{laeken} were already shown in Section~\ref{sec:indicators}. These functions are now discussed in detail. As a reminder, the following indicators are implemented in the package: % \begin{description} \item[\code{arpr()}] for the at-risk-of-poverty rate, as well as the dispersion around the at-risk-of-poverty threshold. \item[\code{qsr()}] for the quintile share ratio. \item[\code{rmpg()}] for the relative median at-risk-of-poverty gap. \item[\code{gini()}] for the gini coefficient. \item[\code{gpg()}] for the gender pay gap. \end{description} % All these functions have a very similar interface and allow to compute point and variance estimates with a single command, even for different subdomains of the data. Most importantly, the user can supply character strings specifying the household income via the first argument and the sample weights via the \code{weights} argument. The data are then taken from the data frame passed as the \code{data} argument. <<>>= gini("eqIncome", weights = "rb050", data = eusilc) @ Alternatively, the user can supply the data directly as vectors: <<>>= gini(eusilc$eqIncome, weights = eusilc$rb050) @ For a full list of arguments, the reader is referred to the \proglang{R} help page of the corresponding function. Package \pkg{laeken} follows an object-oriented design using \proglang{S3} classes \citep{chambers92}. Thus each of the above functions returns an object of a certain class for the respective indicator. All those classes thereby inherit from the class \code{"indicator"}. Among other information, the basic class \code{"indicator"} contains the following components: % \begin{description} \item[\code{value}:] the point estimate. \item[\code{valueByStratum}:] a data frame containing the point estimates for each domain. \item[\code{var}:] the variance estimate. \item[\code{varByStratum}:] a data frame containing the variance estimates for each domain. \item[\code{ci}:] the confidence interval. \item[\code{ciByStratum}:] a data frame containing the confidence intervals for each domain. \end{description} % All indicators inherit the components of class \code{"indicator"}, as well as the methods that are defined for this basic class, which has the advantage that code can be shared among the set of indicators. However, each indicator also has its own class such that methods unique to the indicator can be defined. Following a common convention for \proglang{S3} classes, the classes for the indicators have the same names as the functions for computing them. Hence the following classes are implemented in package \pkg{laeken}: % \begin{itemize} \item Class \code{"arpr"} with the following additional components: \begin{description} \item[\code{p}:] the percentage of the weighted median used for the at-risk-of-poverty threshold. \item[\code{threshold}:] the at-risk-of-poverty threshold. \end{description} \item Class \code{"qsr"} with no additional components. \item Class \code{"rmpg"} with the following additional components: \begin{description} \item[\code{threshold}:] the at-risk-of-poverty threshold. \end{description} \item Class \code{"gini"} with no additional components. \item Class \code{"gpg"} with no additional components. \end{itemize} % Furthermore, functions to test whether an object is a member of the basic class or one of the subclasses are available. The function to test for the basic class is called \code{is.indicator()}. Similarly, the functions to test for the subclasses are called \code{is.foo()}, where \code{foo} is the name of the corresponding class (e.g., \code{is.arpr()}). % <<>>= % a <- arpr("eqIncome", weights = "rb050", data = eusilc) % is.arpr(a) % is.indicator(a) % class(a) % @ \subsection{Estimating the indicators in subdomains} \label{sec:sub} One of the most important features of \pkg{laeken} is that indicators can easily be evaluated for different subdomains. These can be regions, but also any other breakdown given by a categorical variable, for instance age categories or gender. All the user needs to do is to specify such a categorical variable via the \code{breakdown} argument. Note that for the at-risk-of-poverty rate and relative median at-risk-of-poverty gap, the same overall at-risk-of-poverty threshold is used for all subdomains \citep[see][]{EU-SILC04, EU-SILC09}. In the following example, the overall estimate for the at-risk-of-poverty rate is computed together with more regional estimates. <>= a <- arpr("eqIncome", weights = "rb050", breakdown = "db040", data = eusilc) a @ \subsection[Extracting information using the subset() method]{Extracting information using the \code{subset()} method} \label{sec:subset} If estimates of an indicator have been computed for several subdomains, extracting a subset of the results for some domains of particular interest can be done with the corresponding \code{subset()} method. For example, the following command extracts the estimates of the at-risk-of-poverty rate for the regions Lower Austria and Vienna from the object computed above. <<>>= subset(a, strata = c("Lower Austria", "Vienna")) @ It is thereby worth pointing out that not every indicator needs its own \code{subset()} method due to inheritance from the basic class \code{"indicator"}. % ----------------- % Robust estimation % ----------------- \newpage \section{Robust estimation} \label{sec:rob} In economic data, variables such as income are typically heavy-tailed and may contain outliers. To identify extreme outliers, we model heavy tails with a Pareto distribution. In the survey setting, the upper tail of the population values are assumed to follow a Pareto distribution. The \pkg{laeken} package includes recently developed methods of \citet{alfons13a} that allow sampling weights to be incorporated into the Pareto model estimation. In the remainder of the section, we briefly outline the methodology and demonstrate how it can be implemented with the \pkg{laeken} package. \subsection{Pareto distribution} \label{sec:Pareto} The \emph{Pareto distribution} is defined in terms of its cumulative distribution function \begin{equation} \label{eq:CDF} F_{\theta}(x) = 1 - \left( \frac{x}{x_{0}} \right) ^{-\theta}, \qquad x \geq x_{0}, \end{equation} where $x_{0} > 0$ is the scale parameter and $\theta > 0$ is the shape parameter \citep{kleiber03}. Furthermore, its density function is given by \begin{equation} f_{\theta}(x) = \frac{\theta x_{0}^{\theta}}{x^{\theta + 1}}, \qquad x \geq x_{0}. \end{equation} Clearly, the Pareto distribution is a highly right-skewed distribution with a heavy tail. In Pareto tail modeling, the cumulative distribution function on the whole range of $x$ is then modeled as \begin{equation} \label{eq:tail} F(x) = \left\{ \begin{array}{ll} G(x), & \quad \text{if } x \leq x_{0}, \\ G(x_{0}) + (1 - G(x_{0})) F_{\theta}(x), & \quad \text{if } x > x_{0}, \end{array} \right. \end{equation} where $G$ is an unknown distribution function \citep{dupuis06}. For a given survey sample, let $\boldsymbol{x} = (x_{1}, \ldots, x_{n})^{\top}$ be the observed values of the variable of interest with $x_{1} \leq \ldots \leq x_{n}$ and $\boldsymbol{w} := (w_{i}, \ldots, w_{n})^{\top}$ the corresponding sample weights, where $n$ denotes the total number of observations. In addition, let $k$ denote the number of observations to be used for tail modeling. Note that the estimation of $x_{0}$ and $k$ directly correspond with each other. If $k$ is fixed, the threshold is estimated by $\hat{x}_{0} = x_{n-k}$. If in turn an estimate $\hat{x}_{0}$ is obtained, $k$ is given by the number of observations that are larger than $\hat{x}_{0}$. In this section, we focus on the EU-SILC example data, where the equivalized disposable income is the main variable of interest. To illustrate the robustness of the presented methods, we replace the equivalized disposable income of the household with the highest income with a large outlier. Note that the resulting income vector is stored in a new variable. <<>>= hID <- eusilc$db030[which.max(eusilc$eqIncome)] eqIncomeOut <- eusilc$eqIncome eqIncomeOut[eusilc$db030 == hID] <- 10000000 @ Moreover, since the equivalized disposable income is a form of household income, the Pareto distribution needs to be modeled on the household level rather than the personal level. Thus we create a data set that only contains the equivalized disposable income with the outlier and the sample weights on the household level. <<>>= keep <- !duplicated(eusilc$db030) eusilcH <- data.frame(eqIncome=eqIncomeOut, db090=eusilc$db090)[keep,] @ \subsection{Pareto quantile plot and finding the threshold} \label{sec:threshold} The first step in any practical analysis should be to explore the data with visualization techniques. For our purpose, the \emph{Pareto quantile plot} is a powerful tool to check whether the Pareto model is appropriate. The plot was introduced by \citet{beirlant96a} for the case without sample weights, and adapted to take sample weights into account by \citet{alfons13a}. The idea behind the Pareto quantile plot is that under the Pareto model, there exists a linear relationship between the logarithms of the observed values and the quantiles of the standard exponential distribution. For survey samples, the observed values are therefore plotted against the quantities \begin{equation} \label{eq:quantiles} -\log \left( 1 - \frac{\sum_{j=1}^{i} w_{j}}{\sum_{j=1}^{n} w_{j}} \frac{n}{n+1} \right), \qquad i = 1, \ldots, n. \end{equation} When all sample weights are equal, the correction factor $n/(n+1)$ ensures that Equation~\ref{eq:quantiles} reduces to the theoretical quantiles taken on the $n$ inner grid points from $n+1$ equally sized subsets of the interval $[0,1]$ \citep[see][for details]{alfons13a}. \begin{figure}[t!] \begin{center} \setkeys{Gin}{width=0.65\textwidth} <>= paretoQPlot(eusilcH$eqIncome, w = eusilcH$db090) @ \caption{Pareto quantile plot for the EU-SILC example data on the household level with the largest observation replaced by an outlier.} \label{fig:ParetoQuantile} \end{center} \end{figure} In package \pkg{laeken}, the Pareto quantile plot is implemented in the function \code{paretoQPlot()}. Figure~\ref{fig:ParetoQuantile} shows the resulting plot for the EU-SILC example data on the household level. Since the tail of the data forms almost a straight line, the Pareto tail model is suitable for the data at hand. Moreover, Figure~\ref{fig:ParetoQuantile} illustrates the two main advantages that make the Pareto quantile plot so powerful. First, nonrepresentative outliers (i.e., extremely large observations that deviate from the Pareto model) are clearly visible. In our example, the outlier that we introduced into the data set is located far away from the rest of the data in the top right corner of the plot. Second, the leftmost point of a fitted line in the tail of the data can be used as an estimate of the threshold $x_{0}$ in the Pareto model, i.e., the scale parameter of fitted Pareto distribution. The slope of the fitted line is then in turn an estimate of $1/\theta$, the reciprocal of the shape parameter. A disadvantage of this graphical method to determine the parameters of the fitted Pareto distribution is of course that it is not very exact. Nevertheless, the function \code{paretoQPlot()} allows the user to select the threshold in the Pareto model interactively by clicking on a data point. Information on the selected threshold is thereby printed on the \proglang{R} console. This process can be repeated until the user terminates the interactive session, typically by a secondary mouse click. Then the selected threshold is returned as an object of class \code{"paretoScale"}, which consists of the component \code{x0} for the threshold (scale parameter) and the component \code{k} for the number of observations in the tail (i.e., larger than the threshold). \subsubsection{Van Kerm's rule of thumb} For EU-SILC data, \citet{vankerm07} developed a formula for the threshold $x_{0}$ in the Pareto model that has more of a rule-of-thumb nature. It is given by \begin{equation} \hat{x}_{0} := \min(\max(2.5\bar{x}, \hat{q}_{0.98}), \hat{q}_{0.97}), \end{equation} where $\bar{x}$ is the weighted mean, and $\hat{q}_{0.98}$ and $\hat{q}_{0.97}$ are weighted quantiles as defined in Equation~\ref{eq:wq}. It is important to note that this formula is designed specifically for the equivalized disposable income in EU-SILC data and can withstand a small number of nonrepresentative outliers. In \pkg{laeken}, the function \code{paretoScale()} provides functionality for estimating the threshold via \citeauthor{vankerm07}'s formula. Its argument \code{w} can be used to supply sample weights. <<>>= ts <- paretoScale(eusilcH$eqIncome, w = eusilcH$db090) ts @ The estimated threshold is again returned as an object of class \code{"paretoScale"}. % \subsubsection{Other methods for finding the threshold} % % Many procedures for finding the threshold in the Pareto model have been % introduced in the literature. For instance, \citet*{beirlant96b, beirlant96a} % developed an analytical procedure for finding the optimal number of % observations in the tail for the maximum likelihood estimator of the shape % parameter by minimizing the asymptotic mean squared error (AMSE). This % procedure is available in \pkg{laeken} through function \code{minAMSE()}, but % is not further discussed here since it is not robust. \citet{dupuis06}, on the % other hand, proposed a robust prediction error criterion for choosing the % optimal number of observations in the tail and the shape parameter % simultaneously. Nevertheless, our implementation of this robust criterion is % unstable and is therefore not included in \pkg{laeken}. \subsection{Estimation of the shape parameter} \label{sec:shape} Once the threshold for the Pareto model is determined, the shape parameter $\theta$ can be estimated via the \emph{points over threshold} method, i.e., by fitting the distribution to the $k$ data points that are larger than the threshold. Since our aim is to identify extreme outliers that deviate from the Pareto model, the shape parameter needs to be estimated in a robust way. \subsubsection{Integrated squared error estimator} The integrated squared error (ISE) criterion was first introduced by \citet{terrell90} as a more robust alternative to maximum likelihood estimation. \citet{vandewalle07} proposed to use this criterion in the context of Pareto tail modeling, but they do not consider sample weights. However, the Pareto distribution is modeled in terms of the \emph{relative excesses} \begin{equation} y_{i} := \frac{x_{n-k+i}}{x_{n-k}}, \qquad i = 1, \ldots, k. \end{equation} Now the density function of the Pareto distribution for the relative excesses is approximated by \begin{equation} f_{\theta}(y) = \theta y^{-(1+\theta)}. \end{equation} With this model density, the integrated squared error criterion can be written as \begin{equation} \hat{\theta} = \arg \min_{\theta} \left[ \int f_{\theta}^{2}(y) dy - 2 \mathbb{E}(f_{\theta}(Y)) \right] , \end{equation} see \citet{vandewalle07}. For survey samples, \citet{alfons13a} propose to use the weighted mean as an estimator of $\mathbb{E}(f_{\theta}(Y))$ to obtain the \emph{weighted integrated squared error} (wISE) estimator: \begin{equation} \label{eq:wISE} \hat{\theta}_{\mathrm{wISE}} = \arg \min_{\theta} \left[ \int f_{\theta}^{2}(y) dy - \frac{2}{\sum_{i=1}^{k} w_{n-k+i}} \sum_{i=1}^{k} w_{n-k+i} f_{\theta}(y_{i}) \right] . \end{equation} The wISE estimator can be computed using the function \code{thetaISE()}. The arguments \code{k} and \code{x0} are available to supply either the number of observations in the tail or the threshold, and sample weights can be supplied via the argument \code{w}. <<>>= thetaISE(eusilcH$eqIncome, k = ts$k, w = eusilcH$db090) thetaISE(eusilcH$eqIncome, x0 = ts$x0, w = eusilcH$db090) @ \subsubsection{Partial density component estimator} Following the observation by \citet{scott04} that $f_{\theta}$ in the ISE criterion does not need to be a real density, \citet{vandewalle07} proposed to minimize the ISE criterion based on an incomplete density mixture model $u f_{\theta}$ instead. \citet{alfons13a} generalized their estimator to take sample weights into account, yielding the \emph{weighted partial density component} (wPDC) estimator \begin{equation} \label{eq:wPDC} \hat{\theta}_{\mathrm{wPDC}} = \arg \min_{\theta} \left[ u^{2} \int f_{\theta}^{2}(y) dy - \frac{2u}{\sum_{i=1}^{k} w_{n-k+i}} \sum_{i = 1}^{k} w_{n-k+i} f_{\theta}(y_{i}) \right] \end{equation} with \begin{equation} \hat{u} = \left. \frac{1}{\sum_{i=1}^{k} w_{n-k+i}} \sum_{i = 1}^{k} w_{n-k+i} f_{\hat{\theta}}(y_{i}) \right/ \int f_{\hat{\theta}}^{2}(y) dy. \end{equation} Based on extensive simulation studies, \citet{alfons13a} conclude that the wPDC estimator is favorable over the wISE estimator due to better robustness properties. The function \code{thetaPDC()} is implemented in package \pkg{laeken} to compute the wPDC estimator. As before, it is necessary to supply either the number of observations in the tail via the argument \code{k}, or the threshold via the argument \code{x0}. Sample weights can be supplied using the argument \code{w}. <<>>= thetaPDC(eusilcH$eqIncome, k = ts$k, w = eusilcH$db090) thetaPDC(eusilcH$eqIncome, x0 = ts$x0, w = eusilcH$db090) @ % \subsubsection{Other estimators for the shape parameter} % Many other estimators for the shape parameter are implemented in package % \pkg{laeken}, e.g., the maximum likelihood estimator \citep{hill75} or the more % robust weighted maximum likelihood estimator \citep{dupuis02}. However, those % estimators are either not robust or have not (yet) been adapted for sample % weights and are therefore not further discussed in this paper. \subsection{Robust estimation of the indicators via Pareto tail modeling} \label{sec:fit} The basic idea for robust estimation of the indicators is to first detect nonrepresentative outliers based on the Pareto model. Afterwards their influence on the indicators is reduced by either downweighting the outliers and recalibrating the remaining observations, or by replacing the outlying values with values from the fitted distribution. The main advantage of this general approach is that it can be applied to any indicator. With the fitted Pareto distribution $F_{\hat{\theta}}$, nonrepresentative outliers can now be detected as observations being larger than a certain $F_{\hat{\theta}}^{-1}(1-\alpha)$ quantile. From extensive simulation studies \citep{AMELI-D7.1, alfons13a}, $\alpha = 0.005$ or $\alpha = 0.01$ are seem suitable choices for this tuning parameter. Then the following approaches are implemented in \pkg{laeken} to reduce the influence of the outliers: % \begin{description} \item[Calibration of nonrepresentative outliers (CN):] As nonrepresentative outliers are considered to be somewhat unique to the population data, the sample weights of the corresponding observations are set to 1. The weights of the remaining observations are adjusted accordingly by calibration \citep[see, e.g.,][]{deville93}. \item[Replacement of nonrepresentative outliers (RN):] The outliers are replaced by values drawn from the fitted distribution $F_{\hat{\theta}}$, thereby preserving the order of the original values. \item[Shrinkage of nonrepresentative outliers (SN):] The outliers are shrunken to the theoretical quantile $F_{\hat{\theta}}^{-1}(1-\alpha)$ used for outlier detection. \end{description} % A more mathematical formulation and further details on the CN and RN approaches can be found in \citet{alfons13a}, who advocate the CN approach in combination with the wPDC estimator for fitting the Pareto distribution. For a practical analysis with package \pkg{laeken}, let us first revisit the estimation of the shape parameter. Rather than applying a function such as \code{thetaPDC()} directly as in the previous section, the function \code{paretoTail()} should be used to fit the Pareto distribution to the upper tail of the data. It returns an object of class \code{"paretoTail"}, which contains all necessary information for further analysis with one of the approaches described above. <>= fit <- paretoTail(eqIncomeOut, k = ts$k, w = eusilc$db090, groups = eusilc$db030) @ Note that the household IDs are supplied via the argument \code{groups} such that the Pareto distribution is fitted on the household level rather than the individual level. By default, the wPDC is used to estimate the shape parameter, but other estimators can be specified via the \code{method} argument. In addition, the tuning parameter $\alpha$ for outlier detection can be supplied as argument \code{alpha}. \begin{figure}[t!] \begin{center} \setkeys{Gin}{width=0.65\textwidth} <>= plot(fit) @ \caption{Pareto quantile plot for the EU-SILC example data with additional diagnostic information on the fitted distribution and any detected outliers.} \label{fig:diagnostic} \end{center} \end{figure} Moreover, the \code{plot()} method for \code{"paretoTail"} objects produces a Pareto quantile plot (see Section~\ref{sec:threshold}) with additional diagnostic information. Figure~\ref{fig:diagnostic} contains the resulting plot for the object computed above. The lower horizontal dotted line corresponds to the estimated threshold $\hat{x}_{0}$, whereas the slope of the solid grey line is given by the reciprocal of the estimated shape parameter $\hat{\theta}$. Furthermore, the upper horizontal dotted line represents the theoretical quantile used for outlier detection. In this example, the threshold seems somewhat too high. Nevertheless, the estimate of the shape parameter is accurate and the cutoff point for outlier detection is appropriate, resulting in correct identification of the outlier that we added to the data set. For downweighting nonrepresentative outliers, the function \code{reweightOut()} is available. It returns a vector of the recalibrated weights. In the command below, we use regional information as auxiliary variables for calibration. The function \code{calibVars()} thereby transforms a factor into a matrix of binary variables. The returned recalibrated weights are then simply used to estimate the Gini coefficient with function \code{gini()}. <<>>= w <- reweightOut(fit, calibVars(eusilc$db040)) gini(eqIncomeOut, w) @ To replace the nonrepresentative outliers with values drawn from the fitted distribution, the function \code{replaceOut()} is implemented. For reproducible results, the seed of the random number generator is set beforehand. The returned income vector is then supplied to \code{gini()} to estimate the Gini coefficient. <<>>= set.seed(123) eqIncomeRN <- replaceOut(fit) gini(eqIncomeRN, weights = eusilc$rb050) @ Similarly, the function \code{shrinkOut()} can be used to shrink the nonrepresentative outliers to the theoretical quantile used for outlier detection. <<>>= eqIncomeSN <- shrinkOut(fit) gini(eqIncomeSN, weights = eusilc$rb050) @ All three robust estimates are very close to the original value before the outlying household had been introduced (see Section~\ref{sec:laeken}). For comparison, we compute the standard estimate of Gini coefficient with the income vector including the outlying household. <<>>= gini(eqIncomeOut, weights = eusilc$rb050) @ Clearly, the standard estimate shows an unreasonably large influence of only one outlying household, illustrating the need for the robust methods. % ------------------- % Variance estimation % ------------------- \section{Variance estimation} \label{sec:var} The \pkg{laeken} package uses bootstrap techniques for estimating the variance of complex survey indicators. Bootstrap methods in general provide better estimates for nonsmooth estimators than other other resampling techniques such as jackknifing or balanced repeated replication \citep[e.g.,][]{AMELI-D3.1}. The naive bootstrap in \pkg{laeken} is quite fast to compute and provides reasonable estimates whenever there is not much variation in the sample weights, which is for example typically the case for EU-SILC data. If there is larger variation among the sample weights, a calibrated bootstrap should be applied. We describe both approaches and their implementation in the following sections. \subsection{Naive bootstrap} \label{sec:naive} Let $\tau$ denote a certain indicator of interest and let $\boldsymbol{X} := (\bold{x}_{1}, \ldots, \bold{x}_{n})^{\top}$ be a survey sample with $n$ observations. Then the \emph{naive bootstrap} algorithm for estimating the variance and confidence interval of an estimate $\hat{\tau}(\boldsymbol{X})$ of the indicator can be summarized as follows: \begin{enumerate} \item Draw $R$ independent bootstrap samples $\boldsymbol{X}_{1}^{*}, \ldots, \boldsymbol{X}_{R}^{*}$ from $\boldsymbol{X}$. For stratified sampling designs, resampling is performed within each stratum independently. \item Compute the bootstrap replicate estimates $\hat{\tau}_{r}^{*} := \hat{\tau}(\boldsymbol{X}_{r}^{*})$ for each bootstrap sample $\boldsymbol{X}_{r}^{*}$, $r = 1, \ldots, R$, taking the sample weights from the respective bootstrap samples into account. \item Estimate the variance $V(\hat{\tau})$ by the variance of the $R$ bootstrap replicate estimates: \begin{equation} \hat{V}(\hat{\tau}) := \frac{1}{R-1} \sum_{r=1}^{R} \left( \hat{\tau}_{r}^{*} - \frac{1}{R} \sum_{s=1}^{R} \hat{\tau}_{s}^{*} \right)^{2}. \end{equation} \item Estimate the confidence interval at confidence level $1 - \alpha$ by one of the following methods \citep[for details, see][]{davison97}: \begin{description} \item[Percentile method:] $\left[ \hat{\tau}_{((R+1) \frac{\alpha}{2})}^{*}, \hat{\tau}_{((R+1)(1-\frac{\alpha}{2}))}^{*} \right]$, as suggested by \cite{efron93}. \item[Normal approximation:] $\hat{\tau} \pm z_{1-\frac{\alpha}{2}} \cdot \hat{V}(\hat{\tau})^{1/2}$ with $z_{1-\frac{\alpha}{2}} = \Phi^{-1}(1 - \frac{\alpha}{2})$. \item[Basic bootstrap method:] $\left[ 2\hat{\tau} - \hat{\tau}_{((R+1)(1-\frac{\alpha}{2}))}^{*}, 2\hat{\tau} - \hat{\tau}_{((R+1)\frac{\alpha}{2})}^{*} \right]$. \end{description} For the percentile and the basic bootstrap method, $\hat{\tau}_{(1)}^{*} \leq \ldots \leq \hat{\tau}_{(R)}^{*}$ denote the order statistics of the bootstrap replicate estimates. \end{enumerate} With package \pkg{laeken}, variance estimates and confidence intervals can easily be included in the estimation of an indicator. It is only necessary to specify a few more arguments in the call to the function computing the indicator. The argument \code{var} is available to specify the type of variance estimation, although only the bootstrap is currently implemented. Furthermore, the significance level $\alpha$ for the confidence intervals can be supplied via the argument \code{alpha} (the default is to use \code{alpha=0.05} for 95\% confidence intervals). Additional arguments are then passed to the underlying function for variance estimation. <>= arpr("eqIncome", weights = "rb050", design = "db040", cluster = "db030", data = eusilc, var = "bootstrap", bootType = "naive", seed = 1234) @ For the bootstrap, the function \code{bootVar()} is called internally for variance and confidence interval estimation. Important arguments are \code{design} and \code{cluster} for specifying the strata and clusters in the sampling design, \code{R} for supplying the number of bootstrap replicates, \code{bootType} for specifying the type of bootstrap estimator, and \code{ciType} for specifying the type of confidence interval. For reproducibility, the seed of the random number generator can be set via the argument \code{seed}. An important feature of package \pkg{laeken} is that indicators can be estimated for different subdomains with a single command, which still holds for variance and confidence interval estimation. As for point estimation, only the \code{breakdown} argument needs to be specified (cf. the example in Section~\ref{sec:sub}). \subsection{Calibrated bootstrap} \label{sec:calib} In practice, the initial sample weights from the sampling design are often adjusted by calibration, for instance to account for non-response or to ensure that the sums of the sample weights for all observations within certain subgroups equal the respective known population sizes. However, drawing a bootstrap sample then has the effect that the sample weights in the bootstrap sample no longer sum up to the correct values. As a remedy, the sample weights of each bootstrap sample should be recalibrated. For better accuracy at a higher computational cost, the \emph{calibrated bootstrap} algorithm extends the naive bootstrap algorithm from the previous section by adding the following step between Steps~1 and~2: \begin{itemize} \item[1b.] Calibrate the sample weights for each bootstrap sample $\boldsymbol{X}_{r}^{*}$, $r = 1, \ldots, R$ \citep[see, e.g.,][for details on calibration]{deville92, deville93}. \end{itemize} Using \pkg{laeken}, the function call for including variance and confidence intervals via the calibrated bootstrap is very similar to its counterpart for the naive bootstrap. A matrix of auxiliary calibration variables needs to be supplied via the argument \code{X}. The function \code{calibVars()} can thereby by used to transform a factor into a matrix of binary variables. In the %examples example below, information on region and gender is used for calibration. Furthermore, the argument \code{totals} can be used to supply the corresponding population totals. If the \code{totals} argument is omitted, the population totals are computed from the sample weights of the original sample. This follows the assumption that those weights are already calibrated on the supplied auxiliary variables. <>= aux <- cbind(calibVars(eusilc$db040), calibVars(eusilc$rb090)) arpr("eqIncome", weights = "rb050", design = "db040", cluster = "db030", data = eusilc, var = "bootstrap", X = aux, seed = 1234) @ % ----------- % Conclusions % ----------- \section{Conclusions} \label{sec:conclusions} In this paper, we demonstrate the use of the \proglang{R} package \pkg{laeken} for computing point and variance estimates of indicators from complex surveys. Various commonly used indicators on social exclusion and poverty are thereby implemented. Their estimation is made easy with the package, as the corresponding functions allow to compute point and variance estimates with a single command, even for different subdomains of the data. In addition, we illustrate with a simple example that some of the indicators are highly influenced by extreme outliers in the data \citep[cf.][]{hulliger09a, alfons13a}. As a remedy, a general procedure for robust estimation of the indicators is implemented in \pkg{laeken}. The procedure is based on fitting a Pareto distribution to the upper tail of the data and has the advantage that it can be applied to any indicator. A diagnostic plot thereby allows to check whether the Pareto tail model is appropriate for the data at hand. Concerning variance estimation, further techniques for complex survey samples are available in \proglang{R} through other packages. For instance, package \pkg{EVER} \citep{EVER} provides functionality for the delete-a-group jackknife. Other methods such as balanced repeated replication are implemented in package \pkg{survey} \citep{lumley04, survey}. The incorporation of those packages for additional variance estimation procedures is therefore considered for future work. % --------------------- % computational details % --------------------- % \section*{Computational details} % All computations in this paper were performed using \pkg{Sweave} % \citep{leisch02a} with the following \proglang{R} session: % <>= % toLatex(sessionInfo(), locale=FALSE) % @ % % % The most recent version of package \pkg{laeken} is always available from CRAN % (the Comprehensive \proglang{R} Archive Network, % \url{https://CRAN.R-project.org}), and (an up-to-date version of) this paper is % also included as a package vignette. % --------------- % acknowledgments % --------------- \section*{Acknowledgments} This work was partly funded by the European Union (represented by the European Commission) within the \engordnumber{7} framework programme for research (Theme~8, Socio-Economic Sciences and Humanities, Project AMELI (Advanced Methodology for European Laeken Indicators), Grant Agreement No. 217322). Visit \url{http://ameli.surveystatistics.net} for more information on the project. % ------------ % bibliography % ------------ % \bibliographystyle{jss} \bibliography{laeken} \end{document} laeken/NEWS0000644000176200001440000001225614127307117012221 0ustar liggesusersChanges in laeken version 0.5.2 + Added argument 'threshold' to function 'arpr'. + Fixed DOI's and URL's in documentation. Changes in laeken version 0.5.1 + Fixed documentation of 'minAMSE'. Changes in laeken version 0.5.0 + Order of factors changed in function 'prop'. + Bugfixes in computation of weighted quantiles and quintile share ratio to strictly follow Eurostat definitions. + Package title now in title case. + Packages 'boot' and 'MASS' are now only Imports instead of Depends. Changes in laeken version 0.4.6 + Estimation of a proportion and its variance (class "prop", function 'prop' and related changes). Changes in laeken version 0.4.5 + Added references to JSS paper. Changes in laeken version 0.4.4 + Updated vignette 'laeken-intro'. + Updated some help files. Changes in laeken version 0.4.3 + Bugfix in 'bootVar' when resampling clusters of observations rather than individuals. Changes in laeken version 0.4.2 + Bootstrap variance estimation now contains an argument 'cluster', which allows for resampling clusters of observations rather than individuals. + Bugfix in 'gpg': arguments 'gender' and 'method' are now passed to 'variance'. + Updated vignette 'laeken-intro'. Changes in laeken version 0.4.1 + Reduced run times of examples for 'bootVar' and 'variance'. + Functions 'arpt' and 'arpr' can now take a vector of percentages of the weighted median to be used for the at-risk-of-poverty threshold. + Updated vignette 'laeken-intro'. Changes in laeken version 0.4.0 + Added example data 'ses' for the Structure of Earnings Survey. + Added vignette 'laeken-intro'. + Minimal changes in 'print' method for indicators. + Minimal changes in help files and vignettes. + Removed attributes from variable 'eqIncome' in example data 'eusilc'. + The level names of argument 'gender' in function 'gpg' can now be any character string. Note: the first level of gender should always correspond to females. Changes in laeken version 0.3.3 + New plot method for objects of class "paretoTail". + Package roxygen2 is now used for documentation. + Bugfixes in 'paretoQPlot' and 'meanExcessPlot': order of graphical parameters 'pch', 'cex', 'col' and 'bg' for the data points is now preserved. + Bugfix in 'meanExcessPlot': computation of weighted quantiles no longer throws error. Changes in laeken version 0.3.2 + Package is built with flag --resave-data to avoid warning with R 2.15.0. Changes in laeken version 0.3.1 + New arguments in function 'paretoScale' for generalizations of Van Kerm's formula. + Updated author affiliation. Changes in laeken version 0.3 + New function 'gpg' for estimating the gender pay (wage) gap. + Fixed function 'bootVar' for package 'boot' >= 1.3-1. Changes in laeken version 0.2.3 + Corrected mistake in formula for weighted Hill estimator in vignette 'laeken-pareto'. Changes in laeken version 0.2.2 + New function 'shrinkOut' for Pareto tail modeling to shrink outliers to the theoretical quantile used for outlier detection. Changes in laeken version 0.2.1 + Fixed help files for 'meanExcessPlot' and 'paretoQPlot' on Microsoft Windows systems. + Fixed package reference in vignette 'laeken-variance'. Changes in laeken version 0.2 + Functions for fitting a Pareto distribution now have an additional argument 'x0' to specify threshold directly instead of using number of observations 'k' in the tail. + In the graphical exploration of the data using 'meanExcessPlot' or 'paretoQPlot', sample weights can now be considered and the threshold (scale parameter) can be selected interactively. + Function 'paretoQPlot' now simply uses logarithmic y-axis to show the labels in the scale of the original values. + Changed default axis labels in 'paretoQPlot'. + Sample weights can now be considered when fitting a Pareto distribution using 'thetaHill', 'thetaISE' or 'thetaPDC'. + New function 'calibVars' for convenient construction of binary variables for calibration. + New functions 'paretoTail', 'replaceTail', 'replaceOut' and 'reweightOut' for improved methodology for Pareto tail modeling with a common interface. + Row names of the example data 'eusilc' are now given by 1 to the number of rows. + New wrapper function 'weightedMean' for the (weighted) mean. + New function 'paretoScale' for estimating the threshold for Pareto tail modeling. + Totals for calibrated bootstrap variance are now by default computed from the original data using the Horvitz-Thompson estimator. + Added package vignettes 'laeken-standard', 'laeken-pareto' and 'laeken-variance'. Changes in laeken version 0.1.3 + Minimal changes in help file for 'calibWeights'. Changes in laeken version 0.1.2 + Bugfix in 'thetaWML' for bias correction term with weight function based on standardized residuals. Changes in laeken version 0.1.1 + Bugfix in 'bootVar' in case of breakdown by year and domain. laeken/R/0000755000176200001440000000000014125312655011716 5ustar liggesuserslaeken/R/bootVar.R0000644000176200001440000006057314127253256013473 0ustar liggesusers# --------------------------------------- # Author: Andreas Alfons # Vienna University of Technology # --------------------------------------- ## TODO: support estimators based on semiparametric outlier detection ## FIXME: do not use 'p' as argument name for function passed to 'boot' #' Bootstrap variance and confidence intervals of indicators on social exclusion #' and poverty #' #' Compute variance and confidence interval estimates of indicators on social #' exclusion and poverty based on bootstrap resampling. #' #' @param inc either a numeric vector giving the equivalized disposable income, #' or (if \code{data} is not \code{NULL}) a character string, an integer or a #' logical vector specifying the corresponding column of \code{data}. #' @param weights optional; either a numeric vector giving the personal sample #' weights, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. #' @param years optional; either a numeric vector giving the different years of #' the survey, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. If supplied, values are computed for each year. #' @param breakdown optional; either a numeric vector giving different domains, #' or (if \code{data} is not \code{NULL}) a character string, an integer or a #' logical vector specifying the corresponding column of \code{data}. If #' supplied, the values for each domain are computed in addition to the overall #' value. #' @param design optional; either an integer vector or factor giving different #' strata for stratified sampling designs, or (if \code{data} is not #' \code{NULL}) a character string, an integer or a logical vector specifying #' the corresponding column of \code{data}. If supplied, this is used as #' \code{strata} argument in the call to \code{\link[boot]{boot}}. #' @param cluster optional; either an integer vector or factor giving different #' clusters for cluster sampling designs, or (if \code{data} is not #' \code{NULL}) a character string, an integer or a logical vector specifying #' the corresponding column of \code{data}. #' @param data an optional \code{data.frame}. #' @param indicator an object inheriting from the class \code{"indicator"} that #' contains the point estimates of the indicator (see \code{\link{arpr}}, #' \code{\link{qsr}}, \code{\link{rmpg}} or \code{\link{gini}}). #' @param R a numeric value giving the number of bootstrap replicates. #' @param bootType a character string specifying the type of bootstap to be #' performed. Possible values are \code{"calibrate"} (for calibration of the #' sample weights of the resampled observations in every iteration) and #' \code{"naive"} (for a naive bootstrap without calibration of the sample #' weights). #' @param X if \code{bootType} is \code{"calibrate"}, a matrix of calibration #' variables. #' @param totals numeric; if \code{bootType} is \code{"calibrate"}, this gives #' the population totals. If \code{years} is \code{NULL}, a vector should be #' supplied, otherwise a matrix in which each row contains the population totals #' of the respective year. If this is \code{NULL} (the default), the population #' totals are computed from the sample weights using the Horvitz-Thompson #' estimator. #' @param ciType a character string specifying the type of confidence #' interval(s) to be computed. Possible values are \code{"perc"}, \code{"norm"} #' and \code{"basic"} (see \code{\link[boot]{boot.ci}}). #' @param alpha a numeric value giving the significance level to be used for #' computing the confidence interval(s) (i.e., the confidence level is \eqn{1 - #' }\code{alpha}), or \code{NULL}. #' @param seed optional; an integer value to be used as the seed of the random #' number generator, or an integer vector containing the state of the random #' number generator to be restored. #' @param na.rm a logical indicating whether missing values should be removed. #' @param gender either a numeric vector giving the gender, or (if \code{data} #' is not \code{NULL}) a character string, an integer or a logical vector #' specifying the corresponding column of \code{data}. #' @param method a character string specifying the method to be used (only for #' \code{\link{gpg}}). Possible values are \code{"mean"} for the mean, and #' \code{"median"} for the median. If weights are provided, the weighted mean #' or weighted median is estimated. #' @param \dots if \code{bootType} is \code{"calibrate"}, additional arguments #' to be passed to \code{\link{calibWeights}}. #' #' @return An object of the same class as \code{indicator} is returned. See #' \code{\link{arpr}}, \code{\link{qsr}}, \code{\link{rmpg}} or #' \code{\link{gini}} for details on the components. #' #' @note This function gives reasonable variance estimates for basic sample #' designs such as simple random sampling or stratified simple random sampling. #' #' @author Andreas Alfons #' #' @seealso \code{\link{variance}}, \code{\link{calibWeights}}, #' \code{\link{arpr}}, \code{\link{qsr}}, \code{\link{rmpg}}, \code{\link{gini}} #' #' @references #' A. Alfons and M. Templ (2013) Estimation of Social Exclusion Indicators #' from Complex Surveys: The \R Package \pkg{laeken}. \emph{Journal of #' Statistical Software}, \bold{54}(15), 1--25. \doi{10.18637/jss.v054.i15} #' #' @keywords survey #' #' @examples #' data(eusilc) #' a <- arpr("eqIncome", weights = "rb050", data = eusilc) #' #' ## naive bootstrap #' bootVar("eqIncome", weights = "rb050", design = "db040", #' data = eusilc, indicator = a, R = 50, #' bootType = "naive", seed = 123) #' #' ## bootstrap with calibration #' bootVar("eqIncome", weights = "rb050", design = "db040", #' data = eusilc, indicator = a, R = 50, #' X = calibVars(eusilc$db040), seed = 123) #' #' @importFrom stats runif #' @importFrom boot boot boot.ci #' @export bootVar <- function(inc, weights = NULL, years = NULL, breakdown = NULL, design = NULL, cluster = NULL, data = NULL, indicator, R = 100, bootType = c("calibrate", "naive"), X, totals = NULL, ciType = c("perc", "norm", "basic"), # type "stud" and "bca" are currently not allowed alpha = 0.05, seed = NULL, na.rm = FALSE, gender = NULL, method = NULL, ...) { UseMethod("bootVar", indicator) } ## class "indicator" #' @export bootVar.indicator <- function(inc, weights = NULL, years = NULL, breakdown = NULL, design = NULL, cluster = NULL, data = NULL, indicator, R = 100, bootType = c("calibrate", "naive"), X, totals = NULL, ciType = c("perc", "norm", "basic"), # type "stud" and "bca" are currently not allowed alpha = 0.05, seed = NULL, na.rm = FALSE, gender = NULL, method = NULL, ...) { ## initializations # check whether weights have been supplied haveWeights <- !is.null(weights) haveGender <- !is.null(gender) # check whether indicator is broken down by year # if so, check whether years have been supplied ys <- indicator$years byYear <- !is.null(ys) if(byYear && is.null(years)) stop("'years' must be supplied") # check whether indicator is broken down by stratum # if so, check whether breakdown has been supplied rs <- indicator$strata byStratum <- !is.null(rs) if(byStratum && is.null(breakdown)) stop("'breakdown' must be supplied") haveDesign <- !is.null(design) haveCluster <- !is.null(cluster) # if a data.frame has been supplied, extract the respective vectors if(!is.null(data)) { inc <- data[, inc] # make numeric if indicator is proportion inc <- as.numeric(as.integer(inc)) if(!is.null(weights)) weights <- data[, weights] if(!is.null(gender)) gender <- data[, gender] if(byYear) years <- data[, years] if(byStratum) breakdown <- data[, breakdown] if(haveDesign) design <- data[, design] if(haveCluster) cluster <- data[, cluster] } # check whether the vectors have the correct type # make numeric if indicator is proportion inc <- as.numeric(as.integer(inc)) if(!is.numeric(inc)) stop("'inc' must be a numeric vector") n <- length(inc) if(haveWeights && !is.numeric(weights)) { stop("'weights' must be a numeric vector") } # if(haveGender && !is.numeric(gender)) { # stop("'gender' must be a numeric vector") # } if(byYear && !is.numeric(years)) { stop("'years' must be a numeric vector") } if(byStratum && !is.vector(breakdown) && !is.factor(breakdown)) { stop("'breakdown' must be a vector or factor") } if(haveDesign && !is.integer(design) && !is.factor(design)) { stop("'design' must be an integer vector or factor") } if(haveCluster && !is.integer(cluster) && !is.factor(cluster)) { stop("'cluster' must be an integer vector or factor") } if(is.null(data)) { # check vector lengths if(haveWeights && length(weights) != n) { stop("'weights' must have length ", n) } if(byYear && length(years) != n) { stop("'years' must have length ", n) } if(byStratum && length(breakdown) != n) { stop("'breakdown' must have length ", n) } if(haveDesign && length(design) != n) { stop("'design' must have length ", n) } if(haveCluster && length(cluster) != n) { stop("'cluster' must have length ", n) } } if(!haveDesign) design <- rep.int(1, n) # check other input if(!is.numeric(R) || length(R) == 0) stop("'R' must be numeric") else R <- as.integer(R[1]) if(!is.numeric(alpha) || length(alpha) == 0) stop("'alpha' must be numeric") else alpha <- alpha[1] bootType <- match.arg(bootType) calibrate <- haveWeights && bootType == "calibrate" if(calibrate) { X <- as.matrix(X) if(!is.numeric(X)) stop("'X' must be a numeric matrix") # if(nrow(X) != n) stop("'X' must have ", n, " rows") if(is.null(totals)) { # compute totals from original data with Horvitz-Thompson estimator if(byYear) { totals <- lapply(ys, function(y) { # extract current year from calibration variables and # weights i <- years == y X <- X[i, , drop=FALSE] weights <- weights[i] # compute totals for current year apply(X, 2, function(i) sum(i*weights)) }) totals <- do.call(rbind, totals) # form matrix of totals rownames(totals) <- ys # use years as rownames for totals } else totals <- apply(X, 2, function(i) sum(i*weights)) } else if(byYear) totals <- as.matrix(totals) if(!is.numeric(totals)) stop("'totals' must be of type numeric") } else { X <- NULL totals <- NULL } ciType <- match.arg(ciType) ## preparations data <- data.frame(inc=inc) data$weight <- weights data$year <- years data$stratum <- breakdown data$cluster <- cluster data$gender <- gender data$method <- method # this is a bit of an ugly hack if(inherits(indicator, "arpr")) { p <- indicator$p # percentage of median used for threshold } else p <- NULL byP <- length(p) > 1 if(!is.null(seed)) set.seed(seed) # set seed of random number generator if(!exists(".Random.seed", envir=.GlobalEnv, inherits=FALSE)) runif(1) seed <- .Random.seed # seed is later on added to the object to be returned ## calculations # get basic function for bootstrap replications with definition: # function(x, i, p, X, totals, rs, na.rm) fun <- getFun(indicator, byStratum) bootFun <- getBootFun(calibrate, fun) if(byYear) { # ---------- breakdown by year ---------- # get more complex function for additional with definition # function(y, x, R, p, aux, totals, rs, alpha, ciType, na.rm, ...) funByYear <- getFunByYear(byStratum, calibrate, bootFun) if(byStratum) { # ---------- breakdown by stratum ---------- tmp <- lapply(ys, funByYear, data, R, design, cluster, p, X, totals, ys, rs, alpha, ciType, na.rm, ...) var <- do.call(c, lapply(tmp, function(x) x[[1]])) names(var) <- ys varByStratum <- do.call(rbind, lapply(tmp, function(x) x[[2]])) ci <- do.call(rbind, lapply(tmp, function(x) x[[3]])) rownames(ci) <- ys ciByStratum <- do.call(rbind, lapply(tmp, function(x) x[[4]])) # order 'varByStratum' and 'ciByStratum' according to 'valueByStratum' tmp <- indicator$valueByStratum[, 1:2] tmp <- data.frame(order=1:nrow(tmp), tmp) varByStratum <- merge(varByStratum, tmp, all=TRUE, sort=FALSE) varByStratum <- varByStratum[order(varByStratum$order), -4] ciByStratum <- merge(ciByStratum, tmp, all=TRUE, sort=FALSE) ciByStratum <- ciByStratum[order(ciByStratum$order), -5] } else { # ---------- no breakdown by stratum ---------- tmp <- sapply(ys, funByYear, data, R, design, cluster, p, X, totals, ys, rs, alpha, ciType, na.rm, ...) colnames(tmp) <- ys var <- tmp[1,] ci <- t(tmp[2:3,]) } } else { # ---------- no breakdown by year or threshold ---------- b <- clusterBoot(data, bootFun, R, strata=design, cluster=cluster, p=p, aux=X, totals=totals, rs=rs, na.rm=na.rm, ...) if(byStratum) { # ---------- breakdown by stratum ---------- var <- apply(b$t, 2, var) ci <- lapply(1:length(b$t0), function(i) { ci <- boot.ci(b, conf=1-alpha, type=ciType, index=i) switch(ciType, perc=ci$percent[4:5], norm=ci$normal[2:3], basic=ci$basic[4:5], stud=ci$student[4:5], bca=ci$bca[4:5]) }) ci <- do.call(rbind, ci) colnames(ci) <- c("lower", "upper") if(byP) { overall <- 1:length(p) tmp <- indicator$valueByStratum[, 1:2] varByStratum <- data.frame(tmp, var=var[-overall]) var <- var[overall] ciByStratum <- data.frame(tmp, ci[-overall, , drop=FALSE]) ci <- ci[overall, , drop=FALSE] names(var) <- rownames(ci) <- names(indicator$value) } else { varByStratum <- data.frame(stratum=rs, var=var[-1]) var <- var[1] ciByStratum <- data.frame(stratum=rs, ci[-1, , drop=FALSE]) ci <- ci[1,] } } else { # ---------- no breakdown by stratum ---------- if(byP) { var <- apply(b$t, 2, var) ci <- lapply(1:length(b$t0), function(i) { ci <- boot.ci(b, conf=1-alpha, type=ciType, index=i) switch(ciType, perc=ci$percent[4:5], norm=ci$normal[2:3], basic=ci$basic[4:5], stud=ci$student[4:5], bca=ci$bca[4:5]) }) ci <- do.call(rbind, ci) colnames(ci) <- c("lower", "upper") names(var) <- rownames(ci) <- names(indicator$value) } else { var <- var(b$t[, 1]) ci <- boot.ci(b, conf=1-alpha, type=ciType) ci <- switch(ciType, perc=ci$percent[4:5], norm=ci$normal[2:3], basic=ci$basic[4:5], stud=ci$student[4:5], bca=ci$bca[4:5]) names(ci) <- c("lower", "upper") } } } ## modify and return object indicator$varMethod <- "bootstrap" indicator$var <- var indicator$ci <- ci if(byStratum) { indicator$varByStratum <- varByStratum indicator$ciByStratum <- ciByStratum } indicator$alpha <- alpha indicator$seed <- seed return(indicator) } ## function to perform clustered bootstrap sampling clusterBoot <- function(data, statistic, ..., strata, cluster = NULL) { if(is.null(cluster)) boot(data, statistic, ..., strata=strata) else { fun <- function(cluster, i, ..., .data, .statistic) { # retrieve sampled individuals i <- do.call(c, split(1:nrow(.data), .data$cluster)[i]) # call the original statistic for the sample of individuals .statistic(.data, i, ...) } keep <- !duplicated(cluster) boot(cluster[keep], fun, ..., strata=strata[keep], .data=data, .statistic=statistic) } } ## utility functions: return functions to be used in the bootstrap replications # basic function for breakdown by stratum getFun <- function(indicator, byStratum) UseMethod("getFun") getFun.arpr <- function(indicator, byStratum) { if(byStratum) { function(x, p, rs, na.rm) { threshold <- p * weightedMedian(x$inc, x$weight) value <- weightedRate(x$inc, x$weight, threshold, na.rm=na.rm) valueByStratum <- sapply(rs, function(r, x, t) { i <- x$stratum == r weightedRate(x$inc[i], x$weight[i], t, na.rm=na.rm) }, x=x, t=threshold) c(value, valueByStratum) } } else { function(x, p, rs, na.rm) { threshold <- p * weightedMedian(x$inc, x$weight) weightedRate(x$inc, x$weight, threshold, na.rm=na.rm) } } } # the argument 'p' is not necessary here, but is used so # that we have a unified function call for all indicators getFun.qsr <- function(indicator, byStratum) { if(byStratum) { function(x, p, rs, na.rm) { value <- quintileRatio(x$inc, x$weight, na.rm=na.rm) valueByStratum <- sapply(rs, function(r, x, t) { i <- x$stratum == r quintileRatio(x$inc[i], x$weight[i], na.rm=na.rm) }, x=x) c(value, valueByStratum) } } else { function(x, p, rs, na.rm) { quintileRatio(x$inc, x$weight, na.rm=na.rm) } } } getFun.rmpg <- function(indicator, byStratum) { if(byStratum) { function(x, p, rs, na.rm) { threshold <- 0.6 * weightedMedian(x$inc, x$weight) value <- relativeGap(x$inc, x$weight, threshold=threshold, na.rm=na.rm) valueByStratum <- sapply(rs, function(r, x, t) { i <- x$stratum == r relativeGap(x$inc[i], x$weight[i], threshold=t, na.rm=na.rm) }, x=x, t=threshold) c(value, valueByStratum) } } else { function(x, p, rs, na.rm) { threshold <- 0.6 * weightedMedian(x$inc, x$weight) relativeGap(x$inc, x$weight, threshold=threshold, na.rm=na.rm) } } } # the argument 