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generated with the docstrip utility. %% %% The original source files were: %% %% utf8ienc.dtx (with options: `utf8') %% %% This is a generated file. %% %% Copyright 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 %% The LaTeX3 Project and any individual authors listed elsewhere %% in this file. %% %% This file was generated from file(s) of the LaTeX base system. %% -------------------------------------------------------------- %% %% It may be distributed and/or modified under the %% conditions of the LaTeX Project Public License, either version 1.3 %% of this license or (at your option) any later version. %% The latest version of this license is in %% http://www.latex-project.org/lppl.txt %% and version 1.3 or later is part of all distributions of LaTeX %% version 2003/12/01 or later. %% %% This file has the LPPL maintenance status "maintained". %% %% This file may only be distributed together with a copy of the LaTeX %% base system. You may however distribute the LaTeX base system without %% such generated files. %% %% The list of all files belonging to the LaTeX base distribution is %% given in the file `manifest.txt'. See also `legal.txt' for additional %% information. %% %% The list of derived (unpacked) files belonging to the distribution %% and covered by LPPL is defined by the unpacking scripts (with %% extension .ins) which are part of the distribution. \ProvidesFile{utf8.def} [2004/02/09 v1.1b UTF-8 support for inputenc] \makeatletter \def\UTFviii@two@octets#1#2{\expandafter \UTFviii@defined\csname u8:#1\string#2\endcsname} \def\UTFviii@three@octets#1#2#3{\expandafter \UTFviii@defined\csname u8:#1\string#2\string#3\endcsname} \def\UTFviii@four@octets#1#2#3#4{\expandafter \UTFviii@defined\csname u8:#1\string#2\string#3\string#4\endcsname} \def\UTFviii@defined#1{% \ifx#1\relax \PackageError{inputenc}{Unicode char \string#1 not set up\space for use with LaTeX}\@eha \else\expandafter #1% \fi } \begingroup \catcode`\~13 \catcode`\"12 \def\UTFviii@loop{% \uccode`\~\count@ \uppercase\expandafter{\UTFviii@tmp}% \advance\count@\@ne \ifnum\count@<\@tempcnta \expandafter\UTFviii@loop \fi} \count@"C2 \@tempcnta"E0 \def\UTFviii@tmp{\xdef~{\noexpand\UTFviii@two@octets\string~}} \UTFviii@loop \count@"E0 \@tempcnta"F0 \def\UTFviii@tmp{\xdef~{\noexpand\UTFviii@three@octets\string~}} \UTFviii@loop \count@"F0 \@tempcnta"F4 \def\UTFviii@tmp{\xdef~{\noexpand\UTFviii@four@octets\string~}} \UTFviii@loop \endgroup \@inpenc@test \ifx\@begindocumenthook\@undefined \makeatother \endinput \fi \begingroup \catcode`\"=12 \catcode`\<=12 \catcode`\.=12 \catcode`\,=12 \catcode`\;=12 \catcode`\!=12 \catcode`\~=13 \gdef\DeclareUnicodeCharacter#1#2{% \count@"#1\relax \wlog{ \space\space defining Unicode char U+#1 (decimal \the\count@)}% \begingroup \parse@XML@charref \def\UTFviii@two@octets##1##2{\csname u8:##1\string##2\endcsname}% \def\UTFviii@three@octets##1##2##3{\csname u8:##1% \string##2\string##3\endcsname}% \def\UTFviii@four@octets##1##2##3##4{\csname u8:##1% \string##2\string##3\string##4\endcsname}% \expandafter\expandafter\expandafter \expandafter\expandafter\expandafter \expandafter \gdef\UTFviii@tmp{\IeC{#2}}% \endgroup } \gdef\parse@XML@charref{% \ifnum\count@<"A0\relax \PackageError{inputenc}{Cannot define Unicode char value < 00A0}\@eha \else\ifnum\count@<"800\relax \parse@UTFviii@a,% \parse@UTFviii@b C\UTFviii@two@octets.,% \else\ifnum\count@<"10000\relax \parse@UTFviii@a;% \parse@UTFviii@a,% \parse@UTFviii@b E\UTFviii@three@octets.{,;}% \else \parse@UTFviii@a;% \parse@UTFviii@a,% \parse@UTFviii@a!% \parse@UTFviii@b F\UTFviii@four@octets.{!,;}% \fi \fi \fi } \gdef\parse@UTFviii@a#1{% \@tempcnta\count@ \divide\count@ 64 \@tempcntb\count@ \multiply\count@ 64 \advance\@tempcnta-\count@ \advance\@tempcnta 128 \uccode`#1\@tempcnta \count@\@tempcntb} \gdef\parse@UTFviii@b#1#2#3#4{% \advance\count@ "#10\relax \uccode`#3\count@ \uppercase{\gdef\UTFviii@tmp{#2#3#4}}} \endgroup \@onlypreamble\DeclareUnicodeCharacter \@onlypreamble\parse@XML@charref \@onlypreamble\parse@UTFviii@a \@onlypreamble\parse@UTFviii@b \begingroup \def\cdp@elt#1#2#3#4{% \wlog{Now handling font encoding #1 ...}% \lowercase{% \InputIfFileExists{#1enc.dfu}} {\wlog{... processing UTF-8 mapping file for font encoding #1}}% {\wlog{... no UTF-8 mapping file for font encoding #1}}% } \cdp@list \endgroup \def\DeclareFontEncoding@#1#2#3{% \expandafter \ifx\csname T@#1\endcsname\relax \def\cdp@elt{\noexpand\cdp@elt}% \xdef\cdp@list{\cdp@list\cdp@elt{#1}% {\default@family}{\default@series}% {\default@shape}}% \expandafter\let\csname#1-cmd\endcsname\@changed@cmd \begingroup \wlog{Now handling font encoding #1 ...}% \lowercase{% \InputIfFileExists{#1enc.dfu}}% {\wlog{... processing UTF-8 mapping file for font encoding #1}}% {\wlog{... no UTF-8 mapping file for font encoding #1}}% \endgroup \else \@font@info{Redeclaring font encoding #1}% \fi \global\@namedef{T@#1}{#2}% \global\@namedef{M@#1}{\default@M#3}% \xdef\LastDeclaredEncoding{#1}% } \DeclareUnicodeCharacter{00A9}{\textcopyright} \DeclareUnicodeCharacter{00AA}{\textordfeminine} \DeclareUnicodeCharacter{00AE}{\textregistered} \DeclareUnicodeCharacter{00BA}{\textordmasculine} \DeclareUnicodeCharacter{02C6}{\textasciicircum} \DeclareUnicodeCharacter{02CA}{\textasciitilde} \DeclareUnicodeCharacter{200C}{\textcompwordmark} \DeclareUnicodeCharacter{2026}{\textellipsis} \DeclareUnicodeCharacter{2122}{\texttrademark} \DeclareUnicodeCharacter{2423}{\textvisiblespace} \endinput %% %% End of file `utf8.def'. 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Authors: James Honaker, Gary King, Matthew Blackwell

Amelia II "multiply imputes" missing data in a single cross-section (such as a survey), from a time series (like variables collected for each year in a country), or from a time-series-cross-sectional data set (such as collected by years for each of several countries). Amelia II implements our bootstrapping-based algorithm that gives essentially the same answers as the standard IP or EMis approaches, is usually considerably faster than existing approaches and can handle many more variables. Unlike Amelia I and other statistically rigorous imputation software, it virtually never crashes (but please let us know if you find to the contrary!). The program also generalizes existing approaches by allowing for trends in time series across observations within a cross-sectional unit, as well as priors that allow experts to incorporate beliefs they have about the values of missing cells in their data. Amelia II also includes useful diagnostics of the fit of multiple imputation models. The program works from the R command line or via a graphical user interface that does not require users to know R.


Amelia is named after this famous missing person.

Multiple imputation involves imputing m values for each missing cell in your data matrix and creating m "completed" data sets. (Across these completed data sets, the observed values are the same, but the missing values are filled in with different imputations that reflect our uncertainty about the missing data.) After imputation, Amelia will then save the m data sets. You then apply whatever statistical method you would have used if there had been no missing values to each of the m data sets, and use a simple procedure to combine the results. Under normal circumstances, you only need to impute once and can then analyze the m imputed data sets as many times and for as many purposes as you wish. The advantage of Amelia is that it combines the comparative speed and ease-of-use of our algorithm with the power of multiple imputation, to let you focus on your substantive research questions rather than spending time developing complex application-specific models for nonresponse in each new data set. Unless the rate of missingness is exceptionally high, m=5 (the program default) will usually be adequate. Other methods of dealing with missing data, such as listwise deletion, mean substitution, or single imputation, are in common circumstances biased, inefficient, or both. When multiple imputation works properly, it fills in data in such a way as to not change any relationships in the data but which enables the inclusion of all the observed data in the partially missing rows.

Amelia II is a new program, and follows in the spirit with the same purpose as the first version of Amelia by James Honaker, Anne Joseph, Gary King, Kenneth Scheve, and Naunihal Singh.

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Ȩ$I$gהtegsU QfRaSSaX,SGr umyBVM Pk֮~ (g!6$cue8cBS>hA78Z?YV[qEKq[\L]О$b` ZŔEe&Wgbiy) H$٤!}6DIYm-jrl6C*=̩]D܄ A  Tp- (gzT*K۪\W Z{ن =y3*] %=s^Bާ (gks>g RLg ͧ[g Y@*2kls7@h4ղX]kYRa:KIF6P{&y}7l"}x(I`sz[ݧ'b@*Y@&6;g tlF..o1-KA#C5Q#.=]Q^_J뵥҃/*JsG4\8HF Gdj#sh@i -E_:MA9󙁖*pL;D3.4#O{ZB& ZgC3Jz   fc&v^O%92 lS)j AF\HcJT8lW6kJ*f.b=搟WFP| UcK6lP` H[gO6HfIbj$#3tz fNYAvSeJZE)q77r6n d[@A,U %ī7jK<ڤZܤ&j%.gI/iȯ\Fo fLgE O:/Q"Y-A9F>\VIb+I9ff^(o,FlC2FK V)pdžn/ 0APD!]3ć$PN5rTy~/OH ݶ+77rx^%͂jzSlY uHLfK2#+azX1z$*,vzA  k|eQ/'޾}֬YrpF2l!ʙ{R?F@|ˣ p /^;nj&\D/dړ/Ot\9p1МY?ح0 tӜ39)Oꫯ_C$Zg~"~3 sLS |ܕ`-oO*i|] | VVާmn>>S,[,$Zg>3Is~6ٓ{bh{ \ f0Va 4rHmIcbX-qMNg9鏽Ƭl/UR3P|d's9L5`9L4#|v]$Op`D0W&(,}$-x L͂JP Lt-q'eKK̭i׵3d7_ڒ$L{ qK/a9T+[a| [x'8 pYhNO^vLaGp8^#-xms56V >MIBH.nQdT,΢JdX,đ$⪀5QCD#g oQu ҥald lA9CA9v&m ,A:vG>>E9cw!ە[|'ىmn,7UrGَ.Y\xp_Mn\`',p\wjc6]-}QhP_ǝyvZ 5!S@%-Wgƕv[a޴k onnX'Ɲg䲴9[p;˓{rC(gWT6f* 0 D_"˥2[) 5O\i{QGe8#&/3| OnjfMkU{ڞKww8v02Hk:4v-Zg p-;Q h /C9@PJvv66s|hs/*iRmk=r tcBD #;fh's=3 B\[VZͤ?^۔]cN3 L_J !S'qtB8 hddT u,ʙ`R2A:ZYwvݘ]e h~cOPbəi4LtRb+N6fV(Xϯ\:WNRSۛl휹p BE0\rG1>Ud&ࡿ gZ4e<'S\ѿ` lx'bF *۲v{ b \ ޤ#P!o c;@odG܀QA/J9_j%8>頜!HpYaswW|P$h(rĸnҿ99<<#z5k?;;;]vҥk׮ݾ}j 'Ũ&H.M1R"(}1,o1 JTD >?8`XgK$>@hnnOOQaaǏ7n!>3ll*G$O!!5`Liefy13 |l˔q{'ldxZ2"L 9smεm> ,(C9CHI]h?nk0%Цӳ}v'};`A@dj#hb1mreFXrjsD20K\ AL*A @5@A #3APAPAPduv˟==0o> A$Jc g=,gA n,s]}j_¯w@8F9C&9>UWV:*Svm,hO∖^%M*lb. AC9C&:6~ee[]'/]_D6v8A F;@p- #7k6_ټl6{\oooXE7Z,_~l6 h34>I}YJJX, D}}}}}}.ҍ=ZPάVj ?,O IjEECF,/κ_~7nL&ƍwwP3&*ђ3f5ׇ} 52aһ',,|̘1ǏB,^D䬷,,,l̙{&Swww/Xz{{K[ǏQDc1}坜m޼jj1cƸ6~b6bXK)2DB;su$q Hș3 uT$M4kw1zQ(g E?,IENDB`Amelia/vignettes/amelia.bib0000644000176200001440000001134212107457643015372 0ustar liggesusers @STRING{ jasa = "Journal of the American Statistical Association"} @article{Efron94, author={Bradley Efron}, title={Missing data, imputation, and the bootstrap}, journal={Journal of the American Statistical Association}, volume={89}, year={1994}, pages={463-475}, month={June}, number={426} } @article{Lahlrl03, author={P. Lahlrl}, title={On the impact of boostrapping in survey sampling and small area estimation}, journal={Statistical Science}, volume= 18, year= 2003, pages={199-210}, number= 2 } @book{Rubin87, author={Donald B. Rubin}, title={Multiple imputation for nonresponse in surveys}, publisher={John Wiley \& Sons}, year={1987}, address={New York} } @article{Rubin94, author={Donald B. Rubin}, title={Missing data, imputation, and the bootstrap: Comment}, journal={Journal of the American Statistical Association}, volume={89}, year={1994}, pages={475-478}, month={Jun}, number={426} } @article{RubSch86, author={Donald Rubin and Nathaniel Schenker}, title={Multiple imputation for interval estimation for simple random samples with ignorable nonresponse}, journal= jasa, volume= 81, year= 1986, pages={366-374}, number= 394 } @book{Schafer97, author={Joseph L. Schafer}, title={Analysis of incomplete multivariate data}, publisher={Chapman \& Hall}, year={1997}, address={London} } @article{ShaSit96, author={Jun Shao and Randy R. Sitter}, title={Bootstrap for imputed survey data}, journal={Journal of the American Statistical Association}, volume={91}, year={1996}, pages={1278-1288}, month={September}, number={435} } @article{MilKub05, title = {{Why the move to free trade? Democracy and trade policy in the developing countries}}, author = {Helen Milner and Keiko Kubota}, journal = {International Organization}, volume = {59}, number = {1}, pages = {107--143}, year = {2005} } @book{King89, author = {Gary King}, title = {Unifying political methodology: The likelihood theory of statistical inference}, publisher = {Michigan University Press}, year = 1989, address = {Ann Arbor} } @article{HonKin10, author = {James Honaker and Gary King}, title = {What to do about missing values in time series cross-section data}, journal = {American Journal of Political Science}, year = {2010}, volume = {54}, number = {2}, month = {April}, pages = {561--581}, note = {{http://gking.harvard.edu/files/abs/pr-abs.shtml}} } @article{KinHonJos01, author = {Gary King and James Honaker and Anne Joseph and Kenneth Scheve}, title = {Analyzing incomplete political science data: An alternative algorithm for multiple imputation}, journal = {American Political Science Review}, volume = 95, year = 2001, pages = {49--69}, month = {March}, number = 1 , note = {{http://gking.harvard.edu/files/abs/evil-abs.shtml}} } @article{KinTomWit00, author = {Gary King and Michael Tomz and Jason Wittenberg}, title = {Making the most of statistical analyses: Improving interpretation and presentation}, journal = {American Journal of Political Science}, volume = 44, year = 2000, pages = {341--355}, month = {April}, number = 2, note = {{http://gking.harvard.edu/files/abs/making-abs.shtml}} } @article{SchOls98, author={Joseph L. Schafer and Maren K. Olsen}, title={Multiple imputation for multivariate missing-data problems: A data analyst's perspective}, journal={Multivariate Behavioral Research}, volume={33}, year={1998}, pages={545-571}, number={4} } @article{DemLaiRub77, author={Arthur P. Dempster and N.M. Laird and D.B. Rubin}, title={Maximum likelihood estimation from incomplete data via the em algorithm}, journal={Journal of the Royal Statistical Society B}, volume={39}, year={1977}, pages={1-38} } @article{AbaGelLev08, author = {Kobi Abayomi and Andrew Gelman and Marc Levy}, title = {Diagnostics for multivariate imputations}, journal = {Applied Statistics}, volume = {57}, number = {3}, pages = {273--291}, year = {2008} } @misc{HonJosKin98, author = {James Honaker and Anne Joseph and Gary King and Kenneth Scheve and Naunihal Singh.