'p' is not necessary here, but is used so # that we have a unified function call for all indicators getFun.gini <- function(indicator, byStratum) { if(byStratum) { function(x, p, rs, na.rm) { value <- giniCoeff(x$inc, x$weight, na.rm=na.rm) valueByStratum <- sapply(rs, function(r, x, t) { i <- x$stratum == r giniCoeff(x$inc[i], x$weight[i], na.rm=na.rm) }, x=x) c(value, valueByStratum) } } else { function(x, p, rs, na.rm) { giniCoeff(x$inc, x$weight, na.rm=na.rm) } } } # the argument 'p' is not necessary here, but is used so # that we have a unified function call for all indicators getFun.prop <- function(indicator, byStratum) { if(byStratum) { function(x, p, rs, na.rm) { value <- propCoeff(x$inc, x$weight, na.rm=na.rm) valueByStratum <- sapply(rs, function(r, x, t) { i <- x$stratum == r propCoeff(x$inc[i], x$weight[i], na.rm=na.rm) }, x=x) c(value, valueByStratum) } } else { function(x, p, rs, na.rm) { propCoeff(x$inc, x$weight, na.rm=na.rm) } } } # the argument 'p' is not necessary here, but is used so # that we have a unified function call for all indicators getFun.gpg <- function(indicator, byStratum) { if(byStratum) { function(x, p, rs, na.rm) { value <- genderGap(x$inc, x$gender, x$method[1], x$weight, na.rm=na.rm) valueByStratum <- sapply(rs, function(r, x, t) { i <- x$stratum == r genderGap(x$inc[i], x$gender[i], x$method[1], x$weight[i], na.rm=na.rm) }, x=x) c(value, valueByStratum) } } else { function(x, p, rs, na.rm) { genderGap(x$inc, x$gender, x$method[1], x$weight, na.rm=na.rm) } } } # function that incorporates resampling and (if requested) calibration getBootFun <- function(calibrate, fun) { if(calibrate) { function(x, i, p, aux, totals, rs, na.rm, ...) { x <- x[i, , drop=FALSE] aux <- aux[i, , drop=FALSE] g <- calibWeights(aux, x$weight, totals, ...) x$weight <- g * x$weight fun(x, p, rs, na.rm) } } else { function(x, i, p, aux, totals, rs, na.rm, ...) { x <- x[i, , drop=FALSE] fun(x, p, rs, na.rm) } } } # more complex function for additional breakdown by year getFunByYear <- function(byStratum, calibrate, fun) { if(byStratum) { if(calibrate) { # ---------- breakdown by stratum, calibration ---------- function(y, x, R, design, cluster, p, aux, totals, ys, rs, alpha, ciType, na.rm, ...) { i <- x$year == y x <- x[i, , drop=FALSE] aux <- aux[i, , drop=FALSE] design <- design[i] cluster <- cluster[i] totals <- totals[ys == y,] b <- clusterBoot(x, fun, R, strata=design, cluster=cluster, p=p, aux=aux, totals=totals, rs=rs, na.rm=na.rm, ...) var <- apply(b$t, 2, var) varByStratum <- data.frame(year=y, stratum=rs, var=var[-1]) var <- var[1] ci <- sapply(1:((length(rs) + 1)), function(i) { ci <- boot.ci(b, conf=1-alpha, type=ciType, index=i) switch(ciType, perc=ci$percent[4:5], norm=ci$normal[2:3], basic=ci$basic[4:5], stud=ci$student[4:5], bca=ci$bca[4:5]) }) rownames(ci) <- c("lower", "upper") ciByStratum <- data.frame(year=y, stratum=rs, t(ci[, -1])) ci <- ci[, 1] list(var, varByStratum, ci, ciByStratum) } } else { # ---------- breakdown by stratum, no calibration ---------- function(y, x, R, design, cluster, p, aux, totals, ys, rs, alpha, ciType, na.rm, ...) { i <- x$year == y x <- x[i, , drop=FALSE] design <- design[i] cluster <- cluster[i] b <- clusterBoot(x, fun, R, strata=design, cluster=cluster, p=p, aux=aux, totals=totals, rs=rs, na.rm=na.rm, ...) var <- apply(b$t, 2, var) varByStratum <- data.frame(year=y, stratum=rs, var=var[-1]) var <- var[1] ci <- sapply(1:((length(rs) + 1)), function(i) { ci <- boot.ci(b, conf=1-alpha, type=ciType, index=i) switch(ciType, perc=ci$percent[4:5], norm=ci$normal[2:3], basic=ci$basic[4:5], stud=ci$student[4:5], bca=ci$bca[4:5]) }) rownames(ci) <- c("lower", "upper") ciByStratum <- data.frame(year=y, stratum=rs, t(ci[, -1])) ci <- ci[, 1] list(var, varByStratum, ci, ciByStratum) } } } else { if(calibrate) { # ---------- no breakdown by stratum, calibration ---------- function(y, x, R, design, cluster, p, aux, totals, ys, rs, alpha, ciType, na.rm, ...) { i <- x$year == y x <- x[i, , drop=FALSE] aux <- aux[i, , drop=FALSE] design <- design[i] cluster <- cluster[i] totals <- totals[ys == y,] b <- clusterBoot(x, fun, R, strata=design, cluster=cluster, p=p, aux=aux, totals=totals, rs=rs, na.rm=na.rm, ...) var <- var(b$t[, 1]) ci <- boot.ci(b, conf=1-alpha, type=ciType) ci <- switch(ciType, perc=ci$percent[4:5], norm=ci$normal[2:3], basic=ci$basic[4:5], stud=ci$student[4:5], bca=ci$bca[4:5]) names(ci) <- c("lower", "upper") c(var, ci) } } else { # ---------- no breakdown by stratum, no calibration ---------- function(y, x, R, design, cluster, p, aux, totals, ys, rs, alpha, ciType, na.rm, ...) { i <- x$year == y x <- x[i, , drop=FALSE] design <- design[i] cluster <- cluster[i] b <- clusterBoot(x, fun, R, strata=design, cluster=cluster, p=p, aux=aux, totals=totals, rs=rs, na.rm=na.rm, ...) var <- var(b$t[, 1]) ci <- boot.ci(b, conf=1-alpha, type=ciType) ci <- switch(ciType, perc=ci$percent[4:5], norm=ci$normal[2:3], basic=ci$basic[4:5], stud=ci$student[4:5], bca=ci$bca[4:5]) names(ci) <- c("lower", "upper") c(var, ci) } } } } laeken/R/thetaHill.R0000644000176200001440000001071413616467254013774 0ustar liggesusers# ---------------------------------------- # Authors: Andreas Alfons and Josef Holzer # Vienna University of Technology # ---------------------------------------- #' Hill estimator #' #' The Hill estimator uses the maximum likelihood principle to estimate the #' shape parameter of a Pareto distribution. #' #' The arguments \code{k} and \code{x0} of course correspond with each other. #' If \code{k} is supplied, the threshold \code{x0} is estimated with the \eqn{n #' - k} largest value in \code{x}, where \eqn{n} is the number of observations. #' On the other hand, if the threshold \code{x0} is supplied, \code{k} is given #' by the number of observations in \code{x} larger than \code{x0}. Therefore, #' either \code{k} or \code{x0} needs to be supplied. If both are supplied, #' only \code{k} is used (mainly for back compatibility). #' #' @param x a numeric vector. #' @param k the number of observations in the upper tail to which the Pareto #' distribution is fitted. #' @param x0 the threshold (scale parameter) above which the Pareto distribution #' is fitted. #' @param w an optional numeric vector giving sample weights. #' #' @return The estimated shape parameter. #' #' @note The arguments \code{x0} for the threshold (scale parameter) of the #' Pareto distribution and \code{w} for sample weights were introduced in #' version 0.2. #' #' @author Andreas Alfons and Josef Holzer #' #' @seealso \code{\link{paretoTail}}, \code{\link{fitPareto}}, #' \code{\link{thetaPDC}}, \code{\link{thetaWML}}, \code{\link{thetaISE}}, #' \code{\link{minAMSE}} #' #' @references Hill, B.M. (1975) A simple general approach to inference about #' the tail of a distribution. \emph{The Annals of Statistics}, \bold{3}(5), #' 1163--1174. #' #' @keywords manip #' #' @examples #' data(eusilc) #' # equivalized disposable income is equal for each household #' # member, therefore only one household member is taken #' eusilc <- eusilc[!duplicated(eusilc$db030),] #' #' # estimate threshold #' ts <- paretoScale(eusilc$eqIncome, w = eusilc$db090) #' #' # using number of observations in tail #' thetaHill(eusilc$eqIncome, k = ts$k, w = eusilc$db090) #' #' # using threshold #' thetaHill(eusilc$eqIncome, x0 = ts$x0, w = eusilc$db090) #' #' @export thetaHill <- function (x, k = NULL, x0 = NULL, w = NULL) { ## initializations if(!is.numeric(x) || length(x) == 0) stop("'x' must be a numeric vector") haveK <- !is.null(k) if(haveK) { # if 'k' is supplied, it is always used if(!is.numeric(k) || length(k) == 0 || k[1] < 1) { stop("'k' must be a positive integer") } else k <- k[1] } else if(!is.null(x0)) { # otherwise 'x0' (threshold) is used if(!is.numeric(x0) || length(x0) == 0) stop("'x0' must be numeric") else x0 <- x0[1] } else stop("either 'k' or 'x0' must be supplied") haveW <- !is.null(w) if(haveW) { # sample weights are supplied if(!is.numeric(w) || length(w) != length(x)) { stop("'w' must be numeric vector of the same length as 'x'") } if(any(w < 0)) stop("negative weights in 'w'") if(any(i <- is.na(x))) { # remove missing values x <- x[!i] w <- w[!i] } # sort values and sample weights order <- order(x) x <- x[order] w <- w[order] } else { # no sample weights if(any(i <- is.na(x))) x <- x[!i] # remove missing values x <- sort(x) # sort values } .thetaHill(x, k, x0, w) } .thetaHill <- function (x, k = NULL, x0 = NULL, w = NULL) { n <- length(x) # number of observations haveK <- !is.null(k) haveW <- !is.null(w) if(haveK) { # 'k' is supplied, threshold is determined if(k >= n) stop("'k' must be smaller than the number of observed values") x0 <- x[n-k] # threshold (scale parameter) } else { # 'k' is not supplied, it is determined using threshold # values are already sorted if(x0 >= x[n]) stop("'x0' must be smaller than the maximum of 'x'") k <- length(which(x > x0)) } ## computations if(haveW) { ## return weighted Hill estimate w <- w[(n-k+1):n] sum(w)/sum(w*(log(x[(n-k+1):n]) - log(x0))) } else { ## return Hill estimate # k/(sum(log(x[(n-k+1):n])) - k*log(x0)) k/sum(log(x[(n-k+1):n]) - log(x0)) # should be numerically more stable } } laeken/R/incMean.R0000644000176200001440000000566313616467254013437 0ustar liggesusers# --------------------------------------- # Author: Andreas Alfons # Vienna University of Technology # --------------------------------------- #' Weighted mean income #' #' Compute the weighted mean income. #' #' @param inc either a numeric vector giving the (equivalized disposable) #' income, or (if \code{data} is not \code{NULL}) a character string, an integer #' or a logical vector specifying the corresponding column of \code{data}. #' @param weights optional; either a numeric vector giving the personal sample #' weights, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. #' @param years optional; either a numeric vector giving the different years of #' the survey, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. If supplied, values are computed for each year. #' @param data an optional \code{data.frame}. #' @param na.rm a logical indicating whether missing values should be removed. #' #' @return A numeric vector containing the value(s) of the weighted mean income #' is returned. #' #' @author Andreas Alfons #' #' @seealso \code{\link{weightedMean}} #' #' @keywords survey #' #' @examples #' data(eusilc) #' incMean("eqIncome", weights = "rb050", data = eusilc) #' #' @export incMean <- function(inc, weights = NULL, years = NULL, data = NULL, na.rm = FALSE) { ## initializations if(!is.null(data)) { inc <- data[, inc] if(!is.null(weights)) weights <- data[, weights] if(!is.null(years)) years <- data[, years] } # check vectors if(!is.numeric(inc)) stop("'inc' must be a numeric vector") n <- length(inc) if(!is.null(weights) && !is.numeric(weights)) { stop("'weights' must be a numeric vector") } if(!is.null(years) && !is.numeric(years)) { stop("'years' must be a numeric vector") } if(is.null(data)) { # check vector lengths if(!is.null(weights) && length(weights) != n) { stop("'weights' must have the same length as 'x'") } if(!is.null(years) && length(years) != n) { stop("'years' must have the same length as 'x'") } } ## computations if(is.null(years)) { # no breakdown xn <- weightedMean(inc, weights, na.rm=na.rm) } else { # breakdown by years # define wrapper functions calcMean <- function(y, inc, weights, years, na.rm) { i <- years == y weightedMean(inc[i], weights[i], na.rm=na.rm) } # apply wrapper function ys <- sort(unique(years)) xn <- sapply(ys, calcMean, inc=inc, weights=weights, years=years, na.rm=na.rm) names(xn) <- ys # use years as names } ## return results return(xn) } laeken/R/utils.R0000644000176200001440000002277514127253120013207 0ustar liggesusers# --------------------------------------- # Author: Andreas Alfons # Vienna University of Technology # --------------------------------------- # TODO: error handling #' Utility functions for indicators on social exclusion and poverty #' #' Test for class, print and take subsets of indicators on social exclusion and #' poverty. #' #' @name utils #' #' @param x for \code{is.xyz}, any object to be tested. The \code{print} and #' \code{subset} methods are called by the generic functions if an object of the #' respective class is supplied. #' @param years an optional numeric vector giving the years to be extracted. #' @param strata an optional vector giving the domains of the breakdown to be #' extracted. #' @param \dots additional arguments to be passed to and from methods. #' #' @return \code{is.indicator} returns \code{TRUE} if \code{x} inherits from #' class \code{"indicator"} and \code{FALSE} otherwise. #' #' \code{is.arpr} returns \code{TRUE} if \code{x} inherits from class #' \code{"arpr"} and \code{FALSE} otherwise. #' #' \code{is.qsr} returns \code{TRUE} if \code{x} inherits from class #' \code{"qsr"} and \code{FALSE} otherwise. #' #' \code{is.rmpg} returns \code{TRUE} if \code{x} inherits from class #' \code{"rmpg"} and \code{FALSE} otherwise. #' #' \code{is.gini} returns \code{TRUE} if \code{x} inherits from class #' \code{"gini"} and \code{FALSE} otherwise. #' #' \code{is.gini} returns \code{TRUE} if \code{x} inherits from class #' \code{"gini"} and \code{FALSE} otherwise. #' #' \code{print.indicator}, \code{print.arpr} and \code{print.rmpg} return #' \code{x} invisibly. #' #' \code{subset.indicator}, \code{subset.arpr} and \code{subset.rmpg} return a #' subset of \code{x} of the same class. #' #' @seealso \code{\link{arpr}}, \code{\link{qsr}}, \code{\link{rmpg}}, #' \code{\link{gini}}, \code{\link{gpg}} #' #' @references #' A. Alfons and M. Templ (2013) Estimation of Social Exclusion Indicators #' from Complex Surveys: The \R Package \pkg{laeken}. \emph{Journal of #' Statistical Software}, \bold{54}(15), 1--25. \doi{10.18637/jss.v054.i15} #' #' @keywords survey #' #' @examples #' data(eusilc) #' #' # at-risk-of-poverty rate #' a <- arpr("eqIncome", weights = "rb050", #' breakdown = "db040", data = eusilc) #' print(a) #' is.arpr(a) #' is.indicator(a) #' subset(a, strata = c("Lower Austria", "Vienna")) #' #' # quintile share ratio #' q <- qsr("eqIncome", weights = "rb050", #' breakdown = "db040", data = eusilc) #' print(q) #' is.qsr(q) #' is.indicator(q) #' subset(q, strata = c("Lower Austria", "Vienna")) #' #' # relative median at-risk-of-poverty gap #' r <- rmpg("eqIncome", weights = "rb050", #' breakdown = "db040", data = eusilc) #' print(r) #' is.rmpg(r) #' is.indicator(r) #' subset(r, strata = c("Lower Austria", "Vienna")) #' #' # Gini coefficient #' g <- gini("eqIncome", weights = "rb050", #' breakdown = "db040", data = eusilc) #' print(g) #' is.gini(g) #' is.indicator(g) #' subset(g, strata = c("Lower Austria", "Vienna")) #' NULL ## constructors # class "indicator" constructIndicator <- function(value, valueByStratum = NULL, varMethod = NULL, var = NULL, varByStratum = NULL, ci = NULL, ciByStratum = NULL, alpha = NULL, years = NULL, strata = NULL) { # construct and assign class x <- list(value=value, valueByStratum=valueByStratum, varMethod=varMethod, var=var, varByStratum=varByStratum, ci=ci, ciByStratum=ciByStratum, alpha=alpha, years=years, strata=strata) class(x) <- "indicator" # return object return(x) } # class "arpr" constructArpr <- function(..., p = 0.6, threshold) { x <- constructIndicator(...) # call constructor of superclass x$p <- p # set specific x$threshold <- threshold # attributes class(x) <- c("arpr", class(x)) # assign class return(x) # return result } # class "qsr" constructQsr <- function(...) { x <- constructIndicator(...) # call constructor of superclass class(x) <- c("qsr", class(x)) # assign class return(x) # return result } # class "gpg" constructGpg <- function(...) { x <- constructIndicator(...) # call constructor of superclass class(x) <- c("gpg", class(x)) # assign class return(x) # return result } # class "rmrpg" constructRmpg <- function(..., threshold) { x <- constructIndicator(...) # call constructor of superclass x$threshold <- threshold # set specific attributes class(x) <- c("rmpg", class(x)) # assign class return(x) # return result } # class "gini" constructGini <- function(...) { x <- constructIndicator(...) # call constructor of superclass class(x) <- c("gini", class(x)) # assign class return(x) # return result } # class "prop" constructProp <- function(...) { x <- constructIndicator(...) # call constructor of superclass class(x) <- c("prop", class(x)) # assign class return(x) # return result } ## test for class #' @rdname utils #' @export is.indicator <- function(x) inherits(x, "indicator") #' @rdname utils #' @export is.arpr <- function(x) inherits(x, "arpr") #' @rdname utils #' @export is.qsr <- function(x) inherits(x, "qsr") #' @rdname utils #' @export is.rmpg <- function(x) inherits(x, "rmpg") #' @rdname utils #' @export is.gini <- function(x) inherits(x, "gini") #' @rdname utils #' @export is.prop <- function(x) inherits(x, "prop") #' @rdname utils #' @export is.gpg <- function(x) inherits(x, "gpg") ## print # class "indicator" #' @rdname utils #' @method print indicator #' @export print.indicator <- function(x, ...) { cat("Value:\n") print(x$value, ...) if(!is.null(x$var)) { cat("\nVariance:\n") print(x$var, ...) } if(!is.null(x$ci)) { cat("\nConfidence interval:\n") print(x$ci, ...) } if(!is.null(x$valueByStratum)) { cat("\nValue by domain:\n") print(x$valueByStratum, ...) } if(!is.null(x$varByStratum)) { cat("\nVariance by domain:\n") print(x$varByStratum, ...) } if(!is.null(x$varByStratum)) { cat("\nConfidence interval by domain:\n") print(x$ciByStratum, ...) } invisible(x) } # class "arpr" #' @rdname utils #' @method print arpr #' @export print.arpr <- function(x, ...) { print.indicator(x, ...) cat("\nThreshold:\n") print(x$threshold, ...) invisible(x) } # class "rmpg" #' @rdname utils #' @method print rmpg #' @export print.rmpg <- function(x, ...) { print.indicator(x, ...) cat("\nThreshold:\n") print(x$threshold, ...) invisible(x) } # class "minAMSE" #' @rdname minAMSE #' @method print minAMSE #' @export print.minAMSE <- function(x, ...) { cat("Optimal k:\n") print(x$kopt, ...) cat("\nScale parameter:\n") print(x$x0, ...) cat("\nShape parameter:\n") print(x$theta, ...) invisible(x) } ## subsets of indicators # class "indicator" #' @rdname utils #' @method subset indicator #' @export subset.indicator <- function(x, years = NULL, strata = NULL, ...) { # initializations haveYears <- length(x$years) > 1 haveVar <- !is.null(x$varMethod) haveStrata <- length(x$strata) > 1 subsetYears <- haveYears && !is.null(years) subsetStrata <- haveStrata && !is.null(strata) # error handling if(subsetYears && !is.numeric(years)) { stop("'years' must be of type numeric") } if(subsetStrata && !is.character(strata)) { stop("'years' must be of type character") } # extract years from overall values (if available and requested) if(subsetYears) { ys <- as.character(years) x$value <- x$value[ys] if(haveVar) { x$var <- x$var[ys] x$ci <- x$ci[ys, , drop=FALSE] } x$years <- years #set new years } # extract strata from overall values (if available and requested) if(subsetStrata || (haveStrata && subsetYears)) { n <- nrow(x$valueByStratum) if(subsetStrata) keepStrata <- x$valueByStratum$stratum %in% strata else keepStrata <- rep.int(TRUE, n) if(subsetYears) keepYears <- x$valueByStratum$year %in% years else keepYears <- rep.int(TRUE, n) keep <- keepStrata & keepYears x$valueByStratum <- x$valueByStratum[keep, , drop=FALSE] if(haveVar) { x$varByStratum <- x$varByStratum[keep, , drop=FALSE] x$ciByStratum <- x$ciByStratum[keep, , drop=FALSE] } x$strata <- strata # set new strata } # return result return(x) } # class "arpr" # TODO: allow for subsetting by threshold percentage #' @rdname utils #' @method subset arpr #' @export subset.arpr <- function(x, years = NULL, strata = NULL, ...) { haveYear <- length(x$years) > 1 x <- subset.indicator(x, years, strata, ...) # call method for superclass # subset threshold (if requested and available for multiple years) if(haveYear && !is.null(years)) { x$threshold <- x$threshold[as.character(years)] } # return result return(x) } # class "rmpg" #' @rdname utils #' @method subset rmpg #' @export subset.rmpg <- function(x, years = NULL, strata = NULL, ...) { haveYear <- length(x$years) > 1 x <- subset.indicator(x, years, strata, ...) # call method for superclass # subset threshold (if requested and available for multiple years) if(haveYear && !is.null(years)) { x$threshold <- x$threshold[as.character(years)] } # return result return(x) } ## other utility functions # get argument names of a function argNames <- function(fun, removeDots = TRUE) { nam <- names(formals(fun)) if(removeDots) nam <- setdiff(nam, "...") nam } # check percentages for the ARPT checkP <-function(p) { if(is.numeric(p)) { keep <- !is.na(p) & p >= 0 & p <= 1 p <- p[keep] } else p <- numeric() if(length(p) == 0) stop("'p' must contain numeric values in [0,1]") p } # get labels for percentages of the ARPT getPLabels <-function(p) paste(signif(p*100), "%", sep="") laeken/R/thetaMoment.R0000644000176200001440000000664413616467254014352 0ustar liggesusers# ---------------------------------------- # Authors: Andreas Alfons and Josef Holzer # Vienna University of Technology # ---------------------------------------- #' Moment estimator #' #' Estimate the shape parameter of a Pareto distribution based on moments. #' #' The arguments \code{k} and \code{x0} of course correspond with each other. #' If \code{k} is supplied, the threshold \code{x0} is estimated with the \eqn{n #' - k} largest value in \code{x}, where \eqn{n} is the number of observations. #' On the other hand, if the threshold \code{x0} is supplied, \code{k} is given #' by the number of observations in \code{x} larger than \code{x0}. Therefore, #' either \code{k} or \code{x0} needs to be supplied. If both are supplied, #' only \code{k} is used (mainly for back compatibility). #' #' @param x a numeric vector. #' @param k the number of observations in the upper tail to which the Pareto #' distribution is fitted. #' @param x0 the threshold (scale parameter) above which the Pareto distribution #' is fitted. #' #' @return The estimated shape parameter. #' #' @note The argument \code{x0} for the threshold (scale parameter) of the #' Pareto distribution was introduced in version 0.2. #' #' @author Andreas Alfons and Josef Holzer #' #' @seealso \code{\link{paretoTail}}, \code{\link{fitPareto}} #' #' @references Dekkers, A.L.M., Einmahl, J.H.J. and de Haan, L. (1989) A moment #' estimator for the index of an extreme-value distribution. \emph{The Annals of #' Statistics}, \bold{17}(4), 1833--1855. #' #' @keywords manip #' #' @examples #' data(eusilc) #' # equivalized disposable income is equal for each household #' # member, therefore only one household member is taken #' eusilc <- eusilc[!duplicated(eusilc$db030),] #' #' # estimate threshold #' ts <- paretoScale(eusilc$eqIncome, w = eusilc$db090) #' #' # using number of observations in tail #' thetaMoment(eusilc$eqIncome, k = ts$k) #' #' # using threshold #' thetaMoment(eusilc$eqIncome, x0 = ts$x0) #' #' @export thetaMoment <- function(x, k = NULL, x0 = NULL) { ## initializations if(!is.numeric(x) || length(x) == 0) stop("'x' must be a numeric vector") haveK <- !is.null(k) if(haveK) { # if 'k' is supplied, it is always used if(!is.numeric(k) || length(k) == 0 || k[1] < 1) { stop("'k' must be a positive integer") } else k <- k[1] } else if(!is.null(x0)) { # otherwise 'x0' (threshold) is used if(!is.numeric(x0) || length(x0) == 0) stop("'x0' must be numeric") else x0 <- x0[1] } else stop("either 'k' or 'x0' must be supplied") if(any(i <- is.na(x))) x <- x[!i] # remove missing values x <- sort(x) n <- length(x) if(haveK) { # 'k' is supplied, threshold is determined if(k >= n) stop("'k' must be smaller than the number of observed values") x0 <- x[n-k] # threshold (scale parameter) } else { # 'k' is not supplied, it is determined using threshold # values are already sorted if(x0 >= x[n]) stop("'x0' must be smaller than the maximum of 'x'") k <- length(which(x > x0)) } y <- log(x[(n-k+1):n]/x0) # relative excesses ## moments # M1 <- sum(y)/k # first moment # M2 <- sum(y^2)/k # second moment M1 <- mean(y) # first moment M2 <- mean(y^2) # second moment ## moment estimator 1/(M1 + 1 - 1/(2*(1-M1^2/M2))) } laeken/R/thetaTM.R0000644000176200001440000001060613616467254013424 0ustar liggesusers# ---------------------------------------- # Authors: Andreas Alfons and Josef Holzer # Vienna University of Technology # ---------------------------------------- #' Trimmed mean estimator #' #' Estimate the shape parameter of a Pareto distribution using a trimmed mean #' approach. #' #' The arguments \code{k} and \code{x0} of course correspond with each other. #' If \code{k} is supplied, the threshold \code{x0} is estimated with the \eqn{n #' - k} largest value in \code{x}, where \eqn{n} is the number of observations. #' On the other hand, if the threshold \code{x0} is supplied, \code{k} is given #' by the number of observations in \code{x} larger than \code{x0}. Therefore, #' either \code{k} or \code{x0} needs to be supplied. If both are supplied, #' only \code{k} is used (mainly for back compatibility). #' #' @param x a numeric vector. #' @param k the number of observations in the upper tail to which the Pareto #' distribution is fitted. #' @param x0 the threshold (scale parameter) above which the Pareto distribution #' is fitted. #' @param beta A numeric vector of length two giving the trimming proportions #' for the lower and upper end of the tail, respectively. If a single numeric #' value is supplied, it is recycled. #' #' @return The estimated shape parameter. #' #' @note The argument \code{x0} for the threshold (scale parameter) of the #' Pareto distribution was introduced in version 0.2. #' #' @author Andreas Alfons and Josef Holzer #' #' @seealso \code{\link{paretoTail}}, \code{\link{fitPareto}} #' #' @references Brazauskas, V. and Serfling, R. (2000) Robust estimation of tail #' parameters for two-parameter Pareto and exponential models via generalized #' quantile statistics. \emph{Extremes}, \bold{3}(3), 231--249. #' #' Brazauskas, V. and Serfling, R. (2000) Robust and efficient estimation of the #' tail index of a single-parameter Pareto distribution. \emph{North American #' Actuarial Journal}, \bold{4}(4), 12--27. #' #' @keywords manip #' #' @examples #' data(eusilc) #' # equivalized disposable income is equal for each household #' # member, therefore only one household member is taken #' eusilc <- eusilc[!duplicated(eusilc$db030),] #' #' # estimate threshold #' ts <- paretoScale(eusilc$eqIncome, w = eusilc$db090) #' #' # using number of observations in tail #' thetaTM(eusilc$eqIncome, k = ts$k) #' #' # using threshold #' thetaTM(eusilc$eqIncome, x0 = ts$x0) #' #' @export thetaTM <- function(x, k = NULL, x0 = NULL, beta = 0.05) { ## initializations if(!is.numeric(x) || length(x) == 0) stop("'x' must be a numeric vector") haveK <- !is.null(k) if(haveK) { # if 'k' is supplied, it is always used if(!is.numeric(k) || length(k) == 0 || k[1] < 1) { stop("'k' must be a positive integer") } else k <- k[1] } else if(!is.null(x0)) { # otherwise 'x0' (threshold) is used if(!is.numeric(x0) || length(x0) == 0) stop("'x0' must be numeric") else x0 <- x0[1] } else stop("either 'k' or 'x0' must be supplied") if(any(i <- is.na(x))) x <- x[!i] # remove missing values x <- sort(x) n <- length(x) if(haveK) { # 'k' is supplied, threshold is determined if(k >= n) stop("'k' must be smaller than the number of observed values") x0 <- x[n-k] # threshold (scale parameter) } else { # 'k' is not supplied, it is determined using threshold # values are already sorted if(x0 >= x[n]) stop("'x0' must be smaller than the maximum of 'x'") k <- length(which(x > x0)) } # check trimming proportions if(length(beta) == 0) stop("'beta' must be a numeric vector of length two") else beta <- rep(beta, length.out=2) if(beta[1] < 0 || beta[1] >= 1) { stop("beta[1] (the trimming proportion for the lower end) ", "must be greater or equal to 0 and smaller than 1") } if(beta[2] < 0 || beta[2] >= 1-beta[1]) { stop("beta[2] (the trimming proportion for the upper end) ", "must be greater or equal to 0 and smaller than 1-beta[1] ", "(the trimming proportion for the lower end)") } # trimming kl <- trunc(k*beta[1])+1 kh <- k - trunc(k*beta[2]) i <- kl:kh c <- rep.int(0, k) c[i] <- 1/sum(cumsum(1/(k - 1:kh + 1))[i]) # estimate theta 1/sum(c*log(x[(n-k+1):n]/x0)) } laeken/R/arpt.R0000644000176200001440000000567514125312655013024 0ustar liggesusers# --------------------------------------- # Author: Andreas Alfons # Vienna University of Technology # --------------------------------------- #' At-risk-of-poverty threshold #' #' Estimate the at-risk-of-poverty threshold. The standard definition is to use #' 60\% of the weighted median equivalized disposable income. #' #' The implementation strictly follows the Eurostat definition. #' #' @param inc either a numeric vector giving the equivalized disposable income, #' or (if \code{data} is not \code{NULL}) a character string, an integer or a #' logical vector specifying the corresponding column of \code{data}. #' @param weights optional; either a numeric vector giving the personal sample #' weights, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. #' @param sort optional; either a numeric vector giving the personal IDs to be #' used as tie-breakers for sorting, or (if \code{data} is not \code{NULL}) a #' character string, an integer or a logical vector specifying the corresponding #' column of \code{data}. #' @param years optional; either a numeric vector giving the different years of #' the survey, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. If supplied, values are computed for each year. #' @param data an optional \code{data.frame}. #' @param p a numeric vector of values in \eqn{[0,1]} giving the percentages of #' the weighted median to be used for the at-risk-of-poverty threshold. #' @param na.rm a logical indicating whether missing values should be removed. #' #' @return A numeric vector containing the value(s) of the at-risk-of-poverty #' threshold is returned. #' #' @author Andreas Alfons #' #' @seealso \code{\link{arpr}}, \code{\link{incMedian}}, #' \code{\link{weightedMedian}} #' #' @references Working group on Statistics on Income and Living Conditions #' (2004) Common cross-sectional EU indicators based on EU-SILC; the gender pay #' gap. \emph{EU-SILC 131-rev/04}, Eurostat. #' #' @keywords survey #' #' @examples #' data(eusilc) #' arpt("eqIncome", weights = "rb050", data = eusilc) #' #' @export arpt <- function(inc, weights = NULL, sort = NULL, years = NULL, data = NULL, p = 0.6, na.rm = FALSE) { # check 'p' (other arguments are checked in 'incMedian') # if(!is.numeric(p) || length(p) == 0 || p[1] < 0 || p[1] > 1) { # stop("'p' must be a numeric value in [0,1]") # } else p <- p[1] p <- checkP(p) byP <- length(p) > 1 if(byP) { if(!is.null(years)) { stop("breakdown into years not implemented ", "for different threshold levels") } names(p) <- getPLabels(p) # ensure that result has correct names } # compute at-risk-of-poverty threshold p * incMedian(inc, weights, sort, years, data, na.rm=na.rm) } laeken/R/incQuintile.R0000644000176200001440000001122613616467254014341 0ustar liggesusers# --------------------------------------- # Author: Andreas Alfons # Vienna University of Technology # --------------------------------------- #' Weighted income quintile #' #' Compute weighted income quintiles. #' #' The implementation strictly follows the Eurostat definition. #' #' @param inc either a numeric vector giving the (equivalized disposable) #' income, or (if \code{data} is not \code{NULL}) a character string, an integer #' or a logical vector specifying the corresponding column of \code{data}. #' @param weights optional; either a numeric vector giving the personal sample #' weights, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. #' @param sort optional; either a numeric vector giving the personal IDs to be #' used as tie-breakers for sorting, or (if \code{data} is not \code{NULL}) a #' character string, an integer or a logical vector specifying the corresponding #' column of \code{data}. #' @param years optional; either a numeric vector giving the different years of #' the survey, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. If supplied, values are computed for each year. #' @param k a vector of integers between 0 and 5 specifying the quintiles to be #' computed (0 gives the minimum, 5 the maximum). #' @param data an optional \code{data.frame}. #' @param na.rm a logical indicating whether missing values should be removed. #' #' @return A numeric vector (if \code{years} is \code{NULL}) or matrix (if #' \code{years} is not \code{NULL}) containing the values of the weighted income #' quintiles specified by \code{k} are returned. #' #' @author Andreas Alfons #' #' @seealso \code{\link{qsr}}, \code{\link{weightedQuantile}} #' #' @references Working group on Statistics on Income and Living Conditions #' (2004) Common cross-sectional EU indicators based on EU-SILC; the gender pay #' gap. \emph{EU-SILC 131-rev/04}, Eurostat. #' #' @keywords survey #' #' @examples #' data(eusilc) #' incQuintile("eqIncome", weights = "rb050", data = eusilc) #' #' @export incQuintile <- function(inc, weights = NULL, sort = NULL, years = NULL, k = c(1, 4), data = NULL, na.rm = FALSE) { ## initializations if(!is.null(data)) { inc <- data[, inc] if(!is.null(weights)) weights <- data[, weights] if(!is.null(sort)) sort <- data[, sort] if(!is.null(years)) years <- data[, years] } # check vectors if(!is.numeric(inc)) stop("'inc' must be a numeric vector") n <- length(inc) if(!is.null(weights) && !is.numeric(weights)) { stop("'weights' must be a numeric vector") } if(!is.null(sort) && !is.vector(sort) && !is.ordered(sort)) { stop("'sort' must be a vector or ordered factor") } if(!is.null(years) && !is.numeric(years)) { stop("'years' must be a numeric vector") } if(is.null(data)) { # check vector lengths if(!is.null(weights) && length(weights) != n) { stop("'weights' must have the same length as 'x'") } if(!is.null(sort) && length(sort) != n) { stop("'sort' must have the same length as 'x'") } if(!is.null(years) && length(years) != n) { stop("'years' must have the same length as 'x'") } } if(!is.numeric(k) || length(k) == 0 || any(k < -0.5 | k >= 5.5)) { stop("'k' must be a vector of integers between 0 and 5") } else k <- round(k) ## sort values and weights order <- if(is.null(sort)) order(inc) else order(inc, sort) inc <- inc[order] weights <- weights[order] # also works if 'weights' is NULL ## computations if(is.null(years)) { # no breakdown q <- weightedQuantile(inc, weights, probs=k/5, sorted=TRUE, na.rm=na.rm) names(q) <- k # use quintile numbers as names } else { # breakdown by years years <- years[order] # define wrapper functions calcQuantile <- function(y, inc, weights, years, k, na.rm) { i <- years == y weightedQuantile(inc[i], weights[i], probs=k/5, sorted=TRUE, na.rm=na.rm) } # apply wrapper function ys <- sort(unique(years)) q <- t(sapply(ys, calcQuantile, inc=inc, weights=weights, years=years, k=k, na.rm=na.rm)) rownames(q) <- ys # use years as row names colnames(q) <- k # use quintile numbers as column names } ## return results return(q) } laeken/R/thetaLS.R0000644000176200001440000000777713616467254013441 0ustar liggesusers# ---------------------------------------- # Authors: Andreas Alfons and Josef Holzer # Vienna University of Technology # ---------------------------------------- ## should we return estimate for x0? if so, don't we need to re-estimate theta? ## => iterative procedure until change smaller than a threshold? #' Least squares (LS) estimator #' #' Estimate the shape parameter of a Pareto distribution using a least squares #' (LS) approach. #' #' The arguments \code{k} and \code{x0} of course correspond with each other. #' If \code{k} is supplied, the threshold \code{x0} is estimated with the \eqn{n #' - k} largest value in \code{x}, where \eqn{n} is the number of observations. #' On the other hand, if the threshold \code{x0} is supplied, \code{k} is given #' by the number of observations in \code{x} larger than \code{x0}. Therefore, #' either \code{k} or \code{x0} needs to be supplied. If both are supplied, #' only \code{k} is used (mainly for back compatibility). #' #' @param x a numeric vector. #' @param k the number of observations in the upper tail to which the Pareto #' distribution is fitted. #' @param x0 the threshold (scale parameter) above which the Pareto distribution #' is fitted. #' #' @return The estimated shape parameter. #' #' @note The argument \code{x0} for the threshold (scale parameter) of the #' Pareto distribution was introduced in version 0.2. #' #' @author Andreas Alfons and Josef Holzer #' #' @seealso \code{\link{paretoTail}}, \code{\link{fitPareto}} #' #' @references Brazauskas, V. and Serfling, R. (2000) Robust estimation of tail #' parameters for two-parameter Pareto and exponential models via generalized #' quantile statistics. \emph{Extremes}, \bold{3}(3), 231--249. #' #' Brazauskas, V. and Serfling, R. (2000) Robust and efficient estimation of the #' tail index of a single-parameter Pareto distribution. \emph{North American #' Actuarial Journal}, \bold{4}(4), 12--27. #' #' @keywords manip #' #' @examples #' data(eusilc) #' # equivalized disposable income is equal for each household #' # member, therefore only one household member is taken #' eusilc <- eusilc[!duplicated(eusilc$db030),] #' #' # estimate threshold #' ts <- paretoScale(eusilc$eqIncome, w = eusilc$db090) #' #' # using number of observations in tail #' thetaLS(eusilc$eqIncome, k = ts$k) #' #' # using threshold #' thetaLS(eusilc$eqIncome, x0 = ts$x0) #' #' @export thetaLS <- function(x, k = NULL, x0 = NULL) { ## initializations if(!is.numeric(x) || length(x) == 0) stop("'x' must be a numeric vector") haveK <- !is.null(k) if(haveK) { # if 'k' is supplied, it is always used if(!is.numeric(k) || length(k) == 0 || k[1] < 1) { stop("'k' must be a positive integer") } else k <- k[1] } else if(!is.null(x0)) { # otherwise 'x0' (threshold) is used if(!is.numeric(x0) || length(x0) == 0) stop("'x0' must be numeric") else x0 <- x0[1] } else stop("either 'k' or 'x0' must be supplied") if(any(i <- is.na(x))) x <- x[!i] # remove missing values x <- sort(x) n <- length(x) # if(haveK) { # 'k' is supplied, threshold is determined # if(k >= n) stop("'k' must be smaller than the number of observed values") # x0 <- x[n-k] # threshold (scale parameter) # } else { # 'k' is not supplied, it is determined using threshold # # values are already sorted # if(x0 >= x[n]) stop("'x0' must be smaller than the maximum of 'x'") # k <- length(which(x > x0)) # } if(!haveK) { # 'k' is not supplied, it is determined using threshold # values are already sorted if(x0 >= x[n]) stop("'x0' must be smaller than the maximum of 'x'") k <- length(which(x > x0)) } ## computations z <- log(x[(n-k+1):n]) zm <- mean(z) pk <- c((1:(k-1))/k, k/(k+1)) # regression parameters ck <- -log(1-pk) ckm <- mean(ck) ## LS estimator mean((ck - ckm)^2) / (mean(ck*z) - ckm*zm) } laeken/R/weightedQuantile.R0000644000176200001440000000642213616467254015362 0ustar liggesusers# ------------------------------------------ # Authors: Andreas Alfons and Matthias Templ # Vienna University of Technology # ------------------------------------------ #' Weighted quantiles #' #' Compute weighted quantiles (Eurostat definition). #' #' The implementation strictly follows the Eurostat definition. #' #' @param x a numeric vector. #' @param weights an optional numeric vector giving the sample weights. #' @param probs numeric vector of probabilities with values in \eqn{[0,1]}. #' @param sorted a logical indicating whether the observations in \code{x} are #' already sorted. #' @param na.rm a logical indicating whether missing values in \code{x} should #' be omitted. #' #' @return A numeric vector containing the weighted quantiles of values in #' \code{x} at probabilities \code{probs} is returned. Unlike #' \code{\link[stats]{quantile}}, this returns an unnamed vector. #' #' @author Andreas Alfons and Matthias Templ #' #' @seealso \code{\link{incQuintile}}, \code{\link{weightedMedian}} #' #' @references Working group on Statistics on Income and Living Conditions #' (2004) Common cross-sectional EU indicators based on EU-SILC; the gender pay #' gap. \emph{EU-SILC 131-rev/04}, Eurostat. #' #' @keywords survey #' #' @examples #' data(eusilc) #' weightedQuantile(eusilc$eqIncome, eusilc$rb050) #' #' @export weightedQuantile <- function(x, weights = NULL, probs = seq(0, 1, 0.25), sorted = FALSE, na.rm = FALSE) { # initializations if (!is.numeric(x)) stop("'x' must be a numeric vector") n <- length(x) if (n == 0 || (!isTRUE(na.rm) && any(is.na(x)))) { # zero length or missing values return(rep.int(NA, length(probs))) } if (!is.null(weights)) { if (!is.numeric(weights)) stop("'weights' must be a numeric vector") else if (length(weights) != n) { stop("'weights' must have the same length as 'x'") } else if (!all(is.finite(weights))) stop("missing or infinite weights") if (any(weights < 0)) warning("negative weights") if (!is.numeric(probs) || all(is.na(probs)) || isTRUE(any(probs < 0 | probs > 1))) { stop("'probs' must be a numeric vector with values in [0,1]") } if (all(weights == 0)) { # all zero weights warning("all weights equal to zero") return(rep.int(0, length(probs))) } } # remove NAs (if requested) if(isTRUE(na.rm)){ indices <- !is.na(x) x <- x[indices] if(!is.null(weights)) weights <- weights[indices] } # sort values and weights (if requested) if(!isTRUE(sorted)) { # order <- order(x, na.last=NA) ## too slow order <- order(x) x <- x[order] weights <- weights[order] # also works if 'weights' is NULL } # some preparations if(is.null(weights)) rw <- (1:n)/n else rw <- cumsum(weights)/sum(weights) # obtain quantiles q <- sapply(probs, function(p) { if (p == 0) return(x[1]) else if (p == 1) return(x[n]) select <- min(which(rw > p)) if(rw[select] == p) mean(x[select:(select+1)]) else x[select] }) return(unname(q)) } laeken/R/thetaQQ.R0000644000176200001440000000700413616467254013423 0ustar liggesusers# ---------------------------------------- # Authors: Andreas Alfons and Josef Holzer # Vienna University of Technology # ---------------------------------------- #' QQ-estimator #' #' Estimate the shape parameter of a Pareto distribution using a #' quantile-quantile approach. #' #' The arguments \code{k} and \code{x0} of course correspond with each other. #' If \code{k} is supplied, the threshold \code{x0} is estimated with the \eqn{n #' - k} largest value in \code{x}, where \eqn{n} is the number of observations. #' On the other hand, if the threshold \code{x0} is supplied, \code{k} is given #' by the number of observations in \code{x} larger than \code{x0}. Therefore, #' either \code{k} or \code{x0} needs to be supplied. If both are supplied, #' only \code{k} is used (mainly for back compatibility). #' #' @param x a numeric vector. #' @param k the number of observations in the upper tail to which the Pareto #' distribution is fitted. #' @param x0 the threshold (scale parameter) above which the Pareto distribution #' is fitted. #' #' @return The estimated shape parameter. #' #' @note The argument \code{x0} for the threshold (scale parameter) of the #' Pareto distribution was introduced in version 0.2. #' #' @author Andreas Alfons and Josef Holzer #' #' @seealso \code{\link{paretoTail}}, \code{\link{fitPareto}} #' #' @references Kratz, M.F. and Resnick, S.I. (1996) The QQ-estimator and heavy #' tails. \emph{Stochastic Models}, \bold{12}(4), 699--724. #' #' @keywords manip #' #' @examples #' data(eusilc) #' # equivalized disposable income is equal for each household #' # member, therefore only one household member is taken #' eusilc <- eusilc[!duplicated(eusilc$db030),] #' #' # estimate threshold #' ts <- paretoScale(eusilc$eqIncome, w = eusilc$db090) #' #' # using number of observations in tail #' thetaQQ(eusilc$eqIncome, k = ts$k) #' #' # using threshold #' thetaQQ(eusilc$eqIncome, x0 = ts$x0) #' #' @export thetaQQ <- function(x, k = NULL, x0 = NULL) { ## initializations if(!is.numeric(x) || length(x) == 0) stop("'x' must be a numeric vector") haveK <- !is.null(k) if(haveK) { # if 'k' is supplied, it is always used if(!is.numeric(k) || length(k) == 0 || k[1] < 1) { stop("'k' must be a positive integer") } else k <- k[1] } else if(!is.null(x0)) { # otherwise 'x0' (threshold) is used if(!is.numeric(x0) || length(x0) == 0) stop("'x0' must be numeric") else x0 <- x0[1] } else stop("either 'k' or 'x0' must be supplied") if(any(i <- is.na(x))) x <- x[!i] # remove missing values x <- sort(x) n <- length(x) # if(haveK) { # 'k' is supplied, threshold is determined # if(k >= n) stop("'k' must be smaller than the number of observed values") # x0 <- x[n-k] # threshold (scale parameter) # } else { # 'k' is not supplied, it is determined using threshold # # values are already sorted # if(x0 >= x[n]) stop("'x0' must be smaller than the maximum of 'x'") # k <- length(which(x > x0)) # } if(!haveK) { # 'k' is not supplied, it is determined using threshold # values are already sorted if(x0 >= x[n]) stop("'x0' must be smaller than the maximum of 'x'") k <- length(which(x > x0)) } ## calculations logx <- log(x[n:(n-k+1)]) # lograrithm of reversed tail h <- -log((1:k)/(k+1)) ## QQ-estimator (k*sum(h^2) - sum(h)^2) / sum(h * (k*logx - sum(logx))) } laeken/R/arpr.R0000644000176200001440000002503014127253144013004 0ustar liggesusers# --------------------------------------- # Author: Andreas Alfons # Vienna University of Technology # --------------------------------------- #' At-risk-of-poverty rate #' #' Estimate the at-risk-of-poverty rate, which is defined as the proportion of #' persons with equivalized disposable income below the at-risk-of-poverty #' threshold. #' #' The implementation strictly follows the Eurostat definition. #' #' @param inc either a numeric vector giving the equivalized disposable income, #' or (if \code{data} is not \code{NULL}) a character string, an integer or a #' logical vector specifying the corresponding column of \code{data}. #' @param weights optional; either a numeric vector giving the personal sample #' weights, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. #' @param sort optional; either a numeric vector giving the personal IDs to be #' used as tie-breakers for sorting, or (if \code{data} is not \code{NULL}) a #' character string, an integer or a logical vector specifying the corresponding #' column of \code{data}. #' @param years optional; either a numeric vector giving the different years of #' the survey, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. If supplied, values are computed for each year. #' @param breakdown optional; either a numeric vector giving different domains, #' or (if \code{data} is not \code{NULL}) a character string, an integer or a #' logical vector specifying the corresponding column of \code{data}. If #' supplied, the values for each domain are computed in addition to the overall #' value. Note that the same (overall) threshold is used for all domains. #' @param design optional and only used if \code{var} is not \code{NULL}; either #' an integer vector or factor giving different strata for stratified sampling #' designs, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. #' @param cluster optional and only used if \code{var} is not \code{NULL}; #' either an integer vector or factor giving different clusters for cluster #' sampling designs, or (if \code{data} is not \code{NULL}) a character string, #' an integer or a logical vector specifying the corresponding column of #' \code{data}. #' @param data an optional \code{data.frame}. #' @param p a numeric vector of values in \eqn{[0,1]} giving the percentages of #' the weighted median to be used for the at-risk-of-poverty threshold (see #' \code{\link{arpt}}). #' @param var a character string specifying the type of variance estimation to #' be used, or \code{NULL} to omit variance estimation. See #' \code{\link{variance}} for possible values. #' @param alpha numeric; if \code{var} is not \code{NULL}, this gives the #' significance level to be used for computing the confidence interval (i.e., #' the confidence level is \eqn{1 - }\code{alpha}). #' @param threshold if `NULL`, the at-risk-at-poverty threshold is estimated from the data. #' @param na.rm a logical indicating whether missing values should be removed. #' @param \dots if \code{var} is not \code{NULL}, additional arguments to be #' passed to \code{\link{variance}}. #' #' @return A list of class \code{"arpr"} (which inherits from the class #' \code{"indicator"}) with the following components: #' \item{value}{a numeric vector containing the overall value(s).} #' \item{valueByStratum}{a \code{data.frame} containing the values by #' domain, or \code{NULL}.} #' \item{varMethod}{a character string specifying the type of variance #' estimation used, or \code{NULL} if variance estimation was omitted.} #' \item{var}{a numeric vector containing the variance estimate(s), or #' \code{NULL}.} #' \item{varByStratum}{a \code{data.frame} containing the variance #' estimates by domain, or \code{NULL}.} #' \item{ci}{a numeric vector or matrix containing the lower and upper #' endpoints of the confidence interval(s), or \code{NULL}.} #' \item{ciByStratum}{a \code{data.frame} containing the lower and upper #' endpoints of the confidence intervals by domain, or \code{NULL}.} #' \item{alpha}{a numeric value giving the significance level used for #' computing the confidence interval(s) (i.e., the confidence level is \eqn{1 - #' }\code{alpha}), or \code{NULL}.} #' \item{years}{a numeric vector containing the different years of the #' survey.} #' \item{strata}{a character vector containing the different domains of the #' breakdown.} #' \item{p}{a numeric giving the percentage of the weighted median used for #' the at-risk-of-poverty threshold.} #' \item{threshold}{a numeric vector containing the at-risk-of-poverty #' threshold(s).