}, title = {\pkg{AMELIA}: A program for missing data}, year = {1998-2002}, note = {http://gking.harvard.edu/amelia} } @Manual{R11, title = {\proglang{R}: A language and environment for statistical computing}, author = {{\proglang{R} Development Core Team}}, organization = {\proglang{R} Foundation for Statistical Computing}, address = {Vienna, Austria}, year = 2011, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org} }Amelia/vignettes/amelia.Rnw0000644000176200001440000030731012110743614015375 0ustar liggesusers\documentclass[12pt,letterpaper]{article} \usepackage[T1]{fontenc} \usepackage{Rd} \usepackage{Sweave} \usepackage{upquote} % === for bad lm output === % this wasn't working with rbuild %% so i took out the signficance stars %\usepackage[utf8]{inputenc} %\usepackage{Rd/Rd} %\usepackage{Rd/Sweave} %\usepackage{Rd/upquote} \SweaveOpts{width=70} % === graphic packages === \usepackage{graphicx} \usepackage{epstopdf} \DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png} % === bibliography package === \usepackage{natbib} % === margin and formatting === \usepackage{setspace} \usepackage{fullpage} \usepackage{caption} % === math packages === \usepackage[reqno]{amsmath} \usepackage{amsthm} \usepackage{amssymb,enumerate} \newtheorem{Com} {Comment} \newtheorem{prop}{Proposition} \newtheorem{thm}{Theorem} % === dcolumn package === \usepackage{dcolumn} \newcolumntype{.}{D{.}{.}{-1}} \newcolumntype{d}[1]{D{.}{.}{#1}} % === additional packages === \usepackage{url} \setcounter{tocdepth}{4} \usepackage{tikz} %\VignetteIndexEntry{Amelia II: A Package for Missing Data} %\VignetteDepends{Zelig} %\VignetteKeyWords{} %\VignettePackage{Amelia} \usepackage{color} \usepackage[pdftex, bookmarksopen=true, bookmarksnumbered=true, pdfstartview=FitH, breaklinks=true, urlbordercolor={0 1 0}, citebordercolor={0 0 1}]{hyperref} \newcommand{\obs}{{\text{obs}}} \newcommand{\mis}{{\text{mis}}} %\newcommand{\Amelia}{\texttt{Amelia}} \renewcommand{\R}{\textsf{R}} \newcommand{\Amelia}{\ensuremath{\mathbb A}melia} \newcommand{\AmeliaII}{\ensuremath{\mathbb A}melia~\ensuremath{\mathbb{II}}} \newcommand{\AmeliaView}{\ensuremath{\mathbb A}melia\ensuremath{\mathbb{V}}iew} \newcommand{\adjparbox}[2]{\parbox{#1}{\vspace{.35em}#2\vspace{.35em}}} \usepackage{vmargin} \topmargin=0in \title{${\mathbb{AMELIA~II}}$: A Program for Missing Data} % rbuild: replace 'Version ' '\\' Version \author{James Honaker, Gary King, and Matthew Blackwell} \begin{document} <>= ameliaVersion <- packageVersion("Amelia") @ \date{Version \Sexpr{ameliaVersion}\\ \today} \maketitle \tableofcontents \newpage \section{Introduction} \label{sec:intro} Missing data is a ubiquitous problem in social science data. Respondents do not answer every question, countries do not collect statistics every year, archives are incomplete, subjects drop out of panels. Most statistical analysis methods, however, assume the absence of missing data, and are only able to include observations for which every variable is measured. \AmeliaII\ allows users to impute (``fill in'' or rectangularize) incomplete data sets so that analyses which require complete observations can appropriately use all the information present in a dataset with missingness, and avoid the biases, inefficiencies, and incorrect uncertainty estimates that can result from dropping all partially observed observations from the analysis. \AmeliaII\ performs \emph{multiple imputation}, a general-purpose approach to data with missing values. Multiple imputation has been shown to reduce bias and increase efficiency compared to listwise deletion. Furthermore, ad-hoc methods of imputation, such as mean imputation, can lead to serious biases in variances and covariances. Unfortunately, creating multiple imputations can be a burdensome process due to the technical nature of algorithms involved. \Amelia\ provides users with a simple way to create and implement an imputation model, generate imputed datasets, and check its fit using diagnostics. The \AmeliaII\ program goes several significant steps beyond the capabilities of the first version of \Amelia\ \citep*{HonJosKin98}. For one, the bootstrap-based EMB algorithm included in \AmeliaII\ can impute many more variables, with many more observations, in much less time. The great simplicity and power of the EMB algorithm made it possible to write \AmeliaII\ so that it virtually never crashes --- which to our knowledge makes it unique among all existing multiple imputation software --- and is much faster than the alternatives too. \AmeliaII\ also has features to make valid and much more accurate imputations for cross-sectional, time-series, and time-series-cross-section data, and allows the incorporation of observation and data-matrix-cell level prior information. In addition to all of this, \AmeliaII\ provides many diagnostic functions that help users check the validity of their imputation model. This software implements the ideas developed in \citet*{HonKin10}. \section[What Amelia Does]{What \Amelia\ Does} \label{sec:what} Multiple imputation involves imputing $m$ values for each missing cell in your data matrix and creating $m$ ``completed'' data sets. Across these completed data sets, the observed values are the same, but the missing values are filled in with a distribution of imputations that reflect the uncertainty about the missing data. After imputation with \AmeliaII's EMB algorithm, you can apply whatever statistical method you would have used if there had been no missing values to each of the $m$ data sets, and use a simple procedure, described below, to combine the results\footnote{You can combine the results automatically by doing your data analyses within Zelig for R, or within Clarify for Stata; see \url{http://gking.harvard.edu/stats.shtml}.}. Under normal circumstances, you only need to impute once and can then analyze the $m$ imputed data sets as many times and for as many purposes as you wish. The advantage of \AmeliaII\ is that it combines the comparative speed and ease-of-use of our algorithm with the power of multiple imputation, to let you focus on your substantive research questions rather than spending time developing complex application-specific models for nonresponse in each new data set. Unless the rate of missingness is very high, $m = 5$ (the program default) is probably adequate. \subsection{Assumptions} The imputation model in \AmeliaII\ assumes that the complete data (that is, both observed and unobserved) are multivariate normal. If we denote the $(n \times k)$ dataset as $D$ (with observed part $D^{\obs}$ and unobserved part $D^{\mis}$), then this assumption is \begin{equation} \label{eq:norm} D \sim \mathcal{N}_k(\mu, \Sigma), \end{equation} which states that $D$ has a multivariate normal distribution with mean vector $\mu$ and covariance matrix $\Sigma$. The multivariate normal distribution is often a crude approximation to the true distribution of the data, yet there is evidence that this model works as well as other, more complicated models even in the face of categorical or mixed data \citep[see][]{Schafer97, SchOls98}. Furthermore, transformations of many types of variables can often make this normality assumption more plausible (see \ref{sec:trans} for more information on how to implement this in \Amelia). The essential problem of imputation is that we only observe $D^{\obs}$, not the entirety of $D$. In order to gain traction, we need to make the usual assumption in multiple imputation that the data are \emph{missing at random} (MAR). This assumption means that the pattern of missingness only depends on the observed data $D^{\obs}$, not the unobserved data $D^{\mis}$. Let $M$ to be the missingness matrix, with cells $m_{ij} = 1$ if $d_{ij} \in D^{\mis}$ and $m_{ij} = 0$ otherwise. Put simply, $M$ is a matrix that indicates whether or not a cell is missing in the data. With this, we can define the MAR assumption as \begin{equation} \label{eq:mar} p(M|D) = p(M|D^{\obs}). \end{equation} Note that MAR includes the case when missing values are created randomly by, say, coin flips, but it also includes many more sophisticated missingness models. When missingness is not dependent on the data at all, we say that the data are \emph{missing completely at random} (MCAR). \Amelia\ requires both the multivariate normality and the MAR assumption (or the simpler special case of MCAR). Note that the MAR assumption can be made more plausible by including additional variables in the dataset $D$ in the imputation dataset than just those eventually envisioned to be used in the analysis model. \subsection{Algorithm} In multiple imputation, we are concerned with the complete-data parameters, $\theta = (\mu, \Sigma)$. When writing down a model of the data, it is clear that our observed data is actually $D^{\obs}$ and $M$, the missingness matrix. Thus, the likelihood of our observed data is $p(D^{\obs}, M|\theta)$. Using the MAR assumption\footnote{There is an additional assumption hidden here that $M$ does not depend on the complete-data parameters.}, we can break this up, \begin{align} p(D^{\obs},M|\theta) = p(M|D^{\obs})p(D^{\obs}|\theta). \end{align} As we only care about inference on the complete data parameters, we can write the likelihood as \begin{align} L(\theta|D^{\obs}) &\propto p(D^{\obs}|\theta), \end{align} which we can rewrite using the law of iterated expectations as \begin{align} p(D^{\obs}|\theta) &= \int p(D|\theta) dD^{\mis}. \end{align} With this likelihood and a flat prior on $\theta$, we can see that the posterior is \begin{equation} p(\theta | D^{\obs}) \propto p(D^{\obs}|\theta) = \int p(D|\theta) dD^{\mis}. \end{equation} The main computational difficulty in the analysis of incomplete data is taking draws from this posterior. The EM algorithm \citep*{DemLaiRub77} is a simple computational approach to finding the mode of the posterior. Our EMB algorithm combines the classic EM algorithm with a bootstrap approach to take draws from this posterior. For each draw, we bootstrap the data to simulate estimation uncertainty and then run the EM algorithm to find the mode of the posterior for the bootstrapped data, which gives us fundamental uncertainty too (see \citet{HonKin10} for details of the EMB algorithm). Once we have draws of the posterior of the complete-data parameters, we make imputations by drawing values of $D^{\mis}$ from its distribution conditional on $D^{\obs}$ and the draws of $\theta$, which is a linear regression with parameters that can be calculated directly from $\theta$. \subsection{Analysis} In order to combine the results across $m$ data sets, first decide on the quantity of interest to compute, such as a univariate mean, regression coefficient, predicted probability, or first difference. Then, the easiest way is to draw $1/m$ simulations of $q$ from each of the $m$ data sets, combine them into one set of $m$ simulations, and then to use the standard simulation-based methods of interpretation common for single data sets \citep{KinTomWit00}. Alternatively, you can combine directly and use as the multiple imputation estimate of this parameter, $\bar{q}$, the average of the $m$ separate estimates, $q_j$ $(j=1,\dots,m)$: \begin{equation} \bar{q}=\frac{1}{m}\sum^{m}_{j=1}q_j. \end{equation} The variance of the point estimate is the average of the estimated variances from \emph{within} each completed data set, plus the sample variance in the point estimates \emph{across} the data sets (multiplied by a factor that corrects for the bias because $m<\infty$). Let $SE(q_j)^2$ denote the estimated variance (squared standard error) of $q_j$ from the data set $j$, and $S^{2}_{q}=\Sigma^{m}_{j=1}(q_j-\bar{q})^2/(m-1)$ be the sample variance across the $m$ point estimates. The standard error of the multiple imputation point estimate is the square root of \begin{equation} SE(q)^2=\frac{1}{m}\sum^{m}_{j=1}SE(q_j)^2+S^2_q(1+1/m). \end{equation} \begin{figure} \centering \begin{tikzpicture} [inner sep = 2.5mm, missdata/.style={rectangle, draw=red!50, fill=red!20, thick}, impdata/.style={rectangle, draw=blue!50, fill=blue!20, thick}, results/.style= {circle, draw=blue!50, fill=blue!20, thick}] \node (odata) at (3,3) [missdata] {}; \node [red!80,right] at (5.25,3) {incomplete data}; \node (boot1) at (1,1) [missdata] {} edge [<-, bend left=40] (odata); \node (boot2) at (2,1) [missdata] {} edge [<-, bend left=20] (odata); \node (boot3) at (3,1) [missdata] {} edge [<-] (odata); \node (boot4) at (4,1) [missdata] {} edge [<-, bend right=20] (odata); \node (boot5) at (5,1) [missdata] {} edge [<-, bend right=40] (odata); %% add "random question marks" \node[red!50!black] at (2.95,3.1) {\tiny ?}; \node[red!50!black] at (2.85,2.9) {\tiny ?}; \node[red!50!black] at (3.15,2.95) {\tiny ?}; \node[red!50!black] at (2.85,1.1) {\tiny ?}; \node[red!50!black] at (2.95,0.9) {\tiny ?}; \node[red!50!black] at (3.15,1.1) {\tiny ?}; \node[red!50!black] at (0.85,0.9) {\tiny ?}; \node[red!50!black] at (1.15,0.95) {\tiny ?}; \node[red!50!black] at (1.85,1.1) {\tiny ?}; \node[red!50!black] at (1.95,0.9) {\tiny ?}; \node[red!50!black] at (3.85,1.1) {\tiny ?}; \node[red!50!black] at (4.95,0.9) {\tiny ?}; \node[red!50!black] at (5.15,1.1) {\tiny ?}; \node [black!80,right] at (5.25,2) {bootstrap}; \node [red!80, right] at (5.25, 1) {bootstrapped data}; \node (imp1) at (1,0) [impdata] {} edge [<-] (boot1); \node (imp2) at (2,0) [impdata] {} edge [<-] (boot2); \node (imp3) at (3,0) [impdata] {} edge [<-] (boot3); \node (imp4) at (4,0) [impdata] {} edge [<-] (boot4); \node (imp5) at (5,0) [impdata] {} edge [<-] (boot5); \node [blue!80,right] at (5.25,0) {imputed datasets}; \node [black!80, right] at (5.25, 0.5) {EM}; \node (res1) at (1,-1) [results] {} edge [<-] (imp1); \node (res2) at (2,-1) [results] {} edge [<-] (imp2); \node (res3) at (3,-1) [results] {} edge [<-] (imp3); \node (res4) at (4,-1) [results] {} edge [<-] (imp4); \node (res5) at (5,-1) [results] {} edge [<-] (imp5); \node [black!80, right] at (5.25, -0.5) {analysis}; \node [blue!80, right] at (5.25, -1) {separate results}; \node[circle, draw=green!50!black!50, fill=green!20, thick] at (3,-3) {} edge [<-, bend right=40] (res5) edge [<-, bend left=40] (res1) edge [<-, bend right=20] (res4) edge [<-, bend left=20] (res2) edge [<-] (res3); \node [black!80, right] at (5.25, -2) {combination}; \node [green!80!black!80, right] at (5.25, -3) { final results}; \end{tikzpicture} \caption{A schematic of our approach to multiple imputation with the EMB algorithm. } \end{figure} \section[Versions of Amelia]{Versions of \Amelia\ } \label{sec:versions} Two versions of \AmeliaII\ are available, each with its own advantages and drawbacks, but both of which use the same underlying code and algorithms. First, \AmeliaII\ exists as a package for the \R\ statistical software package. Users can utilize their knowledge of the \R\ language to run \AmeliaII\ at the command line or to create scripts that will run \AmeliaII\ and preserve the commands for future use. Alternatively, you may prefer \AmeliaView, where an interactive Graphical User Interface (GUI) allows you to set options and run \Amelia\ without any knowledge of the \R\ programming language. Both versions of \AmeliaII\ are available on the Windows, Mac OS X, and Linux platforms and \AmeliaII\ for \R\ runs in any environment that \R\ can. All versions of \Amelia\ require the \R\ software, which is freely available at \url{http://www.r-project.org/}. Before installing \AmeliaII, you must have installed \R\ version 2.1.0 or higher, which is freely available at \url{http://www.r-project.org/}. \subsection{Installation and Updates from R} \label{sec:install} To install the \Amelia\ package on any platform, simply type the following at the \R\ command prompt, \begin{verbatim} > install.packages("Amelia") \end{verbatim} and \R\ will automatically install the package to your system from CRAN. If you wish to use the most current beta version of \Amelia\, feel free to install the test version, \begin{verbatim} > install.packages("Amelia", repos = "http://gking.harvard.edu") \end{verbatim} In order to keep your copy of \Amelia\ completely up to date, you should use the command \begin{verbatim} > update.packages() \end{verbatim} \subsection[Installation in Windows of AmeliaView as a Standalone Program]{Installation in Windows of \AmeliaView\ as a Standalone Program} \label{sec:win-install} To install a standalone version of \AmeliaView\ in the Windows environment, simply download the installer \texttt{setup.exe} from \url{http://gking.harvard.edu/amelia/} and run it. The installer will ask you to choose a location to install \AmeliaII. If you have installed \R\ with the default options, \AmeliaII\ will automatically find the location of \R. If the installer cannot find \R, it will ask you to locate the directory of the most current version of \R. Make sure you choose the directory name that includes the version number of \R\ (e.g. C:/Program Files/R/R-2.9.0) and contains a subdirectory named \texttt{bin}. The installer will also put shortcuts on your Desktop and Start Menu. Even users familiar with the \R\ language may find it useful to utilize \AmeliaView\ to set options on variables, change arguments, or run diagnostics. From the command line, \AmeliaView\ can be brought up with the call: \begin{verbatim} > library(Amelia) > AmeliaView() \end{verbatim} \subsection{Linux (local installation)} \label{sec:lin-install} Installing \Amelia\ on a Linux system is slightly more complicated due to user permissions. If you are running \R\ with root access, you can simply run the above installation procedure. If you do not have root access, you can install \Amelia\ to a local library. First, create a local directory to house the packages, \begin{verbatim} w4:mblackwell [~]: mkdir ~/myrlibrary \end{verbatim} and then, in an \R\ session, install the package directing \R\ to this location: \begin{verbatim} > install.packages("Amelia", lib = "~/myrlibrary") \end{verbatim} Once this is complete you need to edit or create your \R\ profile. Locate or create \texttt{\~/.Rprofile} in your home directory and add this line: \begin{verbatim} .libPath("~/myrlibrary") \end{verbatim} This will add your local library to the list of library paths that \R\ searches in when you load libraries. Linux users can use \AmeliaView\ in the same way as Windows users of \Amelia\ for \R. From the command line, \AmeliaView\ can be brought up with the call: \begin{verbatim} > AmeliaView() \end{verbatim} \section{A User's Guide} \label{sec:guide} \subsection{Data and Initial Results} We now demonstrate how to use \Amelia\ using data from \citet{MilKub05} which studies the effect of democracy on trade policy. For the purposes of this user's guide, we will use a subset restricted to nine developing countries in Asia from 1980 to 1999\footnote{We have artificially added some missingness to these data for presentational purposes. You can access the original data at \url{http://www.princeton.edu/~hmilner/Research.htm}}. This dataset includes 9 variables: year (\code{year}), country (\code{country}), average tariff rates (\code{tariff}), Polity IV score\footnote{The Polity score is a number between -10 and 10 indicating how democratic a country is. A fully autocratic country would be a -10 while a fully democratic country would be 1 10.} (\code{polity}), total population (\code{pop}), gross domestic product per capita (\code{gdp.pc}), gross international reserves (\code{intresmi}), a dummy variable signifying whether the country had signed an IMF agreement in that year (\code{signed}), a measure of financial openness (\code{fivop}), and a measure of US hegemony\footnote{This measure of US hegemony is the US imports and exports as a percent of the world total imports and exports.} (\code{usheg}). These variables correspond to the variables used in the analysis model of \citet{MilKub05} in table 2. <>= options("digits"=4) options("width"=70) options("show.signif.stars" = FALSE) set.seed(12345) @ We first load the \Amelia\ and the data: <>= require(Amelia) data(freetrade) @ We can check the summary statistics of the data to see that there is missingness on many of the variables: <<>= summary(freetrade) @ In the presence of missing data, most statistical packages use \emph{listwise deletion}, which removes any row that contains a missing value from the analysis. Using the base model of \citet{MilKub05} table 2, we run a simple linear model in \R, which uses listwise deletion: <>= summary(lm(tariff ~ polity + pop + gdp.pc + year + country, data = freetrade)) @ Note that 60 of the 171 original observations are deleted due to missingness. These observations, however, are partially observed, and contain valuable information about the relationships between those variables which are present in the partially completed observations. Multiple imputation will help us retrieve that information and make better, more efficient, inferences. \subsection{Multiple Imputation} When performing multiple imputation, the first step is to identify the variables to include in the imputation model. It is crucial to include at least as much information as will be used in the analysis model. That is, any variable that will be in the analysis model should also be in the imputation model. This includes any transformations or interactions of variables that will appear in the analysis model. In fact, it is often useful to add more information to the imputation model than will be present when the analysis is run. Since imputation is predictive, any variables that would increase predictive power should be included in the model, even if including them in the analysis model would produce bias in estimating a causal effect (such as for post-treatment variables) or collinearity would preclude determining which variable had a relationship with the dependent variable (such as including multiple alternate measures of GDP). In our case, we include all the variables in \code{freetrade} in the imputation model, even though our analysis model focuses on \code{polity}, \code{pop} and \code{gdp.pc}\footnote{Note that this specification does not utilize time or spatial data yet. The \texttt{ts} and \texttt{cs} arguments only have force when we also include \texttt{polytime} or \texttt{intercs}, discussed in section \ref{sec:tscs}}. To