} #' #' @author Andreas Alfons #' #' @seealso \code{\link{arpt}}, \code{\link{variance}} #' #' @references #' A. Alfons and M. Templ (2013) Estimation of Social Exclusion Indicators #' from Complex Surveys: The \R Package \pkg{laeken}. \emph{Journal of #' Statistical Software}, \bold{54}(15), 1--25. \doi{10.18637/jss.v054.i15} #' #' Working group on Statistics on Income and Living Conditions (2004) #' Common cross-sectional EU indicators based on EU-SILC; the gender #' pay gap. \emph{EU-SILC 131-rev/04}, Eurostat, Luxembourg. #' #' @keywords survey #' #' @examples #' data(eusilc) #' #' # overall value #' arpr("eqIncome", weights = "rb050", data = eusilc) #' #' # values by region #' arpr("eqIncome", weights = "rb050", #' breakdown = "db040", data = eusilc) #' #' @importFrom stats aggregate #' @export arpr <- function(inc, weights = NULL, sort = NULL, years = NULL, breakdown = NULL, design = NULL, cluster = NULL, data = NULL, p = 0.6, var = NULL, alpha = 0.05, threshold = NULL, na.rm = FALSE, ...) { ## initializations byYear <- !is.null(years) byStratum <- !is.null(breakdown) p <- checkP(p) byP <- length(p) > 1 # prepare data if(!is.null(data)) { inc <- data[, inc] if(!is.null(weights)) weights <- data[, weights] if(!is.null(sort)) sort <- data[, sort] if(byYear) years <- data[, years] if(byStratum) breakdown <- data[, breakdown] if(!is.null(var)) { if(!is.null(design)) design <- data[, design] if(!is.null(cluster)) cluster <- data[, cluster] } } # check vectors if(!is.numeric(inc)) stop("'inc' must be a numeric vector") n <- length(inc) if(is.null(weights)) weights <- weights <- rep.int(1, n) else if(!is.numeric(weights)) stop("'weights' must be a numeric vector") if(!is.null(sort) && !is.vector(sort) && !is.ordered(sort)) { stop("'sort' must be a vector or ordered factor") } if(byYear && !is.numeric(years)) { stop("'years' must be a numeric vector") } if(byStratum) { if(!is.vector(breakdown) && !is.factor(breakdown)) { stop("'breakdown' must be a vector or factor") } else breakdown <- as.factor(breakdown) } if(is.null(data)) { # check vector lengths if(length(weights) != n) { stop("'weights' must have the same length as 'x'") } if(!is.null(sort) && length(sort) != n) { stop("'sort' must have the same length as 'x'") } if(byYear && length(years) != n) { stop("'years' must have the same length as 'x'") } if(byStratum && length(breakdown) != n) { stop("'breakdown' must have the same length as 'x'") } } ## computations rs <- levels(breakdown) # unique strata (also works if 'breakdown' is NULL) if(byYear) { # ARPR by year ys <- sort(unique(years)) if(is.null(threshold)){ ts <- arpt(inc, weights, sort, years, p=p, na.rm=na.rm) # thresholds } else { ts <- threshold } wr <- function(y, t, inc, weights, years, na.rm) { i <- years == y weightedRate(inc[i], weights[i], t, na.rm=na.rm) } value <- mapply(wr, y=ys, t=ts, MoreArgs=list(inc=inc, weights=weights, years=years, na.rm=na.rm)) names(value) <- ys # use years as names if(byStratum) { wr1 <- function(i, inc, weights, years, ts, na.rm) { y <- years[i[1]] t <- ts[as.character(y)] weightedRate(inc[i], weights[i], t, na.rm=na.rm) } valueByStratum <- aggregate(1:n, list(year=years, stratum=breakdown), wr1, inc=inc, weights=weights, years=years, ts=ts, na.rm=na.rm) names(valueByStratum)[3] <- "value" } else valueByStratum <- NULL } else { # ARPR for only one year ys <- NULL if(is.null(threshold)){ ts <- arpt(inc, weights, sort, p=p, na.rm=na.rm) # threshold } else{ ts <- threshold } value <- weightedRate(inc, weights, ts, na.rm=na.rm) if(byP) names(value) <- getPLabels(p) if(byStratum) { wr2 <- function(i, inc, weights, ts, na.rm) { weightedRate(inc[i], weights[i], ts, na.rm=na.rm) } valueByStratum <- aggregate(1:n, list(stratum=breakdown), wr2, inc=inc, weights=weights, ts=ts, na.rm=na.rm) if(byP) { # correction for data.frame necessary nam <- c("p", names(valueByStratum)[1], "value") valueByStratum <- data.frame(rep.int(p, length(rs)), rep(rs, each=length(p)), as.vector(t(valueByStratum[, -1]))) names(valueByStratum) <- nam # nam <- c(names(valueByStratum)[1], names(value)) # valueByStratum <- data.frame(valueByStratum[, 1, drop=FALSE], # valueByStratum[, -1]) # names(valueByStratum) <- nam } else names(valueByStratum)[2] <- "value" } else valueByStratum <- NULL } ## create object of class "arpr" res <- constructArpr(value=value, valueByStratum=valueByStratum, years=ys, strata=rs, p=p, threshold=ts) # variance estimation (if requested) if(!is.null(var)) { res <- variance(inc, weights, years, breakdown, design, cluster, indicator=res, alpha=alpha, na.rm=na.rm, type=var, ...) } # return results return(res) } ## workhorse weightedRate <- function(x, weights = NULL, threshold, na.rm = FALSE) { ## initializations if(is.null(weights)) weights <- rep.int(1, length(x)) # equal weights if(isTRUE(na.rm)){ indices <- !is.na(x) x <- x[indices] weights <- weights[indices] } else if(any(is.na(x))) return(NA) ## calculations # estimate population total sw <- sum(weights) # percentage of persons below threshold sapply(threshold, function(t) sum(weights[x < t]))*100/sw } laeken/R/weightedMedian.R0000644000176200001440000000250713616467254014775 0ustar liggesusers# ------------------------------------------ # Authors: Andreas Alfons and Matthias Templ # Vienna University of Technology # ------------------------------------------ #' Weighted median #' #' Compute the weighted median (Eurostat definition). #' #' The implementation strictly follows the Eurostat definition. #' #' @param x a numeric vector. #' @param weights an optional numeric vector giving the sample weights. #' @param sorted a logical indicating whether the observations in \code{x} are #' already sorted. #' @param na.rm a logical indicating whether missing values in \code{x} should #' be omitted. #' @return The weighted median of values in \code{x} is returned. #' #' @author Andreas Alfons and Matthias Templ #' #' @seealso \code{\link{arpt}}, \code{\link{incMedian}}, #' \code{\link{weightedQuantile}} #' #' @references Working group on Statistics on Income and Living Conditions #' (2004) Common cross-sectional EU indicators based on EU-SILC; the gender pay #' gap. \emph{EU-SILC 131-rev/04}, Eurostat. #' #' @keywords survey #' #' @examples #' data(eusilc) #' weightedMedian(eusilc$eqIncome, eusilc$rb050) #' #' @export weightedMedian <- function(x, weights = NULL, sorted = FALSE, na.rm = FALSE) { weightedQuantile(x, weights, probs=0.5, sorted=sorted, na.rm=na.rm) } laeken/R/incMedian.R0000644000176200001440000001006313616467254013742 0ustar liggesusers# --------------------------------------- # Author: Andreas Alfons # Vienna University of Technology # --------------------------------------- #' Weighted median income #' #' Compute the weighted median income. #' #' The implementation strictly follows the Eurostat definition. #' #' @param inc either a numeric vector giving the (equivalized disposable) #' income, or (if \code{data} is not \code{NULL}) a character string, an integer #' or a logical vector specifying the corresponding column of \code{data}. #' @param weights optional; either a numeric vector giving the personal sample #' weights, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. #' @param sort optional; either a numeric vector giving the personal IDs to be #' used as tie-breakers for sorting, or (if \code{data} is not \code{NULL}) a #' character string, an integer or a logical vector specifying the corresponding #' column of \code{data}. #' @param years optional; either a numeric vector giving the different years of #' the survey, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. If supplied, values are computed for each year. #' @param data an optional \code{data.frame}. #' @param na.rm a logical indicating whether missing values should be removed. #' #' @return A numeric vector containing the value(s) of the weighted median #' income is returned. #' #' @author Andreas Alfons #' #' @seealso \code{\link{arpt}}, \code{\link{weightedMedian}} #' #' @references Working group on Statistics on Income and Living Conditions #' (2004) Common cross-sectional EU indicators based on EU-SILC; the gender pay #' gap. \emph{EU-SILC 131-rev/04}, Eurostat. #' #' @keywords survey #' #' @examples #' data(eusilc) #' incMedian("eqIncome", weights = "rb050", data = eusilc) #' #' @export incMedian <- function(inc, weights = NULL, sort = NULL, years = NULL, data = NULL, na.rm = FALSE) { ## initializations if(!is.null(data)) { inc <- data[, inc] if(!is.null(weights)) weights <- data[, weights] if(!is.null(sort)) sort <- data[, sort] if(!is.null(years)) years <- data[, years] } # check vectors if(!is.numeric(inc)) stop("'inc' must be a numeric vector") n <- length(inc) if(!is.null(weights) && !is.numeric(weights)) { stop("'weights' must be a numeric vector") } if(!is.null(sort) && !is.vector(sort) && !is.ordered(sort)) { stop("'sort' must be a vector or ordered factor") } if(!is.null(years) && !is.numeric(years)) { stop("'years' must be a numeric vector") } if(is.null(data)) { # check vector lengths if(!is.null(weights) && length(weights) != n) { stop("'weights' must have the same length as 'x'") } if(!is.null(sort) && length(sort) != n) { stop("'sort' must have the same length as 'x'") } if(!is.null(years) && length(years) != n) { stop("'years' must have the same length as 'x'") } } ## sort values and weights order <- if(is.null(sort)) order(inc) else order(inc, sort) inc <- inc[order] weights <- weights[order] # also works if 'weights' is NULL ## computations if(is.null(years)) { # no breakdown med <- weightedMedian(inc, weights, sorted=TRUE, na.rm=na.rm) } else { # breakdown by years years <- years[order] # define wrapper functions calcMedian <- function(y, inc, weights, years, na.rm) { i <- years == y weightedMedian(inc[i], weights[i], sorted=TRUE, na.rm=na.rm) } # apply wrapper function ys <- sort(unique(years)) med <- sapply(ys, calcMedian, inc=inc, weights=weights, years=years, na.rm=na.rm) names(med) <- ys # use years as names } ## return results return(med) } laeken/R/qsr.R0000755000176200001440000002201514127253127012651 0ustar liggesusers# --------------------------------------- # Author: Andreas Alfons # Vienna University of Technology # --------------------------------------- #' Quintile share ratio #' #' Estimate the quintile share ratio, which is defined as the ratio of the sum #' of equivalized disposable income received by the top 20\% to the sum of #' equivalized disposable income received by the bottom 20\%. #' #' The implementation strictly follows the Eurostat definition. #' #' @param inc either a numeric vector giving the equivalized disposable income, #' or (if \code{data} is not \code{NULL}) a character string, an integer or a #' logical vector specifying the corresponding column of \code{data}. #' @param weights optional; either a numeric vector giving the personal sample #' weights, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. #' @param sort optional; either a numeric vector giving the personal IDs to be #' used as tie-breakers for sorting, or (if \code{data} is not \code{NULL}) a #' character string, an integer or a logical vector specifying the corresponding #' column of \code{data}. #' @param years optional; either a numeric vector giving the different years of #' the survey, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. If supplied, values are computed for each year. #' @param breakdown optional; either a numeric vector giving different domains, #' or (if \code{data} is not \code{NULL}) a character string, an integer or a #' logical vector specifying the corresponding column of \code{data}. If #' supplied, the values for each domain are computed in addition to the overall #' value. #' @param design optional and only used if \code{var} is not \code{NULL}; either #' an integer vector or factor giving different strata for stratified sampling #' designs, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. #' @param cluster optional and only used if \code{var} is not \code{NULL}; #' either an integer vector or factor giving different clusters for cluster #' sampling designs, or (if \code{data} is not \code{NULL}) a character string, #' an integer or a logical vector specifying the corresponding column of #' \code{data}. #' @param data an optional \code{data.frame}. #' @param var a character string specifying the type of variance estimation to #' be used, or \code{NULL} to omit variance estimation. See #' \code{\link{variance}} for possible values. #' @param alpha numeric; if \code{var} is not \code{NULL}, this gives the #' significance level to be used for computing the confidence interval (i.e., #' the confidence level is \eqn{1 - }\code{alpha}). #' @param na.rm a logical indicating whether missing values should be removed. #' @param \dots if \code{var} is not \code{NULL}, additional arguments to be #' passed to \code{\link{variance}}. #' #' @return A list of class \code{"qsr"} (which inherits from the class #' \code{"indicator"}) with the following components: #' \item{value}{a numeric vector containing the overall value(s).} #' \item{valueByStratum}{a \code{data.frame} containing the values by #' domain, or \code{NULL}.} #' \item{varMethod}{a character string specifying the type of variance #' estimation used, or \code{NULL} if variance estimation was omitted.} #' \item{var}{a numeric vector containing the variance estimate(s), or #' \code{NULL}.} #' \item{varByStratum}{a \code{data.frame} containing the variance #' estimates by domain, or \code{NULL}.} #' \item{ci}{a numeric vector or matrix containing the lower and upper #' endpoints of the confidence interval(s), or \code{NULL}.} #' \item{ciByStratum}{a \code{data.frame} containing the lower and upper #' endpoints of the confidence intervals by domain, or \code{NULL}.} #' \item{alpha}{a numeric value giving the significance level used for #' computing the confidence interval(s) (i.e., the confidence level is \eqn{1 - #' }\code{alpha}), or \code{NULL}.} #' \item{years}{a numeric vector containing the different years of the #' survey.} #' \item{strata}{a character vector containing the different domains of the #' breakdown.} #' #' @author Andreas Alfons #' #' @seealso \code{\link{incQuintile}}, \code{\link{variance}}, #' \code{\link{gini}} #' #' @references #' A. Alfons and M. Templ (2013) Estimation of Social Exclusion Indicators #' from Complex Surveys: The \R Package \pkg{laeken}. \emph{Journal of #' Statistical Software}, \bold{54}(15), 1--25. \doi{10.18637/jss.v054.i15} #' #' Working group on Statistics on Income and Living Conditions (2004) #' Common cross-sectional EU indicators based on EU-SILC; the gender #' pay gap. \emph{EU-SILC 131-rev/04}, Eurostat, Luxembourg. #' #' @keywords survey #' #' @examples #' data(eusilc) #' #' # overall value #' qsr("eqIncome", weights = "rb050", data = eusilc) #' #' # values by region #' qsr("eqIncome", weights = "rb050", #' breakdown = "db040", data = eusilc) #' #' @importFrom stats aggregate #' @export qsr <- function(inc, weights = NULL, sort = NULL, years = NULL, breakdown = NULL, design = NULL, cluster = NULL, data = NULL, var = NULL, alpha = 0.05, na.rm = FALSE, ...) { ## initializations byYear <- !is.null(years) byStratum <- !is.null(breakdown) if(!is.null(data)) { inc <- data[, inc] if(!is.null(weights)) weights <- data[, weights] if(!is.null(sort)) sort <- data[, sort] if(byYear) years <- data[, years] if(byStratum) breakdown <- data[, breakdown] if(!is.null(var)) { if(!is.null(design)) design <- data[, design] if(!is.null(cluster)) cluster <- data[, cluster] } } # check vectors if(!is.numeric(inc)) stop("'inc' must be a numeric vector") n <- length(inc) if(is.null(weights)) weights <- weights <- rep.int(1, n) else if(!is.numeric(weights)) stop("'weights' must be a numeric vector") if(!is.null(sort) && !is.vector(sort) && !is.ordered(sort)) { stop("'sort' must be a vector or ordered factor") } if(byYear && !is.numeric(years)) { stop("'years' must be a numeric vector") } if(byStratum) { if(!is.vector(breakdown) && !is.factor(breakdown)) { stop("'breakdown' must be a vector or factor") } else breakdown <- as.factor(breakdown) } if(is.null(data)) { # check vector lengths if(length(weights) != n) { stop("'weights' must have the same length as 'x'") } if(!is.null(sort) && length(sort) != n) { stop("'sort' must have the same length as 'x'") } if(byYear && length(years) != n) { stop("'years' must have the same length as 'x'") } if(byStratum && length(breakdown) != n) { stop("'breakdown' must have the same length as 'x'") } } ## computations # QSR by year (if requested) if(byYear) { ys <- sort(unique(years)) # unique years qr <- function(y, inc, weights, sort, years, na.rm) { i <- years == y quintileRatio(inc[i], weights[i], sort[i], na.rm=na.rm) } value <- sapply(ys, qr, inc=inc, weights=weights, sort=sort, years=years, na.rm=na.rm) names(value) <- ys # use years as names } else { ys <- NULL value <- quintileRatio(inc, weights, sort, na.rm=na.rm) } # QSR by stratum (if requested) if(byStratum) { qrR <- function(i, inc, weights, sort, na.rm) { quintileRatio(inc[i], weights[i], sort[i], na.rm=na.rm) } valueByStratum <- aggregate(1:n, if(byYear) list(year=years, stratum=breakdown) else list(stratum=breakdown), qrR, inc=inc, weights=weights, sort=sort, na.rm=na.rm) names(valueByStratum)[ncol(valueByStratum)] <- "value" rs <- levels(breakdown) # unique strata } else valueByStratum <- rs <- NULL ## create object of class "qsr" res <- constructQsr(value=value, valueByStratum=valueByStratum, years=ys, strata=rs) # variance estimation (if requested) if(!is.null(var)) { res <- variance(inc, weights, years, breakdown, design, cluster, indicator=res, alpha=alpha, na.rm=na.rm, type=var, ...) } ## return result return(res) } ## workhorse quintileRatio <- function(x, weights = NULL, sort = NULL, na.rm = FALSE) { # initializations if(isTRUE(na.rm)){ indices <- !is.na(x) x <- x[indices] if(!is.null(weights)) weights <- weights[indices] if(!is.null(sort)) sort <- sort[indices] } else if(any(is.na(x))) return(NA) if(is.null(weights)) weights <- rep.int(1, length(x)) # equal weights # indices of observations in bottom and top quintile q <- incQuintile(x, weights, sort) # quintiles iq1 <- x <= q[1] # in bottom quintile iq4 <- x > q[2] # in top quintile # calculations # (sum(weights[iq4] * x[iq4]) / sum(weights[iq4])) / # (sum(weights[iq1] * x[iq1]) / sum(weights[iq1])) sum(weights[iq4] * x[iq4]) / sum(weights[iq1] * x[iq1]) } laeken/R/gini.R0000755000176200001440000002154514127253135013000 0ustar liggesusers# --------------------------------------- # Author: Andreas Alfons # Vienna University of Technology # --------------------------------------- #' Gini coefficient #' #' Estimate the Gini coefficient, which is a measure for inequality. #' #' The implementation strictly follows the Eurostat definition. #' #' @param inc either a numeric vector giving the equivalized disposable income, #' or (if \code{data} is not \code{NULL}) a character string, an integer or a #' logical vector specifying the corresponding column of \code{data}. #' @param weights optional; either a numeric vector giving the personal sample #' weights, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. #' @param sort optional; either a numeric vector giving the personal IDs to be #' used as tie-breakers for sorting, or (if \code{data} is not \code{NULL}) a #' character string, an integer or a logical vector specifying the corresponding #' column of \code{data}. #' @param years optional; either a numeric vector giving the different years of #' the survey, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. If supplied, values are computed for each year. #' @param breakdown optional; either a numeric vector giving different domains, #' or (if \code{data} is not \code{NULL}) a character string, an integer or a #' logical vector specifying the corresponding column of \code{data}. If #' supplied, the values for each domain are computed in addition to the overall #' value. #' @param design optional and only used if \code{var} is not \code{NULL}; either #' an integer vector or factor giving different domains for stratified sampling #' designs, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. #' @param cluster optional and only used if \code{var} is not \code{NULL}; #' either an integer vector or factor giving different clusters for cluster #' sampling designs, or (if \code{data} is not \code{NULL}) a character string, #' an integer or a logical vector specifying the corresponding column of #' \code{data}. #' @param data an optional \code{data.frame}. #' @param var a character string specifying the type of variance estimation to #' be used, or \code{NULL} to omit variance estimation. See #' \code{\link{variance}} for possible values. #' @param alpha numeric; if \code{var} is not \code{NULL}, this gives the #' significance level to be used for computing the confidence interval (i.e., #' the confidence level is \eqn{1 - }\code{alpha}). #' @param na.rm a logical indicating whether missing values should be removed. #' @param \dots if \code{var} is not \code{NULL}, additional arguments to be #' passed to \code{\link{variance}}. #' #' @return A list of class \code{"gini"} (which inherits from the class #' \code{"indicator"}) with the following components: #' \item{value}{a numeric vector containing the overall value(s).} #' \item{valueByStratum}{a \code{data.frame} containing the values by #' domain, or \code{NULL}.} #' \item{varMethod}{a character string specifying the type of variance #' estimation used, or \code{NULL} if variance estimation was omitted.} #' \item{var}{a numeric vector containing the variance estimate(s), or #' \code{NULL}.} #' \item{varByStratum}{a \code{data.frame} containing the variance #' estimates by domain, or \code{NULL}.} #' \item{ci}{a numeric vector or matrix containing the lower and upper #' endpoints of the confidence interval(s), or \code{NULL}.} #' \item{ciByStratum}{a \code{data.frame} containing the lower and upper #' endpoints of the confidence intervals by domain, or \code{NULL}.} #' \item{alpha}{a numeric value giving the significance level used for #' computing the confidence interval(s) (i.e., the confidence level is \eqn{1 - #' }\code{alpha}), or \code{NULL}.} #' \item{years}{a numeric vector containing the different years of the #' survey.} #' \item{strata}{a character vector containing the different domains of the #' breakdown.} #' #' @author Andreas Alfons #' #' @seealso \code{\link{variance}}, \code{\link{qsr}} #' #' @references #' A. Alfons and M. Templ (2013) Estimation of Social Exclusion Indicators #' from Complex Surveys: The \R Package \pkg{laeken}. \emph{Journal of #' Statistical Software}, \bold{54}(15), 1--25. \doi{10.18637/jss.v054.i15} #' #' Working group on Statistics on Income and Living Conditions (2004) #' Common cross-sectional EU indicators based on EU-SILC; the gender #' pay gap. \emph{EU-SILC 131-rev/04}, Eurostat, Luxembourg. #' #' @keywords survey #' #' @examples #' data(eusilc) #' #' # overall value #' gini("eqIncome", weights = "rb050", data = eusilc) #' #' # values by region #' gini("eqIncome", weights = "rb050", #' breakdown = "db040", data = eusilc) #' #' @importFrom stats aggregate #' @export gini <- function(inc, weights = NULL, sort = NULL, years = NULL, breakdown = NULL, design = NULL, cluster = NULL, data = NULL, var = NULL, alpha = 0.05, na.rm = FALSE, ...) { ## initializations byYear <- !is.null(years) byStratum <- !is.null(breakdown) if(!is.null(data)) { inc <- data[, inc] if(!is.null(weights)) weights <- data[, weights] if(!is.null(sort)) sort <- data[, sort] if(byYear) years <- data[, years] if(byStratum) breakdown <- data[, breakdown] if(!is.null(var)) { if(!is.null(design)) design <- data[, design] if(!is.null(cluster)) cluster <- data[, cluster] } } # check vectors if(!is.numeric(inc)) stop("'inc' must be a numeric vector") n <- length(inc) if(is.null(weights)) weights <- weights <- rep.int(1, n) else if(!is.numeric(weights)) stop("'weights' must be a numeric vector") if(!is.null(sort) && !is.vector(sort) && !is.ordered(sort)) { stop("'sort' must be a vector or ordered factor") } if(byYear && !is.numeric(years)) { stop("'years' must be a numeric vector") } if(byStratum) { if(!is.vector(breakdown) && !is.factor(breakdown)) { stop("'breakdown' must be a vector or factor") } else breakdown <- as.factor(breakdown) } if(is.null(data)) { # check vector lengths if(length(weights) != n) { stop("'weights' must have the same length as 'x'") } if(!is.null(sort) && length(sort) != n) { stop("'sort' must have the same length as 'x'") } if(byYear && length(years) != n) { stop("'years' must have the same length as 'x'") } if(byStratum && length(breakdown) != n) { stop("'breakdown' must have the same length as 'x'") } } ## computations # Gini by year (if requested) if(byYear) { ys <- sort(unique(years)) # unique years gc <- function(y, inc, weights, sort, years, na.rm) { i <- years == y giniCoeff(inc[i], weights[i], sort[i], na.rm=na.rm) } value <- sapply(ys, gc, inc=inc, weights=weights, sort=sort, years=years, na.rm=na.rm) names(value) <- ys # use years as names } else { ys <- NULL value <- giniCoeff(inc, weights, sort, na.rm=na.rm) } # Gini by stratum (if requested) if(byStratum) { gcR <- function(i, inc, weights, sort, na.rm) { giniCoeff(inc[i], weights[i], sort[i], na.rm=na.rm) } valueByStratum <- aggregate(1:n, if(byYear) list(year=years, stratum=breakdown) else list(stratum=breakdown), gcR, inc=inc, weights=weights, sort=sort, na.rm=na.rm) names(valueByStratum)[ncol(valueByStratum)] <- "value" rs <- levels(breakdown) # unique strata } else valueByStratum <- rs <- NULL ## create object of class "qsr" res <- constructGini(value=value, valueByStratum=valueByStratum, years=ys, strata=rs) # variance estimation (if requested) if(!is.null(var)) { res <- variance(inc, weights, years, breakdown, design, cluster, indicator=res, alpha=alpha, na.rm=na.rm, type=var, ...) } ## return result return(res) } ## workhorse giniCoeff <- function(x, weights = NULL, sort = NULL, na.rm = FALSE) { # initializations if(isTRUE(na.rm)){ indices <- !is.na(x) x <- x[indices] if(!is.null(weights)) weights <- weights[indices] if(!is.null(sort)) sort <- sort[indices] } else if(any(is.na(x))) return(NA) # sort values and weights order <- if(is.null(sort)) order(x) else order(x, sort) x <- x[order] # order values if(is.null(weights)) weights <- rep.int(1, length(x)) # equal weights else weights <- weights[order] # order weights ## calculations wx <- weights * x # weighted values sw <- sum(weights) # sum of weights cw <- cumsum(weights) # cumulative sum of weights 100 * ((2 * sum(wx*cw) - sum(weights^2 * x)) / (sw * sum(wx)) - 1) } laeken/R/paretoScale.R0000644000176200001440000001117614127253151014306 0ustar liggesusers# --------------------------------------- # Author: Andreas Alfons # Vienna University of Technology # --------------------------------------- #' Estimate the scale parameter of a Pareto distribution #' #' Estimate the scale parameter of a Pareto distribution, i.e., the threshold #' for Pareto tail modeling. #' #' Van Kerm's formula is given by \deqn{\min(\max(2.5 \bar{x}, q(0.98), #' q(0.97))),}{min(max(2.5 m(x), q(0.98)), q(0.97)),} where \eqn{\bar{x}}{m(x)} #' denotes the weighted mean and \eqn{q(.)} denotes weighted quantiles. This #' function allows to compute generalizations of Van Kerm's formula, where the #' mean can be replaced by the median and different quantiles can be used. #' #' @aliases print.paretoScale #' #' @param x a numeric vector. #' @param w an optional numeric vector giving sample weights. #' @param groups an optional vector or factor specifying groups of elements of #' \code{x} (e.g., households). If supplied, each group of observations is #' expected to have the same value in \code{x} (e.g., household income). Only #' the values of every first group member to appear are used for estimating the #' threshold (scale parameter). #' @param method a character string specifying the estimation method. If #' \code{"VanKerm"}, Van Kerm's method is used, which is a rule of thumb #' specifically designed for the equivalized disposable income in EU-SILC data #' (currently the only method implemented). #' @param center a character string specifying the estimation method for the #' center of the distribution. Possible values are \code{"mean"} for the #' weighted mean and \code{"median"} for the weighted median. This is used if #' \code{method} is \code{"VanKerm"} (currently the only method implemented). #' @param probs a numeric vector of length two giving probabilities to be used #' for computing weighted quantiles of the distribution. Values should be close #' to 1 such that the quantiles correspond to the upper tail. This is used if #' \code{method} is \code{"VanKerm"} (currently the only method implemented). #' @param na.rm a logical indicating whether missing values in \code{x} should #' be omitted. #' #' @return An object of class \code{"paretoScale"} with the following #' components: #' \item{x0}{the threshold (scale parameter).} #' \item{k}{the number of observations in the tail (i.e., larger than the #' threshold).} #' #' @author Andreas Alfons #' #' @seealso \code{\link{minAMSE}}, \code{\link{paretoQPlot}}, #' \code{\link{meanExcessPlot}} #' #' @references #' A. Alfons and M. Templ (2013) Estimation of Social Exclusion Indicators #' from Complex Surveys: The \R Package \pkg{laeken}. \emph{Journal of #' Statistical Software}, \bold{54}(15), 1--25. \doi{10.18637/jss.v054.i15} #' #' Van Kerm, P. (2007) Extreme incomes and the estimation of poverty and #' inequality indicators from EU-SILC. IRISS Working Paper Series 2007-01, #' CEPS/INSTEAD. #' #' @keywords manip #' #' @examples #' data(eusilc) #' paretoScale(eusilc$eqIncome, eusilc$db090, groups = eusilc$db030) #' #' @export paretoScale <- function(x, w = NULL, groups = NULL, method = "VanKerm", center = c("mean", "median"), probs = c(0.97, 0.98), na.rm = FALSE) { ## initializations if(!is.numeric(x) || length(x) == 0) stop("'x' must be a numeric vector") useW <- !is.null(w) if(useW && (!is.numeric(w) || length(w) != length(x))) { stop("'w' must be numeric vector of the same length as 'x'") } haveGroups <- !is.null(groups) if(haveGroups) { if(!is.vector(groups) && !is.factor(groups)) { stop("'groups' must be a vector or factor") } if(length(groups) != length(x)) { stop("'groups' must have the same length as 'x'") } if(any(is.na(groups))) stop("'groups' contains missing values") unique <- !duplicated(groups) x <- x[unique] if(useW) w <- w[unique] } # method <- match.arg(method) # only van Kerm's method currently implemented center <- match.arg(center) probs <- rep(probs, length.out=2) na.rm <- isTRUE(na.rm) # estimate threshold with van Kerm's method if(center == "mean") { mu <- weightedMean(x, w, na.rm=na.rm) q <- weightedQuantile(x, w, probs=probs, na.rm=na.rm) } else { q <- weightedQuantile(x, w, probs=c(0.5, probs), na.rm=na.rm) mu <- q[1] q <- q[-1] } x0 <- max(min(2.5*mu, q[2]), q[1]) res <- list(x0=x0, k=length(which(x > x0))) class(res) <- "paretoScale" res } ## print method for class "paretoScale" #' @export print.paretoScale <- function(x, ...) { cat("Threshold: ") cat(x$x0, ...) cat("\nNumber of observations in the tail: ") cat(x$k, ...) cat("\n") } laeken/R/meanExcessPlot.R0000644000176200001440000001432513616467254015012 0ustar liggesusers# ---------------------------------------- # Authors: Josef Holzer and Andreas Alfons # Vienna University of Technology # ---------------------------------------- #' Mean excess plot #' #' The Mean Excess plot is a graphical method for detecting the threshold (scale #' parameter) of a Pareto distribution. #' #' The corresponding mean excesses are plotted against the values of \code{x} #' (if supplied, only those specified by \code{probs}). If the tail of the data #' follows a Pareto distribution, these observations show a positive linear #' trend. The leftmost point of a fitted line can thus be used as an estimate of #' the threshold (scale parameter). #' #' The interactive selection of the threshold (scale parameter) is implemented #' using \code{\link[graphics]{identify}}. For the usual \code{X11} device, the #' selection process is thus terminated by pressing any mouse button other than #' the first. For the \code{quartz} device (on Mac OS X systems), the process #' is terminated either by a secondary click (usually second mouse button or #' \code{Ctrl}-click) or by pressing the \code{ESC} key. #' #' @param x a numeric vector. #' @param w an optional numeric vector giving sample weights. #' @param probs an optional numeric vector of probabilities with values in #' \eqn{[0,1]}, defining the quantiles to be plotted. This is useful for large #' data sets, when it may not be desirable to plot every single point. #' @param interactive a logical indicating whether the threshold (scale #' parameter) can be selected interactively by clicking on points. Information #' on the selected threshold is then printed on the console. #' @param pch,cex,col,bg graphical parameters for the plot symbol of each data #' point or quantile (see \code{\link[graphics]{points}}). #' @param \dots additional arguments to be passed to #' \code{\link[graphics]{plot.default}}. #' #' @return If \code{interactive} is \code{TRUE}, the last selection for the #' threshold is returned invisibly as an object of class \code{"paretoScale"}, #' which consists of the following components: #' \item{x0}{the selected threshold (scale parameter).} #' \item{k}{the number of observations in the tail (i.e., larger than the #' threshold).} #' #' @note The functionality to account for sample weights and to select the #' threshold (scale parameter) interactively was introduced in version 0.2. #' #' @author Andreas Alfons and Josef Holzer #' #' @seealso \code{\link{paretoScale}}, \code{\link{paretoTail}}, #' \code{\link{minAMSE}}, \code{\link{paretoQPlot}}, #' \code{\link[graphics]{identify}} #' #' @keywords hplot #' #' @examples #' data(eusilc) #' # equivalized disposable income is equal for each household #' # member, therefore only one household member is taken #' eusilc <- eusilc[!duplicated(eusilc$db030),] #' #' # with sample weights #' meanExcessPlot(eusilc$eqIncome, w = eusilc$db090) #' #' # without sample weights #' meanExcessPlot(eusilc$eqIncome) #' #' @importFrom graphics identify abline par plot #' @importFrom stats quantile weighted.mean #' @export meanExcessPlot <- function(x, w = NULL, probs = NULL, interactive = TRUE, pch = par("pch"), cex = par("cex"), col = par("col"), bg = "transparent", ...) { ## initializations n <- length(x) if(!is.numeric(x) || n == 0) stop("'x' must be a numeric vector") if(!is.null(w)) { if(!is.numeric(w) || length(w) != n) { stop("'w' must be numeric vector of the same length as 'x'") } if(any(w < 0)) stop("negative weights in 'w'") } haveProbs <- !is.null(probs) if(haveProbs) n <- length(probs) if(length(pch) > 1) pch <- rep(pch, length.out=n) if(length(cex) > 1) cex <- rep(cex, length.out=n) if(length(col) > 1) col <- rep(col, length.out=n) if(length(bg) > 1) bg <- rep(bg, length.out=n) if(any(i <- is.na(x))) { # remove missing values x <- x[!i] if(!is.null(w)) w <- w[!i] if(!haveProbs) { if(length(pch) > 1) pch <- pch[!i] if(length(cex) > 1) cex <- cex[!i] if(length(col) > 1) col <- col[!i] if(length(bg) > 1) bg <- bg[!i] n <- length(x) } if(length(x) == 0) stop("no observed values") } ## use observed values or quantiles as thresholds if(haveProbs) { if(is.null(w)) { # no weights mu <- quantile(x, probs, names=FALSE, type=1) # compute quantiles } else { # weights are supplied mu <- weightedQuantile(x, w, probs) # compute weighted quantiles } if(max(mu) >= max(x)) stop("largest threshold too high") } else { order <- order(x) keep <- seq_len(n-sqrt(n)) mu <- unname(x[order][keep]) if(length(pch) > 1) pch <- pch[order][keep] if(length(cex) > 1) cex <- cex[order][keep] if(length(col) > 1) col <- col[order][keep] if(length(bg) > 1) bg <- bg[order][keep] } ## compute mean excesses for the different thresholds # this could be done much faster with C (incremental computation) if(is.null(w)) meanExcess <- function(mu) mean(x[x > mu] - mu) else { meanExcess <- function(mu) { i <- x > mu weighted.mean(x[i] - mu, w[i]) } } me <- sapply(mu, meanExcess) ## create plot localPlot <- function(x, y, main = "Mean excess plot", xlab = "Threshold", ylab = "Mean excess", ...) { plot(x, y, main=main, xlab=xlab, ylab=ylab, ...) } localPlot(mu, me, pch=pch, cex=cex, col=col, bg=bg, ...) ## interactive identification of threshold res <- NULL if(isTRUE(interactive)) { nextIndex <- identify(mu, me, n=1, plot=FALSE) i <- 1 while(!identical(nextIndex, integer())) { index <- nextIndex x0 <- mu[index] res <- list(x0=x0, k=length(which(x > x0))) class(res) <- "paretoScale" if(i > 1) cat("\n") print(res) nextIndex <- identify(mu, me, n=1, plot=FALSE) i <- i + 1 } # indicate selected threshold by horizontal and vertical lines if(!is.null(res)) { abline(h=me[index], lty=3) abline(v=x0, lty=3) } } ## return result invisibly invisible(res) } laeken/R/thetaISE.R0000644000176200001440000001415114127253205013506 0ustar liggesusers# ---------------------------------------- # Authors: Andreas Alfons and Josef Holzer # Vienna University of Technology # ---------------------------------------- #' Integrated squared error (ISE) estimator #' #' The integrated squared error (ISE) estimator estimates the shape parameter of #' a Pareto distribution based on the relative excesses of observations above a #' certain threshold. #' #' The arguments \code{k} and \code{x0} of course correspond with each other. #' If \code{k} is supplied, the threshold \code{x0} is estimated with the \eqn{n #' - k} largest value in \code{x}, where \eqn{n} is the number of observations. #' On the other hand, if the threshold \code{x0} is supplied, \code{k} is given #' by the number of observations in \code{x} larger than \code{x0}. Therefore, #' either \code{k} or \code{x0} needs to be supplied. If both are supplied, #' only \code{k} is used (mainly for back compatibility). #' #' The ISE estimator minimizes the integrated squared error (ISE) criterion with #' a complete density model. The minimization is carried out using % #' \code{\link[stats]{nlm}}. By default, the starting value is obtained % with #' the Hill estimator (see \code{\link{thetaHill}}). #' \code{\link[stats]{optimize}}. #' #' @param x a numeric vector. #' @param k the number of observations in the upper tail to which the Pareto #' distribution is fitted. #' @param x0 the threshold (scale parameter) above which the Pareto distribution #' is fitted. #' @param w an optional numeric vector giving sample weights. #' @param \dots additional arguments to be passed to #' \code{\link[stats]{optimize}} (see \dQuote{Details}). #' #' @return The estimated shape parameter. #' #' @note The arguments \code{x0} for the threshold (scale parameter) of the #' Pareto distribution and \code{w} for sample weights were introduced in #' version 0.2. #' #' @author Andreas Alfons and Josef Holzer #' #' @seealso \code{\link{paretoTail}}, \code{\link{fitPareto}}, #' \code{\link{thetaPDC}}, \code{\link{thetaHill}} #' #' @references #' A. Alfons and M. Templ (2013) Estimation of Social Exclusion Indicators #' from Complex Surveys: The \R Package \pkg{laeken}. \emph{Journal of #' Statistical Software}, \bold{54}(15), 1--25. \doi{10.18637/jss.v054.i15} #' #' A. Alfons, M. Templ, P. Filzmoser (2013) Robust estimation of economic #' indicators from survey samples based on Pareto tail modeling. \emph{Journal #' of the Royal Statistical Society, Series C}, \bold{62}(2), 271--286. #' #' Vandewalle, B., Beirlant, J., Christmann, A., and Hubert, M. #' (2007) A robust estimator for the tail index of Pareto-type #' distributions. \emph{Computational Statistics & Data Analysis}, #' \bold{51}(12), 6252--6268. #' #' @keywords manip #' #' @examples #' data(eusilc) #' # equivalized disposable income is equal for each household #' # member, therefore only one household member is taken #' eusilc <- eusilc[!duplicated(eusilc$db030),] #' #' # estimate threshold #' ts <- paretoScale(eusilc$eqIncome, w = eusilc$db090) #' #' # using number of observations in tail #' thetaISE(eusilc$eqIncome, k = ts$k, w = eusilc$db090) #' #' # using threshold #' thetaISE(eusilc$eqIncome, x0 = ts$x0, w = eusilc$db090) #' #' @export thetaISE <- function(x, k = NULL, x0 = NULL, w = NULL, ...) { ## initializations if(!is.numeric(x) || length(x) == 0) stop("'x' must be a numeric vector") haveK <- !is.null(k) if(haveK) { # if 'k' is supplied, it is always used if(!is.numeric(k) || length(k) == 0 || k[1] < 1) { stop("'k' must be a positive integer") } else k <- k[1] } else if(!is.null(x0)) { # otherwise 'x0' (threshold) is used if(!is.numeric(x0) || length(x0) == 0) stop("'x0' must be numeric") else x0 <- x0[1] } else stop("either 'k' or 'x0' must be supplied") haveW <- !is.null(w) if(haveW) { # sample weights are supplied if(!is.numeric(w) || length(w) != length(x)) { stop("'w' must be numeric vector of the same length as 'x'") } if(any(w < 0)) stop("negative weights in 'w'") if(any(i <- is.na(x))) { # remove missing values x <- x[!i] w <- w[!i] } # sort values and sample weights order <- order(x) x <- x[order] w <- w[order] } else { # no sample weights if(any(i <- is.na(x))) x <- x[!i] # remove missing values x <- sort(x) # sort values } .thetaISE(x, k, x0, w, ...) } # internal function that assumes that data are ok and sorted .thetaISE <- function(x, k = NULL, x0 = NULL, w = NULL, tol = .Machine$double.eps^0.25, ...) { n <- length(x) # number of observations haveK <- !is.null(k) haveW <- !is.null(w) if(haveK) { # 'k' is supplied, threshold is determined if(k >= n) stop("'k' must be smaller than the number of observed values") x0 <- x[n-k] # threshold (scale parameter) } else { # 'k' is not supplied, it is determined using threshold # values are already sorted if(x0 >= x[n]) stop("'x0' must be smaller than the maximum of 'x'") k <- length(which(x > x0)) } ## computations y <- x[(n-k+1):n]/x0 # relative excesses if(haveW) { wTail <- w[(n-k+1):n] ## weighted integrated squared error distance criterion # w ... sample weights ISE <- function(theta, y, w) { f <- theta*y^(-1-theta) weighted.mean(theta^2/(2*theta+1) - 2*f, w) } } else { wTail <- NULL ## integrated squared error distance criterion # w ... sample weights (not needed here, only available to have the # same function definition) ISE <- function(theta, y, w) { f <- theta*y^(-1-theta) mean(theta^2/(2*theta+1) - 2*f) } } ## optimize localOptimize <- function(f, interval = NULL, tol, ...) { if(is.null(interval)) { p <- if(haveK) .thetaHill(x, k, w=w) else .thetaHill(x, x0=x0, w=w) interval <- c(0 + tol, 3 * p) # default interval } optimize(f, interval, ...) } localOptimize(ISE, y=y, w=wTail, tol=tol, ...)$minimum } laeken/R/calibWeights.R0000644000176200001440000002213414127273134014450 0ustar liggesusers# --------------------------------------- # Author: Andreas Alfons # Vienna University of Technology # --------------------------------------- #' Calibrate sample weights #' #' Calibrate sample weights according to known marginal population totals. #' Based on initial sample weights, the so-called \emph{g}-weights are computed #' by generalized raking procedures. #' #' The final sample weights need to be computed by multiplying the resulting #' \emph{g}-weights with the initial sample weights. #' #' @encoding utf8 #' #' @param X a matrix of binary calibration variables (see #' \code{\link{calibVars}}). #' @param d a numeric vector giving the initial sample weights. #' @param totals a numeric vector of population totals corresponding to the #' calibration variables in \code{X}. #' @param q a numeric vector of positive values accounting for #' heteroscedasticity. Small values reduce the variation of the #' \emph{g}-weights. #' @param method a character string specifying the calibration method to be #' used. Possible values are \code{"linear"} for the linear method, #' \code{"raking"} for the multiplicative method known as raking and #' \code{"logit"} for the logit method. #' @param bounds a numeric vector of length two giving bounds for the g-weights #' to be used in the logit method. The first value gives the lower bound (which #' must be smaller than or equal to 1) and the second value gives the upper #' bound (which must be larger than or equal to 1). #' @param maxit a numeric value giving the maximum number of iterations. #' @param tol the desired accuracy for the iterative procedure. #' @param eps the desired accuracy for computing the Moore-Penrose generalized #' inverse (see \code{\link[MASS]{ginv}}). #' #' @return A numeric vector containing the \emph{g}-weights. #' #' @note This is a faster implementation of parts of \code{calib} from #' package \code{sampling}. Note that the default calibration method is #' raking and that the truncated linear method is not yet implemented. #' #' @author Andreas Alfons #' #' @seealso \code{\link{calibVars}}, \code{\link{bootVar}} #' #' @references Deville, J.-C. and \enc{Särndal}{Saerndal}, C.-E. (1992) #' Calibration estimators in survey sampling. \emph{Journal of the American #' Statistical Association}, \bold{87}(418), 376--382. #' #' Deville, J.-C., \enc{Särndal}{Saerndal}, C.-E. and Sautory, O. (1993) #' Generalized raking procedures in survey sampling. \emph{Journal of the #' American Statistical Association}, \bold{88}(423), 1013--1020. #' #' @keywords survey #' #' @examples #' data(eusilc) #' # construct auxiliary 0/1 variables for genders #' aux <- calibVars(eusilc$rb090) #' # population totals #' totals <- c(3990798, 4191431) #' # compute g-weights #' g <- calibWeights(aux, eusilc$rb050, totals) #' # compute final weights #' weights <- g * eusilc$rb050 #' summary(weights) #' #' @importFrom MASS ginv #' @export calibWeights <- function(X, d, totals, q = NULL, method = c("raking", "linear", "logit"), bounds = c(0, 10), maxit = 500, tol = 1e-06, eps = .Machine$double.eps) { ## initializations and error handling X <- as.matrix(X) d <- as.numeric(d) totals <- as.numeric(totals) haveNA <- c(any(is.na(X)), any(is.na(d)), any(is.na(totals)), !is.null(q) && any(is.na(q))) if(any(haveNA)) { argsNA <- c("'X'", "'d'", "'totals'", "'q'")[haveNA] stop("missing values in the following arguments", paste(argsNA, collapse=", ")) } n <- nrow(X) # number of rows if(length(d) != n) stop("length of 'd' not equal to number of rows in 'X'") p <- ncol(X) # number of columns if(length(totals) != p) { stop("length of 'totals' not equal to number of columns in 'X'") } if(is.null(q)) q <- rep.int(1, n) else { q <- as.numeric(q) if(length(q) != n) { stop("length of 'q' not equal to number of rows in 'X'") } if(any(is.infinite(q))) stop("infinite values in 'q'") } method <- match.arg(method) ## computation of g-weights if(method == "linear") { ## linear method (no iteration!) lambda <- ginv(t(X * d * q) %*% X, tol=eps) %*% (totals - as.vector(t(d) %*% X)) g <- 1 + q * as.vector(X %*% lambda) # g-weights } else { ## multiplicative method (raking) or logit method lambda <- matrix(0, nrow=p) # initial values # function to determine whether teh desired accuracy has # not yet been reached (to be used in the 'while' loop) tolNotReached <- function(X, w, totals, tol) { max(abs(crossprod(X, w) - totals)/totals) >= tol } if(method == "raking") { ## multiplicative method (raking) # some initial values g <- rep.int(1, n) # g-weights w <- d # sample weights ## iterations i <- 1 while(!any(is.na(g)) && tolNotReached(X, w, totals, tol) && i <= maxit) { # here 'phi' describes more than the phi function in Deville, # Saerndal and Sautory (1993); it is the whole last term of # equation (11.1) phi <- t(X) %*% w - totals T <- t(X * w) dphi <- T %*% X # derivative of phi function (to be inverted) lambda <- lambda - ginv(dphi, tol=eps) %*% phi # update 'lambda' g <- exp(as.vector(X %*% lambda) * q) # update g-weights w <- g * d # update sample weights i <- i + 1 # increase iterator } ## check wether procedure converged if(any(is.na(g)) || i > maxit) { warning("no convergence") g <- NULL } } else { ## logit (L, U) method ## error handling for bounds if(length(bounds) < 2) stop("'bounds' must be a vector of length 2") else bounds <- bounds[1:2] if(bounds[1] >= 1) stop("the lower bound must be smaller than 1") if(bounds[2] <= 1) stop("the lower bound must be larger than 1") ## some preparations A <- diff(bounds)/((1 - bounds[1]) * (bounds[2] - 1)) # function to bound g-weights getG <- function(u, bounds) { (bounds[1] * (bounds[2]-1) + bounds[2] * (1-bounds[1]) * u) / (bounds[2]-1 + (1-bounds[1]) * u) } ## some initial values g <- getG(rep.int(1, n), bounds) # g-weights # in the procedure, g-weights outside the bounds are moved to the # bounds and only the g-weights within the bounds are adjusted. # these duplicates are needed since in general they are changed in # each iteration while the original values are also needed X1 <- X d1 <- d totals1 <- totals q1 <- q g1 <- g indices <- 1:n # function to determine which g-weights are outside the bounds anyOutOfBounds <- function(g, bounds) { any(g < bounds[1]) || any(g > bounds[2]) } ## iterations i <- 1 while(!any(is.na(g)) && (tolNotReached(X, g*d, totals, tol) || anyOutOfBounds(g, bounds)) && i <= maxit) { # if some of the g-weights are outside the bounds, these values # are moved to the bounds and only the g-weights within the # bounds are adjusted if(anyOutOfBounds(g, bounds)) { g[g < bounds[1]] <- bounds[1] g[g > bounds[2]] <- bounds[2] # values within the bounds tmp <- which(g > bounds[1] & g < bounds[2]) if(length(tmp) > 0) { indices <- tmp X1 <- X[indices,] d1 <- d[indices] if(length(indices) < n) { totals1 <- totals - as.vector(t(g[-indices] * d[-indices]) %*% X[-indices, , drop=FALSE]) } q1 <- q[indices] g1 <- g[indices] } } w1 <- g1 * d1 # current sample weights # here 'phi' describes more than the phi function in Deville, # Saerndal and Sautory (1993); it is the whole last term of # equation (11.1) phi <- t(X1) %*% w1 - totals1 T <- t(X1 * w1) dphi <- T %*% X1 # derivative of phi function (to be inverted) lambda <- lambda - ginv(dphi, tol=eps) %*% phi # update 'lambda' # update g-weights u <- exp(A * as.vector(X1 %*% lambda) * q1) g1 <- getG(u, bounds) g[indices] <- g1 i <- i+1 # increase iterator } ## check wether procedure converged if(any(is.na(g)) || i > maxit) { warning("no convergence") g <- NULL } } } ## return g-weights return(g) } laeken/R/paretoQPlot.R0000644000176200001440000001705514127253177014330 0ustar liggesusers# ---------------------------------------- # Authors: Andreas Alfons and Josef Holzer # Vienna University of Technology # ---------------------------------------- #' Pareto quantile plot #' #' The Pareto quantile plot is a graphical method for inspecting the parameters #' of a Pareto distribution. #' #' If the Pareto model holds, there exists a linear relationship between the #' lograrithms of the observed values and the quantiles of the standard #' exponential distribution, since the logarithm of a Pareto distributed random #' variable follows an exponential distribution. Hence the logarithms of the #' observed values are plotted against the corresponding theoretical quantiles. #' If the tail of the data follows a Pareto distribution, these observations #' form almost a straight line. The leftmost point of a fitted line can thus be #' used as an estimate of the threshold (scale parameter). The slope of the #' fitted line is in turn an estimate of \eqn{\frac{1}{\theta}}{1/theta}, the #' reciprocal of the shape parameter. #' #' The interactive selection of the threshold (scale parameter) is implemented #' using \code{\link[graphics]{identify}}. For the usual \code{X11} device, the #' selection process is thus terminated by pressing any mouse button other than #' the first. For the \code{quartz} device (on Mac OS X systems), the process #' is terminated either by a secondary click (usually second mouse button or #' \code{Ctrl}-click) or by pressing the \code{ESC} key. #' #' @param x a numeric vector. #' @param w an optional numeric vector giving sample weights. #' @param xlab,ylab axis labels. #' @param interactive a logical indicating whether the threshold (scale #' parameter) can be selected interactively by clicking on points. Information #' on the selected threshold is then printed on the console. #' @param x0,theta optional; if estimates of the threshold (scale parameter) #' and the shape parameter have already been obtained, they can be passed #' through the corresponding argument (\code{x0} for the threshold, #' \code{theta} for the shape parameter). If both arguments are supplied and #' \code{interactive} is not \code{TRUE}, reference lines are drawn to indicate #' the parameter estimates. #' @param pch,cex,col,bg graphical parameters for the plot symbol of each data #' point (see \code{\link[graphics]{points}}). #' @param \dots additional arguments to be passed to #' \code{\link[graphics]{plot.default}}. #' #' @return If \code{interactive} is \code{TRUE}, the last selection for the #' threshold is returned invisibly as an object of class \code{"paretoScale"}, #' which consists of the following components: #' \item{x0}{the selected threshold (scale parameter).} #' \item{k}{the number of observations in the tail (i.e., larger than the #' threshold).} #' #' @note The functionality to account for sample weights and to select the #' threshold (scale parameter) interactively was introduced in version 0.2. #' Also starting with version 0.2, a logarithmic y-axis is now used to display #' the axis labels in the scale of the original values. #' #' @author Andreas Alfons and Josef Holzer #' #' @seealso \code{\link{paretoScale}}, \code{\link{paretoTail}}, #' \code{\link{minAMSE}}, \code{\link{meanExcessPlot}}, #' \code{\link[graphics]{identify}} #' #' @references #' A. Alfons and M. Templ (2013) Estimation of Social Exclusion Indicators #' from Complex Surveys: The \R Package \pkg{laeken}. \emph{Journal of #' Statistical Software}, \bold{54}(15), 1--25. \doi{10.18637/jss.v054.i15} #' #' A. Alfons, M. Templ, P. Filzmoser (2013) Robust estimation of economic #' indicators from survey samples based on Pareto tail modeling. \emph{Journal #' of the Royal Statistical Society, Series C}, \bold{62}(2), 271--286. #' #' Beirlant, J., Vynckier, P. and Teugels, J.L. (1996) Tail index estimation, #' Pareto quantile plots, and regression diagnostics. \emph{Journal of the #' American Statistical Association}, \bold{91}(436), 1659--1667. #' #' @keywords hplot #' #' @examples #' data(eusilc) #' # equivalized disposable income is equal for each household #' # member, therefore only one household member is taken #' eusilc <- eusilc[!duplicated(eusilc$db030),] #' #' # with sample weights #' paretoQPlot(eusilc$eqIncome, w = eusilc$db090) #' #' # without sample weights #' paretoQPlot(eusilc$eqIncome) #' #' @importFrom graphics identify abline par plot #' @export paretoQPlot <- function(x, w = NULL, xlab = NULL, ylab = NULL, interactive = TRUE, x0 = NULL, theta = NULL, pch = par("pch"), cex = par("cex"), col = par("col"), bg = "transparent", ...) { ## initializations n <- length(x) if(!is.numeric(x) || n == 0) stop("'x' must be a numeric vector") if(!is.null(w)) { if(!is.numeric(w) || length(w) != n) { stop("'w' must be numeric vector of the same length as 'x'") } if(any(w < 0)) stop("negative weights in 'w'") } if(length(pch) > 1) pch <- rep(pch, length.out=n) if(length(cex) > 1) cex <- rep(cex, length.out=n) if(length(col) > 1) col <- rep(col, length.out=n) if(length(bg) > 1) bg <- rep(bg, length.out=n) if(any(i <- is.na(x))) { # remove missing values x <- x[!i] if(!is.null(w)) w <- w[!i] if(length(pch) > 1) pch <- pch[!i] if(length(cex) > 1) cex <- cex[!i] if(length(col) > 1) col <- col[!i] if(length(bg) > 1) bg <- bg[!i] n <- length(x) if(n == 0) stop("no observed values") } # sort values and weights order <- order(x) x <- x[order] if(!is.null(w)) w <- w[order] if(length(pch) > 1) pch <- pch[order] if(length(cex) > 1) cex <- cex[order] if(length(col) > 1) col <- col[order] if(length(bg) > 1) bg <- bg[order] ## computation of theoretical quantiles if(is.null(w)) { y <- -log((n:1)/(n+1)) } else { cw <- cumsum(w) y <- -log(1 - cw/(cw[n]*(n+1)/n)) } ## create plot if(is.null(xlab)) xlab <- "Theoretical quantiles" if(is.null(ylab)) ylab <- "" localPlot <- function(x, y, main = "Pareto quantile plot", log, xlog, ylog, ...) { suppressWarnings(plot(x, y, main=main, log="y", ...)) } localPlot(y, x, xlab=xlab, ylab=ylab, pch=pch, cex=cex, col=col, bg=bg, ...) ## interactive identification of threshold res <- NULL if(isTRUE(interactive)) { nextIndex <- identify(y, x, n=1, plot=FALSE) i <- 1 while(!identical(nextIndex, integer())) { index <- nextIndex x0 <- unname(x[index]) res <- list(x0=x0, k=length(which(x > x0))) class(res) <- "paretoScale" if(i > 1) cat("\n") print(res) nextIndex <- identify(y, x, n=1, plot=FALSE) i <- i + 1 } # indicate selected threshold by horizontal and vertical lines if(!is.null(res)) { abline(h=x0, col="darkgrey", lty=3) abline(v=y[index], col="darkgrey", lty=3) } } else if(!is.null(x0) && !is.null(theta)) { k <- length(which(x > x0)) index <- n - k # add line for estimate of shape parameter usr <- par("usr") par(ylog=FALSE, usr=c(usr[1:2], log(10^usr[3:4]))) # change coordinate system on.exit(par(ylog=TRUE, usr=usr)) # change coordinate system back on exit slope <- 1/theta intercept <- log(x[index]) - slope * y[index] abline(intercept, slope, col="darkgrey") # indicate scale parameter by horizontal line abline(h=log(x0), col="darkgrey", lty=3) } ## return result invisibly invisible(res) } laeken/R/calibVars.R0000644000176200001440000000320613616467254013762 0ustar liggesusers# --------------------------------------- # Author: Andreas Alfons # Vienna University of Technology # --------------------------------------- #' Construct a matrix of binary variables for calibration #' #' Construct a matrix of binary variables for calibration of sample weights #' according to known marginal population totals. #' #' @name calibVars #' @param x a vector that can be interpreted as factor, or a matrix or #' \code{data.frame} consisting of such variables. #' #' @return A matrix of binary variables that indicate membership to the #' corresponding factor levels. #' #' @author Andreas Alfons #' #' @seealso \code{\link{calibWeights}} #' #' @keywords survey #' #' @examples #' data(eusilc) #' # default method #' aux <- calibVars(eusilc$rb090) #' head(aux) #' # data.frame method #' aux <- calibVars(eusilc[, c("db040", "rb090")]) #' head(aux) #' #' @export calibVars <- function(x) UseMethod("calibVars") #' @export calibVars.default <- function(x) { if(length(x) == 0) matrix(integer(), 0, 0) x <- as.factor(x) res <- sapply(levels(x), function(l) as.integer(x == l)) rownames(res) <- names(x) # set rownames from original vector res } #' @export calibVars.matrix <- function(x) calibVars(as.data.frame(x)) #' @export calibVars.data.frame <- function(x) { res <- lapply(x, calibVars) # list of matrices for each variable res <- mapply(function(x, nam) { colnames(x) <- paste(nam, colnames(x), sep=".") x }, res, names(x), SIMPLIFY=FALSE) res <- do.call("cbind", res) # combine matrices rownames(res) <- row.names(x) # set rownames from original data.frame res } laeken/R/weightedMean.R0000644000176200001440000000267013616467254014461 0ustar liggesusers# ------------------------------------------ # Authors: Andreas Alfons and Matthias Templ # Vienna University of Technology # ------------------------------------------ #' Weighted mean #' #' Compute the weighted mean. #' #' This is a simple wrapper function calling \code{\link[stats]{weighted.mean}} #' if sample weights are supplied and \code{\link{mean}} otherwise. #' #' @param x a numeric vector. #' @param weights an optional numeric vector giving the sample weights. #' @param na.rm a logical indicating whether missing values in \code{x} should #' be omitted. #' #' @return The weighted mean of values in \code{x} is returned. #' #' @author Andreas Alfons #' #' @seealso \code{\link{incMean}} #' #' @keywords survey #' #' @examples #' data(eusilc) #' weightedMean(eusilc$eqIncome, eusilc$rb050) #' #' @importFrom stats weighted.mean #' @export weightedMean <- function(x, weights = NULL, na.rm = FALSE) { # initializations if (!is.numeric(x)) stop("'x' must be a numeric vector") if (is.null(weights)) mean(x, na.rm=na.rm) else { n <- length(x) if (!is.numeric(weights)) stop("'weights' must be a numeric vector") else if (length(weights) != n) { stop("'weights' must have the same length as 'x'") } else if (!all(is.finite(weights))) stop("missing or infinite weights") if (any(weights < 0)) warning("negative weights") weighted.mean(x, weights, na.rm=na.rm) } } laeken/R/prop.R0000644000176200001440000002362414127253271013030 0ustar liggesusers# --------------------------------------- # Author: Matthias Templ # Vienna University of Technology # --------------------------------------- #' Proportion of an alternative distribution #' #' Estimate the proportion of an alternative distribution. #' #' If weights are provided, the weighted proportion is estimated. #' #' @param bin either a factor vector giving the values, #' or (if \code{data} is not \code{NULL}) a character string, an integer or a #' logical vector specifying the corresponding column of \code{data}. #' @param weights optional; either a numeric vector giving the personal sample #' weights, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. #' @param sort optional; either a numeric vector giving the personal IDs to be #' used as tie-breakers for sorting, or (if \code{data} is not \code{NULL}) a #' character string, an integer or a logical vector specifying the corresponding #' column of \code{data}. #' @param years optional; either a numeric vector giving the different years of #' the survey, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. If supplied, values are computed for each year. #' @param breakdown optional; either a numeric vector giving different domains, #' or (if \code{data} is not \code{NULL}) a character string, an integer or a #' logical vector specifying the corresponding column of \code{data}. If #' supplied, the values for each domain are computed in addition to the overall #' value. #' @param design optional and only used if \code{var} is not \code{NULL}; either #' an integer vector or factor giving different domains for stratified sampling #' designs, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. #' @param cluster optional and only used if \code{var} is not \code{NULL}; #' either an integer vector or factor giving different clusters for cluster #' sampling designs, or (if \code{data} is not \code{NULL}) a character string, #' an integer or a logical vector specifying the corresponding column of #' \code{data}. #' @param data an optional \code{data.frame}. #' @param var a character string specifying the type of variance estimation to #' be used, or \code{NULL} to omit variance estimation. See #' \code{\link{variance}} for possible values. #' @param alpha numeric; if \code{var} is not \code{NULL}, this gives the #' significance level to be used for computing the confidence interval (i.e., #' the confidence level is \eqn{1 - }\code{alpha}). #' @param na.rm a logical indicating whether missing values should be removed. #' @param \dots if \code{var} is not \code{NULL}, additional arguments to be #' passed to \code{\link{variance}}. #' #' @return A list of class \code{"prop"} (which inherits from the class #' \code{"indicator"}) with the following components: #' \item{value}{a numeric vector containing the overall value(s).} #' \item{valueByStratum}{a \code{data.frame} containing the values by #' domain, or \code{NULL}.} #' \item{varMethod}{a character string specifying the type of variance #' estimation used, or \code{NULL} if variance estimation was omitted.} #' \item{var}{a numeric vector containing the variance estimate(s), or #' \code{NULL}.} #' \item{varByStratum}{a \code{data.frame} containing the variance #' estimates by domain, or \code{NULL}.} #' \item{ci}{a numeric vector or matrix containing the lower and upper #' endpoints of the confidence interval(s), or \code{NULL}.} #' \item{ciByStratum}{a \code{data.frame} containing the lower and upper #' endpoints of the confidence intervals by domain, or \code{NULL}.} #' \item{alpha}{a numeric value giving the significance level used for #' computing the confidence interval(s) (i.e., the confidence level is \eqn{1 - #' }\code{alpha}), or \code{NULL}.} #' \item{years}{a numeric vector containing the different years of the #' survey.} #' \item{strata}{a character vector containing the different domains of the #' breakdown.} #' #' @author Matthias Templ, using code for breaking down #' estimation by Andreas Alfons #' #' @seealso \code{\link{variance}} #' #' @references #' A. Alfons and M. Templ (2013) Estimation of Social Exclusion Indicators #' from Complex Surveys: The \R Package \pkg{laeken}. \emph{Journal of #' Statistical Software}, \bold{54}(15), 1--25. \doi{10.18637/jss.v054.i15} #' #' Working group on Statistics on Income and Living Conditions (2004) #' Common cross-sectional EU indicators based on EU-SILC; the gender #' pay gap. \emph{EU-SILC 131-rev/04}, Eurostat, Luxembourg. #' #' @keywords survey #' #' @examples #' data(eusilc) #' #' # overall value #' prop("rb090", weights = "rb050", data = eusilc) #' #' # values by region #' p1 <- prop("rb090", weights = "rb050", #' breakdown = "db040", cluster = "db030", #' data = eusilc) #' #' p1 #' #' \dontrun{ #' variance("rb090", weights = "rb050", #' breakdown = "db040", data = eusilc, indicator=p1, #' cluster="db030", X = calibVars(eusilc$db040)) #' } #' #' #' eusilc$agecut <- cut(eusilc$age, 2) #' p1 <- prop("agecut", weights = "rb050", #' breakdown = "db040", #' cluster="db030", data = eusilc) #' p1 #' #' \dontrun{ #' variance("agecut", weights = "rb050", #' breakdown = "db040", data = eusilc, indicator=p1, #' X = calibVars(eusilc$db040), cluster="db030") #' } #' #' #' eusilc$eqIncomeCat <- factor(ifelse(eusilc$eqIncome < quantile(eusilc$eqIncome,0.2), "one", "two")) #' p1 <- prop("eqIncomeCat", weights = "rb050", #' breakdown = "db040", data = eusilc, cluster="db030") #' p1 #' #' \dontrun{ #' variance("eqIncomeCat", weights = "rb050", #' breakdown = "db040", data = eusilc, indicator=p1, #' X = calibVars(eusilc$db040), cluster="db030") #' } #' #' #' @importFrom stats aggregate #' @export prop <- function(bin, weights = NULL, sort = NULL, years = NULL, breakdown = NULL, design = NULL, cluster = NULL, data = NULL, var = NULL, alpha = 0.05, na.rm = FALSE, ...) { ## initializations byYear <- !is.null(years) byStratum <- !is.null(breakdown) if(!is.null(data)) { bin <- data[, bin] if(!is.null(weights)) weights <- data[, weights] if(!is.null(sort)) sort <- data[, sort] if(byYear) years <- data[, years] if(byStratum) breakdown <- data[, breakdown] if(!is.null(var)) { if(!is.null(design)) design <- data[, design] if(!is.null(cluster)) cluster <- data[, cluster] } } # check vectors if(!is.factor(bin)) stop("'bin' must be a vector of class factor") if(length(levels(bin)) != 2) stop(paste("'bin' has not exactly 2 levels")) n <- length(bin) if(is.null(weights)) weights <- weights <- rep.int(1, n) else if(!is.numeric(weights)) stop("'weights' must be a numeric vector") if(!is.null(sort) && !is.vector(sort) && !is.ordered(sort)) { stop("'sort' must be a vector or ordered factor") } if(byYear && !is.numeric(years)) { stop("'years' must be a numeric vector") } if(byStratum) { if(!is.vector(breakdown) && !is.factor(breakdown)) { stop("'breakdown' must be a vector or factor") } else breakdown <- as.factor(breakdown) } if(is.null(data)) { # check vector lengths if(length(weights) != n) { stop("'weights' must have the same length as 'x'") } if(!is.null(sort) && length(sort) != n) { stop("'sort' must have the same length as 'x'") } if(byYear && length(years) != n) { stop("'years' must have the same length as 'x'") } if(byStratum && length(breakdown) != n) { stop("'breakdown' must have the same length as 'x'") } } ## computations # prop by year (if requested) if(byYear) { ys <- sort(unique(years)) # unique years gc <- function(y, bin, weights, sort, years, na.rm) { i <- years == y propCoeff(bin[i], weights[i], sort[i], na.rm=na.rm) } value <- sapply(ys, gc, bin=bin, weights=weights, sort=sort, years=years, na.rm=na.rm) names(value) <- ys # use years as names } else { ys <- NULL value <- propCoeff(bin, weights, sort, na.rm=na.rm) } # prop by stratum (if requested) if(byStratum) { gcR <- function(i, bin, weights, sort, na.rm) { propCoeff(bin[i], weights[i], sort[i], na.rm=na.rm) } valueByStratum <- aggregate(1:n, if(byYear) list(year=years, stratum=breakdown) else list(stratum=breakdown), gcR, bin=bin, weights=weights, sort=sort, na.rm=na.rm) names(valueByStratum)[ncol(valueByStratum)] <- "value" rs <- levels(breakdown) # unique strata } else valueByStratum <- rs <- NULL ## create object of class "qsr" res <- constructProp(value=value, valueByStratum=valueByStratum, years=ys, strata=rs) # variance estimation (if requested) if(!is.null(var)) { # bin <- ifelse(as.numeric(as.integer(bin)), 0,1) bin <- as.numeric(as.integer(bin)) res <- variance(bin, weights, years, breakdown, design, cluster, indicator=res, alpha=alpha, na.rm=na.rm, type=var, ...) } ## return result return(res) } ## workhorse propCoeff <- function(x, weights = NULL, sort = NULL, na.rm = FALSE) { # initializations if(isTRUE(na.rm)){ indices <- !is.na(x) x <- x[indices] if(!is.null(weights)) weights <- weights[indices] if(!is.null(sort)) sort <- sort[indices] } else if(any(is.na(x))) return(NA) # sort values and weights # order <- if(is.null(sort)) order(x) else order(x, sort) # x <- x[order] # order values if(is.null(weights)) weights <- rep.int(1, length(x)) # equal weights # else weights <- weights[order] # order weights ## calculations ## bin to numeric x <- as.integer(x) 1-weightedMean(x-1, weights) } laeken/R/variance.R0000644000176200001440000001175714127253303013640 0ustar liggesusers# --------------------------------------- # Author: Andreas Alfons # Vienna University of Technology # --------------------------------------- #' Variance and confidence intervals of indicators on social exclusion and #' poverty #' #' Compute variance and confidence interval estimates of indicators on social #' exclusion and poverty. #' #' This is a wrapper function for computing variance and confidence interval #' estimates of indicators on social exclusion and poverty. #' #' @param inc either a numeric vector giving the equivalized disposable income, #' or (if \code{data} is not \code{NULL}) a character string, an integer or a #' logical vector specifying the corresponding column of \code{data}. #' @param weights optional; either a numeric vector giving the personal sample #' weights, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. #' @param years optional; either a numeric vector giving the different years of #' the survey, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. If supplied, values are computed for each year. #' @param breakdown optional; either a numeric vector giving different domains, #' or (if \code{data} is not \code{NULL}) a character string, an integer or a #' logical vector specifying the corresponding column of \code{data}. If #' supplied, the values for each domain are computed in addition to the overall #' value. #' @param design optional; either an integer vector or factor giving different #' strata for stratified sampling designs, or (if \code{data} is not #' \code{NULL}) a character string, an integer or a logical vector specifying #' the corresponding column of \code{data}. #' @param cluster optional; either an integer vector or factor giving different #' clusters for cluster sampling designs, or (if \code{data} is not #' \code{NULL}) a character string, an integer or a logical vector specifying #' the corresponding column of \code{data}. #' @param data an optional \code{data.frame}. #' @param indicator an object inheriting from the class \code{"indicator"} that #' contains the point estimates of the indicator (see \code{\link{arpr}}, #' \code{\link{qsr}}, \code{\link{rmpg}} or \code{\link{gini}}). #' @param alpha a numeric value giving the significance level to be used for #' computing the confidence interval(s) (i.e., the confidence level is \eqn{1 - #' }\code{alpha}), or \code{NULL}. #' @param na.rm a logical indicating whether missing values should be removed. #' @param type a character string specifying the type of variance estimation to #' be used. Currently, only \code{"bootstrap"} is implemented for variance #' estimation based on bootstrap resampling (see \code{\link{bootVar}}). #' @param gender either a numeric vector giving the gender, or (if \code{data} #' is not \code{NULL}) a character string, an integer or a logical vector #' specifying the corresponding column of \code{data}. #' @param method a character string specifying the method to be used (only for #' \code{\link{gpg}}). Possible values are \code{"mean"} for the mean, and #' \code{"median"} for the median. If weights are provided, the weighted mean #' or weighted median is estimated. #' @param \dots additional arguments to be passed to \code{\link{bootVar}}. #' #' @return An object of the same class as \code{indicator} is returned. See #' \code{\link{arpr}}, \code{\link{qsr}}, \code{\link{rmpg}} or #' \code{\link{gini}} for details on the components. #' #' @author Andreas Alfons #' #' @seealso \code{\link{bootVar}}, \code{\link{arpr}}, \code{\link{qsr}}, #' \code{\link{rmpg}}, \code{\link{gini}} #' #' @references #' A. Alfons and M. Templ (2013) Estimation of Social Exclusion Indicators #' from Complex Surveys: The \R Package \pkg{laeken}. \emph{Journal of #' Statistical Software}, \bold{54}(15), 1--25. \doi{10.18637/jss.v054.i15} #' #' @keywords survey #' #' @examples #' data(eusilc) #' a <- arpr("eqIncome", weights = "rb050", data = eusilc) #' #' ## naive bootstrap #' variance("eqIncome", weights = "rb050", design = "db040", #' data = eusilc, indicator = a, R = 50, #' bootType = "naive", seed = 123) #' #' ## bootstrap with calibration #' variance("eqIncome", weights = "rb050", design = "db040", #' data = eusilc, indicator = a, R = 50, #' X = calibVars(eusilc$db040), seed = 123) #' #' @export variance <- function(inc, weights = NULL, years = NULL, breakdown = NULL, design = NULL, cluster = NULL, data = NULL, indicator, alpha = 0.05, na.rm = FALSE, type = "bootstrap", gender = NULL, method = NULL, ...) { # initializations type <- match.arg(type) # call function corresponding to 'type' switch(type, bootstrap = bootVar(inc, weights, years, breakdown, design, cluster, data, indicator, alpha=alpha, na.rm=na.rm, gender=gender, method=method, ...)) } laeken/R/eqInc.R0000755000176200001440000001104213616467254013113 0ustar liggesusers# --------------------------------------- # Author: Andreas Alfons # Vienna University of Technology # --------------------------------------- # TODO: error handling # TODO: account for inflation-adjustment #' Equivalized disposable income #' #' Compute the equivalized disposable income from household and personal income #' variables. #' #' All income components should already be imputed, otherwise \code{NA}s are #' simply removed before the calculations. #' #' @param hid if \code{data=NULL}, a vector containing the household ID. #' Otherwise a character string specifying the column of \code{data} that #' contains the household ID. #' @param hplus if \code{data=NULL}, a \code{data.frame} containing the #' household income components that have to be added. Otherwise a character #' vector specifying the columns of \code{data} that contain these income #' components. #' @param hminus if \code{data=NULL}, a \code{data.frame} containing the #' household income components that have to be subtracted. Otherwise a #' character vector specifying the columns of \code{data} that contain these #' income components. #' @param pplus if \code{data=NULL}, a \code{data.frame} containing the personal #' income components that have to be added. Otherwise a character vector #' specifying the columns of \code{data} that contain these income components. #' @param pminus if \code{data=NULL}, a \code{data.frame} containing the #' personal income components that have to be subtracted. Otherwise a character #' vector specifying the columns of \code{data} that contain these income #' components. #' @param eqSS if \code{data=NULL}, a vector containing the equivalized #' household size. Otherwise a character string specifying the column of #' \code{data} that contains the equivalized household size. See #' \code{\link{eqSS}} for more details. #' @param year if \code{data=NULL}, a vector containing the year of the survey. #' Otherwise a character string specifying the column of \code{data} that #' contains the year. #' @param data a \code{data.frame} containing EU-SILC survey data, or #' \code{NULL}. #' #' @return A numeric vector containing the equivalized disposable income for #' every individual in \code{data}. #' #' @author Andreas Alfons #' #' @seealso \code{\link{eqSS}} #' #' @references Working group on Statistics on Income and Living Conditions #' (2004) Common cross-sectional EU indicators based on EU-SILC; the gender pay #' gap. \emph{EU-SILC 131-rev/04}, Eurostat. #' @keywords survey #' #' @examples #' data(eusilc) #' #' # compute a simplified version of the equivalized disposable income #' # (not all income components are available in the synthetic data) #' hplus <- c("hy040n", "hy050n", "hy070n", "hy080n", "hy090n", "hy110n") #' hminus <- c("hy130n", "hy145n") #' pplus <- c("py010n", "py050n", "py090n", "py100n", #' "py110n", "py120n", "py130n", "py140n") #' eqIncome <- eqInc("db030", hplus, hminus, #' pplus, character(), "eqSS", data=eusilc) #' #' # combine with household ID and equivalized household size #' tmp <- cbind(eusilc[, c("db030", "eqSS")], eqIncome) #' #' # show the first 8 rows #' head(tmp, 8) #' #' @importFrom stats aggregate #' @export eqInc <- function(hid, hplus, hminus, pplus, pminus, eqSS, year = NULL, data = NULL) { ## initializations if(is.null(data)) { data <- data.frame(hid=hid) hid <- "hid" if(!is.null(year)) { data <- cbind(year=year, data) year <- "year" } npplus <- names(pplus) npminus <- names(pminus) } else { hplus <- data[, hplus] hminus <- data[, hminus] npplus <- pplus pplus <- data[, npplus] npminus <- pminus pminus <- data[, npminus] eqSS <- data[, eqSS] data <- data[, c(year, hid), drop=FALSE] } ## calculations hy020h <- rowSums(hplus, na.rm=TRUE) - rowSums(hminus, na.rm=TRUE) tmp <- aggregate(data.frame(pplus,pminus), data, sum, na.rm=TRUE) hy020p <- rowSums(tmp[,npplus], na.rm=TRUE) - rowSums(tmp[,npminus], na.rm=TRUE) if(is.null(year)) { names(hy020p) <- tmp[, hid] hy020p <- unname(hy020p[as.character(data[, hid])]) } else { tmp <- cbind(tmp[, c(year, hid), drop=FALSE], .hy020p=hy020p) data <- cbind(data, .ID=1:nrow(data)) # add ID to original data data <- merge(data, tmp, sort=FALSE) # merge with original data set ## order according to original data and extract hy020p hy020p <- data$.hy020p[order(data$.ID)] } ## return result (hy020h + hy020p) / eqSS } laeken/R/rmpg.R0000755000176200001440000002350514127253264013020 0ustar liggesusers# --------------------------------------- # Author: Andreas Alfons # Vienna University of Technology # --------------------------------------- #' Relative median at-risk-of-poverty gap #' #' Estimate the relative median at-risk-of-poverty gap, which is defined as the #' relative difference between the median equivalized disposable income of #' persons below the at-risk-of-poverty threshold and the at-risk-of-poverty #' threshold itself (expressed as a percentage of the at-risk-of-poverty #' threshold). #' #' The implementation strictly follows the Eurostat definition. #' #' @param inc either a numeric vector giving the equivalized disposable income, #' or (if \code{data} is not \code{NULL}) a character string, an integer or a #' logical vector specifying the corresponding column of \code{data}. #' @param weights optional; either a numeric vector giving the personal sample #' weights, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. #' @param sort optional; either a numeric vector giving the personal IDs to be #' used as tie-breakers for sorting, or (if \code{data} is not \code{NULL}) a #' character string, an integer or a logical vector specifying the corresponding #' column of \code{data}. #' @param years optional; either a numeric vector giving the different years of #' the survey, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. If supplied, values are computed for each year. #' @param breakdown optional; either a numeric vector giving different domains, #' or (if \code{data} is not \code{NULL}) a character string, an integer or a #' logical vector specifying the corresponding column of \code{data}. If #' supplied, the values for each domain are computed in addition to the overall #' value. Note that the same (overall) threshold is used for all domains. #' @param design optional and only used if \code{var} is not \code{NULL}; either #' an integer vector or factor giving different strata for stratified sampling #' designs, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. #' @param cluster optional and only used if \code{var} is not \code{NULL}; #' either an integer vector or factor giving different clusters for cluster #' sampling designs, or (if \code{data} is not \code{NULL}) a character string, #' an integer or a logical vector specifying the corresponding column of #' \code{data}. #' @param data an optional \code{data.frame}. #' @param var a character string specifying the type of variance estimation to #' be used, or \code{NULL} to omit variance estimation. See #' \code{\link{variance}} for possible values. #' @param alpha numeric; if \code{var} is not \code{NULL}, this gives the #' significance level to be used for computing the confidence interval (i.e., #' the confidence level is \eqn{1 - }\code{alpha}). #' @param na.rm a logical indicating whether missing values should be removed. #' @param \dots if \code{var} is not \code{NULL}, additional arguments to be #' passed to \code{\link{variance}}. #' #' @return A list of class \code{"rmpg"} (which inherits from the class #' \code{"indicator"}) with the following components: #' \item{value}{a numeric vector containing the overall value(s).} #' \item{valueByStratum}{a \code{data.frame} containing the values by #' domain, or \code{NULL}.} #' \item{varMethod}{a character string specifying the type of variance #' estimation used, or \code{NULL} if variance estimation was omitted.} #' \item{var}{a numeric vector containing the variance estimate(s), or #' \code{NULL}.} #' \item{varByStratum}{a \code{data.frame} containing the variance #' estimates by domain, or \code{NULL}.} #' \item{ci}{a numeric vector or matrix containing the lower and upper #' endpoints of the confidence interval(s), or \code{NULL}.} #' \item{ciByStratum}{a \code{data.frame} containing the lower and upper #' endpoints of the confidence intervals by domain, or \code{NULL}.} #' \item{alpha}{a numeric value giving the significance level used for #' computing the confidence interval(s) (i.e., the confidence level is \eqn{1 - #' }\code{alpha}), or \code{NULL}.} #' \item{years}{a numeric vector containing the different years of the #' survey.} #' \item{strata}{a character vector containing the different domains of the #' breakdown.} #' \item{threshold}{a numeric vector containing the at-risk-of-poverty #' threshold(s).} #' #' @author Andreas Alfons #' #' @seealso \code{\link{arpt}}, \code{\link{variance}} #' #' @references #' A. Alfons and M. Templ (2013) Estimation of Social Exclusion Indicators #' from Complex Surveys: The \R Package \pkg{laeken}. \emph{Journal of #' Statistical Software}, \bold{54}(15), 1--25. \doi{10.18637/jss.v054.i15} #' #' Working group on Statistics on Income and Living Conditions (2004) #' Common cross-sectional EU indicators based on EU-SILC; the gender #' pay gap. \emph{EU-SILC 131-rev/04}, Eurostat, Luxembourg. #' #' @keywords survey #' #' @examples #' data(eusilc) #' #' # overall value #' rmpg("eqIncome", weights = "rb050", data = eusilc) #' #' # values by region #' rmpg("eqIncome", weights = "rb050", #' breakdown = "db040", data = eusilc) #' #' @importFrom stats aggregate #' @export rmpg <- function(inc, weights = NULL, sort = NULL, years = NULL, breakdown = NULL, design = NULL, cluster = NULL, data = NULL, var = NULL, alpha = 0.05, na.rm = FALSE, ...) { ## initializations byYear <- !is.null(years) byStratum <- !is.null(breakdown) if(!is.null(data)) { inc <- data[, inc] if(!is.null(weights)) weights <- data[, weights] if(!is.null(sort)) sort <- data[, sort] if(byYear) years <- data[, years] if(byStratum) breakdown <- data[, breakdown] if(!is.null(var)) { if(!is.null(design)) design <- data[, design] if(!is.null(cluster)) cluster <- data[, cluster] } } # check vectors if(!is.numeric(inc)) stop("'inc' must be a numeric vector") n <- length(inc) if(is.null(weights)) weights <- weights <- rep.int(1, n) else if(!is.numeric(weights)) stop("'weights' must be a numeric vector") if(!is.null(sort) && !is.vector(sort) && !is.ordered(sort)) { stop("'sort' must be a vector or ordered factor") } if(byYear && !is.numeric(years)) { stop("'years' must be a numeric vector") } if(byStratum) { if(!is.vector(breakdown) && !is.factor(breakdown)) { stop("'breakdown' must be a vector or factor") } else breakdown <- as.factor(breakdown) } if(is.null(data)) { # check vector lengths if(length(weights) != n) { stop("'weights' must have the same length as 'x'") } if(!is.null(sort) && length(sort) != n) { stop("'sort' must have the same length as 'x'") } if(byYear && length(years) != n) { stop("'years' must have the same length as 'x'") } if(byStratum && length(breakdown) != n) { stop("'breakdown' must have the same length as 'x'") } } ## computations if(byYear) { # RMPG by year ys <- sort(unique(years)) ts <- arpt(inc, weights, sort, years, na.rm=na.rm) # thresholds rg <- function(y, t, inc, weights, sort, years, na.rm) { i <- years == y relativeGap(inc[i], weights[i], sort[i], t, na.rm=na.rm) } value <- mapply(rg, y=ys, t=ts, MoreArgs=list(inc=inc, weights=weights, sort=sort, years=years, na.rm=na.rm)) names(value) <- ys # use years as names if(byStratum) { rg1 <- function(i, inc, weights, sort, years, ts, na.rm) { y <- years[i[1]] t <- ts[as.character(y)] relativeGap(inc[i], weights[i], sort[i], t, na.rm=na.rm) } valueByStratum <- aggregate(1:n, list(year=years, stratum=breakdown), rg1, inc=inc, weights=weights, sort=sort, years=years, ts=ts, na.rm=na.rm) names(valueByStratum)[3] <- "value" } else valueByStratum <- NULL } else { # RMPG for only one year ys <- NULL ts <- arpt(inc, weights, sort, na.rm=na.rm) # threshold value <- relativeGap(inc, weights, sort, ts, na.rm=na.rm) if(byStratum) { rg2 <- function(i, inc, weights, sort, ts, na.rm) { relativeGap(inc[i], weights[i], sort[i], ts, na.rm=na.rm) } valueByStratum <- aggregate(1:n, list(stratum=breakdown), rg2, inc=inc, weights=weights, sort=sort, ts=ts, na.rm=na.rm) names(valueByStratum)[2] <- "value" } else valueByStratum <- NULL } rs <- levels(breakdown) # unique strata (also works if 'breakdown' is NULL) ## create object of class "arpr" res <- constructRmpg(value=value, valueByStratum=valueByStratum, years=ys, strata=rs, threshold=ts) # variance estimation (if requested) if(!is.null(var)) { res <- variance(inc, weights, years, breakdown, design, cluster, indicator=res, alpha=alpha, na.rm=na.rm, type=var, ...) } ## return result return(res) } ## workhorse relativeGap <- function(x, weights = NULL, sort = NULL, threshold, na.rm = FALSE) { ## initializations if(is.null(weights)) weights <- rep.int(1, length(x)) # equal weights if(isTRUE(na.rm)){ indices <- !is.na(x) x <- x[indices] if(!is.null(weights)) weights <- weights[indices] if(!is.null(sort)) sort <- sort[indices] } else if(any(is.na(x))) return(NA) if(length(x) == 0) return(NA) # preparations isPoor <- x < threshold # individuals below threshold x <- x[isPoor] if(!is.null(weights)) weights <- weights[isPoor] if(!is.null(sort)) sort <- sort[isPoor] # calculations medianPoor <- incMedian(x, weights, sort) (threshold - medianPoor) * 100 / threshold } laeken/R/thetaPDC.R0000644000176200001440000001522414127253166013504 0ustar liggesusers# ---------------------------------------- # Authors: Andreas Alfons and Josef Holzer # Vienna University of Technology # ---------------------------------------- #' Partial density component (PDC) estimator #' #' The partial density component (PDC) estimator estimates the shape parameter #' of a Pareto distribution based on the relative excesses of observations above #' a certain threshold. #' #' The arguments \code{k} and \code{x0} of course correspond with each other. #' If \code{k} is supplied, the threshold \code{x0} is estimated with the \eqn{n #' - k} largest value in \code{x}, where \eqn{n} is the number of observations. #' On the other hand, if the threshold \code{x0} is supplied, \code{k} is given #' by the number of observations in \code{x} larger than \code{x0}. Therefore, #' either \code{k} or \code{x0} needs to be supplied. If both are supplied, #' only \code{k} is used (mainly for back compatibility). #' #' The PDC estimator minimizes the integrated squared error (ISE) criterion with #' an incomplete density mixture model. The minimization is carried out using % #' \code{\link[stats]{nlm}}. By default, the starting value is obtained with % #' the Hill estimator (see \code{\link{thetaHill}}). #' \code{\link[stats]{optimize}}. #' #' @param x a numeric vector. #' @param k the number of observations in the upper tail to which the Pareto #' distribution is fitted. #' @param x0 the threshold (scale parameter) above which the Pareto distribution #' is fitted. #' @param w an optional numeric vector giving sample weights. #' @param \dots additional arguments to be passed to #' \code{\link[stats]{optimize}} (see \dQuote{Details}). #' #' @return The estimated shape parameter. #' #' @note The arguments \code{x0} for the threshold (scale parameter) of the #' Pareto distribution and \code{w} for sample weights were introduced in #' version 0.2. #' #' @author Andreas Alfons and Josef Holzer #' #' @seealso \code{\link{paretoTail}}, \code{\link{fitPareto}}, #' \code{\link{thetaISE}}, \code{\link{thetaHill}} #' #' @references #' A. Alfons and M. Templ (2013) Estimation of Social Exclusion Indicators #' from Complex Surveys: The \R Package \pkg{laeken}. \emph{Journal of #' Statistical Software}, \bold{54}(15), 1--25. \doi{10.18637/jss.v054.i15} #' #' A. Alfons, M. Templ, P. Filzmoser (2013) Robust estimation of economic #' indicators from survey samples based on Pareto tail modeling. \emph{Journal #' of the Royal Statistical Society, Series C}, \bold{62}(2), 271--286. #' #' Vandewalle, B., Beirlant, J., Christmann, A., and Hubert, M. #' (2007) A robust estimator for the tail index of Pareto-type #' distributions. \emph{Computational Statistics & Data Analysis}, #' \bold{51}(12), 6252--6268. #' #' @keywords manip #' #' @examples #' data(eusilc) #' # equivalized disposable income is equal for each household #' # member, therefore only one household member is taken #' eusilc <- eusilc[!duplicated(eusilc$db030),] #' #' # estimate threshold #' ts <- paretoScale(eusilc$eqIncome, w = eusilc$db090) #' #' # using number of observations in tail #' thetaPDC(eusilc$eqIncome, k = ts$k, w = eusilc$db090) #' #' # using threshold #' thetaPDC(eusilc$eqIncome, x0 = ts$x0, w = eusilc$db090) #' #' @export thetaPDC <- function(x, k = NULL, x0 = NULL, w = NULL, ...) { ## initializations if(!is.numeric(x) || length(x) == 0) stop("'x' must be a numeric vector") haveK <- !is.null(k) if(haveK) { # if 'k' is supplied, it is always used if(!is.numeric(k) || length(k) == 0 || k[1] < 1) { stop("'k' must be a positive integer") } else k <- k[1] } else if(!is.null(x0)) { # otherwise 'x0' (threshold) is used if(!is.numeric(x0) || length(x0) == 0) stop("'x0' must be numeric") else x0 <- x0[1] } else stop("either 'k' or 'x0' must be supplied") haveW <- !is.null(w) if(haveW) { # sample weights are supplied if(!is.numeric(w) || length(w) != length(x)) { stop("'w' must be numeric vector of the same length as 'x'") } if(any(w < 0)) stop("negative weights in 'w'") if(any(i <- is.na(x))) { # remove missing values x <- x[!i] w <- w[!i] } # sort values and sample weights order <- order(x) x <- x[order] w <- w[order] } else { # no sample weights if(any(i <- is.na(x))) x <- x[!i] # remove missing values x <- sort(x) # sort values } .thetaPDC(x, k, x0, w, ...) } # internal function that assumes that data are ok and sorted .thetaPDC <- function(x, k = NULL, x0 = NULL, w = NULL, tol = .Machine$double.eps^0.25, ...) { n <- length(x) # number of observations haveK <- !is.null(k) haveW <- !is.null(w) if(haveK) { # 'k' is supplied, threshold is determined if(k >= n) stop("'k' must be smaller than the number of observed values") x0 <- x[n-k] # threshold (scale parameter) } else { # 'k' is not supplied, it is determined using threshold # values are already sorted if(x0 >= x[n]) stop("'x0' must be smaller than the maximum of 'x'") k <- length(which(x > x0)) } ## computations y <- x[(n-k+1):n]/x0 # relative excesses if(haveW) { wTail <- w[(n-k+1):n] ## weighted integrated squared error distance criterion with incomplete ## density mixture model # w ... sample weights # u ... robustness weights (from incomplete density mixture model) ISE <- function(theta, y, w) { f <- theta*y^(-1-theta) wm <- weighted.mean(f, w) # weighted mean as unbiased estimator of expectation of f pf2 <- theta^2/(2*theta+1) # primitive of f^2 u <- wm/pf2 u^2*pf2 - 2*u*wm } } else { wTail <- NULL ## integrated squared error distance criterion with incomplete density ## mixture model # w ... sample weights (not needed here, only available to have the # same function definition) # u ... robustness weights (from incomplete density mixture model) ISE <- function(theta, y, w) { f <- theta*y^(-1-theta) m <- mean(f) # mean as unbiased estimator of expectation of f pf2 <- theta^2/(2*theta+1) # primitive of f^2 u <- m/pf2 u^2*pf2 - 2*u*m } } ## optimize localOptimize <- function(f, interval = NULL, tol, ...) { if(is.null(interval)) { p <- if(haveK) .thetaHill(x, k, w=w) else .thetaHill(x, x0=x0, w=w) interval <- c(0 + tol, 3 * p) # default interval } optimize(f, interval, ...) } localOptimize(ISE, y=y, w=wTail, tol=tol, ...)$minimum } laeken/R/plot.R0000644000176200001440000001011014127253044013006 0ustar liggesusers# ---------------------- # Author: Andreas Alfons # KU Leuven # ---------------------- #' Diagnostic plot for the Pareto tail model #' #' Produce a diagnostic Pareto quantile plot for evaluating the fitted Pareto #' distribution. Reference lines indicating the estimates of the threshold #' (scale parameter) and the shape parameter are added to the plot, and any #' detected outliers are highlighted. #' #' While the first horizontal line indicates the estimated threshold (scale #' parameter), the estimated shape parameter is indicated by a line whose slope #' is given by the reciprocal of the estimate. In addition, the second #' horizontal line represents the theoretical quantile of the fitted #' distribution that is used for outlier detection. Thus all values above that #' line are the detected outliers. #' #' @method plot paretoTail #' #' @param x an object of class \code{"paretoTail"} as returned by #' \code{\link{paretoTail}}. #' @param pch,cex,col,bg graphical parameters. Each can be a vector of length #' two, with the first and second element giving the graphical parameter for #' the good data points and the outliers, respectively. #' @param \dots additional arguments to be passed to #' \code{\link{paretoQPlot}}. #' #' @author Andreas Alfons #' #' @seealso \code{\link{paretoTail}}, \code{\link{paretoQPlot}} #' #' @references #' A. Alfons and M. Templ (2013) Estimation of Social Exclusion Indicators #' from Complex Surveys: The \R Package \pkg{laeken}. \emph{Journal of #' Statistical Software}, \bold{54}(15), 1--25. \doi{10.18637/jss.v054.i15} #' #' @keywords hplot #' #' @examples #' data(eusilc) #' #' # estimate threshold #' ts <- paretoScale(eusilc$eqIncome, w = eusilc$db090, #' groups = eusilc$db030) #' #' # estimate shape parameter #' fit <- paretoTail(eusilc$eqIncome, k = ts$k, #' w = eusilc$db090, groups = eusilc$db030) #' #' # produce plot #' plot(fit) #' #' @importFrom graphics abline #' @export plot.paretoTail <- function(x, pch = c(1, 3), cex = 1, col = c("black", "red"), bg = "transparent", ...) { ## initializations values <- x$x n <- length(values) pch <- rep(pch, length.out=2) cex <- rep(cex, length.out=2) col <- rep(col, length.out=2) bg <- rep(bg, length.out=2) ## extract data weights <- x$w haveWeights <- !is.null(weights) groups <- x$groups haveGroups <- !is.null(groups) if(haveGroups) { unique <- !duplicated(groups) values <- values[unique] if(haveWeights) weights <- weights[unique] groups <- groups[unique] } ## define graphical parameters for each data point out <- x$out if(length(out) == 0) { # no outliers pchs <- pch[1] cexs <- cex[1] cols <- col[1] bgs <- bg[1] } else { # allow for cluster effect if(haveGroups) out <- which(groups %in% out) # initialize graphical parameters pchs <- vector(mode=storage.mode(pch), length=n) cexs <- vector(mode=storage.mode(cex), length=n) cols <- vector(mode=storage.mode(col), length=n) bgs <- vector(mode=storage.mode(bg), length=n) # graphical parameters for good data points pchs[-out] <- pch[1] cexs[-out] <- cex[1] cols[-out] <- col[1] bgs[-out] <- bg[1] # graphical parameters for outliers pchs[out] <- pch[2] cexs[out] <- cex[2] cols[out] <- col[2] bgs[out] <- bg[2] } ## create diagnostic plot xOut <- qpareto(1-x$alpha, x0=x$x0, theta=x$theta) localParetoQPlot <- function(x, w, interactive, x0, theta, type, ylim = NULL, ...) { if(is.null(ylim)) { ylim <- range(values[which(values > 0)], xOut, finite=TRUE) } paretoQPlot(values, w=weights, interactive=FALSE, x0=x$x0, theta=x$theta, ylim=ylim, ...) } localParetoQPlot(x, pch=pchs, cex=cexs, col=cols, bg=bgs, ...) # add horizontal line for outlier identification # observations above that line are outliers abline(h=xOut, col="darkgrey", lty=3) # invisible return NULL invisible() } laeken/R/thetaWML.R0000644000176200001440000002017313616467254013543 0ustar liggesusers# ---------------------------------------- # Authors: Andreas Alfons and Josef Holzer # Vienna University of Technology # ---------------------------------------- #' Weighted maximum likelihood estimator #' #' Estimate the shape parameter of a Pareto distribution using a weighted #' maximum likelihood approach. #' #' The arguments \code{k} and \code{x0} of course correspond with each other. #' If \code{k} is supplied, the threshold \code{x0} is estimated with the \eqn{n #' - k} largest value in \code{x}, where \eqn{n} is the number of observations. #' On the other hand, if the threshold \code{x0} is supplied, \code{k} is given #' by the number of observations in \code{x} larger than \code{x0}. Therefore, #' either \code{k} or \code{x0} needs to be supplied. If both are supplied, #' only \code{k} is used (mainly for back compatibility). #' #' The weighted maximum likelihood estimator belongs to the class of #' M-estimators. In order to obtain the estimate, the root of a certain #' function needs to be found, which is implemented using #' \code{\link[stats]{uniroot}}. #' #' @param x a numeric vector. #' @param k the number of observations in the upper tail to which the Pareto #' distribution is fitted. #' @param x0 the threshold (scale parameter) above which the Pareto distribution #' is fitted. #' @param weight a character string specifying the weight function to be used. #' If \code{"residuals"} (the default), the weight function is based on #' standardized residuals. If \code{"probability"}, probability based weighting #' is used. Partial string matching allows these names to be abbreviated. #' @param const Tuning constant(s) that control the robustness of the method. #' If \code{weight="residuals"}, a single numeric value is required (the default #' is 2.5). If \code{weight="probability"}, a numeric vector of length two must #' be supplied (a single numeric value is recycled; the default is 0.005 for #' both tuning parameters). See the references for more details. #' @param bias a logical indicating whether bias correction should be applied. #' @param \dots additional arguments to be passed to #' \code{\link[stats]{uniroot}} (see \dQuote{Details}). #' #' @return The estimated shape parameter. #' #' @note The argument \code{x0} for the threshold (scale parameter) of the #' Pareto distribution was introduced in version 0.2. #' #' @author Andreas Alfons and Josef Holzer #' #' @seealso \code{\link{paretoTail}}, \code{\link{fitPareto}} #' #' @references Dupuis, D.J. and Morgenthaler, S. (2002) Robust weighted #' likelihood estimators with an application to bivariate extreme value #' problems. \emph{The Canadian Journal of Statistics}, \bold{30}(1), 17--36. #' #' Dupuis, D.J. and Victoria-Feser, M.-P. (2006) A robust prediction error #' criterion for Pareto modelling of upper tails. \emph{The Canadian Journal of #' Statistics}, \bold{34}(4), 639--658. #' #' @keywords manip #' #' @examples #' data(eusilc) #' # equivalized disposable income is equal for each household #' # member, therefore only one household member is taken #' eusilc <- eusilc[!duplicated(eusilc$db030),] #' #' # estimate threshold #' ts <- paretoScale(eusilc$eqIncome, w = eusilc$db090) #' #' # using number of observations in tail #' thetaWML(eusilc$eqIncome, k = ts$k) #' #' # using threshold #' thetaWML(eusilc$eqIncome, x0 = ts$x0) #' #' @export thetaWML <- function(x, k = NULL, x0 = NULL, weight = c("residuals", "probability"), const, bias = TRUE, ...) { ## initializations if(!is.numeric(x) || length(x) == 0) stop("'x' must be a numeric vector") haveK <- !is.null(k) if(haveK) { # if 'k' is supplied, it is always used if(!is.numeric(k) || length(k) == 0 || k[1] < 1) { stop("'k' must be a positive integer") } else k <- k[1] } else if(!is.null(x0)) { # otherwise 'x0' (threshold) is used if(!is.numeric(x0) || length(x0) == 0) stop("'x0' must be numeric") else x0 <- x0[1] } else stop("either 'k' or 'x0' must be supplied") if(any(i <- is.na(x))) x <- x[!i] # remove missing values x <- sort(x) # sort values if(missing(const)) .thetaWML(x, k, x0, weight, bias=bias, ...) else .thetaWML(x, k, x0, weight, const, bias, ...) } # internal function that assumes that data are ok and sorted .thetaWML <- function(x, k = NULL, x0 = NULL, weight = c("residuals", "probability"), const, bias = TRUE, tol = .Machine$double.eps^0.25, ...) { n <- length(x) # number of observations haveK <- !is.null(k) if(haveK) { # 'k' is supplied, threshold is determined if(k >= n) stop("'k' must be smaller than the number of observed values") x0 <- x[n-k] # threshold (scale parameter) } else { # 'k' is not supplied, it is determined using threshold # values are already sorted if(x0 >= x[n]) stop("'x0' must be smaller than the maximum of 'x'") k <- length(which(x > x0)) } xt <- x[(n-k+1):n] # tail (values larger than threshold) y <- log(xt/x0) # relative excesses weight <- match.arg(weight) # check type of robustness weights ## define robustness weight function and function for root finding ## derivative of log(f) with respect to theta: 1/theta - log(xt/x0) if(weight == "residuals") { ## check tuning constant if(missing(const)) const <- 2.5 else if(!is.numeric(const) || length(const) == 0) { stop("'const' must be a numeric value") } else const <- const[1] ## some temporary values h <- k:1 hy <- log(h/(k+1)) hsig <- sqrt(cumsum(1/h^2)) ## objective function zeroTheta <- function(theta) { r <- (theta*y + hy) / hsig # standardized residuals u <- pmin(1, const/abs(r)) # robustness weights dlogf <- 1/theta - y # derivative of log(f) sum(u * dlogf) } } else { ## check tuning constants if(missing(const)) const <- rep.int(0.005, 2) else if(!is.numeric(const) || length(const) == 0) { stop("'const' must be a numeric vector of length two") } else const <- rep(const, length.out=2) p1 <- const[1] p2 <- const[2] ## objective function zeroTheta <- function(theta) { F <- 1 - (xt/x0)^(-theta) # distribution function u <- ifelse(F < p1, F/p1, ifelse(F <= 1-p2, 1, (1-F)/p2)) # robustness weights dlogf <- 1/theta - y # derivative of log(f) sum(u * dlogf) } } ## solving sum(phi(xt,theta))=0 localUniroot <- function(f, interval = NULL, tol, ...) { if(is.null(interval)) { p <- if(haveK) .thetaHill(x, k) else .thetaHill(x, x0=x0) interval <- c(0 + tol, 5 * p) # default interval } uniroot(f, interval, ...) } theta <- localUniroot(zeroTheta, tol=tol, ...)$root ## optional bias correction if(bias) { if(weight == "residuals") { r <- (theta*y + hy) / hsig # standardized residuals u <- pmin(1, const/abs(r)) # robustness weights F <- 1 - (xt/x0)^(-theta) # distribution function deltaF <- diff(c(0, F)) # difference operator applied to F dlogf <- 1/theta - y # derivative of log(f) d2logf <- -1/theta^2 # second derivative of log(f) # derivative of robustness weight function du <- ifelse(u == 1, 0, (-const)*y*hsig / (theta*y + hy)^2) # bias correction term bcorr <- -sum(u*dlogf*deltaF)/sum((du*dlogf + u*d2logf) * deltaF) } else { cp1 <- 1-p1 cp2 <- 1-p2 # bias correction term bcorr <- (theta/2) * (2*cp1^2*log(cp1) + p1*cp1 + p1*cp2 + 2*p1*p2*log(p2)) / ((cp1*log(cp1))^2 - p1*cp1 - p1*cp2 + p1*p2*(log(p2))^2) } ## apply bias correction to theta theta <- theta - bcorr } ## return WML-estimate theta } laeken/R/paretoTail.R0000644000176200001440000004626014127253237014157 0ustar liggesusers# ---------------------------------------- # Authors: Andreas Alfons and Josef Holzer # Vienna University of Technology # ---------------------------------------- #' Pareto tail modeling for income distributions #' #' Fit a Pareto distribution to the upper tail of income data. Since a #' theoretical distribution is used for the upper tail, this is a semiparametric #' approach. #' #' The arguments \code{k} and \code{x0} of course correspond with each other. #' If \code{k} is supplied, the threshold \code{x0} is estimated with the \eqn{n #' - k} largest value in \code{x}, where \eqn{n} is the number of observations. #' On the other hand, if the threshold \code{x0} is supplied, \code{k} is given #' by the number of observations in \code{x} larger than \code{x0}. Therefore, #' either \code{k} or \code{x0} needs to be supplied. If both are supplied, #' only \code{k} is used. #' #' The function supplied to \code{method} should take a numeric vector (the #' observations) as its first argument. If \code{k} is supplied, it will be #' passed on (in this case, the function is required to have an argument called #' \code{k}). Similarly, if the threshold \code{x0} is supplied, it will be #' passed on (in this case, the function is required to have an argument called #' \code{x0}). As above, only \code{k} is passed on if both are supplied. If #' the function specified by \code{method} can handle sample weights, the #' corresponding argument should be called \code{w}. Additional arguments are #' passed via the \dots{} argument. #' #' @aliases print.paretoTail #' #' @param x a numeric vector. #' @param k the number of observations in the upper tail to which the Pareto #' distribution is fitted. #' @param x0 the threshold (scale parameter) above which the Pareto distribution #' is fitted. #' @param method either a function or a character string specifying the function #' to be used to estimate the shape parameter of the Pareto distibution, such as #' \code{\link{thetaPDC}} (the default). See \dQuote{Details} for requirements #' for such a function and \dQuote{See also} for available functions. #' @param groups an optional vector or factor specifying groups of elements of #' \code{x} (e.g., households). If supplied, each group of observations is #' expected to have the same value in \code{x} (e.g., household income). Only #' the values of every first group member to appear are used for fitting the #' Pareto distribution. #' @param w an optional numeric vector giving sample weights. #' @param alpha numeric; values above the theoretical \eqn{1 - }\code{alpha} #' quantile of the fitted Pareto distribution will be flagged as outliers for #' further treatment with \code{\link{reweightOut}} or \code{\link{replaceOut}}. #' @param \dots addtional arguments to be passed to the specified method. #' #' @return An object of class \code{"paretoTail"} with the following #' components: #' \item{x}{the supplied numeric vector.} #' \item{k}{the number of observations in the upper tail to which the #' Pareto distribution has been fitted.} #' \item{groups}{if supplied, the vector or factor specifying groups of #' elements.} #' \item{w}{if supplied, the numeric vector of sample weights.} #' \item{method}{the function used to estimate the shape parameter, or the #' name of the function.} #' \item{x0}{the scale parameter.} #' \item{theta}{the estimated shape parameter.} #' \item{tail}{if \code{groups} is not \code{NULL}, this gives the groups #' with values larger than the threshold (scale parameter), otherwise the #' indices of observations in the upper tail.} #' \item{alpha}{the tuning parameter \code{alpha} used for flagging #' outliers.} #' \item{out}{if \code{groups} is not \code{NULL}, this gives the groups #' that are flagged as outliers, otherwise the indices of the flagged #' observations.