create multiple imputations in \Amelia, we can simply run <>= a.out <- amelia(freetrade, m = 5, ts = "year", cs = "country") a.out @ Note that our example dataset is deliberately small both in variables and in cross-sectional elements. Typical datasets may often have hundreds or possibly a couple thousand steps to the EM algorithm. Long chains should remind the analyst to consider whether transformations of the variables would more closely fit the multivariate normal assumptions of the model (correct but omitted transformations will shorten the number of steps and improve the fit of the imputations), but do not necessarily denote problems with the imputation model. The output gives some information about how the algorithm ran. Each of the imputed datasets is now in the list \code{a.out\$imputations}. Thus, we could plot a histogram of the \code{tariff} variable from the 3rd imputation, <>= hist(a.out$imputations[[3]]$tariff, col="grey", border="white") @ %$ <>= options(SweaveHooks = list(fig = function() par(mfrow=c(1,1)))) @ \begin{figure} \begin{center} <>= <> @ \caption{Histogram of the \texttt{tariff} variable from the 3rd imputed dataset.} \label{fig:hist1} \end{center} \end{figure} \subsubsection{Saving imputed datasets} \label{sec:saving} If you need to save your imputed datasets, one direct method is to save the output list from \code{amelia}, \begin{verbatim} > save(a.out, file = "imputations.RData") \end{verbatim} As in the previous example, the \emph{i}th imputed datasets can be retrieved from this list as \code{a.out\$imputations[[i]]}. In addition, you can save each of the imputed datasets to its own file using the \code{write.amelia} command, \begin{verbatim} > write.amelia(obj=a.out, file.stem = "outdata") \end{verbatim} This will create one comma-separated value file for each imputed dataset in the following manner: \begin{verbatim} outdata1.csv outdata2.csv outdata3.csv outdata4.csv outdata5.csv \end{verbatim} The \code{write.amelia} function can also save files in tab-delimited and Stata (\code{.dta}) file formats. For instance, to save Stata files, simply change the \code{format} argument to \code{"dta"}, \begin{verbatim} > write.amelia(obj=a.out, file.stem = "outdata", format = "dta") \end{verbatim} Additionally, \code{write.amelia} can create a ``stacked'' version of the imputed dataset which stacks each imputed dataset on top of one another. This can be done by setting the \code{separate} argument to \code{FALSE}. The resulting matrix is of size $(N \cdot m) \times p$ if the original dataset is excluded (\code{orig.data = FALSE}) and of size $(N \cdot (m+1))\times p$ if it is included (\code{orig.data = TRUE}). The stacked dataset will include a variable (set with \code{impvar}) that indicates to which imputed dataset the observation belongs. See Section~\ref{sec:analysis} for a description of how to use this stacked dataset with the \code{mi} commands in Stata. \subsubsection[Combining Multiple Amelia Runs]{Combining Multiple \Amelia\ Runs} The EMB algorithm is what computer scientists call \emph{embarrassingly parallel}, meaning that it is simple to separate each imputation into parallel processes. With \Amelia\ it is simple to run subsets of the imputations on different machines and then combine them after the imputation for use in analysis model. This allows for a huge increase in the speed of the algorithm. Output lists from different \Amelia\ runs can be combined together into a new list. For instance, suppose that we wanted to add another ten imputed datasets to our earlier call to \code{amelia}. First, run the function to get these additional imputations, <<>>= a.out.more <- amelia(freetrade, m = 10, ts = "year", cs = "country", p2s=0) a.out.more @ then combine this output with our original output using the \code{ameliabind} function, <<>>= a.out.more <- ameliabind(a.out, a.out.more) a.out.more @ This function binds the two outputs into the same output so that you can pass the combined imputations easily to analysis models and diagnostics. Note that \code{a.out.more} now has a total of 15 imputations. A simple way to execute a parallel processing scheme with \Amelia\ would be to run \code{amelia} with \code{m} set to 1 on $m$ different machines or processors, save each output using the \code{save} function, load them all on the same \R\ session using \code{load} command and then combine them using \code{ameliabind}. In order to do this, however, make sure to name each of the outputs a different name so that they do not overwrite each other when loading into the same \R\ session. Also, some parallel environments will dump all generated files into a common directory, where they may overwrite each other. If it is convenient in a parallel environment to run a large number of \code{amelia} calls from a single piece of code, one useful way to avoid overwriting is to create the \code{file.stem} with a random suffix. For example: \begin{verbatim} > b<-round(runif(1,min=1111,max=9999)) > random.name<-paste("am",b,sep="") > amelia <- write.amelia(obj=a.out, file.stem = random.name) \end{verbatim} \subsubsection{Screen Output} Screen output can be adjusted with the ``print to screen'' argument, \code{p2s}. At a value of 0, no screen printing will occur. This may be useful in large jobs or simulations where a very large number of imputation models may be required. The default value of 1, lists each bootstrap, and displays the number of iterations required to reach convergence in that bootstrapped dataset. The value of 2 gives more thorough screen output, including, at each iteration, the number of parameters that have significantly changed since the last iteration. This may be useful when the EM chain length is very long, as it can provide an intuition for many parameters still need to converge in the EM chain, and a sense of the time remaining. However, it is worth noting that the last several parameters can often take a significant fraction of the total number of iterations to converge. Setting \code{p2s} to 2 will also generate information on how EM algorithm is behaving, such as a \code{!} when the current estimated complete data covariance matrix is not invertible and a \code{*} when the likelihood has not monotonically increased in that step. Having many of these two symbols in the screen output is an indication of a problematic imputation model\footnote{Problems of non-invertible matrices often mean that current guess for the covariance matrix is singular. This is a sign that there may be two highly correlated variables in the model. One way to resolve is to use a ridge prior (see \ref{sec:prior})}. An example of the output when \code{p2s} is 2 would be <>= amelia(freetrade, m = 1, ts = "year", cs = "country", p2s = 2) @ \subsection{Parallel Imputation Using Multicore CPUs} \label{sec:parallel} Each imputation in the above EMB algorithm is completely independent of any other imputation, a property called embarrassingly parallel. This type of approach can take advantage of the multiple -core infrastructure of modern CPUs. Each core in a multi-core processor can execute independent operations in parallel. \Amelia\ can utilize this parallel processing internally \emph{via} the \code{parallel} and the \code{ncpus} arguments. The \code{parallel} argument sets the parallel processing backend, either with \code{"multicore"} or \code{"snow"} (or \code{"no"} for no parallel