} #' #' @author Andreas Alfons #' #' @seealso \code{\link{reweightOut}}, \code{\link{shrinkOut}}, #' \code{\link{replaceOut}}, \code{\link{replaceTail}}, \code{\link{fitPareto}} #' #' \code{\link{thetaPDC}}, \code{\link{thetaWML}}, \code{\link{thetaHill}}, #' \code{\link{thetaISE}}, \code{\link{thetaLS}}, \code{\link{thetaMoment}}, #' \code{\link{thetaQQ}}, \code{\link{thetaTM}} #' #' @references #' A. Alfons and M. Templ (2013) Estimation of Social Exclusion Indicators #' from Complex Surveys: The \R Package \pkg{laeken}. \emph{Journal of #' Statistical Software}, \bold{54}(15), 1--25. \doi{10.18637/jss.v054.i15} #' #' A. Alfons, M. Templ, P. Filzmoser (2013) Robust estimation of economic #' indicators from survey samples based on Pareto tail modeling. \emph{Journal #' of the Royal Statistical Society, Series C}, \bold{62}(2), 271--286. #' #' @keywords manip #' #' @examples #' data(eusilc) #' #' #' ## gini coefficient without Pareto tail modeling #' gini("eqIncome", weights = "rb050", data = eusilc) #' #' #' ## gini coefficient with Pareto tail modeling #' #' # estimate threshold #' ts <- paretoScale(eusilc$eqIncome, w = eusilc$db090, #' groups = eusilc$db030) #' #' # estimate shape parameter #' fit <- paretoTail(eusilc$eqIncome, k = ts$k, #' w = eusilc$db090, groups = eusilc$db030) #' #' # calibration of outliers #' w <- reweightOut(fit, calibVars(eusilc$db040)) #' gini(eusilc$eqIncome, w) #' #' # winsorization of outliers #' eqIncome <- shrinkOut(fit) #' gini(eqIncome, weights = eusilc$rb050) #' #' # replacement of outliers #' eqIncome <- replaceOut(fit) #' gini(eqIncome, weights = eusilc$rb050) #' #' # replacement of whole tail #' eqIncome <- replaceTail(fit) #' gini(eqIncome, weights = eusilc$rb050) #' #' @importFrom stats qexp runif #' @export paretoTail <- function(x, k = NULL, x0 = NULL, method = "thetaPDC", groups = NULL, w = NULL, alpha = 0.01, ...) { ## initializations if(!is.numeric(x) || length(x) == 0) stop("'x' must be a numeric vector") haveK <- !is.null(k) if(haveK) { # if 'k' is supplied, it is always used if(!is.numeric(k) || length(k) == 0 || k[1] < 1) { stop("'k' must be a positive integer") } else k <- k[1] } else if(!is.null(x0)) { # otherwise 'x0' (threshold) is used if(!is.numeric(x0) || length(x0) == 0) stop("'x0' must be numeric") else x0 <- x0[1] } else stop("either 'k' or 'x0' must be supplied") if(is.character(method)) method <- getDotTheta(method) nam <- argNames(method) useW <- !is.null(w) && ("w" %in% nam) if(useW && (!is.numeric(w) || length(w) != length(x))) { stop("'w' must be numeric vector of the same length as 'x'") } ## allow for cluster effect haveGroups <- !is.null(groups) if(haveGroups) { if(!is.vector(groups) && !is.factor(groups)) { stop("'groups' must be a vector or factor") } if(length(groups) != length(x)) { stop("'groups' must have the same length as 'x'") } if(any(is.na(groups))) stop("'groups' contains missing values") unique <- !duplicated(groups) xx <- x[unique] if(useW) ww <- w[unique] } else { xx <- x if(useW) ww <- w } xx <- unname(xx) ## check for missing values if(any(i <- is.na(xx))) { xx <- xx[!i] if(useW) ww <- ww[!i] } ## order of observed values order <- order(xx) xx <- xx[order] if(useW) ww <- ww[order] n <- length(xx) ## start constructing call to 'method' for estimation of shape parameter dots <- list(xx, ...) if(haveK) { # 'k' is supplied, threshold is determined if(k >= n) stop("'k' must be smaller than the number of observed values") x0 <- xx[n-k] # threshold (scale parameter) dots$k <- k # 'method' is expected to have 'k' as argument } else { # 'k' is not supplied, it is determined using threshold if(x0 >= xx[n]) { # compare to sorted values stop("'x0' must be smaller than the largest value") } k <- length(which(xx > x0)) # number of observations in tail dots$x0 <- x0 # 'method' is expected to have threshold 'x0' as argument } ## estimate shape parameter if(useW) dots$w <- ww theta <- do.call(method, dots) ## indicate observations in tail if(haveGroups) { tail <- groups[unique] tail <- tail[!i] tail <- tail[order] tail <- tail[(n-k+1):n] } else tail <- order[(n-k+1):n] ## flag suspicious observations (nonrepresentative outliers) if(!is.numeric(alpha) || length(alpha) == 0 || alpha < 0 || alpha > 1) { stop("'alpha' must be a numeric value in [0,1]") } else alpha <- alpha[1] q <- qpareto(1-alpha, x0, theta) # quantile of the Pareto distribution if(haveGroups) { out <- which(xx[(n-k+1):n] > q) out <- tail[out] } else { out <- unname(which(x > q)) out <- out[order(x[out])] } ## return object res <- list(x=x, k=k, groups=groups, w=w, method=method, x0=x0, theta=theta, tail=tail, alpha=alpha, out=out) class(res) <- "paretoTail" res } #' Replace observations under a Pareto model #' #' Replace observations under a Pareto model for the upper tail with values #' drawn from the fitted distribution. #' #' \code{replaceOut(x, \dots{})} is a simple wrapper for \code{replaceTail(x, #' all = FALSE, \dots{})}. #' #' @param x an object of class \code{"paretoTail"} (see #' \code{\link{paretoTail}}). #' @param all a logical indicating whether all observations in the upper tail #' should be replaced or only those flagged as outliers. #' @param \dots additional arguments to be passed down. #' #' @return A numeric vector consisting mostly of the original values, but with #' observations in the upper tail replaced with values from the fitted Pareto #' distribution. #' #' @author Andreas Alfons #' #' @seealso \code{\link{paretoTail}}, \code{\link{reweightOut}}, #' \code{\link{shrinkOut}} #' #' @references #' A. Alfons and M. Templ (2013) Estimation of Social Exclusion Indicators #' from Complex Surveys: The \R Package \pkg{laeken}. \emph{Journal of #' Statistical Software}, \bold{54}(15), 1--25. \doi{10.18637/jss.v054.i15} #' #' A. Alfons, M. Templ, P. Filzmoser (2013) Robust estimation of economic #' indicators from survey samples based on Pareto tail modeling. \emph{Journal #' of the Royal Statistical Society, Series C}, \bold{62}(2), 271--286. #' #' @keywords manip #' #' @examples #' data(eusilc) #' #' #' ## gini coefficient without Pareto tail modeling #' gini("eqIncome", weights = "rb050", data = eusilc) #' #' #' ## gini coefficient with Pareto tail modeling #' #' # estimate threshold #' ts <- paretoScale(eusilc$eqIncome, w = eusilc$db090, #' groups = eusilc$db030) #' #' # estimate shape parameter #' fit <- paretoTail(eusilc$eqIncome, k = ts$k, #' w = eusilc$db090, groups = eusilc$db030) #' #' # replacement of outliers #' eqIncome <- replaceOut(fit) #' gini(eqIncome, weights = eusilc$rb050) #' #' # replacement of whole tail #' eqIncome <- replaceTail(fit) #' gini(eqIncome, weights = eusilc$rb050) #' #' @export replaceTail <- function(x, ...) UseMethod("replaceTail") #' @rdname replaceTail #' @method replaceTail paretoTail #' @export replaceTail.paretoTail <- function(x, all = TRUE, ...) { which <- if(isTRUE(all)) x$tail else x$out k <- length(which) # number of observations to be replaced res <- x$x if(k > 0) { new <- sort(rpareto(k, x$x0, x$theta)) groups <- x$groups if(is.null(groups)) res[which] <- new else { groups <- as.character(groups) which <- as.character(which) replace <- which(groups %in% which) names(new) <- which new <- new[groups[replace]] names(new) <- names(res[replace]) res[replace] <- new } } res } #replaceOut <- function(x) UseMethod("replaceOut") # #replaceOut.paretoTail <- function(x) { # out <- x$out # nout <- length(out) # number of nonrepresentative outliers # new <- sort(rpareto(nout, x$x0, x$theta)) # res <- x$x # if(nout > 0) { # groups <- x$groups # if(is.null(groups)) res[out] <- new # else { # groups <- as.character(groups) # out <- as.character(out) # replace <- which(groups %in% out) # names(new) <- out # new <- new[groups[replace]] # names(new) <- names(res[replace]) # res[replace] <- new # } # } # res #} #replaceOut <- function(x) replaceTail(x, all=FALSE) #' @rdname replaceTail #' @export replaceOut <- function(x, ...) { localReplaceTail <- function(x, all, ...) replaceTail(x, all=FALSE, ...) localReplaceTail(x, ...) } #' Reweight outliers in the Pareto model #' #' Reweight observations that are flagged as outliers in a Pareto model for the #' upper tail of the distribution. #' #' If the data contain sample weights, the weights of the outlying observations #' are set to \eqn{1} and the weights of the remaining observations are #' calibrated according to auxiliary variables. Otherwise, weight \eqn{0} is #' assigned to outliers and weight \eqn{1} to other observations. #' #' @param x an object of class \code{"paretoTail"} (see #' \code{\link{paretoTail}}). #' @param X a matrix of binary calibration variables (see #' \code{\link{calibVars}}). This is only used if \code{x} contains sample #' weights or if \code{w} is supplied. #' @param w a numeric vector of sample weights. This is only used if \code{x} #' does not contain sample weights, i.e., if sample weights were not considered #' in estimating the shape parameter of the Pareto distribution. #' @param \dots additional arguments to be passed down. #' #' @return If the data contain sample weights, a numeric containing the #' recalibrated weights is returned, otherwise a numeric vector assigning weight #' \eqn{0} to outliers and weight \eqn{1} to other observations. #' #' @author Andreas Alfons #' #' @seealso \code{\link{paretoTail}}, \code{\link{shrinkOut}} , #' \code{\link{replaceOut}}, \code{\link{replaceTail}} #' #' @references #' A. Alfons and M. Templ (2013) Estimation of Social Exclusion Indicators #' from Complex Surveys: The \R Package \pkg{laeken}. \emph{Journal of #' Statistical Software}, \bold{54}(15), 1--25. \doi{10.18637/jss.v054.i15} #' #' A. Alfons, M. Templ, P. Filzmoser (2013) Robust estimation of economic #' indicators from survey samples based on Pareto tail modeling. \emph{Journal #' of the Royal Statistical Society, Series C}, \bold{62}(2), 271--286. #' #' @keywords manip #' #' @examples #' data(eusilc) #' #' ## gini coefficient without Pareto tail modeling #' gini("eqIncome", weights = "rb050", data = eusilc) #' #' ## gini coefficient with Pareto tail modeling #' # estimate threshold #' ts <- paretoScale(eusilc$eqIncome, w = eusilc$db090, #' groups = eusilc$db030) #' # estimate shape parameter #' fit <- paretoTail(eusilc$eqIncome, k = ts$k, #' w = eusilc$db090, groups = eusilc$db030) #' # calibration of outliers #' w <- reweightOut(fit, calibVars(eusilc$db040)) #' gini(eusilc$eqIncome, w) #' #' @export reweightOut <- function(x, ...) UseMethod("reweightOut") #' @rdname reweightOut #' @method reweightOut paretoTail #' @export reweightOut.paretoTail <- function(x, X, w = NULL, ...) { # in case of sample weights, set weights of outliers to one and calibrate # other observations # otherwise, set weights of outliers to zero and weights of other # observations to one out <- x$out n <- length(x$x) # number of observations if(is.null(x$w)) { # check supplied weights if(!is.null(w) && (!is.numeric(w) || length(w) != n)) { stop(sprintf("'w' must be numeric vector of length %d", n)) } } else w <- x$w if(length(out) > 0) { # nonrepresentative outliers groups <- x$groups if(!is.null(groups)) out <- which(groups %in% out) if(is.null(w)) { w <- rep.int(1, n) w[out] <- 0 } else { totals <- apply(X, 2, function(i) sum(i*w)) args <- list(...) args$X <- X[-out, , drop=FALSE] args$d <- w[-out] w[out] <- 1 # set weight of nonrepresentative outliers to 1 totalsOut <- apply(X[out, , drop=FALSE], 2, sum) args$totals <- totals - totalsOut g <- do.call("calibWeights", args) w[-out] <- g * args$d } } w } #' Shrink outliers in the Pareto model #' #' Shrink observations that are flagged as outliers in a Pareto model for the #' upper tail of the distribution to the theoretical quantile used for outlier #' detection. #' #' @param x an object of class \code{"paretoTail"} (see #' \code{\link{paretoTail}}). #' @param \dots additional arguments to be passed down (currently ignored as #' there are no additional arguments in the only method implemented). #' @return A numeric vector consisting mostly of the original values, but with #' outlying observations in the upper tail shrunken to the corresponding #' theoretical quantile of the fitted Pareto distribution. #' #' @author Andreas Alfons #' #' @seealso \code{\link{paretoTail}}, \code{\link{reweightOut}}, #' \code{\link{replaceOut}}, \code{\link{replaceTail}} #' #' @references #' A. Alfons and M. Templ (2013) Estimation of Social Exclusion Indicators #' from Complex Surveys: The \R Package \pkg{laeken}. \emph{Journal of #' Statistical Software}, \bold{54}(15), 1--25. \doi{10.18637/jss.v054.i15} #' #' @keywords manip #' #' @examples #' data(eusilc) #' #' ## gini coefficient without Pareto tail modeling #' gini("eqIncome", weights = "rb050", data = eusilc) #' #' ## gini coefficient with Pareto tail modeling #' # estimate threshold #' ts <- paretoScale(eusilc$eqIncome, w = eusilc$db090, #' groups = eusilc$db030) #' # estimate shape parameter #' fit <- paretoTail(eusilc$eqIncome, k = ts$k, #' w = eusilc$db090, groups = eusilc$db030) #' # shrink outliers #' eqIncome <- shrinkOut(fit) #' gini(eqIncome, weights = eusilc$rb050) #' #' @export shrinkOut <- function(x, ...) UseMethod("shrinkOut") #' @rdname shrinkOut #' @method shrinkOut paretoTail #' @export shrinkOut.paretoTail <- function(x, ...) { # winsorize outliers in the upper tail out <- x$out res <- x$x if(length(out) > 0) { # nonrepresentative outliers new <- qpareto(1-x$alpha, x$x0, x$theta) # quantile of the Pareto distribution groups <- x$groups if(!is.null(groups)) out <- which(groups %in% out) res[out] <- new } res } ## print method for class "paretoTail" #' @export print.paretoTail <- function(x, ...) { cat("Threshold: ") cat(x$x0, ...) items <- if(is.null(x$groups)) "observations" else "groups" cat(sprintf("\nNumber of %s in the tail: ", items)) cat(x$k, ...) cat("\nShape parameter: ") cat(x$theta, ...) cat(sprintf("\n\nOutlying %s:\n", items)) print(x$out, ...) } ## utility functions for Pareto distribution dpareto <- function(x, x0 = 1, theta = 1) theta*x0^theta / x^(theta+1) ppareto <- function(q, x0 = 1, theta = 1) 1 - (q/x0)^(-theta) qpareto <- function(p, x0 = 1, theta = 1) unname(x0*exp(qexp(p)/theta)) rpareto <- function(n, x0 = 1, theta = 1) x0/runif(n)^(1/theta) ## other utility functions getDotTheta <- function(method) { if(length(method) == 0) stop("'method' has length 0") else method <- method[1] if(method %in% c("thetaPDC", "thetaISE", "thetaWML", "thetaHill")) { method <- paste(".", method, sep="") } method } laeken/R/fitPareto.R0000644000176200001440000001533113616467254014013 0ustar liggesusers# ---------------------------------------- # Authors: Andreas Alfons and Josef Holzer # Vienna University of Technology # ---------------------------------------- #' Fit income distribution models with the Pareto distribution #' #' Fit a Pareto distribution to the upper tail of income data. Since a #' theoretical distribution is used for the upper tail, this is a semiparametric #' approach. #' #' The arguments \code{k} and \code{x0} of course correspond with each other. #' If \code{k} is supplied, the threshold \code{x0} is estimated with the \eqn{n #' - k} largest value in \code{x}, where \eqn{n} is the number of observations. #' On the other hand, if the threshold \code{x0} is supplied, \code{k} is given #' by the number of observations in \code{x} larger than \code{x0}. Therefore, #' either \code{k} or \code{x0} needs to be supplied. If both are supplied, #' only \code{k} is used (mainly for back compatibility). #' #' The function supplied to \code{method} should take a numeric vector (the #' observations) as its first argument. If \code{k} is supplied, it will be #' passed on (in this case, the function is required to have an argument called #' \code{k}). Similarly, if the threshold \code{x0} is supplied, it will be #' passed on (in this case, the function is required to have an argument called #' \code{x0}). As above, only \code{k} is passed on if both are supplied. If #' the function specified by \code{method} can handle sample weights, the #' corresponding argument should be called \code{w}. Additional arguments are #' passed via the \dots{} argument. #' #' @param x a numeric vector. #' @param k the number of observations in the upper tail to which the Pareto #' distribution is fitted. #' @param x0 the threshold (scale parameter) above which the Pareto distribution #' is fitted. #' @param method either a function or a character string specifying the function #' to be used to estimate the shape parameter of the Pareto distibution, such as #' \code{\link{thetaPDC}} (the default). See \dQuote{Details} for requirements #' for such a function and \dQuote{See also} for available functions. #' @param groups an optional vector or factor specifying groups of elements of #' \code{x} (e.g., households). If supplied, each group of observations is #' expected to have the same value in \code{x} (e.g., household income). Only #' the values of every first group member to appear are used for fitting the #' Pareto distribution. For each group above the threshold, every group member #' is assigned the same value. #' @param w an optional numeric vector giving sample weights. #' @param \dots addtional arguments to be passed to the specified method. #' #' @return A numeric vector with a Pareto distribution fit to the upper tail. #' #' @note The arguments \code{x0} for the threshold (scale parameter) of the #' Pareto distribution and \code{w} for sample weights were introduced in #' version 0.2. This results in slightly different behavior regarding the #' function calls to \code{method} compared to prior versions. #' #' @author Andreas Alfons and Josef Holzer #' #' @seealso \code{\link{paretoTail}}, \code{\link{replaceTail}} #' #' \code{\link{thetaPDC}}, \code{\link{thetaWML}}, \code{\link{thetaHill}}, #' \code{\link{thetaISE}}, \code{\link{thetaLS}}, \code{\link{thetaMoment}}, #' \code{\link{thetaQQ}}, \code{\link{thetaTM}} #' #' @keywords manip #' #' @examples #' data(eusilc) #' #' #' ## gini coefficient without Pareto tail modeling #' gini("eqIncome", weights = "rb050", data = eusilc) #' #' #' ## gini coefficient with Pareto tail modeling #' #' # using number of observations in tail #' eqIncome <- fitPareto(eusilc$eqIncome, k = 175, #' w = eusilc$db090, groups = eusilc$db030) #' gini(eqIncome, weights = eusilc$rb050) #' #' # using threshold #' eqIncome <- fitPareto(eusilc$eqIncome, x0 = 44150, #' w = eusilc$db090, groups = eusilc$db030) #' gini(eqIncome, weights = eusilc$rb050) #' #' @importFrom stats optimize runif uniroot #' @export fitPareto <- function(x, k = NULL, x0 = NULL, method = "thetaPDC", groups = NULL, w = NULL, ...) { ## initializations if(!is.numeric(x) || length(x) == 0) stop("'x' must be a numeric vector") haveK <- !is.null(k) if(haveK) { # if 'k' is supplied, it is always used if(!is.numeric(k) || length(k) == 0 || k[1] < 1) { stop("'k' must be a positive integer") } else k <- k[1] } else if(!is.null(x0)) { # otherwise 'x0' (threshold) is used if(!is.numeric(x0) || length(x0) == 0) stop("'x0' must be numeric") else x0 <- x0[1] } else stop("either 'k' or 'x0' must be supplied") nam <- argNames(method) useW <- !is.null(w) && ("w" %in% nam) if(useW && (!is.numeric(w) || length(w) != length(x))) { stop("'w' must be numeric vector of the same length as 'x'") } haveGroups <- !is.null(groups) if(haveGroups) { if(!is.vector(groups) && !is.factor(groups)) { stop("'groups' must be a vector or factor") } if(length(groups) != length(x)) { stop("'groups' must have the same length as 'x'") } if(any(is.na(groups))) stop("'groups' contains missing values") unique <- !duplicated(groups) values <- x[unique] if(useW) w <- w[unique] } else values <- x ## check for missing values indices <- 1:length(values) if(any(i <- is.na(values))) indices <- indices[!i] ## order of observed values order <- order(values[indices]) indicesSorted <- indices[order] # indices of sorted vector n <- length(indicesSorted) ## start constructing call to 'method' for estimation of shape parameter dots <- list(values[indices], ...) if(haveK) { # 'k' is supplied, threshold is determined if(k >= n) stop("'k' must be smaller than the number of observed values") x0 <- values[indicesSorted[n-k]] # threshold (scale parameter) dots$k <- k # 'method' is expected to have 'k' as argument } else { # 'k' is not supplied, it is determined using threshold if(x0 >= values[indicesSorted[n]]) { # compare to sorted values stop("'x0' must be smaller than the largest value") } k <- length(which(values[indices] > x0)) # number of observations in tail dots$x0 <- x0 # 'method' is expected to have threshold 'x0' as argument } ## estimate shape parameter if(useW) dots$w <- w[indices] theta <- do.call(method, dots) ## fit Pareto distribution valuesPareto <- x0/runif(k)^(1/theta) values[indicesSorted[(n-k+1):n]] <- sort(valuesPareto) ## return values if(haveGroups) { groups <- as.character(groups) names(values) <- groups[unique] values <- values[groups] names(values) <- names(x) } values } laeken/R/eqSS.R0000644000176200001440000000515513616467254012734 0ustar liggesusers# --------------------------------------- # Author: Andreas Alfons # Vienna University of Technology # --------------------------------------- # TODO: error handling #' Equivalized household size #' #' Compute the equivalized household size according to the modified OECD scale #' adopted in 1994. #' #' @param hid if \code{data=NULL}, a vector containing the household ID. #' Otherwise a character string specifying the column of \code{data} that #' contains the household ID. #' @param age if \code{data=NULL}, a vector containing the age of the #' individuals. Otherwise a character string specifying the column of #' \code{data} that contains the age. #' @param year if \code{data=NULL}, a vector containing the year of the survey. #' Otherwise a character string specifying the column of \code{data} that #' contains the year. #' @param data a \code{data.frame} containing EU-SILC survey data, or #' \code{NULL}. #' #' @return A numeric vector containing the equivalized household size for every #' observation in \code{data}. #' #' @author Andreas Alfons #' #' @seealso \code{\link{eqInc}} #' #' @references Working group on Statistics on Income and Living Conditions #' (2004) Common cross-sectional EU indicators based on EU-SILC; the gender pay #' gap. \emph{EU-SILC 131-rev/04}, Eurostat. #' #' @keywords survey #' #' @examples #' data(eusilc) #' #' # calculate equivalized household size #' eqSS <- eqSS("db030", "age", data=eusilc) #' #' # combine with household ID and household size #' tmp <- cbind(eusilc[, c("db030", "hsize")], eqSS) #' #' # show the first 8 rows #' head(tmp, 8) #' #' @export eqSS <- function(hid, age, year = NULL, data = NULL) { ## initializations if(is.null(data)) { data <- data.frame(hid=hid) hid <- "hid" if(!is.null(year)) { data <- cbind(year=year, data) year <- "year" } } else { age <- data[, age] data <- data[, c(year, hid), drop=FALSE] } ## calculations i <- if(is.null(year)) 2 else 3 tmp <- as.data.frame(table(data)) # number of household members hm14p <- as.data.frame(table(data[age >= 14,]))[, i] # at least 14 years hm13m <- tmp[, i] - hm14p # younger than 14 tmp[, i] <- 1 + 0.5*(hm14p-1) + 0.3*hm13m # eqSS for househoulds names(tmp) <- c(year, hid, ".eqSS") data <- cbind(data, .ID=1:nrow(data)) # add ID to original data data <- merge(data, tmp, sort=FALSE) # merge with original data set ## order according to original data and extract eqSS data$.eqSS[order(data$.ID)] } laeken/R/minAMSE.R0000644000176200001440000002157413616475005013306 0ustar liggesusers# ---------------------------------------- # Authors: Josef Holzer and Andreas Alfons # Vienna University of Technology # ---------------------------------------- ## nonlinear integer minimization is done by brute force ## it is strongly recommended to set bounds 'kmax' and 'mmax' #' Weighted asymptotic mean squared error (AMSE) estimator #' #' Estimate the scale and shape parameters of a Pareto distribution with an #' iterative procedure based on minimizing the weighted asymptotic mean squared #' error (AMSE) of the Hill estimator. #' #' The weights used in the weighted AMSE depend on a nuisance parameter #' \eqn{\rho}{rho}. Both the optimal number of observations in the tail and the #' nuisance parameter \eqn{\rho}{rho} are estimated iteratively using nonlinear #' integer minimization. This is currently done by a brute force algorithm, #' hence it is stronly recommended to supply upper bounds \code{kmax} and #' \code{mmax}. #' #' See the references for more details on the iterative algorithm. #' #' @param x for \code{minAMSE}, a numeric vector. The \code{print} method is #' called by the generic function if an object of class \code{"minAMSE"} is #' supplied. #' @param weight a character vector specifying the weighting scheme to be used #' in the procedure. If \code{"Bernoulli"}, the weight functions as described #' in the \emph{Bernoulli} paper are applied. If \code{"JASA"}, the weight #' functions as described in the \emph{Journal of the Americal Statistical #' Association} are used. #' @param kmin An optional integer giving the lower bound for finding the #' optimal number of observations in the tail. It defaults to #' \eqn{[\frac{n}{100}]}{[n/100]}, where \eqn{n} denotes the number of #' observations in \code{x} (see the references). #' @param kmax An optional integer giving the upper bound for finding the #' optimal number of observations in the tail (see \dQuote{Details}). #' @param mmax An optional integer giving the upper bound for finding the #' optimal number of observations for computing the nuisance parameter #' \eqn{\rho}{rho} (see \dQuote{Details} and the references). #' @param tol an integer giving the desired tolerance level for finding the #' optimal number of observations in the tail. #' @param maxit a positive integer giving the maximum number of iterations. #' @param \dots additional arguments to be passed to #' \code{\link[base]{print.default}}. #' #' @return An object of class \code{"minAMSE"} with the following components: #' \item{kopt}{the optimal number of observations in the tail.} #' \item{x0}{the corresponding threshold.} #' \item{theta}{the estimated shape parameter of the Pareto distribution.} #' \item{MSEmin}{the minimal MSE.} #' \item{rho}{the estimated nuisance parameter.} #' \item{k}{the examined range for the number of observations in the tail.} #' \item{MSE}{the corresponding MSEs.} #' #' @author Josef Holzer and Andreas Alfons #' #' @seealso \code{\link{thetaHill}} #' #' @references Beirlant, J., Vynckier, P. and Teugels, J.L. (1996) Tail index #' estimation, Pareto quantile plots, and regression diagnostics. \emph{Journal #' of the American Statistical Association}, \bold{91}(436), 1659--1667. #' #' Beirlant, J., Vynckier, P. and Teugels, J.L. (1996) Excess functions and #' estimation of the extreme-value index. \emph{Bernoulli}, \bold{2}(4), #' 293--318. #' #' Dupuis, D.J. and Victoria-Feser, M.-P. (2006) A robust prediction error #' criterion for Pareto modelling of upper tails. \emph{The Canadian Journal of #' Statistics}, \bold{34}(4), 639--658. #' #' @keywords manip #' #' @examples #' data(eusilc) #' # equivalized disposable income is equal for each household #' # member, therefore only one household member is taken #' minAMSE(eusilc$eqIncome[!duplicated(eusilc$db030)], #' kmin = 60, kmax = 150, mmax = 250) #' #' @export minAMSE <- function(x, weight = c("Bernoulli", "JASA"), kmin, kmax, mmax, tol = 0, maxit = 100) { ## initializations if(!is.numeric(x) || length(x) == 0) stop("'x' must be a numeric vector") if(any(i <- is.na(x))) x <- x[!i] x <- sort(x) n <- length(x) if(n == 0) stop("no observed values") weight <- match.arg(weight) kbounds <- c(trunc(n/100), n-2) mbound <- n-1 if(missing(kmin)) kmin <- kbounds[1] if(missing(kmax)) kmax <- kbounds[2] if(missing(mmax)) mmax <- mbound if(!is.numeric(kmin) || length(kmin) == 0 || kmin[1] < 1) { stop("'kmin' must be a single positive integer") } else kmin <- kmin[1] if(!is.numeric(kmax) || length(kmax) == 0 || kmax[1] <= kmin) { stop("'kmax' must be a single positive integer larger than 'kmin'") } else kmax <- kmax[1] if(!is.numeric(mmax) || length(mmax) == 0 || mmax[1] <= kmax) { stop("'mmax' must be a single positive integer larger than 'kmax'") } else mmax <- mmax[1] if(!is.numeric(maxit) || length(maxit) == 0 || maxit[1] < 1) { stop("'maxit' must be a single positive integer") } else maxit <- maxit[1] ## check bounds for k if(kmin < kbounds[1]) { kmin <- kbounds[1] warning("'kmin' is set to ", kbounds[1], ", as this is the suggested minumum") } if(kmax > kbounds[2]) { kmax <- kbounds[2] warning("'kmax' is set to ", kbounds[2], ", as this is the allowed maximum") } ## check bound for m if(mmax > mbound) { mmax <- mbound warning("'mmax' is set to ", mbound, ", as this is the allowed maximum") } ## Hill estimates of theta for range of k kl <- trunc(kmin/2) theta <- rep.int(NA, kmax) theta[kl:mmax] <- sapply(kl:mmax, function(k) thetaHill(x, k)) # shape ## initial estimate of k k <- kmin:kmax # range of k to search for minimum MSE <- mapply(function(k, theta) MSEinit(x, k, theta), k, theta[(kmin:kmax)-kl+1]) k0 <- k[which.min(MSE)] theta0 <- theta[k0] ## initial estimate of rho m <- (k0+1):mmax rho <- sapply(m, function(m) Rm(x, theta, m, k0)) cr <- mapply(function(m, rho) critRm(x, rho, theta, m, k0), m, rho) rho0 <- rho[which.min(cr)] ## iterative procedure for(i in 1:maxit) { ## estimate k MSE <- sapply(k, function(k) MSEopt(x, k, theta0, rho0, weight)) tmp <- which.min(MSE) MSEmin <- MSE[tmp] kn <- k[tmp] thetan <- theta[kn] ## estimate rho m <- (kn+1):mmax rho <- sapply(m, function(m) Rm(x, theta, m, kn)) cr <- mapply(function(m, rho) critRm(x, rho, theta, m, k0), m, rho) rhon <- rho[which.min(cr)] if(abs(kn-k0) <= tol) break else { k0 <- kn theta0 <- thetan rho0 <- rhon } } ## return results res <- list(kopt=kn, x0=x[n-kn], theta=thetan, MSEmin=MSEmin, rho=rhon, k=k, MSE=MSE) class(res) <- "minAMSE" res } ## internal functions for the evaluation of the MSEopt criterion ## x is not expected to contain missing values and is assumed to be sorted MSEinit <- function(x, k, theta) { n <- length(x) x0 <- x[n-k] # threshold (scale parameter) y <- log(x[(n-k+1):n]/x0) # relative excesses nyhat <- log((k:1)/(k+1))/theta # negative predicted values ## MSE 1/k * sum((y + nyhat)^2) } MSEopt <- function(x, k, theta, rho, weight = c("Bernoulli", "JASA")) { n <- length(x) x0 <- x[n-k] # threshold (scale parameter) y <- log(x[(n-k+1):n]/x0) # relative excesses h <- k:1 nyhat <- log(h/(k+1))/theta # negative predicted values ## weight functions according to paper in Bernoulli or JASA i <- 1:k hv <- i/(k+1) if(weight == "Bernoulli") { wk1 <- hv wk2 <- -log(i/(k+1)) } else { wk1 <- rep.int(1, k) wk2 <- h/(k+1) # second weight function (first is identical to 1) } ## define delta functions tmp1 <- hv^(-1)-1 tmp2 <- (1-rho)^2 tmp3 <- ((hv^(-rho)-1)/rho)^2 ak1 <- mean(wk1*tmp1) ak2 <- mean(wk2*tmp1) bk1 <- tmp2 * mean(wk1*tmp3) bk2 <- tmp2 * mean(wk2*tmp3) den <- (ak1*bk2 - bk1*ak2) # denomitator for delta functions delta1 <- (bk2 - ak2) / den delta2 <- (ak1 - bk1) / den ## define optimal weight function woptk <- delta1*wk1 + delta2*wk2 ## WMSE mean(woptk * (y + nyhat)^2) } ## internal functions for estimating rho ## requirements for m and k are assumed to be fulfilled Rm <- function(x, theta, m, k) { mk <- m+k # denominators 2 and 4 do not cause problems with floating point arithmetic Hmk4 <- theta[trunc(mk/4)] Hmk2 <- theta[trunc(mk/2)] Hm2 <- theta[trunc(m/2)] Hm <- theta[m] log(abs((Hmk4-Hmk2)/(Hm2-Hm))) / (log(2*m/(m-k))) } ## internal functions for the evaluation of the criterion for rho ## x is not expected to contain missing values and is assumed to be sorted critRm <- function(x, rho, theta, m, k) { j <- k:(m-1) l <- log(abs((theta[trunc(j/2)] - theta[j])/(theta[trunc(m/2)] - theta[m]))) mean((l - rho*log(m/j))^2) } laeken/R/gpg.R0000755000176200001440000002576014127253245012634 0ustar liggesusers# --------------------------------------- # Author: Matthias Templ # Vienna University of Technology # --------------------------------------- #' Gender pay (wage) gap. #' #' Estimate the gender pay (wage) gap. #' #' The implementation strictly follows the Eurostat definition (with default #' method \code{"mean"} and alternative method \code{"median"}). If weights are #' provided, the weighted mean or weighted median is estimated. #' #' @param inc either a numeric vector giving the equivalized disposable income, #' or (if \code{data} is not \code{NULL}) a character string, an integer or a #' logical vector specifying the corresponding column of \code{data}. #' @param gender either a factor giving the gender, or (if \code{data} is not #' \code{NULL}) a character string, an integer or a logical vector specifying #' the corresponding column of \code{data}. #' @param method a character string specifying the method to be used. Possible #' values are \code{"mean"} for the mean, and \code{"median"} for the median. #' If weights are provided, the weighted mean or weighted median is estimated. #' @param weights optional; either a numeric vector giving the personal sample #' weights, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. #' @param sort optional; either a numeric vector giving the personal IDs to be #' used as tie-breakers for sorting, or (if \code{data} is not \code{NULL}) a #' character string, an integer or a logical vector specifying the corresponding #' column of \code{data}. #' @param years optional; either a numeric vector giving the different years of #' the survey, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. If supplied, values are computed for each year. #' @param breakdown optional; either a numeric vector giving different domains, #' or (if \code{data} is not \code{NULL}) a character string, an integer or a #' logical vector specifying the corresponding column of \code{data}. If #' supplied, the values for each domain are computed in addition to the overall #' value. #' @param design optional and only used if \code{var} is not \code{NULL}; either #' an integer vector or factor giving different strata for stratified sampling #' designs, or (if \code{data} is not \code{NULL}) a character string, an #' integer or a logical vector specifying the corresponding column of #' \code{data}. #' @param cluster optional and only used if \code{var} is not \code{NULL}; #' either an integer vector or factor giving different clusters for cluster #' sampling designs, or (if \code{data} is not \code{NULL}) a character string, #' an integer or a logical vector specifying the corresponding column of #' \code{data}. #' @param data an optional \code{data.frame}. #' @param var a character string specifying the type of variance estimation to #' be used, or \code{NULL} to omit variance estimation. See #' \code{\link{variance}} for possible values. #' @param alpha numeric; if \code{var} is not \code{NULL}, this gives the #' significance level to be used for computing the confidence interval (i.e., #' the confidence level is \eqn{1 - }\code{alpha}). #' @param na.rm a logical indicating whether missing values should be removed. #' @param \dots if \code{var} is not \code{NULL}, additional arguments to be #' passed to \code{\link{variance}}. #' #' @return A list of class \code{"gpg"} (which inherits from the class #' \code{"indicator"}) with the following components: #' \item{value}{a numeric vector containing the overall value(s).} #' \item{valueByStratum}{a \code{data.frame} containing the values by #' domain, or \code{NULL}.} #' \item{varMethod}{a character string specifying the type of variance #' estimation used, or \code{NULL} if variance estimation was omitted.} #' \item{var}{a numeric vector containing the variance estimate(s), or #' \code{NULL}.} #' \item{varByStratum}{a \code{data.frame} containing the variance #' estimates by domain, or \code{NULL}.} #' \item{ci}{a numeric vector or matrix containing the lower and upper #' endpoints of the confidence interval(s), or \code{NULL}.} #' \item{ciByStratum}{a \code{data.frame} containing the lower and upper #' endpoints of the confidence intervals by domain, or \code{NULL}.} #' \item{alpha}{a numeric value giving the significance level used for #' computing the confidence interv al(s) (i.e., the confidence level is \eqn{1 - #' }\code{alpha}), or \code{NULL}.} #' \item{years}{a numeric vector containing the different years of the #' survey.} #' \item{strata}{a character vector containing the different domains of the #' breakdown.} #' #' @author Matthias Templ and Alexander Haider, using code for breaking down #' estimation by Andreas Alfons #' #' @seealso \code{\link{variance}}, \code{\link{qsr}}, \code{\link{gini}} #' #' @references #' A. Alfons and M. Templ (2013) Estimation of Social Exclusion Indicators #' from Complex Surveys: The \R Package \pkg{laeken}. \emph{Journal of #' Statistical Software}, \bold{54}(15), 1--25. \doi{10.18637/jss.v054.i15} #' #' Working group on Statistics on Income and Living Conditions (2004) #' Common cross-sectional EU indicators based on EU-SILC; the gender #' pay gap. \emph{EU-SILC 131-rev/04}, Eurostat, Luxembourg. #' #' @keywords survey #' #' @examples #' data(ses) #' #' # overall value with mean #' gpg("earningsHour", gender = "sex", weigths = "weights", #' data = ses) #' #' # overall value with median #' gpg("earningsHour", gender = "sex", weigths = "weights", #' data = ses, method = "median") #' #' # values by education with mean #' gpg("earningsHour", gender = "sex", weigths = "weights", #' breakdown = "education", data = ses) #' #' # values by education with median #' gpg("earningsHour", gender = "sex", weigths = "weights", #' breakdown = "education", data = ses, method = "median") #' #' @importFrom stats aggregate weighted.mean #' @export gpg <- function(inc, gender = NULL, method = c("mean", "median"), weights = NULL, sort = NULL, years = NULL, breakdown = NULL, design = NULL, cluster = NULL, data = NULL, var = NULL, alpha = 0.05, na.rm = FALSE, ...) { ## initializations if(is.null(gender)) stop("'gender' must be supplied") byYear <- !is.null(years) byStratum <- !is.null(breakdown) if(!is.null(data)) { inc <- data[, inc] gender <- data[, gender] if(!is.null(weights)) weights <- data[, weights] if(!is.null(sort)) sort <- data[, sort] if(byYear) years <- data[, years] if(byStratum) breakdown <- data[, breakdown] if(!is.null(var)) { if(!is.null(design)) design <- data[, design] if(!is.null(cluster)) cluster <- data[, cluster] } } # check vectors if(!is.numeric(inc)) stop("'inc' must be a numeric vector") method <- match.arg(method) if(!is.factor(gender)) stop("'gender' must be a factor.") if(length(levels(gender)) != 2) stop("'gender' must have exactly two levels") if(!all(levels(gender) == c("female", "male"))) { gender <- factor(gender, labels=c("female","male")) warning("The levels of gender were internally recoded - your first level has to correspond to females") } if(!is.null(years)) { if(!is.factor(years)) stop("'years' should be a factor") nage <- length(levels(years)) if(n > 12) warning(paste("Too small sample sizes may occur by using ", n," age classes")) } n <- length(inc) if(is.null(weights)) weights <- weights <- rep.int(1, n) else if(!is.numeric(weights)) stop("'weights' must be a numeric vector") if(!is.null(sort) && !is.vector(sort) && !is.ordered(sort)) { stop("'sort' must be a vector or ordered factor") } if(byYear && !is.numeric(years)) { stop("'years' must be a numeric vector") } if(byStratum) { if(!is.vector(breakdown) && !is.factor(breakdown)) { stop("'breakdown' must be a vector or factor") } else breakdown <- as.factor(breakdown) } if(is.null(data)) { # check vector lengths if(length(weights) != n) { stop("'weights' must have the same length as 'x'") } if(!is.null(sort) && length(sort) != n) { stop("'sort' must have the same length as 'x'") } if(byYear && length(years) != n) { stop("'years' must have the same length as 'x'") } if(byStratum && length(breakdown) != n) { stop("'breakdown' must have the same length as 'x'") } } ## computations # GPG by year (if requested) if(byYear) { ys <- sort(unique(years)) # unique years gp <- function(y, inc, weights, sort, years, na.rm) { i <- years == y genderGap(inc[i], gender[i], method, weights[i], sort[i], na.rm=na.rm) } value <- sapply(ys, gp, inc=inc, weights=weights, sort=sort, years=years, na.rm=na.rm) names(value) <- ys # use years as names } else { ys <- NULL value <- genderGap(inc, gender, method, weights, sort, na.rm=na.rm) } # GPG by stratum (if requested) if(byStratum) { gpR <- function(i, inc, weights, sort, na.rm) { genderGap(inc[i], gender[i], method, weights[i], sort[i], na.rm=na.rm) } valueByStratum <- aggregate(1:n, if(byYear) list(year=years, stratum=breakdown) else list(stratum=breakdown), gpR, inc=inc, weights=weights, sort=sort, na.rm=na.rm) names(valueByStratum)[ncol(valueByStratum)] <- "value" rs <- levels(breakdown) # unique strata } else valueByStratum <- rs <- NULL ## create object of class "gpg" res <- constructGpg(value=value, valueByStratum=valueByStratum, years=ys, strata=rs) # variance estimation (if requested) if(!is.null(var)) { res <- variance(inc, weights, years, breakdown, design, cluster, indicator=res, alpha=alpha, na.rm=na.rm, type=var, gender=gender, method=method, ...) } ## return result return(res) } ## workhorse genderGap <- function(x, gend, method = 'mean', weights = NULL, sort = NULL, na.rm = FALSE) { if(is.null(gend)) stop("'gender' must be supplied") # initializations if(isTRUE(na.rm)){ indices <- !is.na(x) x <- x[indices] gend <- gend[indices] if(!is.null(weights)) weights <- weights[indices] if(!is.null(sort)) sort <- sort[indices] } else if(any(is.na(x))) return(NA) male <- levels(gend)[1] 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liggesuserslaeken/inst/doc/0000755000176200001440000000000014127307300013230 5ustar liggesuserslaeken/inst/doc/laeken-variance.R0000644000176200001440000000347014127307300016404 0ustar liggesusers### R code from vignette source 'laeken-variance.Rnw' ################################################### ### code chunk number 1: laeken-variance.Rnw:52-53 ################################################### options(prompt="R> ") ################################################### ### code chunk number 2: laeken-variance.Rnw:93-95 (eval = FALSE) ################################################### ## vignette("laeken-standard") ## vignette("laeken-pareto") ################################################### ### code chunk number 3: laeken-variance.Rnw:111-113 ################################################### library("laeken") data("eusilc") ################################################### ### code chunk number 4: laeken-variance.Rnw:140-141 ################################################### args(variance) ################################################### ### code chunk number 5: laeken-variance.Rnw:226-229 ################################################### a <- arpr("eqIncome", weights = "rb050", data = eusilc) variance("eqIncome", weights = "rb050", design = "db040", data = eusilc, indicator = a, bootType = "naive", seed = 123) ################################################### ### code chunk number 6: laeken-variance.Rnw:237-240 ################################################### b <- arpr("eqIncome", weights = "rb050", breakdown = "db040", data = eusilc) variance("eqIncome", weights = "rb050", breakdown = "db040", design = "db040", data = eusilc, indicator = b, bootType = "naive", seed = 123) ################################################### ### code chunk number 7: laeken-variance.Rnw:309-312 ################################################### variance("eqIncome", weights = "rb050", design = "db040", data = 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J# $#qH+UXnT^+mpt?ЈoF8,̈4Qt4#rAO"\u> stream xR=o0+8>~YdˢсH8-%; Y ݻw8}sӓJ8MOCh3|'{QhjbiOݧ&M3Xm<רּ8Δ3?Ϙ_Ca\.ۘb]Jg_ݯYLޓ?2endstream endobj 403 0 obj << /Type /XRef /Length 280 /Filter /FlateDecode /DecodeParms << /Columns 5 /Predictor 12 >> /W [ 1 3 1 ] /Info 3 0 R /Root 2 0 R /Size 404 /ID [] >> stream xӱjPsoԄZV#T8C/Jv(-΂`: BХC$>~{n-8hPӓfkT={3]3T=`6p'ɛ|qgSUhd` 3xW躾1?+JW~{wߡZr3 endstream endobj startxref 710811 %%EOF laeken/inst/doc/laeken-variance.Rnw0000644000176200001440000004121514127275501016757 0ustar liggesusers\documentclass[a4paper,10pt]{scrartcl} \usepackage[OT1]{fontenc} \usepackage{Sweave} %% additional packages \usepackage{natbib} \bibpunct{(}{)}{,}{a}{}{,} \usepackage{amsmath, amssymb} \usepackage{hyperref} \hypersetup{colorlinks, citecolor=blue, linkcolor=blue, urlcolor=blue} \usepackage[top=30mm, bottom=30mm, left=30mm, right=30mm]{geometry} \usepackage{enumerate} \usepackage{engord} %% additional commands \newcommand{\code}[1]{\texttt{#1}} \newcommand{\pkg}[1]{\mbox{\textbf{#1}}} \newcommand{\proglang}[1]{\mbox{\textsf{#1}}} %%\VignetteIndexEntry{Variance Estimation of Indicators on Social Exclusion and Poverty using the R Package laeken} %%\VignetteDepends{laeken} %%\VignetteKeywords{social exclusion, poverty, indicators, variance estimation} %%\VignettePackage{laeken} \begin{document} \title{Variance Estimation of Indicators on Social Exclusion and Poverty using the \proglang{R} Package \pkg{laeken}} \author{Matthias Templ$^{1}$, Andreas Alfons$^{2}$} \date{} \maketitle \setlength{\footnotesep}{11pt} \footnotetext[1]{ \begin{tabular}[t]{l} Zurich University of Applied Sciences\\ E-mail: \href{mailto:matthias.templ@zhaw.ch}{matthias.templ@zhaw.ch} \end{tabular} } \footnotetext[2]{ \begin{tabular}[t]{l} Erasmus School of Economics, Erasmus University Rotterdam\\ E-mail: \href{mailto:alfons@ese.eur.nl}{alfons@ese.eur.nl} \end{tabular} } % change R prompt <>= options(prompt="R> ") @ \paragraph{Abstract} This vignette illustrates the application of variance estimation procedures to indicators on social exclusion and poverty using the \proglang{R} package \pkg{laeken}. To be more precise, it describes a general framework for estimating variance and confidence intervals of indicators under complex sampling designs. Currently, the package is focused on bootstrap approaches. While the naive bootstrap does not modify the weights of the bootstrap samples, a calibrated version allows to calibrate each bootstrap sample on auxiliary information before deriving the bootstrap replicate estimate. % ------------ % introduction % ------------ \section{Introduction} When point estimates of indicators are computed from samples, it is important to also obtain variance estimates and confidence intervals in order to account for variability due to sampling. Other sources of variability such as data editing or imputation may need to be considered as well, but this is not further discussed in this paper. While this vignette targets the topic of variance and confidence interval estimation for the indicators on social exclusion and poverty according to \citet{EU-SILC04, EU-SILC09}, the aim is not to describe and evaluate the different approaches that have been proposed to date. Instead, the aim is to present the functionality for the statistical environment \proglang{R} \citep{RDev} implemented in the add-on package \pkg{laeken} \citep{laeken}. It should be noted that the basic design of the package, as well as standard point estimation of the indicators on social exclusion and poverty, is discussed in detail in vignette \code{laeken-standard} \citep{templ11a}. In addition, vignette \code{laeken-pareto} \citep{alfons11a} presents more sophisticated methods for point estimation of the indicators, which are less influenced by outliers. Those documents can be viewed from within \proglang{R} with the following commands: <>= vignette("laeken-standard") vignette("laeken-pareto") @ Morover, a general introduction to package \pkg{laeken} is published as \citet{alfons13b}. The data basis for the estimation of the indicators on social exclusion and poverty is the \emph{European Union Statistics on Income and Living Conditions} (EU-SILC), which is an annual panel survey conducted in EU member states and other European countries. Package \pkg{laeken} provides the synthetic example data \code{eusilc} consisting of $14\,827$ observations from $6\,000$ households. Furthermore, the data were generated from Austrian EU-SILC survey data from 2006 using the data simulation methodology proposed by \citet{alfons11c} and implemented in the \proglang{R} package \pkg{simPopulation} \citep{simPopulation}. The data set \code{eusilc} is used in the code examples throughout the paper. % ----- <<>>= library("laeken") data("eusilc") @ The rest of the paper is organized as follows. Section~\ref{sec:variance} presents the general wrapper function for estimating variance and confidence intervals of indicators in package \pkg{laeken}. The naive and calibrated bootstrap approaches are discussed in Sections~\ref{sec:naive} and~\ref{sec:calib}, respectively. Section~\ref{sec:concl} concludes. % --------------- % general wrapper % --------------- \section{General wrapper function for variance estimation} \label{sec:variance} The function \code{variance()} provides a flexible framework for estimating the variance and confidence intervals of indicators such as the \emph{at-risk-of-poverty rate}, the \emph{Gini coefficient}, the \emph{quintile share ratio} and the \emph{relative median at-risk-of-poverty gap}. For a mathematical description and details on the implementation of these indicators in the \proglang{R} package \pkg{laeken}, the reader is referred to vignette \code{laeken-standard} \citep{templ11a}. In any case, \code{variance()} acts as a general wrapper function for computing variance and confidence interval estimates of indicators on social exclusion and poverty with package \pkg{laeken}. The arguments of function \code{variance()} are shown in the following: <<>>= args(variance) @ All these arguments are fully described in the \proglang{R} help page of function \code{variance()}. The most important arguments are: \begin{description} \item[inc:] the income vector. \item[weights:] an optional vector of sample weights. \item[breakdown:] an optional vector giving different domains in which variances and confidence intervals should be computed. \item[design:] an optional vector or factor giving different strata for stratified sampling designs. \item[data:] an optional \code{data.frame}. If supplied, each of the above arguments should be specified as a character string or an integer or logical vector specifying the corresponding column. \item[indicator:] an object inheriting from the class \code{"indicator"} that contains the point estimates of the indicator, such as \code{"arpr"} for the at-risk-of-poverty rate, \code{"qsr"} for the quintile share ratio, \code{"rmpg"} for the relative median at-risk-of-poverty gap, or \code{"gini"} for the Gini coefficient. \item[type:] a character string specifying the type of variance estimation to be used. Currently, only \code{"bootstrap"} is implemented for variance estimation based on bootstrap resampling. \end{description} In the following sections, two bootstrap methods for estimating the variance and confidence intervals of point estimates for complex survey data are described. Furthermore, their application using the function \code{variance()} from package \pkg{laeken} is demonstrated. % --------------- % naive bootstrap % --------------- \section{Naive bootstrap} \label{sec:naive} Let $\boldsymbol{X} := (\boldsymbol{x}_{1}, \ldots, \boldsymbol{x}_{n})'$ denote a survey sample with $n$ observations and $p$ variables. Then the \emph{naive bootstrap algorithm} for estimating the variance and confidence interval of an indicator can be summarized as follows: \begin{enumerate} \item Draw $R$ independent bootstrap samples $\boldsymbol{X}_{1}^{*}, \ldots, \boldsymbol{X}_{R}^{*}$ from $\boldsymbol{X}$. \item Compute the bootstrap replicate estimates $\hat{\theta}_{r}^{*} := \hat{\theta}(\boldsymbol{X}_{r}^{*})$ for each bootstrap sample $\boldsymbol{X}_{r}^{*}$, $r = 1, \ldots, R$, where $\hat{\theta}$ denotes an estimator for a certain indicator of interest. Of course the sample weights always need to be considered for the computation of the bootstrap replicate estimates. \item Estimate the variance $V(\hat{\theta})$ by the variance of the $R$ bootstrap replicate estimates: \begin{equation} \hat{V}(\hat{\theta}) := \frac{1}{R-1} \sum_{r=1}^{R} \left( \hat{\theta}_{r}^{*} - \frac{1}{R} \sum_{s=1}^{R} \hat{\theta}_{s}^{*} \right)^{2}. \end{equation} \item Estimate the confidence interval at confidence level $1 - \alpha$ by one of the following methods \citep[for details, see][]{davison97}: \begin{description} \item[Percentile method:] $\left[ \hat{\theta}_{((R+1) \frac{\alpha}{2})}^{*}, \hat{\theta}_{((R+1)(1-\frac{\alpha}{2}))}^{*} \right]$, as suggested by \cite{efron93}. \item[Normal approximation:] $\hat{\theta} \pm z_{1-\frac{\alpha}{2}} \cdot \hat{V}(\hat{\theta})^{1/2}$ with $z_{1-\frac{\alpha}{2}} = \Phi^{-1}(1 - \frac{\alpha}{2})$. \item[Basic bootstrap method:] $\left[ 2\hat{\theta} - \hat{\theta}_{((R+1)(1-\frac{\alpha}{2}))}^{*}, 2\hat{\theta} - \hat{\theta}_{((R+1)\frac{\alpha}{2})}^{*} \right]$. \end{description} For the percentile and the basic bootstrap method, $\hat{\theta}_{(1)}^{*} \leq \ldots \leq \hat{\theta}_{(R)}^{*}$ denote the order statistics of the bootstrap replicate estimates. \end{enumerate} In the following example, the variance and confidence interval of the at-risk-of-poverty rate are estimated with the naive bootstrap procedure. The output of function \code{variance()} is an object of the same class as the point estimate supplied as the \code{indicator} argument, but with additional components for the variance and confidence interval. In addition to the point estimate, the income and the sample weights need to be supplied. Furthermore, a stratified sampling design can be considered by specifying the \code{design} argument, in which case observations are resampled separately within the strata. To ensure reproducibility of the results, the seed of the random number generator is set. <<>>= a <- arpr("eqIncome", weights = "rb050", data = eusilc) variance("eqIncome", weights = "rb050", design = "db040", data = eusilc, indicator = a, bootType = "naive", seed = 123) @ One of the most convenient features of package \pkg{laeken} is that indicators can be evaluated for different subdomains using a single command. This also holds for variance estimation. Using the \code{breakdown} argument, the example below produces variance and confidence interval estimates for each NUTS2 region in addition to the overall values. <<>>= b <- arpr("eqIncome", weights = "rb050", breakdown = "db040", data = eusilc) variance("eqIncome", weights = "rb050", breakdown = "db040", design = "db040", data = eusilc, indicator = b, bootType = "naive", seed = 123) @ It should be noted that the workhorse function \code{bootVar()} is called internally by \code{variance()} for bootstrap variance and confidence interval estimation. The function \code{bootVar()} could also be called directly by the user in exactly the same manner. Moreover, variance and confidence interval estimation for any other indicator implemented in package \pkg{laeken} is straightforward---the application using function \code{variance()} or \code{bootVar()} remains the same. % -------------------- % calibrated bootstrap % -------------------- \section{Calibrated bootstrap} \label{sec:calib} \cite{rao88} showed that the naive bootstrap is biased when used in the complex survey context. They propose to increase the variance estimate in the $h$-th stratum by a factor of $\frac{n_{h} - 1}{n_{h}}$ (if the bootstrap sample is of the same size). In addition, they describe extensions to sampling without replacement, unequal probability sampling, and two-stage cluster sampling with equal probabilities and without replacement. \cite{deville92} and \cite{deville93} provide a general description on how to calibrate sample weights to account for known population totals. The naive bootstrap does not include the recalibration of bootstrap samples in order to fit known population totals and therefore is, strictly formulated, not suitable for many practical applications. However, even though a bias might be introduced, the naive bootstrap works well in many situations and is faster to compute than the calibrated version. Hence it is a popular method often used in practice. In real-world data, the inclusion probabilities for observations in the population are in general not all equal, resulting in different \emph{design weights} for the observations in the sample. Furthermore, the initial design weights are in practice often adjusted by calibration, e.g., to account for non-response or so that certain known population totals can be precisely estimated from the survey sample. To give a simplified example, if the population sizes in different regions are known, the sample weights may be calibrated so that the Horvitz-Thompson estimates \citep{horvitz52} of the population sizes equal the known true values. However, when bootstrap samples are drawn from survey data, resampling observations has the effect that such known population totals can no longer be precisely estimated. As a remedy, the sample weights of each bootstrap sample should be calibrated. The calibrated version of the bootstrap thus results in more precise variance and confidence interval estimation, but comes with higher computational costs than the naive approach. In any case, the \emph{calibrated bootstrap algorithm} is obtained by adding the following step between Steps~1 and~2 of the naive bootstrap algorithm from Section~\ref{sec:naive}: \begin{itemize} \item[1b.] Calibrate the sample weights for each bootstrap sample $\boldsymbol{X}_{r}^{*}$, $r = 1, \ldots, R$. Generalized raking procedures are thereby used for calibration: either a multiplicative method known as \emph{raking}, an additive method or a logit method \citep[see][]{deville92, deville93}. \end{itemize} The function call to \code{variance()} for the calibrated bootstrap is very similar to its counterpart for the naive bootstrap. A matrix of auxiliary calibration variables needs to be supplied via the argument \code{X}. In addition, the argument \code{totals} can be used to supply the corresponding population totals. If the \code{totals} argument is omitted, as in the following example, the population totals are computed from the sample weights of the original sample. This follows the assumption that those weights are already calibrated on the supplied auxiliary variables. % ----- <<>>= variance("eqIncome", weights = "rb050", design = "db040", data = eusilc, indicator = a, X = calibVars(eusilc$db040), seed = 123) @ % ----- Note that the function \code{calibVars()} transforms a factor into a matrix of binary variables, as required by the calibration function \code{calibWeights()}, which is called internally. While the default is to use raking for calibration, other methods can be specified via the \code{method} argument. % ----------- % conclusions % ----------- \section{Conclusions} \label{sec:concl} Both bootstrap procedures for variance and confidence interval estimation of indicators on social exclusion and poverty currently implemented in the \proglang{R} package \pkg{laeken} have their strengths. While the naive bootstrap is faster to compute, the calibrated bootstrap in general leads to more precise results. The implementation of other procedures such as linearization techniques \citep{kovacevic97, deville99, hulliger06, osier09} or the delete-a-group jackknife \citep{kott01} is future work. Furthermore, \citet{alfons09} demonstrated how the variance of indicators computed from data with imputed values may be underestimated in bootstrap procedures, depending on the indicator itself and the imputation procedure used. They proposed to use the method described in \cite{little02}, which consists of drawing bootstrap samples from the original data with missing values, and to impute the missing data for each bootstrap sample before computing the corresponding bootstrap replicate estimate. Of course, this results in an additional increase of the computation time. The implementation of this procedure in package \pkg{laeken} is future work. It should also be noted that multiple imputation is a further possibility to consider the additional uncertainty from imputation when estimating the variance of an indicator \citep[see][]{little02}. % --------------- % acknowledgments % --------------- \section*{Acknowledgments} This work was partly funded by the European Union (represented by the European Commission) within the 7$^{\mathrm{th}}$ framework programme for research (Theme~8, Socio-Economic Sciences and Humanities, Project AMELI (Advanced Methodology for European Laeken Indicators), Grant Agreement No. 217322). Visit \url{http://ameli.surveystatistics.net} for more information on the project. % ------------ % bibliography % ------------ \bibliographystyle{plainnat} \bibliography{laeken} \end{document} laeken/inst/doc/laeken-pareto.Rnw0000644000176200001440000011610214127275361016463 0ustar liggesusers\documentclass[a4paper,10pt]{scrartcl} \usepackage[OT1]{fontenc} \usepackage{Sweave} %% additional packages \usepackage{natbib} \bibpunct{(}{)}{,}{a}{}{,} \usepackage{amsmath, amssymb} \usepackage{hyperref} \hypersetup{colorlinks, citecolor=blue, linkcolor=blue, urlcolor=blue} \usepackage[top=30mm, bottom=30mm, left=30mm, right=30mm]{geometry} %% additional commands \newcommand{\code}[1]{\texttt{#1}} \newcommand{\pkg}[1]{\mbox{\textbf{#1}}} \newcommand{\proglang}[1]{\mbox{\textsf{#1}}} %%\VignetteIndexEntry{Robust Pareto Tail Modeling for the Estimation of Indicators on Social Exclusion using the R Package laeken} %%\VignetteDepends{laeken} %%\VignetteKeywords{social exclusion, indicators, robust estimation, Pareto distribution} %%\VignettePackage{laeken} \begin{document} \title{Robust Pareto Tail Modeling for the Estimation of Indicators on Social Exclusion using the \proglang{R} Package \pkg{laeken}} %\author{ % Andreas Alfons\footnote{Vienna University of Technology, % \href{mailto:alfons@statistik.tuwien.ac.at}{alfons@statistik.tuwien.ac.at}}, % Matthias Templ\footnote{Vienna University of Technology \& Statistics Austria, % \href{mailto:templ@tuwien.ac.at}{templ@tuwien.ac.at}}, % Peter Filzmoser\footnote{Vienna University of Technology, % \href{mailto:p.filzmoser@tuwien.ac.at}{p.filzmoser@tuwien.ac.at}}, % Josef Holzer\footnote{Landesstatistik Steiermark, % \href{mailto:josef.holzer@stmk.gv.at}{josef.holzer@stmk.gv.at}} %} \author{ Andreas Alfons$^{1}$, Matthias Templ$^{2}$, Peter Filzmoser$^{3}$, Josef Holzer$^{4}$ } \date{} \maketitle \setlength{\footnotesep}{11pt} \footnotetext[1]{ \begin{tabular}[t]{l} Erasmus School of Economics, Erasmus University Rotterdam\\ E-mail: \href{mailto:alfons@ese.eur.nl}{alfons@ese.eur.nl} \end{tabular} } \footnotetext[2]{ \begin{tabular}[t]{l} Zurich University of Applied Sciences\\ E-mail: \href{mailto:matthias.templ@zhaw.ch}{matthias.templ@zhaw.ch} \end{tabular} } \footnotetext[3]{ \begin{tabular}[t]{l} Vienna University of Technology\\ E-mail: \href{mailto:p.filzmoser@tuwien.ac.at}{p.filzmoser@tuwien.ac.at} \end{tabular} } \footnotetext[4]{ \begin{tabular}[t]{l} Landesstatistik Steiermark\\ E-mail: \href{mailto:josef.holzer@stmk.gv.at}{josef.holzer@stmk.gv.at} \end{tabular} } % change R prompt <>= options(prompt="R> ") @ %% specify folder and name for Sweave graphics %\SweaveOpts{prefix.string=figures-pareto/fig} \paragraph{Abstract} In this vignette, robust semiparametric estimation of social exclusion indicators using the \proglang{R} package \pkg{laeken} is discussed. Special emphasis is thereby given to income inequality indicators, as the standard estimates for these indicators are highly influenced by outliers in the upper tail of the income distribution. This influence can be reduced by modeling the upper tail with a Pareto distribution in a robust manner. While the focus of the paper is to demonstrate the functionality of \pkg{laeken} beyond the standard estimation techniques, a brief mathematical description of the implemented procedures is given as well. % ------------ % introduction % ------------ \section{Introduction} From a robustness point of view, the standard estimators for some of the social exclusion indicators defined by \citet{EU-SILC04, EU-SILC09} are problematic. In particular the income inequality indicators \emph{quintile share ratio} (QSR) and \emph{Gini coefficient} suffer from a lack of robustness. Consider, e.g., the QSR, which is estimated as the ratio of estimated totals or means (see Section~\ref{sec:QSR} for an exact definition). It is well known that the classical estimates for totals or means have a breakdown point of 0, meaning that even a single outlier can distort the results to an arbitrary extent. In fact, the influence of a single observation in the upper tail of the income distribution on the estimation of the QSR is linear and therefore unbounded. For practical purposes, the standard QSR estimator thus cannot be recommended in many situations \citep[cf.][]{hulliger09a}. It is also important to note that the behavior of the Gini coefficient is similar to the behavior of the QSR. The data basis for the estimation of the social exclusion indicators according to \citet{EU-SILC04, EU-SILC09} is the \emph{European Union Statistics on Income and Living Conditions} (EU-SILC), which is an annual panel survey conducted in EU member states and other European countries. On the one hand, EU-SILC data typically contain a considerable amount of \emph{representative} outliers in the upper tail of the income distribution, i.e., correct observations that behave differently from the main part of the data, but that are not unique in the population and hence need to be considered for computing estimates of the indicators. On the other hand, EU-SILC data frequently contain some even more extreme \emph{nonrepresentative} outliers, i.e., observations that are either incorrect or can be considered unique in the population. Consequently, such nonrepresentative outliers need to be excluded from the estimation process or downweighted. As a remedy, the upper tail of the income distribution may be modeled with a \emph{Pareto distribution} in order to recalibrate the sample weights or use fitted income values for observations in the upper tail when estimating the indicators (see Section~\ref{sec:fit}). %This is highly applicable because the upper tail of the income distribution in %EU-SILC data virtually always contains a considerable amount of representative %outliers. Nevertheless, classical estimators for the parameters of the Pareto distribution are highly influenced by the nonrepresentative outliers themselves. Using robust methods reduces the influence on fitting the Pareto distribution to the representative outliers and therefore on the estimation of the indicators. Rather than evaluating these methods, the paper concentrates on showing how they can be applied in the statistical environment \proglang{R} \citep{RDev} with the add-on package \pkg{laeken} \citep{laeken}. The basic design of the package, as well as standard estimation of the social exclusion indicators is discussed in detail in vignette \code{laeken-standard} \citep{templ11a}. Furthermore, the general framework for variance estimation is illustrated in vignette \code{laeken-variance} \citep{templ11b}. Those documents can be viewed from within \proglang{R} with the following commands: <>= vignette("laeken-standard") vignette("laeken-variance") @ Morover, a general introduction to package \pkg{laeken} is published as \citet{alfons13b}. Throughout the paper, the example data from package \pkg{laeken} is used. The data set is called \code{eusilc} and consists of $14\,827$ observations from $6\,000$ households. In addition, it was synthetically generated from Austrian EU-SILC survey data from 2006 using the data simulation methodology proposed by \citet{alfons11c} and implemented in the \proglang{R} package \pkg{simPopulation} \citep{simPopulation}. More information on the example data can be found in vignette \code{laeken-standard} or in the corresponding \proglang{R} help page. <<>>= library("laeken") data("eusilc") @ The rest of the paper is organized as follows. Section~\ref{sec:laeken} gives a mathematical description of the Eurostat definitions of the social exclusion indicators QSR and Gini coefficient. In Section~\ref{sec:Pareto}, the Pareto distribution is briefly discussed. Section~\ref{sec:threshold} discusses a rule of thumb for estimating the threshold for the upper tail of the distribution, and illustrates graphical methods for exploring the data in order to find the threshold. Classical and robust estimators for the shape parameter of the Pareto distribution are described in Section~\ref{sec:shape}. How to use Pareto tail modeling to estimate the social exclusion indicators is then shown in Section~\ref{sec:fit}. Finally, Section~\ref{sec:concl} concludes. % ------------------- % selected indicators % ------------------- \section{Social exclusion indicators} \label{sec:laeken} This paper is focused on the inequality indicators \emph{quintile share ratio} (QSR) and \emph{Gini coefficient}, which are both highly influenced by outliers in the upper tail of the distribution. Note that for the estimation of the social exclusion indicators, each person in a household is assigned the same \emph{eqivalized disposable income}. See vignette \code{laeken-standard} \citep{templ11a} for the computation of the equivalized disposable income with the \proglang{R} package \pkg{laeken}. For the following definitions, let $\boldsymbol{x} := (x_{1}, \ldots, x_{n})'$ be the equivalized disposable income with $x_{1} \leq \ldots \leq x_{n}$ and let $\boldsymbol{w} := (w_{i}, \ldots, w_{n})'$ be the corresponding personal sample weights, where $n$ denotes the number of observations. \subsection{Quintile share ratio (QSR)} \label{sec:QSR} The income \emph{quintile share ratio} (QSR) is defined as the ratio of the sum of the equivalized disposable income received by the 20\% of the population with the highest equivalized disposable income to that received by the 20\% of the population with the lowest equivalized disposable income \citep{EU-SILC04, EU-SILC09}. For the estimation of the quintile share ratio from a sample, let $\hat{q}_{0.2}$ and $\hat{q}_{0.8}$ denote the weighted 20\% and 80\% quantiles, respectively. With $0 \leq p \leq 1$, these weighted quantiles are given by \begin{equation} \label{eq:wq} \hat{q}_{p} = \hat{q}_{p} (\boldsymbol{x}, \boldsymbol{w}) := \begin{cases} \frac{1}{2} (x_{j} + x_{j+1}), & \quad \text{if } \sum_{i=1}^{j} w_{i} = p \sum_{i=1}^{n} w_{i}, \\ x_{j+1}, & \quad \text{if } \sum_{i=1}^{j} w_{i} < p \sum_{i=1}^{n} w_{i} < \sum_{i=1}^{j+1} w_{i}. \end{cases} \end{equation} %See also vignette \code{laeken-standard} \citep{templ11a} for the computation %of these quantiles with package \pkg{laeken}. Using index sets \mbox{$I_{\leq \hat{q}_{0.2}} := \{ i \in \{ 1, \ldots, n \} : x_{i} \leq \hat{q}_{0.2} \}$} and \mbox{$I_{> \hat{q}_{0.8}} := \{ i \in \{ 1, \ldots, n \} : x_{i} > \hat{q}_{0.8} \}$}, the quintile share ratio is estimated by \begin{equation} \widehat{QSR} := \frac{\sum_{i \in I_{> \hat{q}_{0.8}}} w_{i} x_{i}}{\sum_{i \in I_{\leq \hat{q}_{0.2}}} w_{i} x_{i}}. \end{equation} With package \pkg{laeken}, the quintile share ratio can be estimated using the function \code{qsr()}. Sample weights can thereby be supplied via the \code{weights} argument. <<>>= qsr("eqIncome", weights = "rb050", data = eusilc) @ \subsection{Gini coefficient} \label{sec:Gini} The \emph{Gini coefficient} is defined as the relationship of cumulative shares of the population arranged according to the level of equivalized disposable income, to the cumulative share of the equivalized total disposable income received by them \citep{EU-SILC04, EU-SILC09}. For the estimation of the Gini coefficient from a sample, the sample weights need to be taken into account. In mathematical terms, the Gini coefficient is estimated by \begin{equation} \widehat{Gini} := 100 \left[ \frac{2 \sum_{i=1}^{n} \left( w_{i} x_{i} \sum_{j=1}^{i} w_{j} \right) - \sum_{i=1}^{n} w_{i}^{\phantom{i}2} x_{i}}{\left( \sum_{i=1}^{n} w_{i} \right) \sum_{i=1}^{n} \left(w_{i} x_{i} \right)} - 1 \right]. \end{equation} The function \code{gini()} is available in \pkg{laeken} to estimate the Gini coefficient. As before, sample weights can be specified with the \code{weights} argument. <<>>= gini("eqIncome", weights = "rb050", data = eusilc) @ % ------------------- % Pareto distribution % ------------------- \section{The Pareto distribution} \label{sec:Pareto} The \emph{Pareto distribution} is well studied in the literature and is defined in terms of its cumulative distribution function \begin{equation} \label{eq:CDF} F_{\theta}(x) = 1 - \left( \frac{x}{x_{0}} \right) ^{-\theta}, \qquad x \geq x_{0}, \end{equation} where $x_{0} > 0$ is the scale parameter and $\theta > 0$ is the shape parameter \citep{kleiber03}. Furthermore, its density function is given by \begin{equation} f_{\theta}(x) = \frac{\theta x_{0}^{\theta}}{x^{\theta + 1}}, \qquad x \geq x_{0}. \end{equation} Figure~\ref{fig:Pareto} visualizes the Pareto probability density function with scale parameter $x_{0} = 1$ and different values of the shape parameter $\theta$. Clearly, the Pareto distribution is a highly right-skewed distribution with a heavy tail. It is therefore reasonable to assume that a random variable following a Pareto distribution contains extreme values. The effect of changing the shape parameter $\theta$ is visible in the probability mass at the scale parameter $x_{0}$: the higher $\theta$, the higher the probability mass at $x_{0}$. <>= x <- seq(1, 6, length.out=1000) dpareto <- function(x, x0 = 1, theta = 1) theta*x0^theta / x^(theta+1) y1 <- dpareto(x, theta=1) y2 <- dpareto(x, theta=2) y3 <- dpareto(x, theta=3) @ \begin{figure} \begin{center} <>= par(mar = c(4, 4, 0.5, 0.5) + 0.1) plot(x, y3, type = "l", lty = 3, ylab = "f(x)", xlim = c(0.75, 6), panel.first = { abline(h = 0, col = grey(0.75)) abline(v = 1, col = grey(0.75)) }) lines(x, y2, lty = 2) lines(x, y1, lty = 1) leg <- expression(paste(theta, " = 1"), paste(theta, " = 2"), paste(theta, " = 3")) legend("topright", legend = leg, lty = 1:3) @ \caption{Pareto probability density functions with parameters $x_{0} = 1$ and $\theta = 1, 2, 3$.} \label{fig:Pareto} \end{center} \end{figure} In Pareto tail modeling, the cumulative distribution function on the whole range of $x$ is modeled as \begin{equation} \label{eq:tail} F(x) = \left\{ \begin{array}{ll} G(x), & \quad \text{if } x \leq x_{0}, \\ G(x_{0}) + (1 - G(x_{0})) F_{\theta}(x), & \quad \text{if } x > x_{0}, \end{array} \right. \end{equation} where $G$ is an unknown distribution function \citep{dupuis06}. Let $n$ be the number of observations and let $\boldsymbol{x} = (x_{1}, \ldots, x_{n})'$ denote the observed values with $x_{1} \leq \ldots \leq x_{n}$. In addition, let $k$ be the number of observations to be used for tail modeling. In this scenario, the threshold $x_{0}$ is estimated by % Let $k$ be the number of observations to be used for tail modeling and let % $x_{(1)} \leq \ldots \leq x_{(n)}$, denote the sorted observations. In this % scenario, the threshold $x_{0}$ is estimated by \begin{equation} \hat{x}_{0} := x_{n-k}. \end{equation} If an estimate $\hat{x}_{0}$ for the scale parameter of the Pareto distribution has been obtained, $k$ is given by the number of observations larger than $\hat{x}_{0}$. Thus estimating $x_{0}$ and $k$ directly corresponds with each other. In the remainder of this package vignette, the equivalized disposable income of the EU-SILC example data is of main interest. Consequently, the Pareto distribution will be modeled at the household level rather than the individual level. Moreover, the focus of this vignette is on robust estimation of the social exclusion indicators. Hence the equivalized disposable income of the household with the largest income is replaced by a large outlier. <<>>= hID <- eusilc$db030[which.max(eusilc$eqIncome)] eusilc[eusilc$db030 == hID, "eqIncome"] <- 10000000 @ Since the aim is to model a Pareto distribution at the household level, the following command creates a data set that contains only the equivalized disposable income and the sample weights on the household level. This data set will be used in Sections~\ref{sec:threshold} and~\ref{sec:shape} to estimate the parameters of the Pareto distribution. <<>>= eusilcH <- eusilc[!duplicated(eusilc$db030), c("eqIncome", "db090")] @ % --------- % threshold % --------- \section{Finding the threshold} \label{sec:threshold} The aim of the methods presented in this sections is to find the threshold $x_{0}$ for modeling the Pareto distribution. Several methods for the estimation of the threshold $x_{0}$ or the number of observations $k$ in the tail have been proposed in the literature, but those proposals typically do not consider sample weights. \citet{beirlant96a, beirlant96b} developed a procedure that analytically determines the optimal choice of $k$ for the Hill estimator of the shape parameter \citep[see also Section~\ref{sec:Hill} of this paper]{hill75} by minimizing the asymptotic mean squared error (AMSE). In package \pkg{laeken}, this approach is implemented in the function \code{minAMSE()}. However, the procedure is designed for the non-robust Hill estimator and is therefore not further discussed in this paper. Furthermore, \citet{danielsson01} proposed a bootstrap method to find the optimal $k$ for the Hill estimator with respect to the AMSE, which has less analytical requirements than the approach by \citet{beirlant96a, beirlant96b}. Please note that this method is not robust either and that it is currently not available in package \pkg{laeken}. A robust prediction error criterion for choosing the number of observations $k$ in the tail and estimating the shape parameter $\theta$ was developed by \citet{dupuis06}. Nevertheless, our implementation of this robust criterion was unstable and is therefore not included in \pkg{laeken}. In any case, \citet{holzer09} concludes that graphical methods for finding the threshold outperform those analytical approaches in the case of EU-SILC data. While this section is thus focused graphical methods, a simple rule of thumb designed specifically for the equivalized disposable income in EU-SILC data is described in the following as well. \subsection{Van Kerm's rule of thumb} \label{sec:vanKerm} \citet{vankerm07} presented a formula that is more of a rule of thumb for the threshold of the equivalized disposable income in EU-SILC data. Is is given by \begin{equation} \hat{x}_{0} := \min(\max(2.5\bar{x}, q_{0.98}), q_{0.97}), \end{equation} where $\bar{x}$ is the weighted mean, and $q_{0.98}$ and $q_{0.97}$ are weighted quantiles as defined in Equation~(\ref{eq:wq}). In package \pkg{laeken}, the function \code{paretoScale()} provides functionality for computing the threshold with van Kerm's rule of thumb. The argument \code{w} is available to supply sample weights. %In the example below, the household IDs are supplied via the argument %\code{groups} to estimate the threshold on the houshold level rather than the %personal level. %<<>>= %paretoScale(eusilc$eqIncome, eusilc$db090, groups = eusilc$db030) %@ <<>>= ts <- paretoScale(eusilcH$eqIncome, w = eusilcH$db090) ts @ It should be noted that the function returns an object of class \code{"paretoScale"}, which consists of a component \code{x0} for the threshold (scale parameter) and a component \code{k} for the number of observations in the tail of the distribution, i.e., that are larger than the threshold. \subsection{Pareto quantile plot} The \emph{Pareto quantile plot} is a graphical method for inspecting the parameters of a Pareto distribution. For the case without sample weights, it is described in detail in \citet{beirlant96a}. If the Pareto model holds, there exists a linear relationship between the lograrithms of the observed values and the quantiles of the standard exponential distribution, since the logarithm of a Pareto distributed random variable follows an exponential distribution. Hence the logarithms of the observed values, $\log (x_{i})$, $i = 1, \ldots, n$, are plotted against the theoretical quantiles. In the case without sample weights, the theoretical quantiles of the standard exponential distribution are given by \begin{equation} \label{eq:quantiles} -\log \left( 1 - \frac{i}{n+1} \right), \qquad i = 1, \ldots, n, \end{equation} i.e., by dividing the range into $n + 1$ equally sized subsets and using the resulting $n$ inner gridpoints as probabilities for the quantiles. If the data contain sample weights, the range of the exponential distribution needs to be divided according to the weights of the $n$ observations. The Pareto quantile plot is thus generalized by using the theoretical quantiles \begin{equation} -\log \left( 1 - \frac{\sum_{j=1}^{i} w_{j}}{\sum_{j=1}^{n} w_{j}} \frac{n}{n+1} \right), \qquad i = 1, \ldots, n, \end{equation} where the correction factor $\frac{n}{n+1}$ ensures that the quantiles reduce to (\ref{eq:quantiles}) if all sample weights are equal. If the tail of the data follows a Pareto distribution, those observations form almost a straight line. The leftmost point of a fitted line can thus be used as an estimate of the threshold $x_{0}$, the scale parameter. All values starting from the point after the threshold may be modeled by a Pareto distribution, but this point cannot be determined exactly. Furthermore, the slope of the fitted line is in turn an estimate of $\frac{1}{\theta}$, the reciprocal of the shape parameter. Figure~\ref{fig:ParetoQuantile} displays the Pareto quantile plot for the example data \code{eusilc} on the household level with the largest observation replaced by an outlier. The plot is generated using the function \code{paretoQPlot()}, which allows to supply sample weights via the argument \code{w}. In addition, the threshold can be selected interactively by clicking on a data point. Information on the selected threshold is then printed on the \proglang{R} console. When the interactive selection is terminated, which is typically done by a secondary mouse click, the selected threshold is returned as an object of class \code{"paretoScale"}. Another advantage of the Pareto quantile plot is also illustrated in Figure~\ref{fig:ParetoQuantile}. Nonrepresentative outliers such as the large income introduced into the example data in Section~\ref{sec:Pareto}, i.e., extreme observations in the upper tail that deviate from the Pareto model, are clearly visible. \begin{figure} \begin{center} \setkeys{Gin}{width=.75\textwidth} <>= paretoQPlot(eusilcH$eqIncome, w = eusilcH$db090) @ \caption{Pareto Quantile plot for the example data \code{eusilc} on the household level with the largest observation replaced by an outlier.} \label{fig:ParetoQuantile} \end{center} \end{figure} \subsection{Mean excess plot} The \emph{mean excess plot} is another graphical method for inspecting the threshold for Pareto tail modeling, but it does not provide information on the shape parameter. It is based on the excess function \begin{equation} \label{eq:excess} e(x_{0}) := \mathbb{E}(x - x_{0}|x > x_{0}), \qquad x_{0} \geq 0. \end{equation} A detailed description for the case without sample weights can be found in \citet{borkovec00}. For the following definition of the mean excess plot, keep in mind that the observations are sorted such that $x_{1} \leq \ldots \leq x_{n}$. For each observation $x_{i}$, $i = 1, \ldots, \lfloor n-\sqrt{n} \rfloor$, the empirical excess function $e_{n}$ is computed. In the case without sample weights, the expectation in Equation~(\ref{eq:excess}) is replaced by the arithmetic mean, and the empirical excess function is given by \begin{equation} e_{n}(x_{i}) := \frac{1}{n-i} \sum_{j=i+1}^{n} (x_{j} - x_{i}), \qquad i = 1, \ldots, \lfloor n-\sqrt{n} \rfloor. \end{equation} The values of the empirical excess function $e_{n}(x_{i})$ are then plotted against the corresponding $x_{i}$, $i = 1, \ldots, \lfloor n-\sqrt{n} \rfloor$. If sample weights are available in the data, the mean excess plot is simply generalized by using the weighted mean for the empirical excess function: \begin{equation} e_{n}(x_{i}) := \frac{1}{\sum_{j=i+1}^{n} w_{j}} \sum_{j=i+1}^{n} w_{j} (x_{j} - x_{i}), \qquad i = 1, \ldots, \lfloor n-\sqrt{n} \rfloor. \end{equation} If the tail of the data follows a Pareto distribution, those observations show a positive linear trend. The leftmost point of a fitted line can thus be used as an estimate of the threshold $x_{0}$, the scale parameter. As for the Pareto quantile plot, a disadvantage of the mean excess plot is that the threshold cannot be determined exactly. \begin{figure} \begin{center} \setkeys{Gin}{width=.75\textwidth} <>= meanExcessPlot(eusilcH$eqIncome, w = eusilcH$db090) @ \caption{Mean excess plot for the example data \code{eusilc} on the household level with the largest observation replaced by an outlier.} \label{fig:meanExcess} \end{center} \end{figure} Figure~\ref{fig:meanExcess} shows the mean excess plot for the example data \code{eusilc} on the household level with the largest observation replaced by an outlier. The function \code{meanExcessPlot()} is thereby used to produce the plot. Sample weights can be supplied via the argument \code{w}. Interactive selection of the threshold works just like for the Pareto quantile plot. Again, the selected threshold is returned as an object of class \code{"paretoScale"}. % --------------- % shape parameter % --------------- \section{Estimation of the shape parameter} \label{sec:shape} This section is focused on methods for estimating the shape parameter $\theta$ once the threshold $x_0$ is fixed. It should be noted that none of the original proposals takes sample weights into account. Most estimators presented in the following were therefore adjusted for the case of sample weights. \subsection{Hill estimator} \label{sec:Hill} The maximum likelihood estimator for the shape parameter of the Pareto distribution was introduced by \citet{hill75} and is referred to as the \emph{Hill} estimator. If the data do not contain sample weights, it is given by \begin{equation} \label{eq:Hill} \hat{\theta}_{\mathrm{Hill}} = \frac{k}{\sum_{i = 1}^{k} \log x_{n-k+i} - k \log x_{n-k}}. \end{equation} In the case of sample weights, the \emph{weighted Hill} (wHill) estimator is given by generalizing Equation~(\ref{eq:Hill}) to \begin{equation} \label{eq:wHill} \hat{\theta}_{\mathrm{wHill}} = \frac{\sum_{i = 1}^{k} w_{n-k+i}}{\sum_{i = 1}^{k} w_{n-k+i} \left( \log x_{n-k+i} - \log x_{n-k} \right)} . \end{equation} Package \pkg{laeken} provides the function \code{thetaHill()} to compute the Hill estimator. It requires to specify either the number of observations in the tail via the argument \code{k}, or the threshold via the argument \code{x0}. Furthermore, the argument \code{w} can be used to supply sample weights. In the following example, the shape parameter is estimated using the largest observations (first command) and the threshold (second command) as computed with van Kerm's rule of thumb in Section~\ref{sec:vanKerm}. <<>>= thetaHill(eusilcH$eqIncome, k = ts$k, w = eusilcH$db090) thetaHill(eusilcH$eqIncome, x0 = ts$x0, w = eusilcH$db090) @ \subsection{Weighted maximum likelihood estimator} The \emph{weighted maximum likelihood} (WML) estimator \citep{dupuis02, dupuis06} falls into the class of M-estimators and is given by the solution $\hat{\theta}$ of \begin{equation} \sum_{i = 1}^{k} \mathrm{\Psi}(x_{n-k+i}, \theta) = 0 \end{equation} with \begin{equation} \mathrm{\Psi}(x, \theta) := u(x, \theta) \frac{\partial}{\partial \theta} \log f(x, \theta) = u(x, \theta) \left( \frac{1}{\theta} - \log \frac{x}{x_{0}} \right), \end{equation} where $u(x, \theta)$ is a weight function with values in $[0,1]$. In the implementation in package \pkg{laeken}, a Huber type weight function is used by default, as proposed by \citet{dupuis06}. Let the logarithms of the relative excesses be denoted by \begin{equation} z_{i} := \log \left( \frac{x_{n-k+i}}{x_{n-k}} \right), \qquad i = 1, \ldots, k. \end{equation} In the Pareto model, these can be predicted by \begin{equation} \hat{z}_{i} := -\frac{1}{\theta} \log \left( \frac{k+1-i}{k+1} \right), \qquad i = 1, \ldots, k. \end{equation} The variance of $z_{i}$ is given by \begin{equation} \sigma_{i}^{\phantom{i}2} := \sum_{j = 1}^{i} \frac{1}{\theta^{2} (k-i+j)^{2}}, \qquad i = 1, \ldots, k. \end{equation} Using the standardized residuals \begin{equation} r_{i} := \frac{z_{i} - \hat{z}_{i}}{\sigma_{i}}, \end{equation} the Huber type weight function with tuning constant $c$ is defined as \begin{equation} u(x_{n-k+i}, \theta) := \left\{ \begin{array}{cl} 1, & \quad \text{if } |r_{i}| \leq c, \\ \frac{c}{|r_{i}|}, & \quad \text{if } |r_{i}| > c. \end{array} \right. \end{equation} For this choice of weight function, the bias of $\hat{\theta}$ is approximated by \begin{equation} \hat{B}(\hat{\theta}) = - \frac{\sum_{i=1}^{k} \left( u_{i} \frac{\partial}{\partial \theta} \log f_{i} \right) \vert_{\hat{\theta}} \left( F_{\hat{\theta}}(x_{n-k+i}) - F_{\hat{\theta}}(x_{n-k+i-1}) \right)}{\sum_{i=1}^{k} \left( \frac{\partial}{\partial \theta} u_{i} \frac{\partial}{\partial \theta} \log f_{i} + u_{i} \frac{\partial^{2}}{\partial \theta^{2}} \log f_{i} \right) \vert_{\hat{\theta}} \left( F_{\hat{\theta}}(x_{n-k+i}) - F_{\hat{\theta}}(x_{n-k+i-1}) \right)}, \end{equation} where $u_{i} := u(x_{n-k+i}, \theta)$ and $f_{i} := f(x_{n-k+i}, \theta)$. This term is used to obtain a bias-corrected estimator \begin{equation} \tilde{\theta} := \hat{\theta} - \hat{B}(\hat{\theta}). \end{equation} For details and proofs of the above statements, as well as for information on a probability-based weight function $u(x, \theta)$, the reader is referred to \citet{dupuis02} and \citet{dupuis06}. However, note the WML estimator does not consider sample weights. An adjustment of the estimator to take sample weights into account is currently not available due to its complexity. For sampling designs that lead to equal sample weights, the WML estimator may still be useful, though. The function \code{thetaWML()} is available in \pkg{laeken} to compute the WML estimator. Again, either the argument \code{k} or \code{x0} needs to be used to specify the number of observations in the tail or the threshold. Since the sample weights in the example data are not equal, the following example is only included to demonstrate the use of the function. <<>>= thetaWML(eusilcH$eqIncome, k = ts$k) thetaWML(eusilcH$eqIncome, x0 = ts$x0) @ \subsection{Integrated squared error estimator} For the \emph{integrated squared error} (ISE) estimator \citep{vandewalle07}, the Pareto distribution is modeled in terms of the relative excesses \begin{equation} y_{i} := \frac{x_{n-k+i}}{x_{n-k}}, \qquad i = 1, \ldots, k. \end{equation} The density function of the Pareto distribution for the relative excesses is approximated by \begin{equation} f_{\theta}(y) = \theta y^{-(1+\theta)}. \end{equation} The ISE estimator is then given by minimizing the integrated squared error criterion \citep{terrell90}: \begin{equation} \hat{\theta} = \arg \min_{\theta} \left[ \int f_{\theta}^{2}(y) dy - 2 \mathbb{E}(f_{\theta}(Y)) \right] . \end{equation} If there are no sample weights in the data, the mean is used as an unbiased estimator of $\mathbb{E}(f_{\theta}(Y))$ in order to obtain the ISE estimate \begin{equation} \label{eq:ISE} \hat{\theta}_{\mathrm{ISE}} = \arg \min_{\theta} \left[ \int f_{\theta}^{2}(y) dy - \frac{2}{k} \sum_{i=1}^{k} f_{\theta}(y_{i}) \right] . \end{equation} See \citet{vandewalle07} for more information on the ISE estimator for the case without sample weights. If sample weights are available in the data, the mean in Equation~(\ref{eq:ISE}) is simply replaced by a weighted mean to obtain the \emph{weighted integrated squared error} (wISE) estimator: \begin{equation} \label{eq:wISE} \hat{\theta}_{\mathrm{wISE}} = \arg \min_{\theta} \left[ \int f_{\theta}^{2}(y) dy - \frac{2}{\sum_{i=1}^{k} w_{n-k+i}} \sum_{i=1}^{k} w_{n-k+i} f_{\theta}(y_{i}) \right] . \end{equation} With package \pkg{laeken}, the ISE estimator can be computed using the function \code{thetaISE()}. The arguments \code{k} and \code{x0} are available to specify either the number of observations in the tail or the threshold, and sample weights can be supplied via the argument \code{w}. <<>>= thetaISE(eusilcH$eqIncome, k = ts$k, w = eusilcH$db090) thetaISE(eusilcH$eqIncome, x0 = ts$x0, w = eusilcH$db090) @ \subsection{Partial density component estimator} For the \emph{partial density component} (PDC) estimator \cite{vandewalle07} minimizes the integrated squared error criterion using an incomplete density mixture model $u f_{\theta}$. If the data do not contain sample weights, the PDC estimator in is thus given by \begin{equation} \label{eq:PDC} \hat{\theta}_{\mathrm{PDC}} = \arg \min_{\theta} \left[ u^{2} \int f_{\theta}^{2}(y) dy - \frac{2 u}{k} \sum_{i = 1}^{k} f_{\theta}(y_{i}) \right]. \end{equation} The parameter $u$ can be interpreted as a measure of the uncontaminated part of the sample and is estimated by \begin{equation} \label{eq:u} \hat{u} = \frac{\frac{1}{k} \sum_{i = 1}^{k} f_{\hat{\theta}}(y_{i})}{\int f_{\hat{\theta}}^{2}(y) dy}. \end{equation} See \cite{vandewalle07} and references therein for more information on the PDC estimator for the case without sample weights. Taking sample weights into account, the \emph{weighted partial density component} (wPDC) estimator is obtained by generalizing Equations~(\ref{eq:PDC}) and~(\ref{eq:u}) to \begin{align} \label{eq:wPDC} \hat{\theta}_{\mathrm{wPDC}} =& \arg \min_{\theta} \left[ u^{2} \int f_{\theta}^{2}(y) dy - \frac{2u}{\sum_{i=1}^{k} w_{n-k+i}} \sum_{i = 1}^{k} w_{n-k+i} f_{\theta}(y_{i}) \right] , \\ \hat{u} =& \frac{\frac{1}{\sum_{i=1}^{k} w_{n-k+i}} \sum_{i = 1}^{k} w_{n-k+i} f_{\hat{\theta}}(y_{i})}{\int f_{\hat{\theta}}^{2}(y) dy} . \end{align} The function \code{thetaPDC()} is implemented in package \pkg{laeken} to compute the PDC estimator. As for the other estimators, it is necessary to specify either the number of observations in the tail via the argument \code{k}, or the threshold via the argument \code{x0}. Sample weights can be supplied using the argument \code{w}. <<>>= thetaPDC(eusilcH$eqIncome, k = ts$k, w = eusilcH$db090) thetaPDC(eusilcH$eqIncome, x0 = ts$x0, w = eusilcH$db090) @ % ---------------------------- % estimation of the indicators % ---------------------------- \section{Estimation of the indicators using Pareto tail modeling} \label{sec:fit} Three approaches based on Pareto tail modeling for reducing the influence of outliers on the social exclusion indicators are implemented in the \proglang{R} package \pkg{laeken}: \begin{description} \item[Calibration for nonrepresentative outliers (CN):] Values larger than a certain quantile of the fitted distribution are declared as nonrepresentative outliers. Since these are considered to be unique to the population data, the sample weights of the corresponding observations are set to $1$ and the weights of the remaining observations are adjusted accordingly by calibration. \item[Replacement of nonrepresentative outliers (RN):] Values larger than a certain quantile of the fitted distribution are declared as nonrepresentative outliers. Only these nonrepresentative outliers are replaced by values drawn from the fitted distribution, thereby preserving the order of the original values. \item[Replacement of the tail (RT):] All values above the threshold are replaced by values drawn from the fitted distribution. The order of the original values is preserved. \end{description} An evaluation of the RT approach by means of a simulation study can be found in \citet{alfons10b}. Keep in mind that the largest observation in the example data \code{eusilc} was replaced by a large outlier in Section~\ref{sec:Pareto}. With the following command, the Gini coefficient is estimated according to the Eurostat definition to show that even a single outlier can completely distort the results for the standard estimation (see Section~\ref{sec:Gini} for the original value). <<>>= gini("eqIncome", weights = "rb050", data = eusilc) @ For Pareto tail modeling, the function \code{paretoTail()} is implemented in \pkg{laeken}. It returns an object of class \code{"paretoTail"}, which contains all the necessary information for further analysis using the three approaches described above. Note that the household IDs are supplied via the argument \code{groups} such that the Pareto distribution is fitted on the household level rather than the individual level. In addition, the PDC is used by default to estimate the shape parameter. Other estimators can be specified via the \code{method} argument. <<>>= fit <- paretoTail(eusilc$eqIncome, k = ts$k, w = eusilc$db090, groups = eusilc$db030) @ The function \code{reweightOut()} is available for semiparametric estimation with the CN approach. It returns a vector of the recalibrated weights. In this example, regional information is used as auxiliary variables for calibration. The function \code{calibVars()} thereby transforms a factor into a matrix of binary variables, as required by the calibration function \code{calibWeights()}, which is called internally. These recalibrated weights are then simply used to estimate the Gini coefficient with function \code{gini()}. <<>>= w <- reweightOut(fit, calibVars(eusilc$db040)) gini(eusilc$eqIncome, w) @ For the RN approach, the function \code{replaceOut()} is implemented. Since values are drawn from the fitted distribution to replace the observations flagged as outliers, the seed of the random number generator is set first for reproducibility of the results. The returned vector of incomes is then supplied to \code{gini()} to estimate the Gini coefficient. <<>>= set.seed(1234) eqIncome <- replaceOut(fit) gini(eqIncome, weights = eusilc$rb050) @ Similarly, the function \code{replaceTail()} is available for the RT approach. Again, the seed of the random number generator is set beforehand. <<>>= set.seed(1234) eqIncome <- replaceTail(fit) gini(eqIncome, weights = eusilc$rb050) @ It should be noted that \code{replaceTail()} can also be used for the RN approach by setting the argument \code{all} to \code{FALSE}. In fact, \code{replaceOut(x, ...)} is a simple wrapper for \code{replaceTail(x, all = FALSE, ...)}. In any case, the estimates for the semiparametric approaches based on Pareto tail modeling are very close to the original value before the outlier has been introduced (see Section~\ref{sec:Gini}), whereas the standard estimation is corrupted by the outlier. Furthermore, the estimation of other indicators such as the quintile share ratio (see Section~\ref{sec:QSR}) using the semiparametric approaches is straightforward and hence not shown here. % ----------- % conclusions % ----------- \section{Conclusions} \label{sec:concl} This vignette shows the functionality of package \pkg{laeken} for robust semiparametric estimation of social exclusion indicators based on Pareto tail modeling. Most notably, it demonstrates that the functions are easy to use and that the implementation follows an object-oriented design. While the focus of the paper lies on the use of the package, a mathematical description of the methods is given as well. Furthermore, it is shown that the standard estimation of the inequality indicators can be corrupted by a single outlier, thus underlining the need for robust alternatives. Three approaches for robust semiparametric estimation based on Pareto tail modeling are thereby implemented such that the corresponding functions share a common interface for ease of use. % --------------- % acknowledgments % --------------- \section*{Acknowledgments} This work was partly funded by the European Union (represented by the European Commission) within the 7$^{\mathrm{th}}$ framework programme for research (Theme~8, Socio-Economic Sciences and Humanities, Project AMELI (Advanced Methodology for European Laeken Indicators), Grant Agreement No. 217322). Visit \url{http://ameli.surveystatistics.net} for more information on the project. % ------------ % bibliography % ------------ \bibliographystyle{plainnat} \bibliography{laeken} \end{document} laeken/inst/doc/laeken-variance.pdf0000644000176200001440000030333714127307305016766 0ustar liggesusers%PDF-1.5 % 1 0 obj << /Type /ObjStm /Length 3729 /Filter /FlateDecode /N 72 /First 585 >> stream x[r۶}?_&)ANOg8N|lqɃ"Yr%*Igm(RinFy76PR&L3h,3^1ǬOg;&R&Q0b"cHL&52eZ&,ǵyI}{ T%NY&r޳L1-#vf$AD;lf31f%J0X' s)pJ\2gRs91L ^9@$)ɼK-53 I)/Ra1b8R m0<pԂ`H=x \zxU.ڏ_E9؃x2e1<9{,NjiCPw^NS(GlڳY~ήÏ?Ogg@`zvl1AKl4ᬸ*O2Nj2;)qNg0vI zzECv῀ǫ,/}͞ȱ9ɴ$rqU|x\vUϱg!qT|*R񑊏T|#okIO NA~V M$^'Dxfx|>]̆! ɗqIU Fg6%hOK h(fs@y"~w0-I%Ild1Wԝ]rHuUޤ RCb3kYQ{2]FK.YgcQڀYw) 6°'LEqpZG8:G#;בTt#HD*&R18&c58f-8/Hp!E4 ZcA{`j9"FbH&ǒ8KW'}e|!̢M700S TF@ߐ` Oxx}Vyz_`O{p|΁? gd^n ;HbLOUpB,_h l"j鄟+~G_ $W|h< _b$?e8bN!LͣZC/|>-9^[ICU w&pY<(sQ~\旿ANҦ\4DVQVo'޽ -ue+Y=efY?cT3~ZB!!q2J-ܑRcMϗgZk|Z vQ~Cdrޠc;t0@AG"lE9\WGR~]g,1-0zN%C %{:WU*8ziG[Ქ[]O_ϡa/>ik" {AE%b`rp0pqy>οT9^^gE>z`l0|-c`+x#](@\9Rȿ2B/@XL_AeQO 5/h?|6] ĻhԚiI)fC#Ú5 Zl1 kb.Z߲ Ԧa0ۢH(˥(5X3rnZf+ber =ٌ\'_=zo+C7m+zߐb~CzmeY_FgFB)tݝH6n_Fk*xESM6?Kԡ栯*,Й7NNI;q*fHS[v(6Z_O{sw]Qhv>␭K1&eCް:k/xc^SG3lyӫZmŴ>+wc2TEkmbK;ں8`fҝzDZ5Ň'OԿFmjw וpӽ V^x7S:d ۃ@N]?N)PC.ϚDj4omT*I&iT*[i9)zX^V"6hJ7m)ΉkÖVzhֶ2]okU,XMӚ ͂ tDžZ;U!Vm.8L9!# vD, $opF)< &/B2r`o(}[Oܚ$^mE#*R5[2#6<MuևHN)W v6V#U]WR1U$H%eYWX-&a|"4V]" MV'hx|.1i($KLF& A=`,zw>)m{VcRF%ڹԣP/It)gK@iN!%a#IYsP^$HkLRDN =zIуGnRv#j1w#g/MSjC)VXTچZj6;{gwCpxrUlUnCvB]N))s9vwq[mO*1(ukZR8q QVѩF[uy Qf deȴKK%F"&Ŕ͏hքR}Jhe6DZ/WUWZJ_U5BwUfHrRkz_Sk|%vuN\> stream GPL Ghostscript 9.23 2021-10-06T13:55:49+02:00 2021-10-06T13:55:49+02:00 LaTeX with hyperref package endstream endobj 75 0 obj << /Type /ObjStm /Length 2901 /Filter /FlateDecode /N 72 /First 630 >> stream xZks6Yx;N?Ll*'ML>02ms*KZQN=$%ʉQ$ν,e15sIc L9|ʴ fc^2g-L} 2E'˄2L*LX#LXHY&XLJmXPLI1aҢSLzS) )J!1y NSAbZ̪ŴI,Lxbp"Af:H3JόD" x*Dl*/>Ϫx5:f -Ũr9OKStxCxI IYwJ@ E9ClJ㐣&J+"(.(;ATAtP$)͒T=JRYSO3U%l A1tf. 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(vOM-NzҨX6X'DfwPmNP??)1#E_,'Ǒьgp:Tũ}9 ɝ =9_ oW|S] A ߝnKzV*$'⫟~yL컬o{G\=#WՓ>A72ZigDUy=Yx^~/i1I5ZѡɡQV6O̿L<*ȭ*SJ>)F)ґE7b\2 !+DiZVWQE7*jB{t(0mg膀%&lʈ"7΋YJ gJhK4VAt;Fi0.^ww:YЧ5Ϭx%q=gc8$\b&q=m| _^uchY\8[Lhi#{ێr>pPP?oS=ꍻ+XC ;Ͽʤ*=`1+=n;y)0J+XANt[^֧ڠ>tsV>Z@U˫:Z=5r숳/w{Fٳ US  K\1:Ez{~i@\pJrl#SÍ-;zșe&0DE29WGnT[Qr/s l~c"$78na5-xFɏMEM;yN:K˗{X}آUm| :W $4AvpEihXF\{3hy<% iMfb te%xf}\n2c=ݖI^,' 8t[D1HHh΂z<g%*&~עr`mhvc4rev?X ]YcWQg+G1''JjxY,dʇE{HSY/*~u/ =.gqIE+#! @|tk*mhYy6EO‘kX՘)iajh\4a eQNal괫Tr 2p\ms\ .h9:b`a6<6 .M߭`KM-lG\ P,N*~)(E`I Q>]է /&>h͍y;!r]קuE}RM޵o{aG [;65Wj[ҘϦ{/؁Pr LmZmya7mluF_rY~ٶit6̈́1|81Z|817gpx{ިwc.ԧRMyDק4]vndeo-L@WLꓦ&jb}ғ8|W? ҰOD)\N&:_AQ '4+77㺺c#֪c]E_ln$?ȹ䶙>mSۅPdL QGWYۄy=YնҗΔD񹛬v2eZz *:eA.