processing). The \code{"multicore"} backend is not available on Windows systems, but tends to be quicker at parallel processing. On a Windows system, the \code{"snow"} backend provides parallel processing through a cluster of worker processes across the CPUs. You can set the default for this argument using the \code{"amelia.parallel"} option. This allows you to run Amelia in parallel as the default for an entire R session without setting arguments in the \code{amelia} call. For each of the parallel backends, \Amelia\ requires a number of CPUs to use in parallel. This can be set using the \code{ncpus} argument. It can be higher than the number of physical cores in the system if hyperthreading or other technologies are available. You can use the \code{parallel::detectCores} function to determine how many cores are available on your machine. The default for this argument can be set using the \code{"amelia.ncpus"} option. On Unix-alike systems (such as Mac OS X and Linux distributions), the \code{"multicore"} backend automatically sets up and stops the parallel workers by forking the process. On Windows, the \code{"snow"} backend requires more attention. \Amelia\ will attempt to create a parallel cluster of worker processes (since Windows systems cannot fork a process) and will stop this cluster after the imputations are complete. Alternatively, \Amelia\ also has a \code{cl} argument, which accepts a predefined cluster made using the \code{parallel::makePSOCKcluster}. For more information about parallel processing in R, see the documentation for the \pkg{parallel} package that ships along with R. \subsection{Imputation-improving Transformations} \label{sec:trans} Social science data commonly includes variables that fail to fit to a multivariate normal distribution. Indeed, numerous models have been introduced specifically to deal with the problems they present. As it turns out, much evidence in the literature (discussed in \citealt{KinHonJos01}) indicates that the multivariate normal model used in \Amelia\ usually works well for the imputation stage even when discrete or non-normal variables are included and when the analysis stage involves these limited dependent variable models. Nevertheless, \Amelia\ includes some limited capacity to deal directly with ordinal and nominal variables and to modify variables that require other transformations. In general nominal and log transform variables should be declared to \Amelia, whereas ordinal (including dichotomous) variables often need not be, as described below. (For harder cases, see \citep{Schafer97}, for specialized MCMC-based imputation models for discrete variables.)\nocite{KinHonJos01} Although these transformations are taken internally on these variables to better fit the data to the multivariate normal assumptions of the imputation model, all the imputations that are created will be returned in the original untransformed form of the data. If the user has already performed transformations on their data (such as by taking a log or square root prior to feeding the data to \code{amelia}) these do not need to be declared, as that would result in the transformation occurring \emph{doubly} in the imputation model. The fully imputed data sets that are returned will always be in the form of the original data that is passed to the \code{amelia} routine. \subsubsection{Ordinal} \label{sec:ord} In much statistical research, researchers treat independent ordinal (including dichotomous) variables as if they were really continuous. If the analysis model to be employed is of this type, then nothing extra is required of the of the imputation model. Users are advised to allow \Amelia\ to impute non-integer values for any missing data, and to use these non-integer values in their analysis. Sometimes this makes sense, and sometimes this defies intuition. One particular imputation of 2.35 for a missing value on a seven point scale carries the intuition that the respondent is between a 2 and a 3 and most probably would have responded 2 had the data been observed. This is easier to accept than an imputation of 0.79 for a dichotomous variable where a zero represents a male and a one represents a female respondent. However, in both cases the non-integer imputations carry more information about the underlying distribution than would be carried if we were to force the imputations to be integers. Thus whenever the analysis model permits, missing ordinal observations should be allowed to take on continuously valued imputations. In the \code{freetrade} data, one such ordinal variable is \code{polity} which ranges from -10 (full autocracy) to 10 (full democracy). If we tabulate this variable from one of the imputed datasets, <>= table(a.out$imputations[[3]]$polity) @ %$ we can see that there is one imputation between -4 and -3 and one imputation between 6 and 7. Again, the interpretation of these values is rather straightforward even if they are not strictly in the coding of the original Polity data. Often, however, analysis models require some variables to be strictly ordinal, as for example, when the dependent variable will be modeled in a logistical or Poisson regression. Imputations for variables set as ordinal are created by taking the continuously valued imputation and using an appropriately scaled version of this as the probability of success in a binomial distribution. The draw from this binomial distribution is then translated back into one of the ordinal categories. For our data we can simply add \code{polity} to the \code{ords} argument: <>= a.out1 <- amelia(freetrade, m = 5, ts = "year", cs = "country", ords = "polity", p2s = 0) table(a.out1$imputations[[3]]$polity) @ Now, we can see that all of the imputations fall into one of the original polity categories. \subsubsection{Nominal} \label{sec:nom} Nominal variables\footnote{Dichotomous (two category) variables are a special case of nominal variables. For these variables, the nominal and ordinal methods of transformation in \Amelia\ agree.} must be treated quite differently than ordinal variables. Any multinomial variables in the data set (such as religion coded 1 for Catholic, 2 for Jewish, and 3 for Protestant) must be specified to \Amelia. In our \code{freetrade} dataset, we have \code{signed} which is 1 if a country signed an IMF agreement in that year and 0 if it did not. Of course, our first imputation did not limit the imputations to these two categories <<>>= table(a.out1$imputations[[3]]$signed) @ In order to fix this for a $ p$-category multinomial variable,\Amelia\ will determine $ p$ (as long as your data contain at least one value in each category), and substitute $ p-1$ binary variables to specify each possible category. These new $ p-1$ variables will be treated as the other variables in the multivariate normal imputation method chosen, and receive continuous imputations. These continuously valued imputations will then be appropriately scaled into probabilities for each of the $ p$ possible categories, and one of these categories will be drawn, where upon the original $ p$-category multinomial variable will be reconstructed and returned to the user. Thus all imputations will be appropriately multinomial. For our data we can simply add \code{signed} to the \code{noms} argument: <>= a.out2 <- amelia(freetrade, m = 5, ts = "year", cs = "country", noms = "signed", p2s = 0) table(a.out2$imputations[[3]]$signed) @ Note that \Amelia\ can only fit imputations into categories that exist in the original data. Thus, if there was a third category of signed, say 2, that corresponded to a different kind of IMF agreement, but it never occurred in the original data, \Amelia\ could not match imputations to it. Since \Amelia\ properly treats a $ p$-category multinomial variable as $ p-1$ variables, one should understand the number of parameters that are quickly accumulating if many multinomial variables are being used. If the square of the number of real and constructed variables is large relative to the number of observations, it is useful to use a ridge prior as in section \ref{sec:prior}. \subsubsection{Natural Log} \label{sec:log} If one of your variables is heavily skewed or has outliers that may alter the imputation in an unwanted way, you can use a natural logarithm transformation of that variable in order to normalize its distribution. This transformed distribution helps \Amelia\ to avoid imputing values that depend too heavily on outlying data points. Log transformations are common in expenditure and economic variables where we have strong beliefs that the marginal relationship between two variables decreases as we move across the range. For instance, figure \ref{fig:logshist} show the \code{tariff} variable clearly has positive (or, right) skew while its natural log transformation has a roughly normal distribution. <>= hist(freetrade$tariff, col="grey", border="white") hist(log(freetrade$tariff), col="grey", border="white") @ %$ <>= options(SweaveHooks = list(fig = function() par(mfrow=c(1,2)))) @ \begin{figure} \begin{center} <>= <> @ \caption{Histogram of \texttt{tariff} and \texttt{log(tariff)}.} \label{fig:logshist} \end{center} \end{figure} <>= options(SweaveHooks = list(fig = function() par(mfrow=c(1,1)))) @ \subsubsection{Square Root} \label{sec:sqrt} Event count data is often heavily skewed and has nonlinear relationships with other variables. One common transformation to tailor the linear model to count data is to take the square roots of the counts. This is a transformation that can be set as an option in \Amelia. \subsubsection{Logistic} \label{sec:lgstc} Proportional data is sharply bounded between 0 and 1. A logistic transformation is one possible option in \Amelia\ to make the distribution symmetric and relatively unbounded. \subsection{Identification Variables} \label{sec:idvars} Datasets often contain identification variables, such as country names, respondent numbers, or other identification numbers, codes or abbreviations. Sometimes these are text and sometimes these are numeric. Often it is not appropriate to include these variables in the imputation model, but it is useful to have them remain in the imputed datasets (However, there are models that would include the ID variables in the imputation model, such as fixed effects model for data with repeated observations of the same countries). Identification variables which are not to be included in the imputation model can be identified with the argument \code{idvars}. These variables will not be used in the imputation model, but will be kept in the imputed datasets. If the \code{year} and \code{country} contained no information except labels, we could omit them from the imputation: <>= amelia(freetrade, idvars = c("year", "country")) @ Note that Amelia will return with an error if your dataset contains a factor or character variable that is not marked as a nominal or identification variable. Thus, if we were to omit the factor \code{country} from the \code{cs} or \code{idvars} arguments, we would receive an error: <<>>= a.out2 <- amelia(freetrade, idvars = c("year")) @ In order to conserve memory, it is wise to remove unnecessary variables from a data set before loading it into \Amelia. The only variables you should include in your data when running \Amelia\ are variables you will use in the analysis stage and those variables that will help in the imputation model. While it may be tempting to simply mark unneeded variables as IDs, it only serves to waste memory and slow down the imputation procedure. \subsection{Time Series, or Time Series Cross Sectional Data} \label{sec:tscs} Many variables that are recorded over time within a cross-sectional unit are observed to vary smoothly over time. In such cases, knowing the observed values of observations close in time to any missing value may enormously aid the imputation of that value. However, the exact pattern may vary over time within any cross-section. There may be periods of growth, stability, or decline; in each of which the observed values would be used in a different fashion to impute missing values. Also, these patterns may vary enormously across different cross-sections, or may exist in some and not others. \Amelia\ can build a general model of patterns within variables across time by creating a sequence of polynomials of the time index. If, for example, tariffs vary smoothly over time, then we make the modeling assumption that there exists some polynomial that describes the economy in cross-sectional unit $i$ at time $t$ as: \begin{equation} \textrm{tariff}_{ti} = \beta_0 + \beta_1 t + \beta_1 t^2 + \beta_1 t^3 \ldots \end{equation} And thus if we include enough higher order terms of time then the pattern between observed values of the tariff rate can be estimated. \Amelia\ will create polynomials of time up to the user defined $k$-th order, ($k\leq3$). We can implement this with the \code{ts} and \code{polytime} arguments. If we thought that a second-order polynomial would help predict we could run <>= a.out2 <- amelia(freetrade, ts = "year", cs = "country", polytime = 2) @ With this input, \Amelia\ will add covariates to the model that correspond to time and its polynomials. These covariates will help better predict the missing values. If cross-sectional units are specified these polynomials can be interacted with the cross-section unit to allow the patterns over time to vary between cross-sectional units. Unless you strongly believe all units have the same patterns over time in all variables (including the same constant term), this is a reasonable setting. When $k$ is set to 0, this interaction simply results in a model of \emph{fixed effects} where every unit has a uniquely estimated constant term. \Amelia\ does not smooth the observed data, and only uses this functional form, or one you choose, with all the other variables in the analysis and the uncertainty of the prediction, to impute the missing values. In order to impute with trends specific to each cross-sectional unit, we can set \code{intercs} to \code{TRUE}: <>= a.out.time <- amelia(freetrade, ts = "year", cs = "country", polytime = 2, intercs = TRUE, p2s = 2) @ Note that attempting to use \code{polytime} without the \code{ts} argument, or \code{intercs} without the \code{cs} argument will result in an error. Using the \code{tscsPlot} function (discussed below), we can see in figure \ref{fig:tcomp} that we have a much better prediction about the missing values when incorporating time than when we omit it: <>= tscsPlot(a.out, cs = "Malaysia", main = "Malaysia (no time settings)", var = "tariff", ylim = c(-10, 60)) tscsPlot(a.out.time, cs = "Malaysia", main = "Malaysia (with time settings)", var = "tariff", ylim = c(-10, 60)) @ <>= options(SweaveHooks = list(fig = function() par(mfrow=c(1,2)))) @ \begin{figure} \begin{center} <