`麌)IW`18{]2_yo}wbZk7ZPkB6Js7[m2_1Z*`;^qu^w;qFfa;9^QR|_ڧAkίMG): 4cmG*L?!Uvqzan~GbzJ{Hof=@s83Iua%S<8L'1}t~9 tQ] wҎy5]ƷKj+ NJ}g!g/:XJăL{\ʙ)T[ jT͗i/ dKQVYvZ籰Wd6.t7z ʕE|}? ^فh[)ŇΨ_\?p0QkUDfPq^٭x;O˕z_\ p9do:XߨrxLzO9dendstream endobj 176 0 obj << /Filter /FlateDecode /Subtype /Type1C /Length 343 >> stream xcd`ab`ddp 44~H3a!3,l^;nnO}=C9(3=DA#YS\GR17(391O7$#57QOL-Tа())///K-/JQ(,PJ-N-*KMQp+QKMU:UJ;)槤1000103012)ٽgfD+n ~ߍپBߕپ qw1([tᏀ9^;i: r\,y8O00{endstream endobj 177 0 obj << /Type /XRef /Length 155 /Filter /FlateDecode /DecodeParms << /Columns 5 /Predictor 12 >> /W [ 1 3 1 ] /Info 3 0 R /Root 2 0 R /Size 178 /ID [<799221ccea2025f91c5018f65f02ee3c>] >> stream xcb&F~0 $8J??N -w!3(>ŝPXH[ RD1 R!D2 RD |{@lf Dr٧ g)d 6'X<DJHp 9`N0yDʟ72 endstream endobj startxref 99634 %%EOF laeken/inst/doc/laeken-pareto.R0000644000176200001440000001224114127307260016107 0ustar liggesusers### R code from vignette source 'laeken-pareto.Rnw' ################################################### ### code chunk number 1: laeken-pareto.Rnw:74-75 ################################################### options(prompt="R> ") ################################################### ### code chunk number 2: laeken-pareto.Rnw:149-151 (eval = FALSE) ################################################### ## vignette("laeken-standard") ## vignette("laeken-variance") ################################################### ### code chunk number 3: laeken-pareto.Rnw:165-167 ################################################### library("laeken") data("eusilc") ################################################### ### code chunk number 4: laeken-pareto.Rnw:235-236 ################################################### qsr("eqIncome", weights = "rb050", data = eusilc) ################################################### ### code chunk number 5: laeken-pareto.Rnw:260-261 ################################################### gini("eqIncome", weights = "rb050", data = eusilc) ################################################### ### code chunk number 6: laeken-pareto.Rnw:293-298 ################################################### x <- seq(1, 6, length.out=1000) dpareto <- function(x, x0 = 1, theta = 1) theta*x0^theta / x^(theta+1) y1 <- dpareto(x, theta=1) y2 <- dpareto(x, theta=2) y3 <- dpareto(x, theta=3) ################################################### ### code chunk number 7: laeken-pareto.Rnw:303-313 ################################################### par(mar = c(4, 4, 0.5, 0.5) + 0.1) plot(x, y3, type = "l", lty = 3, ylab = "f(x)", xlim = c(0.75, 6), panel.first = { abline(h = 0, col = grey(0.75)) abline(v = 1, col = grey(0.75)) }) lines(x, y2, lty = 2) lines(x, y1, lty = 1) leg <- expression(paste(theta, " = 1"), paste(theta, " = 2"), paste(theta, " = 3")) legend("topright", legend = leg, lty = 1:3) ################################################### ### code chunk number 8: laeken-pareto.Rnw:355-357 ################################################### hID <- eusilc$db030[which.max(eusilc$eqIncome)] eusilc[eusilc$db030 == hID, "eqIncome"] <- 10000000 ################################################### ### code chunk number 9: laeken-pareto.Rnw:366-367 ################################################### eusilcH <- eusilc[!duplicated(eusilc$db030), c("eqIncome", "db090")] ################################################### ### code chunk number 10: laeken-pareto.Rnw:424-426 ################################################### ts <- paretoScale(eusilcH$eqIncome, w = eusilcH$db090) ts ################################################### ### code chunk number 11: laeken-pareto.Rnw:491-492 ################################################### paretoQPlot(eusilcH$eqIncome, w = eusilcH$db090) ################################################### ### code chunk number 12: laeken-pareto.Rnw:539-540 ################################################### meanExcessPlot(eusilcH$eqIncome, w = eusilcH$db090) ################################################### ### code chunk number 13: laeken-pareto.Rnw:592-594 ################################################### thetaHill(eusilcH$eqIncome, k = ts$k, w = eusilcH$db090) thetaHill(eusilcH$eqIncome, x0 = ts$x0, w = eusilcH$db090) ################################################### ### code chunk number 14: laeken-pareto.Rnw:671-673 ################################################### thetaWML(eusilcH$eqIncome, k = ts$k) thetaWML(eusilcH$eqIncome, x0 = ts$x0) ################################################### ### code chunk number 15: laeken-pareto.Rnw:718-720 ################################################### thetaISE(eusilcH$eqIncome, k = ts$k, w = eusilcH$db090) thetaISE(eusilcH$eqIncome, x0 = ts$x0, w = eusilcH$db090) ################################################### ### code chunk number 16: laeken-pareto.Rnw:760-762 ################################################### thetaPDC(eusilcH$eqIncome, k = ts$k, w = eusilcH$db090) thetaPDC(eusilcH$eqIncome, x0 = ts$x0, w = eusilcH$db090) ################################################### ### code chunk number 17: laeken-pareto.Rnw:800-801 ################################################### gini("eqIncome", weights = "rb050", data = eusilc) ################################################### ### code chunk number 18: laeken-pareto.Rnw:812-814 ################################################### fit <- paretoTail(eusilc$eqIncome, k = ts$k, w = eusilc$db090, groups = eusilc$db030) ################################################### ### code chunk number 19: laeken-pareto.Rnw:824-826 ################################################### w <- reweightOut(fit, calibVars(eusilc$db040)) gini(eusilc$eqIncome, w) ################################################### ### code chunk number 20: laeken-pareto.Rnw:834-837 ################################################### set.seed(1234) eqIncome <- replaceOut(fit) gini(eqIncome, weights = eusilc$rb050) ################################################### ### code chunk number 21: laeken-pareto.Rnw:842-845 ################################################### set.seed(1234) eqIncome <- replaceTail(fit) gini(eqIncome, weights = eusilc$rb050) laeken/inst/doc/laeken-intro.R0000644000176200001440000001515714127307252015762 0ustar liggesusers### R code from vignette source 'laeken-intro.Rnw' ################################################### ### code chunk number 1: laeken-intro.Rnw:107-109 ################################################### options(prompt = "R> ", continue = "+ ", width = 72, useFancyQuotes = FALSE) library("laeken") ################################################### ### code chunk number 2: laeken-intro.Rnw:164-165 (eval = FALSE) ################################################### ## vignette(package="laeken") ################################################### ### code chunk number 3: laeken-intro.Rnw:244-246 ################################################### data("eusilc") head(eusilc[, 1:10], 3) ################################################### ### code chunk number 4: laeken-intro.Rnw:269-271 ################################################### data("ses") head(ses[, 1:7], 3) ################################################### ### code chunk number 5: laeken-intro.Rnw:392-393 ################################################### arpr("eqIncome", weights = "rb050", data = eusilc) ################################################### ### code chunk number 6: laeken-intro.Rnw:408-409 ################################################### arpr("eqIncome", weights = "rb050", p = c(0.4, 0.5, 0.7), data = eusilc) ################################################### ### code chunk number 7: laeken-intro.Rnw:431-432 ################################################### qsr("eqIncome", weights = "rb050", data = eusilc) ################################################### ### code chunk number 8: laeken-intro.Rnw:462-463 ################################################### rmpg("eqIncome", weights = "rb050", data = eusilc) ################################################### ### code chunk number 9: laeken-intro.Rnw:483-484 ################################################### gini("eqIncome", weights = "rb050", data = eusilc) ################################################### ### code chunk number 10: laeken-intro.Rnw:526-527 ################################################### gpg("earningsHour", gender = "sex", weigths = "weights", data = ses) ################################################### ### code chunk number 11: laeken-intro.Rnw:550-552 ################################################### gpg("earningsHour", gender = "sex", weigths = "weights", data = ses, method = "median") ################################################### ### code chunk number 12: laeken-intro.Rnw:589-590 ################################################### gini("eqIncome", weights = "rb050", data = eusilc) ################################################### ### code chunk number 13: laeken-intro.Rnw:593-594 ################################################### gini(eusilc$eqIncome, weights = eusilc$rb050) ################################################### ### code chunk number 14: laeken-intro.Rnw:670-672 ################################################### a <- arpr("eqIncome", weights = "rb050", breakdown = "db040", data = eusilc) a ################################################### ### code chunk number 15: laeken-intro.Rnw:686-687 ################################################### subset(a, strata = c("Lower Austria", "Vienna")) ################################################### ### code chunk number 16: laeken-intro.Rnw:755-758 ################################################### hID <- eusilc$db030[which.max(eusilc$eqIncome)] eqIncomeOut <- eusilc$eqIncome eqIncomeOut[eusilc$db030 == hID] <- 10000000 ################################################### ### code chunk number 17: laeken-intro.Rnw:765-767 ################################################### keep <- !duplicated(eusilc$db030) eusilcH <- data.frame(eqIncome=eqIncomeOut, db090=eusilc$db090)[keep,] ################################################### ### code chunk number 18: laeken-intro.Rnw:796-797 ################################################### paretoQPlot(eusilcH$eqIncome, w = eusilcH$db090) ################################################### ### code chunk number 19: laeken-intro.Rnw:852-854 ################################################### ts <- paretoScale(eusilcH$eqIncome, w = eusilcH$db090) ts ################################################### ### code chunk number 20: laeken-intro.Rnw:919-921 ################################################### thetaISE(eusilcH$eqIncome, k = ts$k, w = eusilcH$db090) thetaISE(eusilcH$eqIncome, x0 = ts$x0, w = eusilcH$db090) ################################################### ### code chunk number 21: laeken-intro.Rnw:953-955 ################################################### thetaPDC(eusilcH$eqIncome, k = ts$k, w = eusilcH$db090) thetaPDC(eusilcH$eqIncome, x0 = ts$x0, w = eusilcH$db090) ################################################### ### code chunk number 22: laeken-intro.Rnw:1009-1011 ################################################### fit <- paretoTail(eqIncomeOut, k = ts$k, w = eusilc$db090, groups = eusilc$db030) ################################################### ### code chunk number 23: laeken-intro.Rnw:1023-1024 ################################################### plot(fit) ################################################### ### code chunk number 24: laeken-intro.Rnw:1050-1052 ################################################### w <- reweightOut(fit, calibVars(eusilc$db040)) gini(eqIncomeOut, w) ################################################### ### code chunk number 25: laeken-intro.Rnw:1060-1063 ################################################### set.seed(123) eqIncomeRN <- replaceOut(fit) gini(eqIncomeRN, weights = eusilc$rb050) ################################################### ### code chunk number 26: laeken-intro.Rnw:1069-1071 ################################################### eqIncomeSN <- shrinkOut(fit) gini(eqIncomeSN, weights = eusilc$rb050) ################################################### ### code chunk number 27: laeken-intro.Rnw:1078-1079 ################################################### gini(eqIncomeOut, weights = eusilc$rb050) ################################################### ### code chunk number 28: laeken-intro.Rnw:1152-1154 ################################################### arpr("eqIncome", weights = "rb050", design = "db040", cluster = "db030", data = eusilc, var = "bootstrap", bootType = "naive", seed = 1234) ################################################### ### code chunk number 29: laeken-intro.Rnw:1202-1205 ################################################### aux <- cbind(calibVars(eusilc$db040), calibVars(eusilc$rb090)) arpr("eqIncome", weights = "rb050", design = "db040", cluster = "db030", data = eusilc, var = "bootstrap", X = aux, seed = 1234) laeken/inst/doc/laeken-standard.R0000644000176200001440000001312214127307266016422 0ustar liggesusers### R code from vignette source 'laeken-standard.Rnw' ################################################### ### code chunk number 1: laeken-standard.Rnw:52-53 ################################################### options(prompt="R> ") ################################################### ### code chunk number 2: laeken-standard.Rnw:135-137 (eval = FALSE) ################################################### ## vignette("laeken-pareto") ## vignette("laeken-variance") ################################################### ### code chunk number 3: laeken-standard.Rnw:150-153 ################################################### library("laeken") data("eusilc") head(eusilc, 3) ################################################### ### code chunk number 4: laeken-standard.Rnw:252-253 ################################################### methods(class="indicator") ################################################### ### code chunk number 5: laeken-standard.Rnw:331-333 ################################################### eusilc$eqSS <- eqSS("db030", "age", data=eusilc) head(eusilc[,c("db030", "age", "eqSS")], 8) ################################################### ### code chunk number 6: laeken-standard.Rnw:345-352 ################################################### hplus <- c("hy040n", "hy050n", "hy070n", "hy080n", "hy090n", "hy110n") hminus <- c("hy130n", "hy145n") pplus <- c("py010n", "py050n", "py090n", "py100n", "py110n", "py120n", "py130n", "py140n") eusilc$eqIncome <- eqInc("db030", hplus, hminus, pplus, character(), "eqSS", data=eusilc) head(eusilc[,c("db030", "eqSS", "eqIncome")], 8) ################################################### ### code chunk number 7: laeken-standard.Rnw:408-410 ################################################### weightedQuantile(eusilc$eqIncome, eusilc$rb050, probs = c(0.2, 0.5, 0.8)) ################################################### ### code chunk number 8: laeken-standard.Rnw:416-417 ################################################### weightedMedian(eusilc$eqIncome, eusilc$rb050) ################################################### ### code chunk number 9: laeken-standard.Rnw:429-431 ################################################### incMedian("eqIncome", weights = "rb050", data = eusilc) incQuintile("eqIncome", weights = "rb050", k = c(1, 4), data = eusilc) ################################################### ### code chunk number 10: laeken-standard.Rnw:521-523 ################################################### arpt("eqIncome", weights = "rb050", data = eusilc) arpr("eqIncome", weights = "rb050", data = eusilc) ################################################### ### code chunk number 11: laeken-standard.Rnw:532-535 ################################################### arpr("eqIncome", weights = "rb050", p = 0.4, data = eusilc) arpr("eqIncome", weights = "rb050", p = 0.5, data = eusilc) arpr("eqIncome", weights = "rb050", p = 0.7, data = eusilc) ################################################### ### code chunk number 12: laeken-standard.Rnw:543-544 ################################################### arpr("eqIncome", weights = "rb050", breakdown = "db040", data = eusilc) ################################################### ### code chunk number 13: laeken-standard.Rnw:552-555 ################################################### ageCat <- cut(eusilc$age, c(-1, 16, 25, 50, 65, Inf), right=FALSE) eusilc$breakdown <- paste(ageCat, eusilc$rb090, sep=":") arpr("eqIncome", weights = "rb050", breakdown = "breakdown", data = eusilc) ################################################### ### code chunk number 14: laeken-standard.Rnw:591-592 ################################################### qsr("eqIncome", weights = "rb050", data = eusilc) ################################################### ### code chunk number 15: laeken-standard.Rnw:598-599 ################################################### qsr("eqIncome", weights = "rb050", breakdown = "db040", data = eusilc) ################################################### ### code chunk number 16: laeken-standard.Rnw:650-651 ################################################### rmpg("eqIncome", weights = "rb050", data = eusilc) ################################################### ### code chunk number 17: laeken-standard.Rnw:657-658 ################################################### rmpg("eqIncome", weights = "rb050", breakdown = "db040", data = eusilc) ################################################### ### code chunk number 18: laeken-standard.Rnw:665-668 ################################################### ageCat <- cut(eusilc$age, c(-1, 16, 25, 50, 65, Inf), right=FALSE) eusilc$breakdown <- paste(ageCat, eusilc$rb090, sep=":") rmpg("eqIncome", weights = "rb050", breakdown = "breakdown", data = eusilc) ################################################### ### code chunk number 19: laeken-standard.Rnw:692-693 ################################################### gini("eqIncome", weights = "rb050", data = eusilc) ################################################### ### code chunk number 20: laeken-standard.Rnw:698-699 ################################################### gini("eqIncome", weights = "rb050", breakdown = "db040", data = eusilc) ################################################### ### code chunk number 21: laeken-standard.Rnw:724-729 ################################################### a <- arpr("eqIncome", weights = "rb050", breakdown = "db040", data = eusilc) print(a) is.arpr(a) is.indicator(a) class(a) ################################################### ### code chunk number 22: laeken-standard.Rnw:739-740 ################################################### subset(a, strata = c("Lower Austria", "Vienna")) 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Units sampled from finite populations typically come with different inclusion probabilities. Together with additional preprocessing steps of the raw data, this yields unequal sampling weights of the observations. Whenever indicators are estimated from such complex samples, the corresponding sampling weights have to be taken into account. In addition, many indicators suffer from a strong influence of outliers, which are a common problem in real-world data. The \proglang{R} package \pkg{laeken} is an object-oriented toolkit for the estimation of indicators from complex survey samples via standard or robust methods. In particular the most widely used social exclusion and poverty indicators are implemented in the package. A general calibrated bootstrap method to estimate the variance of indicators for common survey designs is included as well. Furthermore, the package contains synthetically generated close-to-reality data for the European Union Statistics on Income and Living Conditions and the Structure of Earnings Survey, which are used in the code examples throughout the paper. Even though the paper is focused on showing the functionality of package \pkg{laeken}, it also provides a brief mathematical description of the implemented indicator methodology. } \Keywords{indicators, robust estimation, sample weights, survey methodology, \proglang{R}} \Plainkeywords{indicators, robust estimation, sample weights, survey methodology, R} %% without formatting %% at least one keyword must be supplied %% publication information %% NOTE: Typically, this can be left commented and will be filled out by the technical editor %% \Volume{50} %% \Issue{9} %% \Month{June} %% \Year{2012} %% \Submitdate{2012-06-04} %% \Acceptdate{2012-06-04} %% The address of (at least) one author should be given %% in the following format: \Address{ Andreas Alfons \\ Erasmus School of Economics \\ Erasmus University Rotterdam \\ Burgemeester Oudlaan 50 \\ 3062PA Rotterdam, Netherlands \\ E-mail: \email{alfons@ese.eur.nl} \\ URL: \url{https://personal.eur.nl/alfons/} \bigskip Matthias Templ \\ Zurich University of Applied Sciences \\ Rosenstra\ss e 3 \\ 8400 Winterthur, Switzerland \\ E-mail: \email{matthias.templ@zhaw.ch} \\ URL: \url{https://data-analysis.at/} } %% It is also possible to add a telephone and fax number %% before the e-mail in the following format: %% Telephone: +43/512/507-7103 %% Fax: +43/512/507-2851 %% for those who use Sweave please include the following line (with % symbols): %% need no \usepackage{Sweave.sty} %%\VignetteIndexEntry{Estimation of Social Exclusion Indicators From Complex Surveys: The R Package laeken} %%\VignetteDepends{laeken} %%\VignetteKeywords{indicators, robust estimation, sample weights, survey methodology, R} %%\VignettePackage{laeken} %% end of declarations %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% additional packages \usepackage{amsfonts} \usepackage{amsmath} \usepackage{amssymb} \usepackage{engord} \usepackage{enumerate} \usepackage{soul} \begin{document} % \SweaveOpts{concordance=TRUE} %% include your article here, just as usual %% Note that you should use the \pkg{}, \proglang{} and \code{} commands. %% load package "laeken" <>= options(prompt = "R> ", continue = "+ ", width = 72, useFancyQuotes = FALSE) library("laeken") @ %% some references have to many authors to list them in the text \shortcites{AMELI-D7.1} % ------------ % Introduction % ------------ \section{Introduction} Estimation of indicators is one of the main tasks in survey statistics. They are usually estimated from complex surveys with many thousands of observations, conducted in a harmonized manner over many countries. Indicators are designed to reflect major developments in society, for example with respect to poverty, social cohesion or gender inequality, in order to quantify and monitor progress towards policy objectives. Moreover, by implementing a monitoring system across countries via a harmonized set of indicators, different policies can be compared based on quantitative information regarding their impact on society. Thus statistical indicators are an important source of information on which policy makers can base their decisions. Nevertheless, for policy decisions to be effective, the underlying quantitative information from the indicators needs to be reliable. Not only should the variability of the indicators be kept in mind, but also the impact of data collection and preprocessing needs to be considered. Indicators are typically based on complex surveys, in which units are drawn from finite populations, most often with unequal inclusion probabilities. Hence the observations in the sample represent different numbers of units in the population, giving them unequal sample weights. In addition, those initial weights are often modified by preprocessing steps such as calibration for nonresponse. Therefore, sample weights always need to be taken into account in the estimation of indicators from survey samples, otherwise the estimates may be biased. The focus of this paper is on socioeconomic indicators on poverty, social cohesion and gender differences. In economic data, extreme outliers are a common problem. Such outliers can have a disproportionally large influence on the estimates of indicators and may completely distort them. If indicators are corrupted by outliers, wrong conclusions could be drawn by policy makers. Robust estimators that give reliable estimates even in the presence of extreme outliers are therefore necessary. We introduce the add-on package \pkg{laeken} \citep{laeken} for the open source statistical computing environment \proglang{R} \citep{RDev}. It provides functionality for standard and robust estimation of indicators on social exclusion and poverty from complex survey samples. The aim of the paper is to present the most important functionality of the package. A more complete overview of the available functionality is given in additional package vignettes on specialized topics. A list of the available vignettes can be viewed from within \proglang{R} with the following command: <>= vignette(package="laeken") @ Even though official statistical agencies usually rely on commercial software, \proglang{R} has gained some traction in the survey statistics community over the years. Various add-on packages for survey methodology are now available. For instance, an extensive collection of methods for the analysis of survey samples is implemented in package \pkg{survey} \citep{lumley04, survey}. The accompanying book by \citet{lumley10} also serves as an excellent introduction to survey statistics with \proglang{R}. Other examples for more specialized functionality are package \pkg{sampling} \citep{sampling} for finite population sampling, and package \pkg{EVER} \citep{EVER} for variance estimation based on efficient resampling. For the common problem of nonresponse, package \pkg{VIM} \citep{VIM} allows to explore the structure of missing data via visualization techniques \citep[see][]{templ12}, and to impute the missing values via advanced imputation methods \citep[e.g.,][]{templ11}. Even a general framework for simulation studies in survey statistics is available through package \pkg{simFrame} \citep{alfons10c, simFrame}. Package \pkg{laeken} provides functionality for the estimation of indicators that is not available in any of the packages listed above, including a novel approach for robust estimation of indicators. While packages \pkg{survey} and \pkg{EVER} require the generation of certain objects describing the survey design prior to analysis, the methods in \pkg{laeken} can be directly applied to the data. This allows \pkg{laeken} to be used more efficiently in simulations, for instance with the \pkg{simFrame} framework. Furthermore, \pkg{laeken} can easily be used on samples drawn with the \pkg{sampling} package or preprocessed with the \pkg{VIM} package. The rest of the paper is organized as follows. Section~\ref{sec:data} introduces the data sets that are used in the examples throughout the paper. In Section~\ref{sec:indicators}, the most widely used indicators on social exclusion and poverty are briefly described. The basic design of the package and its core functionality are then presented in Section~\ref{sec:design}. More advanced topics such as robust estimation and variance estimation via bootstrap techniques are discussed in Sections~\ref{sec:rob} and~\ref{sec:var}, respectively. The final Section~\ref{sec:conclusions} concludes. % --------- % Data sets % --------- \section{Data sets} \label{sec:data} Package \pkg{laeken} contains example data sets for two well-known surveys: the \emph{European Union Statistics on Income and Living Conditions} (EU-SILC) and the \emph{Structure of Earnings Survey} (SES). Since original data from those surveys are confidential, the example data sets are simulated using the methodology described in \citet{alfons11c} and implemented in the \proglang{R} package \pkg{simPopulation} \citep{simPopulation}. Such close-to-reality data sets provide nearly the same multivariate structure as the confidential original data sets and allow researchers to test and compare methods. However, for policy making purposes and economic interpretation, estimations need to be based on the original data. In any case, the simulated data sets are used in the code examples throughout the paper. \subsection{European Union Statistics on Income and Living Conditions} \label{sec:eusilc} EU-SILC is an annual household survey conducted in EU member states and other European countries. Samples consist of about 450 variables containing information on demographics, income and living conditions \citep[see][]{EU-SILC}. Most notably, EU-SILC serves as data basis for measuring risk-of-poverty and social cohesion in Europe. A subset of the indicators computed from EU-SILC is presented in Section~\ref{sec:laeken}. The EU-SILC example data set in \pkg{laeken} is called \code{eusilc} and contains $14\,827$ observations from $6\,000$ households on the 28 most important variables. The data are synthetically generated from preprocessed Austrian EU-SILC data from 2006 provided by Statistics Austria. A description of all the variables is given in the \proglang{R} help page of the data set. To give an overview of what the data look like, the first three observations of the first ten variables of \code{eusilc} are printed below. <<>>= data("eusilc") head(eusilc[, 1:10], 3) @ For this paper, the variable \code{eqIncome} (equivalized disposable income) is of main interest. Other variables are in some cases used to break down the data into different demographics in order to estimate the indicators on those subsets. \subsection{Structure of Earnings Survey} \label{sec:ses} The Structure of Earnings Survey (SES) \citep{SES} is an enterprise survey that aims at providing harmonized data on earnings for almost all European countries. SES data not only contain information on the enterprise level, but also on the individual employment level from a large sample of employees. The most important indicator on the basis of SES data is the gender pay gap, which is described in Section~\ref{sec:GPG}. The SES example data set in \pkg{laeken} is called \code{ses} and contains information on 27 variables and 15\,691 employees from 500 places of work. It is a subset of synthetic data that are simulated from preprocessed Austrian SES 2006 data provided by Statistics Austria. The first three observations of the first seven variables are shown below. <>= data("ses") head(ses[, 1:7], 3) @ In this paper, the SES data is used to illustrate the estimation of the gender pay gap. Hence the most important variables for our purposes are \code{earningsHour}, \code{sex} and \code{education}. For a description of all the variables in the data set, the reader is referred to its \proglang{R} help page. % ---------- % Indicators % ---------- \section{Indicators} \label{sec:indicators} This section gives a brief description of the most widely used indicators on poverty, social cohesion and gender differences. Unless otherwise stated, the presented definitions strictly follow \citet{EU-SILC04, EU-SILC09}. While quick examples for their computation are provided in this section, a detailed discussion on the respective functions is given later on in Section~\ref{sec:design}. % ------------------ % weighted quantiles % ------------------ \subsection{Weighted median and quantile estimation} \label{sec:w} Nearly all of the indicators considered in the paper require the estimation of the median income or other quantiles of the income distribution. Note that in the analysis of income distributions, the median income is of higher interest than the arithmetic mean, since income distributions typically are strongly right-skewed. In mathematical terms, quantiles are defined as $q_{p} := F^{-1}(p)$, where $F$ is the distribution function on the population level and $0 \leq p \leq 1$. The median as an important special case is given by $p = 0.5$. For the following definitions, let $n$ be the number of observations in the sample, let $\boldsymbol{x} := (x_{1}, \ldots, x_{n})^{\top}$ denote the income with \mbox{$x_{1} \leq \ldots \leq x_{n}$}, and let $\boldsymbol{w} := (w_{i}, \ldots, w_{n})^{\top}$ be the corresponding sample weights. Weighted quantiles for the estimation of the population values are then given by \begin{equation} \label{eq:wq} \hat{q}_{p} = \hat{q}_{p} (\boldsymbol{x}, \boldsymbol{w}) := \begin{cases} \frac{1}{2} (x_{j} + x_{j+1}), & \quad \text{if } \sum_{i=1}^{j} w_{i} = p \sum_{i=1}^{n} w_{i}, \\ x_{j+1}, & \quad \text{if } \sum_{i=1}^{j} w_{i} < p \sum_{i=1}^{n} w_{i} < \sum_{i=1}^{j+1} w_{i}. \end{cases} \end{equation} % ------------------- % selected indicators % ------------------- \subsection{Indicators on social exclusion and poverty} \label{sec:laeken} The indicators described in this section are estimated from EU-SILC data based on household income rather than personal income. For each person, this \emph{equivalized disposable income} is defined as the total household disposable income divided by the equivalized household size. It follows that each person in the same household receives the same equivalized disposable income. The total disposable income of a household is thereby calculated by adding together the personal income received by all of the household members plus the income received at the household level. The equivalized household size is defined according to the modified OECD scale, which gives a weight of 1.0 to the first adult, 0.5 to other household members aged 14 or over, and 0.3 to household members aged less than 14. For the definitions of the following indicators, let $\boldsymbol{x} := (x_{1}, \ldots, x_{n})^{\top}$ be the equivalized disposable income with $x_{1} \leq \ldots \leq x_{n}$ and let $\boldsymbol{w} := (w_{i}, \ldots, w_{n})^{\top}$ be the corresponding sample weights, where $n$ denotes the number of observations. Furthermore, define the following index sets for a certain threshold $t$: \begin{align} I_{< t} &:= \{ i \in \{1, \ldots, n\} : x_{i} < t \},\label{eq:01-Ilt}\\ I_{\leq t} &:= \{ i \in \{ 1,\ldots, n\} : x_{i} \leq t \},\label{eq:01-Ileqt}\\ I_{> t} &:= \{ i \in \{1, \ldots, n\} : x_{i} > t\}\label{eq:01-Igt}. \end{align} \subsubsection{At-risk-at-poverty rate} % \label{sec:ARPR} In order to define the \emph{at-risk-of-poverty rate} (ARPR), the \emph{at-risk-of-poverty threshold} (ARPT) needs to be introduced first, which is set at $60\%$ of the national median equivalized disposable income. Then the at-risk-at-poverty rate is defined as the proportion of persons with an equivalized disposable income below the at-risk-at-poverty threshold. In a more mathematical notation, the at-risk-at-poverty rate is defined as \begin{equation} \label{eq:ARPR} ARPR := P(x < 0.6 \cdot q_{0.5}) \cdot 100,% = F(0.6 \cdot q_{0.5}) \cdot 100, \end{equation} where $q_{0.5} := F^{-1}(0.5)$ denotes the population median (50\% quantile) and $F$ is the distribution function of the equivalized income on the population level. For the estimation of the at-risk-at-poverty rate from a sample, first the at-risk-at-poverty threshold is estimated by \begin{equation} \label{eq:ARPT} \widehat{ARPT} = 0.6 \cdot \hat{q}_{0.5}, \end{equation} where $\hat{q}_{0.5}$ is the weighted median as defined in Equation~\ref{eq:wq}. Then the at-risk-at-poverty rate can be estimated by \begin{equation} \widehat{ARPR} := \frac{\sum_{i \in I_{< \widehat{ARPT}}} w_{i}}{\sum_{i=1}^{n} w_{i}} \cdot 100, \end{equation} where $I_{< \widehat{ARPT}}$ is an index set of persons with an equivalized disposable income below the estimated at-risk-of-poverty threshold as defined in Equation~\ref{eq:01-Ilt}. In package \pkg{laeken}, the function \code{arpr()} is implemented to estimate the at-risk-at-poverty rate. <<>>= arpr("eqIncome", weights = "rb050", data = eusilc) @ Note that the at-risk-of-poverty threshold is computed internally by \code{arpr()}. If necessary, it can also be computed by the user through function \code{arpt()}. % <<>>= % arpt("eqIncome", weights = "rb050", data = eusilc) % @ In addition, a highly related indicator is the \emph{dispersion around the at-risk-of-poverty threshold}, which is defined as the proportion of persons with an equivalized disposable income below $40\%$, $50\%$ and $70\%$ of the national weighted median equivalized disposable income. For the estimation of this indicator with function \code{arpr()}, the proportion of the median equivalized income to be used can easily be adjusted via the argument \code{p}. <<>>= arpr("eqIncome", weights = "rb050", p = c(0.4, 0.5, 0.7), data = eusilc) @ \subsubsection{Quintile share ratio} The income \emph{quintile share ratio} (QSR) is defined as the ratio of the sum of the equivalized disposable income received by the 20\% of the population with the highest equivalized disposable income to that received by the 20\% of the population with the lowest equivalized disposable income. For a given sample, let $\hat{q}_{0.2}$ and $\hat{q}_{0.8}$ denote the weighted 20\% and 80\% quantiles, respectively, as defined in Equation~\ref{eq:wq}. Using index sets $I_{\leq \hat{q}_{0.2}}$ and $I_{> \hat{q}_{0.8}}$ as defined in Equations~\ref{eq:01-Ileqt} and~\ref{eq:01-Igt}, respectively, the quintile share ratio is estimated by \begin{equation} \widehat{QSR} := \frac{\sum_{i \in I_{> \hat{q}_{0.8}}} w_{i} x_{i}}{\sum_{i \in I_{\leq \hat{q}_{0.2}}} w_{i} x_{i}}. \end{equation} To estimate the quintile share ratio, the function \code{qsr()} is available. <<>>= qsr("eqIncome", weights = "rb050", data = eusilc) @ \subsubsection{Relative median at-risk-of-poverty gap} The \emph{relative median at-risk-of-poverty gap} (RMPG) is given by the difference between the median equivalized disposable income of persons below the at-risk-of-poverty threshold and the at-risk of poverty threshold itself, expressed as a percentage of the at-risk-of-poverty threshold. For the estimation of the relative median at-risk-of-poverty gap from a sample, let $\widehat{ARPT}$ be the estimated at-risk-of-poverty threshold according to Equation~\ref{eq:ARPT}, and let $I_{< \widehat{ARPT}}$ be an index set of persons with an equivalized disposable income below the estimated at-risk-of-poverty threshold as defined in Equation~\ref{eq:01-Ilt}. Using this index set, define $\boldsymbol{x}_{< \widehat{ARPT}} := (x_{i})_{i \in I_{< \widehat{ARPT}}}$ and $\boldsymbol{w}_{< \widehat{ARPT}} := (w_{i})_{i \in I_{< \widehat{ARPT}}}$. Furthermore, let $\hat{q}_{0.5} (\boldsymbol{x}_{< \widehat{ARPT}}, \boldsymbol{w}_{< \widehat{ARPT}})$ be the corresponding weighted median according to the definition in Equation~\ref{eq:wq}. Then the relative median at-risk-of-poverty gap is estimated by \begin{equation} \widehat{RMPG} = \frac{\widehat{ARPT} - \hat{q}_{0.5} (\boldsymbol{x}_{< \widehat{ARPT}}, \boldsymbol{w}_{< \widehat{ARPT}})}{\widehat{ARPT}} \cdot 100. \end{equation} The relative median at-risk-of-poverty gap is implemented in the function \code{rmpg()}. <<>>= rmpg("eqIncome", weights = "rb050", data = eusilc) @ \subsubsection{Gini coefficient} The \emph{Gini coefficient} is defined as the relationship of cumulative shares of the population arranged according to the level of equivalized disposable income, to the cumulative share of the equivalized total disposable income received by them. Mathematically speaking, the Gini coefficient is estimated from a sample by \begin{equation} \widehat{Gini} := 100 \left[ \frac{2 \sum_{i=1}^{n} \left( w_{i} x_{i} \sum_{j=1}^{i} w_{j} \right) - \sum_{i=1}^{n} w_{i}^{\phantom{i}2} x_{i}}{\left( \sum_{i=1}^{n} w_{i} \right) \sum_{i=1}^{n} \left(w_{i} x_{i} \right)} - 1 \right]. \end{equation} For estimating the Gini coefficient, the function \code{gini()} can be used. <<>>= gini("eqIncome", weights = "rb050", data = eusilc) @ % -------------- % gender pay gap % -------------- \newpage \subsection{The gender pay gap} \label{sec:GPG} Probably the most important indicator derived from the SES data is the \textit{gender pay gap} (GPG). The calculation of the gender pay gap is based on each person's hourly earnings, which are given by the gross monthly earnings from employment divided by the number of hours usually worked per week in employment during $4.33$ weeks. The gender pay gap in unadjusted form is then defined as the difference between average gross earnings of male paid employees and of female paid employees divided by the earnings of male paid employees \citep{EU-SILC04}. Further discussion on the gender pay gap in Europe can be found in, e.g., \citet{beblot03}. For the following definitions, let $\boldsymbol{x} := (x_{1}, \ldots, x_{n})^{\top}$ be the hourly earnings with \mbox{$x_{1} \leq \ldots \leq x_{n}$}, where $n$ is the number of observations. As in the previous sections, $\boldsymbol{w} := (w_{i}, \ldots, w_{n})^{\top}$ denotes the corresponding sample weights. Then define the index set \begin{align*} I_{M} := \{ i \in \{ 1, \ldots, n\} : & \ \text{worked as least 1 hour per week} \ \wedge \\ & \ (16 \leq \text{age} \leq 65) \wedge \, \text{person is male} \}, \end{align*} and define $I_{F}$ analogously as the index set which differs from $I_{M}$ in the fact that it includes females instead of males. With these index sets, the gender pay gap in unadjusted form is estimated by \begin{equation} \label{eq:GPGmean} GPG_{(mean)} = \left( \frac{\sum_{i \in I_{M}} w_i x_i}{\sum_{i \in I_{M}} w_i} - \frac{\sum_{i \in I_{F}} w_i x_i}{\sum_{i \in I_{F} w_i}} \right) \Bigg/ \ \frac{\sum_{i \in I_{M}} w_i x_i}{\sum_{i \in I_{M}} w_i}. \end{equation} The function \code{gpg()} is implemented in \pkg{laeken} to estimate the gender pay gap. <>= gpg("earningsHour", gender = "sex", weigths = "weights", data = ses) @ While \citet{EU-SILC04} proposes the weighted mean as a measure for the average in the definition of the gender pay gap, the U.S. Census Bureau uses the weighted median %as a robust alternative to better reflect the average in skewed earnings distributions \citep[see, e.g.,][]{Weinberg07}. In this case, the estimate of the gender pay gap in unadjusted form changes to \begin{equation} GPG_{(med)} = \frac{\hat{q}_{0.5}(\boldsymbol{x}_{I_{M}}) - \hat{q}_{0.5}(\boldsymbol{x}_{I_{F}})} {\hat{q}_{0.5}(\boldsymbol{x}_{I_{M}})}, \end{equation} where $\boldsymbol{x}_{I_{M}} := (x_{i})_{i \in I_{M}}$ and $\boldsymbol{x}_{I_{F}} := (x_{i})_{i \in I_{F}}$. It should be noted that even though Eurostat proposes to estimate the gender pay gap via weighted means, Statistics Austria for example uses the variant based on weighted medians as well. In function \code{gpg()}, using the weighted median rather than the weighted mean can be specified via the \code{method} argument. <>= gpg("earningsHour", gender = "sex", weigths = "weights", data = ses, method = "median") @ % ------------ % basic design % ------------ \section{Basic design and core functionality} \label{sec:design} This section discusses the basic design of package \pkg{laeken} and its core functions for the estimation of indicators. \subsection{Indicators and class structure} \label{sec:class} Small examples for computing the social exclusion and poverty indicators with package \pkg{laeken} were already shown in Section~\ref{sec:indicators}. These functions are now discussed in detail. As a reminder, the following indicators are implemented in the package: % \begin{description} \item[\code{arpr()}] for the at-risk-of-poverty rate, as well as the dispersion around the at-risk-of-poverty threshold. \item[\code{qsr()}] for the quintile share ratio. \item[\code{rmpg()}] for the relative median at-risk-of-poverty gap. \item[\code{gini()}] for the gini coefficient. \item[\code{gpg()}] for the gender pay gap. \end{description} % All these functions have a very similar interface and allow to compute point and variance estimates with a single command, even for different subdomains of the data. Most importantly, the user can supply character strings specifying the household income via the first argument and the sample weights via the \code{weights} argument. The data are then taken from the data frame passed as the \code{data} argument. <<>>= gini("eqIncome", weights = "rb050", data = eusilc) @ Alternatively, the user can supply the data directly as vectors: <<>>= gini(eusilc$eqIncome, weights = eusilc$rb050) @ For a full list of arguments, the reader is referred to the \proglang{R} help page of the corresponding function. Package \pkg{laeken} follows an object-oriented design using \proglang{S3} classes \citep{chambers92}. Thus each of the above functions returns an object of a certain class for the respective indicator. All those classes thereby inherit from the class \code{"indicator"}. Among other information, the basic class \code{"indicator"} contains the following components: % \begin{description} \item[\code{value}:] the point estimate. \item[\code{valueByStratum}:] a data frame containing the point estimates for each domain. \item[\code{var}:] the variance estimate. \item[\code{varByStratum}:] a data frame containing the variance estimates for each domain. \item[\code{ci}:] the confidence interval. \item[\code{ciByStratum}:] a data frame containing the confidence intervals for each domain. \end{description} % All indicators inherit the components of class \code{"indicator"}, as well as the methods that are defined for this basic class, which has the advantage that code can be shared among the set of indicators. However, each indicator also has its own class such that methods unique to the indicator can be defined. Following a common convention for \proglang{S3} classes, the classes for the indicators have the same names as the functions for computing them. Hence the following classes are implemented in package \pkg{laeken}: % \begin{itemize} \item Class \code{"arpr"} with the following additional components: \begin{description} \item[\code{p}:] the percentage of the weighted median used for the at-risk-of-poverty threshold. \item[\code{threshold}:] the at-risk-of-poverty threshold. \end{description} \item Class \code{"qsr"} with no additional components. \item Class \code{"rmpg"} with the following additional components: \begin{description} \item[\code{threshold}:] the at-risk-of-poverty threshold. \end{description} \item Class \code{"gini"} with no additional components. \item Class \code{"gpg"} with no additional components. \end{itemize} % Furthermore, functions to test whether an object is a member of the basic class or one of the subclasses are available. The function to test for the basic class is called \code{is.indicator()}. Similarly, the functions to test for the subclasses are called \code{is.foo()}, where \code{foo} is the name of the corresponding class (e.g., \code{is.arpr()}). % <<>>= % a <- arpr("eqIncome", weights = "rb050", data = eusilc) % is.arpr(a) % is.indicator(a) % class(a) % @ \subsection{Estimating the indicators in subdomains} \label{sec:sub} One of the most important features of \pkg{laeken} is that indicators can easily be evaluated for different subdomains. These can be regions, but also any other breakdown given by a categorical variable, for instance age categories or gender. All the user needs to do is to specify such a categorical variable via the \code{breakdown} argument. Note that for the at-risk-of-poverty rate and relative median at-risk-of-poverty gap, the same overall at-risk-of-poverty threshold is used for all subdomains \citep[see][]{EU-SILC04, EU-SILC09}. In the following example, the overall estimate for the at-risk-of-poverty rate is computed together with more regional estimates. <>= a <- arpr("eqIncome", weights = "rb050", breakdown = "db040", data = eusilc) a @ \subsection[Extracting information using the subset() method]{Extracting information using the \code{subset()} method} \label{sec:subset} If estimates of an indicator have been computed for several subdomains, extracting a subset of the results for some domains of particular interest can be done with the corresponding \code{subset()} method. For example, the following command extracts the estimates of the at-risk-of-poverty rate for the regions Lower Austria and Vienna from the object computed above. <<>>= subset(a, strata = c("Lower Austria", "Vienna")) @ It is thereby worth pointing out that not every indicator needs its own \code{subset()} method due to inheritance from the basic class \code{"indicator"}. % ----------------- % Robust estimation % ----------------- \newpage \section{Robust estimation} \label{sec:rob} In economic data, variables such as income are typically heavy-tailed and may contain outliers. To identify extreme outliers, we model heavy tails with a Pareto distribution. In the survey setting, the upper tail of the population values are assumed to follow a Pareto distribution. The \pkg{laeken} package includes recently developed methods of \citet{alfons13a} that allow sampling weights to be incorporated into the Pareto model estimation. In the remainder of the section, we briefly outline the methodology and demonstrate how it can be implemented with the \pkg{laeken} package. \subsection{Pareto distribution} \label{sec:Pareto} The \emph{Pareto distribution} is defined in terms of its cumulative distribution function \begin{equation} \label{eq:CDF} F_{\theta}(x) = 1 - \left( \frac{x}{x_{0}} \right) ^{-\theta}, \qquad x \geq x_{0}, \end{equation} where $x_{0} > 0$ is the scale parameter and $\theta > 0$ is the shape parameter \citep{kleiber03}. Furthermore, its density function is given by \begin{equation} f_{\theta}(x) = \frac{\theta x_{0}^{\theta}}{x^{\theta + 1}}, \qquad x \geq x_{0}. \end{equation} Clearly, the Pareto distribution is a highly right-skewed distribution with a heavy tail. In Pareto tail modeling, the cumulative distribution function on the whole range of $x$ is then modeled as \begin{equation} \label{eq:tail} F(x) = \left\{ \begin{array}{ll} G(x), & \quad \text{if } x \leq x_{0}, \\ G(x_{0}) + (1 - G(x_{0})) F_{\theta}(x), & \quad \text{if } x > x_{0}, \end{array} \right. \end{equation} where $G$ is an unknown distribution function \citep{dupuis06}. For a given survey sample, let $\boldsymbol{x} = (x_{1}, \ldots, x_{n})^{\top}$ be the observed values of the variable of interest with $x_{1} \leq \ldots \leq x_{n}$ and $\boldsymbol{w} := (w_{i}, \ldots, w_{n})^{\top}$ the corresponding sample weights, where $n$ denotes the total number of observations. In addition, let $k$ denote the number of observations to be used for tail modeling. Note that the estimation of $x_{0}$ and $k$ directly correspond with each other. If $k$ is fixed, the threshold is estimated by $\hat{x}_{0} = x_{n-k}$. If in turn an estimate $\hat{x}_{0}$ is obtained, $k$ is given by the number of observations that are larger than $\hat{x}_{0}$. In this section, we focus on the EU-SILC example data, where the equivalized disposable income is the main variable of interest. To illustrate the robustness of the presented methods, we replace the equivalized disposable income of the household with the highest income with a large outlier. Note that the resulting income vector is stored in a new variable. <<>>= hID <- eusilc$db030[which.max(eusilc$eqIncome)] eqIncomeOut <- eusilc$eqIncome eqIncomeOut[eusilc$db030 == hID] <- 10000000 @ Moreover, since the equivalized disposable income is a form of household income, the Pareto distribution needs to be modeled on the household level rather than the personal level. Thus we create a data set that only contains the equivalized disposable income with the outlier and the sample weights on the household level. <<>>= keep <- !duplicated(eusilc$db030) eusilcH <- data.frame(eqIncome=eqIncomeOut, db090=eusilc$db090)[keep,] @ \subsection{Pareto quantile plot and finding the threshold} \label{sec:threshold} The first step in any practical analysis should be to explore the data with visualization techniques. For our purpose, the \emph{Pareto quantile plot} is a powerful tool to check whether the Pareto model is appropriate. The plot was introduced by \citet{beirlant96a} for the case without sample weights, and adapted to take sample weights into account by \citet{alfons13a}. The idea behind the Pareto quantile plot is that under the Pareto model, there exists a linear relationship between the logarithms of the observed values and the quantiles of the standard exponential distribution. For survey samples, the observed values are therefore plotted against the quantities \begin{equation} \label{eq:quantiles} -\log \left( 1 - \frac{\sum_{j=1}^{i} w_{j}}{\sum_{j=1}^{n} w_{j}} \frac{n}{n+1} \right), \qquad i = 1, \ldots, n. \end{equation} When all sample weights are equal, the correction factor $n/(n+1)$ ensures that Equation~\ref{eq:quantiles} reduces to the theoretical quantiles taken on the $n$ inner grid points from $n+1$ equally sized subsets of the interval $[0,1]$ \citep[see][for details]{alfons13a}. \begin{figure}[t!] \begin{center} \setkeys{Gin}{width=0.65\textwidth} <>= paretoQPlot(eusilcH$eqIncome, w = eusilcH$db090) @ \caption{Pareto quantile plot for the EU-SILC example data on the household level with the largest observation replaced by an outlier.} \label{fig:ParetoQuantile} \end{center} \end{figure} In package \pkg{laeken}, the Pareto quantile plot is implemented in the function \code{paretoQPlot()}. Figure~\ref{fig:ParetoQuantile} shows the resulting plot for the EU-SILC example data on the household level. Since the tail of the data forms almost a straight line, the Pareto tail model is suitable for the data at hand. Moreover, Figure~\ref{fig:ParetoQuantile} illustrates the two main advantages that make the Pareto quantile plot so powerful. First, nonrepresentative outliers (i.e., extremely large observations that deviate from the Pareto model) are clearly visible. In our example, the outlier that we introduced into the data set is located far away from the rest of the data in the top right corner of the plot. Second, the leftmost point of a fitted line in the tail of the data can be used as an estimate of the threshold $x_{0}$ in the Pareto model, i.e., the scale parameter of fitted Pareto distribution. The slope of the fitted line is then in turn an estimate of $1/\theta$, the reciprocal of the shape parameter. A disadvantage of this graphical method to determine the parameters of the fitted Pareto distribution is of course that it is not very exact. Nevertheless, the function \code{paretoQPlot()} allows the user to select the threshold in the Pareto model interactively by clicking on a data point. Information on the selected threshold is thereby printed on the \proglang{R} console. This process can be repeated until the user terminates the interactive session, typically by a secondary mouse click. Then the selected threshold is returned as an object of class \code{"paretoScale"}, which consists of the component \code{x0} for the threshold (scale parameter) and the component \code{k} for the number of observations in the tail (i.e., larger than the threshold). \subsubsection{Van Kerm's rule of thumb} For EU-SILC data, \citet{vankerm07} developed a formula for the threshold $x_{0}$ in the Pareto model that has more of a rule-of-thumb nature. It is given by \begin{equation} \hat{x}_{0} := \min(\max(2.5\bar{x}, \hat{q}_{0.98}), \hat{q}_{0.97}), \end{equation} where $\bar{x}$ is the weighted mean, and $\hat{q}_{0.98}$ and $\hat{q}_{0.97}$ are weighted quantiles as defined in Equation~\ref{eq:wq}. It is important to note that this formula is designed specifically for the equivalized disposable income in EU-SILC data and can withstand a small number of nonrepresentative outliers. In \pkg{laeken}, the function \code{paretoScale()} provides functionality for estimating the threshold via \citeauthor{vankerm07}'s formula. Its argument \code{w} can be used to supply sample weights. <<>>= ts <- paretoScale(eusilcH$eqIncome, w = eusilcH$db090) ts @ The estimated threshold is again returned as an object of class \code{"paretoScale"}. % \subsubsection{Other methods for finding the threshold} % % Many procedures for finding the threshold in the Pareto model have been % introduced in the literature. For instance, \citet*{beirlant96b, beirlant96a} % developed an analytical procedure for finding the optimal number of % observations in the tail for the maximum likelihood estimator of the shape % parameter by minimizing the asymptotic mean squared error (AMSE). This % procedure is available in \pkg{laeken} through function \code{minAMSE()}, but % is not further discussed here since it is not robust. \citet{dupuis06}, on the % other hand, proposed a robust prediction error criterion for choosing the % optimal number of observations in the tail and the shape parameter % simultaneously. Nevertheless, our implementation of this robust criterion is % unstable and is therefore not included in \pkg{laeken}. \subsection{Estimation of the shape parameter} \label{sec:shape} Once the threshold for the Pareto model is determined, the shape parameter $\theta$ can be estimated via the \emph{points over threshold} method, i.e., by fitting the distribution to the $k$ data points that are larger than the threshold. Since our aim is to identify extreme outliers that deviate from the Pareto model, the shape parameter needs to be estimated in a robust way. \subsubsection{Integrated squared error estimator} The integrated squared error (ISE) criterion was first introduced by \citet{terrell90} as a more robust alternative to maximum likelihood estimation. \citet{vandewalle07} proposed to use this criterion in the context of Pareto tail modeling, but they do not consider sample weights. However, the Pareto distribution is modeled in terms of the \emph{relative excesses} \begin{equation} y_{i} := \frac{x_{n-k+i}}{x_{n-k}}, \qquad i = 1, \ldots, k. \end{equation} Now the density function of the Pareto distribution for the relative excesses is approximated by \begin{equation} f_{\theta}(y) = \theta y^{-(1+\theta)}. \end{equation} With this model density, the integrated squared error criterion can be written as \begin{equation} \hat{\theta} = \arg \min_{\theta} \left[ \int f_{\theta}^{2}(y) dy - 2 \mathbb{E}(f_{\theta}(Y)) \right] , \end{equation} see \citet{vandewalle07}. For survey samples, \citet{alfons13a} propose to use the weighted mean as an estimator of $\mathbb{E}(f_{\theta}(Y))$ to obtain the \emph{weighted integrated squared error} (wISE) estimator: \begin{equation} \label{eq:wISE} \hat{\theta}_{\mathrm{wISE}} = \arg \min_{\theta} \left[ \int f_{\theta}^{2}(y) dy - \frac{2}{\sum_{i=1}^{k} w_{n-k+i}} \sum_{i=1}^{k} w_{n-k+i} f_{\theta}(y_{i}) \right] . \end{equation} The wISE estimator can be computed using the function \code{thetaISE()}. The arguments \code{k} and \code{x0} are available to supply either the number of observations in the tail or the threshold, and sample weights can be supplied via the argument \code{w}. <<>>= thetaISE(eusilcH$eqIncome, k = ts$k, w = eusilcH$db090) thetaISE(eusilcH$eqIncome, x0 = ts$x0, w = eusilcH$db090) @ \subsubsection{Partial density component estimator} Following the observation by \citet{scott04} that $f_{\theta}$ in the ISE criterion does not need to be a real density, \citet{vandewalle07} proposed to minimize the ISE criterion based on an incomplete density mixture model $u f_{\theta}$ instead. \citet{alfons13a} generalized their estimator to take sample weights into account, yielding the \emph{weighted partial density component} (wPDC) estimator \begin{equation} \label{eq:wPDC} \hat{\theta}_{\mathrm{wPDC}} = \arg \min_{\theta} \left[ u^{2} \int f_{\theta}^{2}(y) dy - \frac{2u}{\sum_{i=1}^{k} w_{n-k+i}} \sum_{i = 1}^{k} w_{n-k+i} f_{\theta}(y_{i}) \right] \end{equation} with \begin{equation} \hat{u} = \left. \frac{1}{\sum_{i=1}^{k} w_{n-k+i}} \sum_{i = 1}^{k} w_{n-k+i} f_{\hat{\theta}}(y_{i}) \right/ \int f_{\hat{\theta}}^{2}(y) dy. \end{equation} Based on extensive simulation studies, \citet{alfons13a} conclude that the wPDC estimator is favorable over the wISE estimator due to better robustness properties. The function \code{thetaPDC()} is implemented in package \pkg{laeken} to compute the wPDC estimator. As before, it is necessary to supply either the number of observations in the tail via the argument \code{k}, or the threshold via the argument \code{x0}. Sample weights can be supplied using the argument \code{w}. <<>>= thetaPDC(eusilcH$eqIncome, k = ts$k, w = eusilcH$db090) thetaPDC(eusilcH$eqIncome, x0 = ts$x0, w = eusilcH$db090) @ % \subsubsection{Other estimators for the shape parameter} % Many other estimators for the shape parameter are implemented in package % \pkg{laeken}, e.g., the maximum likelihood estimator \citep{hill75} or the more % robust weighted maximum likelihood estimator \citep{dupuis02}. However, those % estimators are either not robust or have not (yet) been adapted for sample % weights and are therefore not further discussed in this paper. \subsection{Robust estimation of the indicators via Pareto tail modeling} \label{sec:fit} The basic idea for robust estimation of the indicators is to first detect nonrepresentative outliers based on the Pareto model. Afterwards their influence on the indicators is reduced by either downweighting the outliers and recalibrating the remaining observations, or by replacing the outlying values with values from the fitted distribution. The main advantage of this general approach is that it can be applied to any indicator. With the fitted Pareto distribution $F_{\hat{\theta}}$, nonrepresentative outliers can now be detected as observations being larger than a certain $F_{\hat{\theta}}^{-1}(1-\alpha)$ quantile. From extensive simulation studies \citep{AMELI-D7.1, alfons13a}, $\alpha = 0.005$ or $\alpha = 0.01$ are seem suitable choices for this tuning parameter. Then the following approaches are implemented in \pkg{laeken} to reduce the influence of the outliers: % \begin{description} \item[Calibration of nonrepresentative outliers (CN):] As nonrepresentative outliers are considered to be somewhat unique to the population data, the sample weights of the corresponding observations are set to 1. The weights of the remaining observations are adjusted accordingly by calibration \citep[see, e.g.,][]{deville93}. \item[Replacement of nonrepresentative outliers (RN):] The outliers are replaced by values drawn from the fitted distribution $F_{\hat{\theta}}$, thereby preserving the order of the original values. \item[Shrinkage of nonrepresentative outliers (SN):] The outliers are shrunken to the theoretical quantile $F_{\hat{\theta}}^{-1}(1-\alpha)$ used for outlier detection. \end{description} % A more mathematical formulation and further details on the CN and RN approaches can be found in \citet{alfons13a}, who advocate the CN approach in combination with the wPDC estimator for fitting the Pareto distribution. For a practical analysis with package \pkg{laeken}, let us first revisit the estimation of the shape parameter. Rather than applying a function such as \code{thetaPDC()} directly as in the previous section, the function \code{paretoTail()} should be used to fit the Pareto distribution to the upper tail of the data. It returns an object of class \code{"paretoTail"}, which contains all necessary information for further analysis with one of the approaches described above. <>= fit <- paretoTail(eqIncomeOut, k = ts$k, w = eusilc$db090, groups = eusilc$db030) @ Note that the household IDs are supplied via the argument \code{groups} such that the Pareto distribution is fitted on the household level rather than the individual level. By default, the wPDC is used to estimate the shape parameter, but other estimators can be specified via the \code{method} argument. In addition, the tuning parameter $\alpha$ for outlier detection can be supplied as argument \code{alpha}. \begin{figure}[t!] \begin{center} \setkeys{Gin}{width=0.65\textwidth} <>= plot(fit) @ \caption{Pareto quantile plot for the EU-SILC example data with additional diagnostic information on the fitted distribution and any detected outliers.} \label{fig:diagnostic} \end{center} \end{figure} Moreover, the \code{plot()} method for \code{"paretoTail"} objects produces a Pareto quantile plot (see Section~\ref{sec:threshold}) with additional diagnostic information. Figure~\ref{fig:diagnostic} contains the resulting plot for the object computed above. The lower horizontal dotted line corresponds to the estimated threshold $\hat{x}_{0}$, whereas the slope of the solid grey line is given by the reciprocal of the estimated shape parameter $\hat{\theta}$. Furthermore, the upper horizontal dotted line represents the theoretical quantile used for outlier detection. In this example, the threshold seems somewhat too high. Nevertheless, the estimate of the shape parameter is accurate and the cutoff point for outlier detection is appropriate, resulting in correct identification of the outlier that we added to the data set. For downweighting nonrepresentative outliers, the function \code{reweightOut()} is available. It returns a vector of the recalibrated weights. In the command below, we use regional information as auxiliary variables for calibration. The function \code{calibVars()} thereby transforms a factor into a matrix of binary variables. The returned recalibrated weights are then simply used to estimate the Gini coefficient with function \code{gini()}. <<>>= w <- reweightOut(fit, calibVars(eusilc$db040)) gini(eqIncomeOut, w) @ To replace the nonrepresentative outliers with values drawn from the fitted distribution, the function \code{replaceOut()} is implemented. For reproducible results, the seed of the random number generator is set beforehand. The returned income vector is then supplied to \code{gini()} to estimate the Gini coefficient. <<>>= set.seed(123) eqIncomeRN <- replaceOut(fit) gini(eqIncomeRN, weights = eusilc$rb050) @ Similarly, the function \code{shrinkOut()} can be used to shrink the nonrepresentative outliers to the theoretical quantile used for outlier detection. <<>>= eqIncomeSN <- shrinkOut(fit) gini(eqIncomeSN, weights = eusilc$rb050) @ All three robust estimates are very close to the original value before the outlying household had been introduced (see Section~\ref{sec:laeken}). For comparison, we compute the standard estimate of Gini coefficient with the income vector including the outlying household. <<>>= gini(eqIncomeOut, weights = eusilc$rb050) @ Clearly, the standard estimate shows an unreasonably large influence of only one outlying household, illustrating the need for the robust methods. % ------------------- % Variance estimation % ------------------- \section{Variance estimation} \label{sec:var} The \pkg{laeken} package uses bootstrap techniques for estimating the variance of complex survey indicators. Bootstrap methods in general provide better estimates for nonsmooth estimators than other other resampling techniques such as jackknifing or balanced repeated replication \citep[e.g.,][]{AMELI-D3.1}. The naive bootstrap in \pkg{laeken} is quite fast to compute and provides reasonable estimates whenever there is not much variation in the sample weights, which is for example typically the case for EU-SILC data. If there is larger variation among the sample weights, a calibrated bootstrap should be applied. We describe both approaches and their implementation in the following sections. \subsection{Naive bootstrap} \label{sec:naive} Let $\tau$ denote a certain indicator of interest and let $\boldsymbol{X} := (\bold{x}_{1}, \ldots, \bold{x}_{n})^{\top}$ be a survey sample with $n$ observations. Then the \emph{naive bootstrap} algorithm for estimating the variance and confidence interval of an estimate $\hat{\tau}(\boldsymbol{X})$ of the indicator can be summarized as follows: \begin{enumerate} \item Draw $R$ independent bootstrap samples $\boldsymbol{X}_{1}^{*}, \ldots, \boldsymbol{X}_{R}^{*}$ from $\boldsymbol{X}$. For stratified sampling designs, resampling is performed within each stratum independently. \item Compute the bootstrap replicate estimates $\hat{\tau}_{r}^{*} := \hat{\tau}(\boldsymbol{X}_{r}^{*})$ for each bootstrap sample $\boldsymbol{X}_{r}^{*}$, $r = 1, \ldots, R$, taking the sample weights from the respective bootstrap samples into account. \item Estimate the variance $V(\hat{\tau})$ by the variance of the $R$ bootstrap replicate estimates: \begin{equation} \hat{V}(\hat{\tau}) := \frac{1}{R-1} \sum_{r=1}^{R} \left( \hat{\tau}_{r}^{*} - \frac{1}{R} \sum_{s=1}^{R} \hat{\tau}_{s}^{*} \right)^{2}. \end{equation} \item Estimate the confidence interval at confidence level $1 - \alpha$ by one of the following methods \citep[for details, see][]{davison97}: \begin{description} \item[Percentile method:] $\left[ \hat{\tau}_{((R+1) \frac{\alpha}{2})}^{*}, \hat{\tau}_{((R+1)(1-\frac{\alpha}{2}))}^{*} \right]$, as suggested by \cite{efron93}. \item[Normal approximation:] $\hat{\tau} \pm z_{1-\frac{\alpha}{2}} \cdot \hat{V}(\hat{\tau})^{1/2}$ with $z_{1-\frac{\alpha}{2}} = \Phi^{-1}(1 - \frac{\alpha}{2})$. \item[Basic bootstrap method:] $\left[ 2\hat{\tau} - \hat{\tau}_{((R+1)(1-\frac{\alpha}{2}))}^{*}, 2\hat{\tau} - \hat{\tau}_{((R+1)\frac{\alpha}{2})}^{*} \right]$. \end{description} For the percentile and the basic bootstrap method, $\hat{\tau}_{(1)}^{*} \leq \ldots \leq \hat{\tau}_{(R)}^{*}$ denote the order statistics of the bootstrap replicate estimates. \end{enumerate} With package \pkg{laeken}, variance estimates and confidence intervals can easily be included in the estimation of an indicator. It is only necessary to specify a few more arguments in the call to the function computing the indicator. The argument \code{var} is available to specify the type of variance estimation, although only the bootstrap is currently implemented. Furthermore, the significance level $\alpha$ for the confidence intervals can be supplied via the argument \code{alpha} (the default is to use \code{alpha=0.05} for 95\% confidence intervals). Additional arguments are then passed to the underlying function for variance estimation. <>= arpr("eqIncome", weights = "rb050", design = "db040", cluster = "db030", data = eusilc, var = "bootstrap", bootType = "naive", seed = 1234) @ For the bootstrap, the function \code{bootVar()} is called internally for variance and confidence interval estimation. Important arguments are \code{design} and \code{cluster} for specifying the strata and clusters in the sampling design, \code{R} for supplying the number of bootstrap replicates, \code{bootType} for specifying the type of bootstrap estimator, and \code{ciType} for specifying the type of confidence interval. For reproducibility, the seed of the random number generator can be set via the argument \code{seed}. An important feature of package \pkg{laeken} is that indicators can be estimated for different subdomains with a single command, which still holds for variance and confidence interval estimation. As for point estimation, only the \code{breakdown} argument needs to be specified (cf. the example in Section~\ref{sec:sub}). \subsection{Calibrated bootstrap} \label{sec:calib} In practice, the initial sample weights from the sampling design are often adjusted by calibration, for instance to account for non-response or to ensure that the sums of the sample weights for all observations within certain subgroups equal the respective known population sizes. However, drawing a bootstrap sample then has the effect that the sample weights in the bootstrap sample no longer sum up to the correct values. As a remedy, the sample weights of each bootstrap sample should be recalibrated. For better accuracy at a higher computational cost, the \emph{calibrated bootstrap} algorithm extends the naive bootstrap algorithm from the previous section by adding the following step between Steps~1 and~2: \begin{itemize} \item[1b.] Calibrate the sample weights for each bootstrap sample $\boldsymbol{X}_{r}^{*}$, $r = 1, \ldots, R$ \citep[see, e.g.,][for details on calibration]{deville92, deville93}. \end{itemize} Using \pkg{laeken}, the function call for including variance and confidence intervals via the calibrated bootstrap is very similar to its counterpart for the naive bootstrap. A matrix of auxiliary calibration variables needs to be supplied via the argument \code{X}. The function \code{calibVars()} can thereby by used to transform a factor into a matrix of binary variables. In the %examples example below, information on region and gender is used for calibration. Furthermore, the argument \code{totals} can be used to supply the corresponding population totals. If the \code{totals} argument is omitted, the population totals are computed from the sample weights of the original sample. This follows the assumption that those weights are already calibrated on the supplied auxiliary variables. <>= aux <- cbind(calibVars(eusilc$db040), calibVars(eusilc$rb090)) arpr("eqIncome", weights = "rb050", design = "db040", cluster = "db030", data = eusilc, var = "bootstrap", X = aux, seed = 1234) @ % ----------- % Conclusions % ----------- \section{Conclusions} \label{sec:conclusions} In this paper, we demonstrate the use of the \proglang{R} package \pkg{laeken} for computing point and variance estimates of indicators from complex surveys. Various commonly used indicators on social exclusion and poverty are thereby implemented. Their estimation is made easy with the package, as the corresponding functions allow to compute point and variance estimates with a single command, even for different subdomains of the data. In addition, we illustrate with a simple example that some of the indicators are highly influenced by extreme outliers in the data \citep[cf.][]{hulliger09a, alfons13a}. As a remedy, a general procedure for robust estimation of the indicators is implemented in \pkg{laeken}. The procedure is based on fitting a Pareto distribution to the upper tail of the data and has the advantage that it can be applied to any indicator. A diagnostic plot thereby allows to check whether the Pareto tail model is appropriate for the data at hand. Concerning variance estimation, further techniques for complex survey samples are available in \proglang{R} through other packages. For instance, package \pkg{EVER} \citep{EVER} provides functionality for the delete-a-group jackknife. Other methods such as balanced repeated replication are implemented in package \pkg{survey} \citep{lumley04, survey}. The incorporation of those packages for additional variance estimation procedures is therefore considered for future work. % --------------------- % computational details % --------------------- % \section*{Computational details} % All computations in this paper were performed using \pkg{Sweave} % \citep{leisch02a} with the following \proglang{R} session: % <>= % toLatex(sessionInfo(), locale=FALSE) % @ % % % The most recent version of package \pkg{laeken} is always available from CRAN % (the Comprehensive \proglang{R} Archive Network, % \url{https://CRAN.R-project.org}), and (an up-to-date version of) this paper is % also included as a package vignette. % --------------- % acknowledgments % --------------- \section*{Acknowledgments} This work was partly funded by the European Union (represented by the European Commission) within the \engordnumber{7} framework programme for research (Theme~8, Socio-Economic Sciences and Humanities, Project AMELI (Advanced Methodology for European Laeken Indicators), Grant Agreement No. 217322). Visit \url{http://ameli.surveystatistics.net} for more information on the project. % ------------ % bibliography % ------------ % \bibliographystyle{jss} \bibliography{laeken} \end{document} laeken/inst/CITATION0000644000176200001440000000135514127254454013637 0ustar liggesuserscitHeader("To cite laeken in publications use:") citEntry(entry = "Article", title = "Estimation of Social Exclusion Indicators from Complex Surveys: The {R} Package {laeken}", author = personList(as.person("Andreas Alfons"), as.person("Matthias Templ")), journal = "Journal of Statistical Software", year = "2013", volume = "54", number = "15", pages = "1--25", doi = "10.18637/jss.v054.i15", textVersion = paste("Andreas Alfons, Matthias Templ (2013).", "Estimation of Social Exclusion Indicators from Complex Surveys: The R Package laeken.", "Journal of Statistical Software, 54(15), 1-25.", "doi: 10.18637/jss.v054.i15.") )

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(f""9@@ X'X d6`I0 ,^6y:ؖW`qw@$+.>,C J۸DrT ,X?=' endstream endobj startxref 149855 %%EOF laeken/inst/doc/laeken-standard.Rnw0000644000176200001440000010667114127275434017004 0ustar liggesusers\documentclass[a4paper,10pt]{scrartcl} \usepackage[OT1]{fontenc} \usepackage{Sweave} %% additional packages \usepackage{natbib} \bibpunct{(}{)}{,}{a}{}{,} \usepackage{amsmath, amssymb} \usepackage{hyperref} \hypersetup{colorlinks, citecolor=blue, linkcolor=blue, urlcolor=blue} \usepackage[top=30mm, bottom=30mm, left=30mm, right=30mm]{geometry} \usepackage{enumerate} \usepackage{engord} %% additional commands \newcommand{\code}[1]{\texttt{#1}} \newcommand{\pkg}[1]{\mbox{\textbf{#1}}} \newcommand{\proglang}[1]{\mbox{\textsf{#1}}} %%\VignetteIndexEntry{Standard Methods for Point Estimation of Indicators on Social Exclusion and Poverty using the R Package laeken} %%\VignetteDepends{laeken} %%\VignetteKeywords{social exclusion, poverty, indicators, point estimation} %%\VignettePackage{laeken} \begin{document} \title{Standard Methods for Point Estimation of Indicators on Social Exclusion and Poverty using the \proglang{R} Package \pkg{laeken}} \author{Matthias Templ$^{1}$, Andreas Alfons$^{2}$} \date{} \maketitle \setlength{\footnotesep}{11pt} \footnotetext[1]{ \begin{tabular}[t]{l} Zurich University of Applied Sciences\\ E-mail: \href{mailto:matthias.templ@zhaw.ch}{matthias.templ@zhaw.ch} \end{tabular} } \footnotetext[2]{ \begin{tabular}[t]{l} Erasmus School of Economics, Erasmus University Rotterdam\\ E-mail: \href{mailto:alfons@ese.eur.nl}{alfons@ese.eur.nl} \end{tabular} } % change R prompt <>= options(prompt="R> ") @ \paragraph{Abstract} This vignette demonstrates the use of the \proglang{R} package \pkg{laeken} for standard point estimation of indicators on social exclusion and poverty according to the definitions by Eurostat. The package contains synthetically generated data for the European Union Statistics on Income and Living Conditions (EU-SILC), which is used in the code examples throughout the paper. Furthermore, the basic object-oriented design of the package is discussed. Even though the paper is focused on showing the functionality of package \pkg{laeken}, it also provides a brief mathematical description of the implemented indicators. % ------------ % introduction % ------------ \section{Introduction} The \emph{European Union Statistics on Income and Living Conditions} (EU-SILC) is a panel survey conducted in EU member states and other European countries, and serves as basis for measuring risk-of-poverty and social cohesion in Europe. %and for evaluating the Lisbon~2010 strategy and for monitoring the %Europe~2020 goals of the European Union. A short overview of the $11$ most important indicators on social exclusion and poverty according to \cite{EU-SILC04} %and \cite{EU-SILC09} is given in the following. \paragraph{Primary indicators} \begin{enumerate} \item At-risk-of-poverty rate (after social transfers) \begin{enumerate}[a.] \item At-risk-of-poverty rate by age and gender \item At-risk-of-poverty rate by most frequent activity status and gender \item At-risk-of-poverty rate by household type \item At-risk-of-poverty rate by accommodation tenure status \item At-risk-of-poverty rate by work intensity of the household \item At-risk-of-poverty threshold (illustrative values) \end{enumerate} \item Inequality of income distribution: S80/S20 income quintile share ratio \item At-persistent-risk-of-poverty rate by age and gender ($60\%$ median) \item Relative median at-risk-of-poverty gap, by age and gender \newcounter{enumi_last} \setcounter{enumi_last}{\value{enumi}} \end{enumerate} \paragraph{Secondary indicators} \begin{enumerate} \setcounter{enumi}{\value{enumi_last}} \item Dispersion around the at-risk-of-poverty threshold \item At-risk-of-poverty rate anchored at a moment in time \item At-risk-of-poverty rate before social transfers by age and gender \item Inequality of income distribution: Gini coefficient \item At-persistent-risk-of-poverty rate, by age and gender ($50\%$ median) \setcounter{enumi_last}{\value{enumi}} \end{enumerate} \paragraph{Other indicators} \begin{enumerate} \setcounter{enumi}{\value{enumi_last}} \item Mean equivalized disposable income \item The gender pay gap \end{enumerate} \paragraph{} Note that especially the Gini coefficient is very well studied due to its importance in many fields of research. The add-on package \pkg{laeken} \citep{laeken} aims is to bring functionality for the estimation of indicators on social exclusion and poverty to the statistical environment \proglang{R} \citep{RDev}. In the examples in this vignette, standard estimates for the most important indicators are computed according to the Eurostat definitions \citep{EU-SILC04, EU-SILC09}. More sophisticated methods that are less influenced by outliers are described in vignette \code{laeken-pareto} \citep{alfons11a}, while the basic framework for variance estimation is discussed in vignette \code{laeken-variance} \citep{templ11b}. Those documents can be viewed from within \proglang{R} with the following commands: <>= vignette("laeken-pareto") vignette("laeken-variance") @ Morover, a general introduction to package \pkg{laeken} is published as \citet{alfons13b}. The example data set of package \pkg{laeken}, which is called \code{eusilc} and consists of $14\,827$ observations from $6\,000$ households, is used throughout the paper. It was synthetically generated from Austrian EU-SILC survey data from 2006 using the data simulation methodology proposed by \citet{alfons11c} and implemented in the \proglang{R} package \pkg{simPopulation} \citep{simPopulation}. The first three observations of the synthetic data set \code{eusilc} are printed below. <<>>= library("laeken") data("eusilc") head(eusilc, 3) @ Only a few of the large number of variables in the original survey are included in the example data set. The variable names are rather cryptic codes, but these are the standardized names used by the statistical agencies. Furthermore, the variables \code{hsize} (household size), \code{age}, \code{eqSS} (equivalized household size) and \code{eqIncome} (equivalized disposable income) are not included in the standardized format of EU-SILC data, but have been derived from other variables for convenience. Moreover, some very sparse income components were not included in the the generation of this synthetic data set. Thus the equivalized household income is computed from the available income components. For the remainder of the paper, the variable \code{eqIncome} (equivalized disposable income) is of main interest. Other variables are in some cases used to break down the data in order to evaluate the indicators on the resulting subsets. It is important to note that EU-SILC data are in practice conducted through complex sampling designs with different inclusion probabilities for the observations in the population, which results in different weights for the observations in the sample. Furthermore, calibration is typically performed for non-response adjustment of these initial design weights. Therefore, the sample weights have to be considered for all estimates, otherwise biased results are obtained. The rest of the paper is organized as follows. Section \ref{sec:design} briefly illustrates the basic object-oriented design of the package. The calculation of the equivalized household size and the equivalized disposable income is then described in Section \ref{sec:income}. Afterwards, Section~\ref{sec:w} introduces the Eurostat definitions of the weighted median and weighted quantiles, which are required for the estimation of some of the indicators. In Section~\ref{sec:ind}, a mathematical description of the most important indicators on social exclusion and poverty is given and their estimation with package \pkg{laeken} is demonstrated. Section~\ref{sec:sub} discusses a useful subsetting method, and Section~\ref{sec:concl} concludes. % ------------ % basic design % ------------ \section{Basic design of the package} \label{sec:design} The implementation of the package follows an object-oriented design using \proglang{S3} classes \citep{chambers92}. Its aim is to provide functionality for point and variance estimation of Laeken indicators with a single command, even for different years and domains. Currently, the following indicators are available in the \proglang{R} package \pkg{laeken}: \begin{itemize} \item \emph{At-risk-of-poverty rate}: function \code{arpr()} \item \emph{Quintile share ratio}: function \code{qsr()} \item \emph{Relative median at-risk-of-poverty gap}: function \code{rmpg()} \item \emph{Dispersion around the at-risk-of-poverty threshold}: also function \code{arpr()} \item \emph{Gini coefficient}: function \code{gini()} \end{itemize} Note that the implementation strictly follows the Eurostat definitions \citep{EU-SILC04,EU-SILC09}. %In addition, robust estimators are also implemented. Here, the focus is on %Pareto tail modeling. \subsection{Class structure} In this section, the class structure of package \pkg{laeken} is briefly discussed. Section~\ref{sec:indicator} describes the basic class \code{"indicator"}, while the different subclasses for the specific indicators are listed in Section~\ref{sec:classes}. \subsubsection{Class \code{"indicator"}} \label{sec:indicator} The basic class \code{"indicator"} acts as the superclass for all classes in the package corresponding to specific indicators. It consists of the following components: % \begin{description} \item[\code{value}:] A numeric vector containing the point estimate(s). \item[\code{valueByStratum}:] A \code{data.frame} containing the point estimates by domain. \item[\code{varMethod}:] A character string specifying the type of variance estimation used. \item[\code{var}:] A numeric vector containing the variance estimate(s). \item[\code{varByStratum}:] A \code{data.frame} containing the variance estimates by domain. \item[\code{ci}:] A numeric vector or matrix containing the confidence interval(s). \item[\code{ciByStratum}:] A \code{data.frame} containing the confidence intervals by domain. \item[\code{alpha}:] The confidence level is given by $1 - $\code{alpha}. \item[\code{years}:] A numeric vector containing the different years of the survey. \item[\code{strata}:] A character vector containing the different strata of the breakdown. % \item[\code{seed}:] The seed of the random number generator before the computations. \end{description} These list components are inherited by each indicator in the package. One of the most important features of \pkg{laeken} is that indicators can be evaluated for different years and domains. The latter of which can be regions (e.g., NUTS2), but also any other breakdown given by a categorical variable (see the examples in Section~\ref{sec:ind}). In any case, the advantage of the object-oriented implementation is the possibility of sharing code among the indicators. To give an example, the following methods for the basic class \code{"indicator"} are implemented in the package: <<>>= methods(class="indicator") @ The \code{print()} and \code{subset()} methods are called by their respective generic functions if an object inheriting from class \code{"indicator"} is supplied. While the \code{print()} method defines the output of objects inheriting from class \code{"indicator"} shown on the \proglang{R} console, the \code{subset()} method allows to extract subsets of an object inheriting from class \code{"indicator"} and is discussed in detail in Section~\ref{sec:sub}. Furthermore, the function \code{is.indicator()} is available to test whether an object is of class \code{"indicator"}. \subsubsection{Additional classes} \label{sec:classes} For the specific indicators on social exclusion and poverty, the following classes are implemented in package \pkg{laeken}: % \begin{itemize} \item Class \code{"arpr"} with the following additional components: \begin{description} \item[\code{p}:] The percentage of the weighted median used for the at-risk-of-poverty threshold. \item[\code{threshold}:] The at-risk-of-poverty threshold(s). \end{description} \item Class \code{"qsr"} with no additional components. \item Class \code{"rmpg"} with the following additional components: \begin{description} \item[\code{threshold}:] The at-risk-of-poverty threshold(s). \end{description} \item Class \code{"gini"} with no additional components. \end{itemize} % All these classes are subclasses of the basic class \code{"indicator"} and therefore inherit all its components and methods. In addition, functions to test whether an object is a member of one of these subclasses are implemented. Similarly to \code{is.indicator()}, these are called \code{is.foo()}, where \code{foo} is the name of the respective class (e.g., \code{is.arpr()}). % ----------------------------- % equivalized disposable income % ----------------------------- \section{Calculation of the equivalized disposable income} \label{sec:income} For each person, the equivalized disposable income is defined as the total household disposable income divided by the equivalized household size. It follows that each person in the same household receives the same equivalized disposable income. The total disposable income of a household is calculated by adding together the personal income received by all of the household members plus the income received at the household level. The equivalized household size is defined according to the modified OECD scale, which gives a weight of 1.0 to the first adult, 0.5 to other household members aged 14 or over, and 0.3 to household members aged less than 14 \citep{EU-SILC04, EU-SILC09}. In practice, the equivalized disposable income needs to be computed from the income components included in EU-SILC for the estimation of the indicators on social exclusion and poverty. Therefore, this section outlines how to perform this step with package \pkg{laeken}, even though the variable \code{eqIncome} containing the equivalized disposable income is already available in the example data set \code{eusilc}. Note that not all variables that are required for an exact computation of the equivalized income are included in the synthetic example data. However, the functions of the package can be applied in exactly the same manner to real EU-SILC data. First, the equivalized household size according to the modified OECD scale needs to be computed. This can be done with the function \code{eqSS()}, which requires the household ID and the age of the individuals as arguments. In the example data, household~ID and age are stored in the variables \code{db030} and \code{age}, respectively. It should be noted that the variable \code{age} is not in the standardized format of EU-SILC data and needs to be calculated from the data beforehand. Nevertheless, these computations are very simple and are therefore not shown here \citep[for details, see][]{EU-SILC09}. The following two lines of code calculate the equivalized household size, add it to the data set, and print the first eight observations of the variables involved. <<>>= eusilc$eqSS <- eqSS("db030", "age", data=eusilc) head(eusilc[,c("db030", "age", "eqSS")], 8) @ Then the equivalized disposable income can be computed with the function \code{eqInc()}. It requires the following information to be supplied: the household~ID, the household income components to be added and subtracted, respectively, the personal income components to be added and subtracted, respectively, as well as the equivalized household size. With the following commands, the equivalized disposable income is calculated and added to the data set, after which the first eight observations of the important variables in this context are printed. <<>>= hplus <- c("hy040n", "hy050n", "hy070n", "hy080n", "hy090n", "hy110n") hminus <- c("hy130n", "hy145n") pplus <- c("py010n", "py050n", "py090n", "py100n", "py110n", "py120n", "py130n", "py140n") eusilc$eqIncome <- eqInc("db030", hplus, hminus, pplus, character(), "eqSS", data=eusilc) head(eusilc[,c("db030", "eqSS", "eqIncome")], 8) @ % Note that the net income is considered in this example, therefore no personal income component needs to be subtracted \citep[see][]{EU-SILC04, EU-SILC09}. This is reflected in the call to \code{eqInc()} by the use of an empty character vector \code{character()} for the corresponding argument. % ------------------ % weighted quantiles % ------------------ \section{Weighted median and quantile estimation} \label{sec:w} Some of the indicators on social exclusion and poverty require the estimation of the median income or other quantiles of the income distribution. Hence functions that strictly follow the definitions according to \citet{EU-SILC04, EU-SILC09} are implemented in package \pkg{laeken}. They are used internally for the estimation of the respective indicators, but can also be called by the user directly. In the analysis of income distributions, the median income is typically of higher interest than the arithmetic mean. This is because income distributions commonly are strongly right-skewed with a heavy tail of \emph{representative outliers} (correctly measured units that are not unique to the population) and \emph{nonrepresentative outliers} (either measurement errors or correct observations that can be considered unique in the population). Therefore, the center of the distribution is more reliably estimated by a weighted median than by a weighted mean, as the latter is highly influenced by extreme values. In mathematical terms, quantiles are defined as $q_{p} := F^{-1}(p)$, where $F$ is the distribution function on the population level and $0 \leq p \leq 1$. The median as an important special case is given by $p = 0.5$. For the following definitions, let $n$ be the number of observations in the sample, let $\boldsymbol{x} := (x_{1}, \ldots, x_{n})'$ denote the equivalized disposable income with \mbox{$x_{1} \leq \ldots \leq x_{n}$}, and let $\boldsymbol{w} := (w_{i}, \ldots, w_{n})'$ be the corresponding personal sample weights. Weighted quantiles for the estimation of the population values according to \citet{EU-SILC04, EU-SILC09} are then given by \begin{equation} \label{eq:wq} \hat{q}_{p} = \hat{q}_{p} (\boldsymbol{x}, \boldsymbol{w}) := \begin{cases} \frac{1}{2} (x_{j} + x_{j+1}), & \quad \text{if } \sum_{i=1}^{j} w_{i} = p \sum_{i=1}^{n} w_{i}, \\ x_{j+1}, & \quad \text{if } \sum_{i=1}^{j} w_{i} < p \sum_{i=1}^{n} w_{i} < \sum_{i=1}^{j+1} w_{i}. \end{cases} \end{equation} This definition of weighted quantiles is available in \pkg{laeken} through the function \code{weightedQuantile()}. The following command computes the weighed 20\% quantile, the weighted median, and the weighted 80\% quantile. In the context of social exclusion indicators, these are of most importance. % ----- <>= weightedQuantile(eusilc$eqIncome, eusilc$rb050, probs = c(0.2, 0.5, 0.8)) @ % ----- For the important special case of the weighted median, the function \code{weightedMedian()} is available for convenience. % ----- <<>>= weightedMedian(eusilc$eqIncome, eusilc$rb050) @ In addition, the functions \code{incMedian()} and \code{incQuintile()} are more tailored towards application in the case of indicators on social exclusion and poverty and provide a similar interface as the functions for the indicators (see Section~\ref{sec:ind}). In particular, they allow to supply an additional variable to be used as tie-breakers for sorting, and to compute the weighted median and income quintiles, respectively, for several years of the survey. With the following lines of code, the median income as well as the \engordnumber{1} and \engordnumber{4} income quintile (i.e., the weighted 20\% and 80\% quantiles) are estimated. <<>>= incMedian("eqIncome", weights = "rb050", data = eusilc) incQuintile("eqIncome", weights = "rb050", k = c(1, 4), data = eusilc) @ % ------------------- % selected indicators % ------------------- \section{Indicators on social exclusion and poverty} \label{sec:ind} In this section, the most important indicators on social exclusion and poverty are described in detail. Furthermore, the functionality of package \pkg{laeken} to estimate these indicators is demonstrated. It should be noted that all functions for the implemented indicators provide a very similar interface. Most importantly, it is possible to compute estimates for several years of the survey and different subdomains with a single command. Furthermore, the functions allow to supply an additional variable to be used as tie-breakers for sorting. However, not all of the implemented functionality is shown in this vignette. For a complete description of the functions and their arguments, the reader is referred to the corresponding \proglang{R} help pages. In addition, only point estimation of the indicators on social exclusion and poverty is illustrated here, statistical significance of these estimates is not discussed. The functionality for variance estimation of the indicators is described in the package vignette \code{laeken-variance} \citep{templ11b}. For the following definitions of the estimators according to \citet{EU-SILC04, EU-SILC09}, let $\boldsymbol{x} := (x_{1}, \ldots, x_{n})'$ be the equivalized disposable income with $x_{1} \leq \ldots \leq x_{n}$ and let $\boldsymbol{w} := (w_{i}, \ldots, w_{n})'$ be the corresponding personal sample weights, where $n$ denotes the number of observations. Furthermore, define the following index sets for a certain threshold $t$: \begin{align} I_{< t} &:= \{ i \in \{ 1, \ldots, n \} : x_{i} < t \},\label{eq:01-Ilt}\\ I_{\leq t} &:= \{ i \in \{ 1, \ldots, n \} : x_{i} \leq t \},\label{eq:01-Ileqt}\\ I_{> t} &:= \{ i \in \{ 1, \ldots, n \} : x_{i} > t\}\label{eq:01-Igt}. \end{align} \subsection{At-risk-at-poverty rate} \label{sec:ARPR} In order to define the \emph{at-risk-of-poverty rate} (ARPR), the \emph{at-risk-of-poverty threshold} (ARPT) needs to be introduced first, which is set at $60\%$ of the national median equivalized disposable income. Then the at-risk-at-poverty rate is defined as the proportion of persons with an equivalized disposable income below the at-risk-at-poverty threshold \citep{EU-SILC04, EU-SILC09}. In a more mathematical notation, the at-risk-at-poverty rate is defined as \begin{equation} \label{eq:ARPR} ARPR := P(x < 0.6 \cdot q_{0.5}) \cdot 100,% = F(0.6 \cdot q_{0.5}) \cdot 100, \end{equation} where $q_{0.5} := F^{-1}(0.5)$ denotes the population median (50\% quantile) and $F$ is the distribution function of the equivalized income on the population level. For the estimation of the at-risk-at-poverty rate from a sample, the sample weights need to be taken into account. %Let $n$ be the number of observations in the sample, let $\boldsymbol{x} := %(x_{1}, \ldots, x_{n})'$ denote the equivalized disposable income with %\mbox{$x_{1} \leq \ldots \leq x_{n}$}, and let $\boldsymbol{w} := (w_{i}, %\ldots, w_{n})'$ be the corresponding personal sample weights. Then the %at-risk-at-poverty threshold is estimated by First, the at-risk-at-poverty threshold is estimated by \begin{equation} \label{eq:ARPT} \widehat{ARPT} = 0.6 \cdot \hat{q}_{0.5}, \end{equation} where $\hat{q}_{0.5}$ is the weighted median as defined in Equation~(\ref{eq:wq}). %Furthermore, define an index set of observations with an equivalized disposable %income below the estimated at-risk-at-poverty threshold as %\begin{equation} %I_{< \widehat{ARPT}} := \{ i \in \{ 1, \ldots, n \} : x_{i} < \widehat{ARPT} \}. %\end{equation} %With these definitions, the at-risk-at-poverty rate can be estimated by Then the at-risk-at-poverty rate can be estimated by \begin{equation} \widehat{ARPR} := \frac{\sum_{i \in I_{< \widehat{ARPT}}} w_{i}}{\sum_{i=1}^{n} w_{i}} \cdot 100, \end{equation} where $I_{< \widehat{ARPT}}$ is an index set of persons with an equivalized disposable income below the estimated at-risk-of-poverty threshold as defined in Equation~(\ref{eq:01-Ilt}). In package \pkg{laeken}, the functions \code{arpt()} and \code{arpr()} are implemented for the estimation of the at-risk-of-poverty threshold and the at-risk-of-poverty rate. Whenever sample weights are available in the data, they should be supplied as the \code{weights} argument. Even though \code{arpt()} is called internally by \code{arpr()}, it can also be called by the user directly. <<>>= arpt("eqIncome", weights = "rb050", data = eusilc) arpr("eqIncome", weights = "rb050", data = eusilc) @ It is also possible to use these functions for the estimation of the indicator \emph{dispersion around the at-risk-of-poverty threshold}, which is defined as the proportion of persons with an equivalized disposable income below $40\%$, $50\%$ and $70\%$ of the national weighted median equivalized disposable income. The proportion of the median equivalized income to be used can thereby be adjusted via the argument \code{p}. <<>>= arpr("eqIncome", weights = "rb050", p = 0.4, data = eusilc) arpr("eqIncome", weights = "rb050", p = 0.5, data = eusilc) arpr("eqIncome", weights = "rb050", p = 0.7, data = eusilc) @ In order to compute estimates for different subdomains, a breakdown variable simply needs to be supplied as the \code{breakdown} argument. Note that in this case the same overall at-risk-of-poverty threshold is used for all subdomains \citep[see][]{EU-SILC04, EU-SILC09}. The following command computes the overall estimate, as well as estimates for all NUTS2 regions. <<>>= arpr("eqIncome", weights = "rb050", breakdown = "db040", data = eusilc) @ However, any kind of breakdown can be supplied, e.g., the breakdowns defined by \citet{EU-SILC04, EU-SILC09}. With the following lines of code, a breakdown variable with all possible combinations of age categories and gender is defined and added to the data set, before it is used to compute estimates for the corresponding domains. <<>>= ageCat <- cut(eusilc$age, c(-1, 16, 25, 50, 65, Inf), right=FALSE) eusilc$breakdown <- paste(ageCat, eusilc$rb090, sep=":") arpr("eqIncome", weights = "rb050", breakdown = "breakdown", data = eusilc) @ Clearly, the results are even more heterogeneous than for the breakdown into NUTS2 regions. %The results are even more different when considering household size %(\code{hsize}) and citizenship (\code{pb220a}) as the domain level for %estimation. %<<>>= %eusilc$breakdown <- paste(eusilc$hsize, eusilc$pb220a, sep=":") %arpr("eqIncome", weights = "rb050", breakdown = "breakdown", data = eusilc) %@ \subsection{Quintile share ratio} The income \emph{quintile share ratio} (QSR) is defined as the ratio of the sum of the equivalized disposable income received by the 20\% of the population with the highest equivalized disposable income to that received by the 20\% of the population with the lowest equivalized disposable income \citep{EU-SILC04, EU-SILC09}. For the estimation of the quintile share ratio from a sample, let $\hat{q}_{0.2}$ and $\hat{q}_{0.8}$ denote the weighted 20\% and 80\% quantiles, respectively, as defined in Equation~(\ref{eq:wq}). Using index sets $I_{\leq \hat{q}_{0.2}}$ and $I_{> \hat{q}_{0.8}}$ as defined in Equations~(\ref{eq:01-Ileqt}) and~(\ref{eq:01-Igt}), respectively, the quintile share ratio is estimated by \begin{equation} \widehat{QSR} := \frac{\sum_{i \in I_{> \hat{q}_{0.8}}} w_{i} x_{i}}{\sum_{i \in I_{\leq \hat{q}_{0.2}}} w_{i} x_{i}}. \end{equation} With package \pkg{laeken}, the quintile share ratio can be estimated using the function \code{qsr()}. As for the at-risk-of-poverty rate, sample weights can be supplied via the \code{weights} argument. <<>>= qsr("eqIncome", weights = "rb050", data = eusilc) @ Computing estimates for different subdomains is again possible by specifying the \code{breakdown} argument. In the following example, estimates for each NUTS2 region are computed in addition to the overall estimate. <<>>= qsr("eqIncome", weights = "rb050", breakdown = "db040", data = eusilc) @ Nevertheless, it should be noted that the quintile share ratio is highly influenced by outliers \citep[see][]{hulliger09a, alfons10b}. Since the upper tail of income distributions virtually always contains nonrepresentative outliers, robust estimators of the quintile share ratio should preferably be used. Thus robust semi-parametric methods based on Pareto tail modeling are implemented in package \pkg{laeken} as well. Their application is discussed in vignette \code{laeken-pareto} \citep{alfons11a}. \subsection{Relative median at-risk-of-poverty gap (by age and gender)} The \emph{relative median at-risk-of-poverty gap} (RMPG) is defined as the difference between the median equivalized disposable income of persons below the at-risk-of-poverty threshold and the at-risk of poverty threshold itself, expressed as a percentage of the at-risk-of-poverty threshold \citep{EU-SILC04, EU-SILC09}. %Let $wmed_{(poor)}$ the weighted median of the people who having an income %below $ARPR$ defined in Equation~\ref{eq:ARPR}. Then the relative median %at-risk-of-poverty gap is estimated by %\begin{displaymath} %RMPG = \frac{ARPR - wmed_{(poor)}}{ARPR} \cdot 100 %\end{displaymath} For the estimation of the relative median at-risk-of-poverty gap from a sample, let $\widehat{ARPT}$ be the estimated at-risk-of-poverty threshold according to Equation~(\ref{eq:ARPT}), and let $I_{< \widehat{ARPT}}$ be an index set of persons with an equivalized disposable income below the estimated at-risk-of-poverty threshold as defined in Equation~(\ref{eq:01-Ilt}). Using this index set, define $\boldsymbol{x}_{< \widehat{ARPT}} := (x_{i})_{i \in I_{< \widehat{ARPT}}}$ and $\boldsymbol{w}_{< \widehat{ARPT}} := (w_{i})_{i \in I_{< \widehat{ARPT}}}$. Furthermore, let $\hat{q}_{0.5} (\boldsymbol{x}_{< \widehat{ARPT}}, \boldsymbol{w}_{< \widehat{ARPT}})$ be the corresponding weighted median according to the definition in Equation~(\ref{eq:wq}). Then the relative median at-risk-of-poverty gap is estimated by \begin{equation} \widehat{RMPG} = \frac{\widehat{ARPT} - \hat{q}_{0.5} (\boldsymbol{x}_{< \widehat{ARPT}}, \boldsymbol{w}_{< \widehat{ARPT}})}{\widehat{ARPT}} \cdot 100. \end{equation} In package \pkg{laeken}, the function \code{rmpg()} is implemented for the estimation of the relative median at-risk-of-poverty gap. If available in the data, sample weights should be supplied as the \code{weights} argument. Note that the function \code{arpt()} for the estimation of the at-risk-of-poverty threshold is called internally (cf. function \code{arpr()} for the at-risk-of-poverty rate in Section~\ref{sec:ARPR}). <<>>= rmpg("eqIncome", weights = "rb050", data = eusilc) @ Estimates for different subdomains can be computed by making use of the \code{breakdown} argument. With the following command, the overall estimate and estimates for all NUTS2 regions are computed. <<>>= rmpg("eqIncome", weights = "rb050", breakdown = "db040", data = eusilc) @ For the relative median at-risk-of-poverty gap, the breakdown by age and gender is of particular interest. In the following example, a breakdown variable with all possible combinations of age categories and gender is defined and added to the data set. Afterwards, estimates for the corresponding domains are computed. <<>>= ageCat <- cut(eusilc$age, c(-1, 16, 25, 50, 65, Inf), right=FALSE) eusilc$breakdown <- paste(ageCat, eusilc$rb090, sep=":") rmpg("eqIncome", weights = "rb050", breakdown = "breakdown", data = eusilc) @ \subsection{Gini coefficient} The \emph{Gini coefficient} is defined as the relationship of cumulative shares of the population arranged according to the level of equivalized disposable income, to the cumulative share of the equivalized total disposable income received by them \citep{EU-SILC04, EU-SILC09}. For the estimation of the Gini coefficient from a sample, the sample weights need to be taken into account. In mathematical terms, the Gini coefficient is estimated by \begin{equation} \widehat{Gini} := 100 \left[ \frac{2 \sum_{i=1}^{n} \left( w_{i} x_{i} \sum_{j=1}^{i} w_{j} \right) - \sum_{i=1}^{n} w_{i}^{\phantom{i}2} x_{i}}{\left( \sum_{i=1}^{n} w_{i} \right) \sum_{i=1}^{n} \left(w_{i} x_{i} \right)} - 1 \right]. \end{equation} The function \code{gini()} is available in \pkg{laeken} to estimate the Gini coefficient. As for the other indicators, sample weights can be specified with the \code{weights} argument. <<>>= gini("eqIncome", weights = "rb050", data = eusilc) @ Using the \code{breakdown} argument in the following command, estimates for the NUTS2 regions are computed in addition to the overall estimate. <<>>= gini("eqIncome", weights = "rb050", breakdown = "db040", data = eusilc) @ Since outliers have a strong influence on the Gini coefficient, robust estimators are preferred to the standard estimation described above \citep[see][]{alfons10b}. Vignette \code{laeken-pareto} \citep{alfons11a} describes how to apply the robust semi-parametric methods implemented in package \pkg{laeken}. % ------------------ % extracting subsets % ------------------ \section{Extracting information using the \code{subset()} method} \label{sec:sub} If estimates of an indicator have been computed for several subdomains, it may sometimes be desired to extract the results for some domains of particular interest. In package \pkg{laeken}, this is implemented by taking advantage of the object-oriented design of the package. Each of the functions for the indicators described in Section~\ref{sec:ind} returns an object belonging to a class of the same name as the respective function, e.g., function \code{arpr()} returns an object of class \code{"arpr"}. All these classes thereby inherit from the basic class \code{"indicator"} (see Section~\ref{sec:design}). <<>>= a <- arpr("eqIncome", weights = "rb050", breakdown = "db040", data = eusilc) print(a) is.arpr(a) is.indicator(a) class(a) @ To extract a subset of results from such an object, a \code{subset()} method for the class \code{"indicator"} is implemented in \pkg{laeken}. The method \code{subset.indicator()} is hidden from the user and is called internally by the generic function \code{subset()} whenever an object of class \code{"indicator"} is supplied. In the following example, the estimates of the at-risk-of-poverty rate for the regions Lower Austria and Vienna are extracted from the object computed above. <<>>= subset(a, strata = c("Lower Austria", "Vienna")) @ % ----------- % conclusions % ----------- \section{Conclusions} \label{sec:concl} This vignette demonstrates the use of package \pkg{laeken} for point estimation of the European Union indicators on social exclusion and poverty. Since the description of the indicators in \citet{EU-SILC04, EU-SILC09} is weak from a mathematical point of view, a more precise notation is given in this paper. Currently, the most important indicators are implemented in \pkg{laeken}. Their estimation is made easy with the package, as it is even possible to compute estimates for several years and different subdomains with a single command. Concerning the inequality indicators quintile share ratio and Gini coefficient, it is clearly visible from their definitions that the standard estimators are highly influenced by outliers \citep[see also][]{hulliger09a, alfons10b}. Therefore, robust semi-parametric methods are implemented in \pkg{laeken} as well. These are described in vignette \code{laeken-pareto} \citep{alfons11a}, while variance and confidence interval estimation for the indicators on social exclusion and poverty with package \pkg{laeken} is treated in vignette \code{laeken-variance} \citep{templ11b}. % --------------- % acknowledgments % --------------- \section*{Acknowledgments} This work was partly funded by the European Union (represented by the European Commission) within the 7$^{\mathrm{th}}$ framework programme for research (Theme~8, Socio-Economic Sciences and Humanities, Project AMELI (Advanced Methodology for European Laeken Indicators), Grant Agreement No. 217322). Visit \url{http://ameli.surveystatistics.net} for more information on the project. % ------------ % bibliography % ------------ \bibliographystyle{plainnat} \bibliography{laeken} \end{document} laeken/inst/doc/laeken-pareto.pdf0000644000176200001440000261201614127307304016466 0ustar liggesusers%PDF-1.5 % 1 0 obj << /Type /ObjStm /Length 4705 /Filter /FlateDecode /N 84 /First 703 >> stream x\ms6~߮;psי8vriԵ6m'dy%W~= Ȏ&Aža XIf,SL4Sxeyƹ+9g(+WLHfB|7LXdp^gR@֡gJJ4L9ii<1$32# sTgVHdqI 2/ЙkaICTAc#.>2`snAŹ@wL ACS*@ƒ1 ȒD4 h@Z/0%^-V3 JkCl2EbkȠ, )q{͈mZb,`?"I 2Z@[A"nnL F͍,  - iX@9@)0r$7 9@sl-21ozr>' T#x[OYd|8DAɇOQ:d@< ͷ('u5N}OQ΋?G!^oQI+v؋Eyn&gST{Y~x7_/.g%>?gϯzynj3!QYT4YMJPg } 7O;45ܜJH@hw6Dn9kBG Oz'@7SxOث(t O糺S# y5ٙTbA,[x\ c콯+(x L-ge 3|_߷jbg"Ts\+o~ 0F@f)!4=ŋYgDؑxX8qJMEbl[$Ebl]J>F$l$]0.xK`!+KE *JE TJiBnuRG('0_elLIl'bdnLdNL~(#:b#Wu䪎ԱdL&ƥ&67c&AnØO!2iXMGg&7r7H-1PےĨ$ Z~m2P<Ôa0' #Y$bM~C>-/sV<3v^"_r/yWw6+?|? ;A?N.) h\V1_URBE[&'K(T%d>fk~\?y%-(G/>EG-r"?ո-%*hZi ) B)lT]]?jEg+mp E,\rR.z u2H@w<hCe6菎0t"v (Rt'[м)1wz/ahҼ#NJ*mP?BIJQ_'Ue>-'sbrGYULQbZO}6@ϫrQ.e~>GC>*/L%H ك$\}RWm-~,ѧ7Ѹ"kv+#RM{$_r1Ȣ|V6MiѦt~c75(^py?{2[Vk⊡q}76.a5g+0 OnM4M*xJP뉁$Zzr/;S ģo\ũe)}rM=ld87iTDǟ;~N^t&z0M=1PO̚dHm+n]dhU–H<9ۦ: eXH8Y ]3Д#>T N`җUpah[8 ÷ucݹm'~瓎:`?u6c0;8|۞*ը =ݥ Ɓ)4q`2bUMt? gy3JGX3 ˻QǧDr#i5N7c}zyhom'c5K'5r)3 ^p/е:ʍjN5p6Z1Q\3w_M ůZ@iheWiI4L޷5& zȜĈ9EvR9&Dһ>5ϩOϨn>X3S3N' R͠产'.)bi;ituS񳁰(+@AQyzɓ]~uimcD..2_pl ZqP7&MJ1jڮvc})r=%B .("tAT(j"x0EI~m9~b5&b=OAKz ֌)YmR0YHۛ^SoYe5 Ûm+ 2q7]Yuv h#-u&+雑wW#kK庹YU[+?$ְڮ-IOY&~=x#͒ս6q:^\t>_)~L,麚.&d^oN7zXxt]\w *o^_/UpKWHJuG=WH TzD,/B[ٶ'S/#+TiSJN~ZKM'l7/N/$W\_L'V|_=@ZӪ[ly'fkӪW̗ihәMwjd%m Ә U>bd.ԙ n\1je3!7]2FS8P2 )/4YIZ&R!ddjz8BdViZȤ, 3kI*3-4HqkŬH?nd8P~r3B*]8(2L(>@! 7@PLb${P`%2:6&mu6:Pтro  *89ڑSX|t ,bJIGd̈%(=2*^.`XMF{Jr۬pM_']kvh RYlm _dB}rtb`Õ<p! 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