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l 78.35 205.87 l 79.93 210.14 l 81.50 203.20 l 83.08 199.69 l 84.66 196.87 l 86.23 196.29 l 87.81 198.90 l 89.38 200.63 l 90.96 202.03 l 92.53 200.25 l 94.11 198.79 l 95.68 197.17 l 97.26 197.19 l 98.84 196.46 l 100.41 196.80 l 101.99 197.38 l 103.56 196.06 l 105.14 195.08 l 106.71 195.21 l 108.29 195.31 l 109.86 197.11 l 111.44 196.60 l 113.01 196.81 l 114.59 197.97 l 116.17 198.39 l 117.74 201.02 l 119.32 200.10 l 120.89 200.04 l S Q q 0.000 0.000 0.000 RG 0.75 w [] 0 d 1 J 1 j 10.00 M 42.04 192.47 m 120.89 192.47 l S 42.04 192.47 m 42.04 187.72 l S 57.81 192.47 m 57.81 187.72 l S 73.58 192.47 m 73.58 187.72 l S 89.35 192.47 m 89.35 187.72 l S 105.12 192.47 m 105.12 187.72 l S 120.89 192.47 m 120.89 187.72 l S BT 0.000 0.000 0.000 rg /F2 1 Tf 8.00 0.00 -0.00 8.00 39.81 175.36 Tm (0) Tj /F2 1 Tf 8.00 0.00 -0.00 8.00 62.46 175.36 Tm (20000) Tj /F2 1 Tf 8.00 0.00 -0.00 8.00 94.00 175.36 Tm (40000) Tj ET 38.97 194.57 m 38.97 246.94 l S 38.97 194.57 m 34.21 194.57 l S 38.97 207.66 m 34.21 207.66 l S 38.97 220.75 m 34.21 220.75 l S 38.97 233.85 m 34.21 233.85 l S 38.97 246.94 m 34.21 246.94 l S BT /F2 1 Tf 0.00 8.00 -8.00 0.00 27.56 186.78 Tm (1.00) Tj /F2 1 Tf 0.00 8.00 -8.00 0.00 27.56 212.97 Tm (1.10) Tj /F2 1 Tf 0.00 8.00 -8.00 0.00 27.56 239.15 Tm (1.20) Tj ET 38.97 192.47 m 124.04 192.47 l 124.04 249.03 l 38.97 249.03 l 38.97 192.47 l S Q q 0.00 144.00 144.00 144.00 re W n BT 0.000 0.000 0.000 rg /F2 1 Tf 8.00 0.00 -0.00 8.00 45.04 156.36 Tm (last iteration in chain) Tj /F2 1 Tf 0.00 8.00 -8.00 0.00 8.55 198.91 Tm (shrink factor) Tj ET Q q 38.97 192.47 85.08 56.56 re W n 1.000 0.000 0.000 RG 0.75 w [ 3.00 5.00] 0 d 1 J 1 j 10.00 M 46.06 432.00 m 46.84 298.21 l 47.98 432.00 l S 49.01 432.00 m 49.99 346.10 l 51.57 233.65 l 53.15 258.92 l 54.72 220.87 l 56.30 211.65 l 57.87 198.62 l 59.45 203.80 l 61.02 222.44 l 62.60 212.19 l 64.17 227.61 l 65.75 213.40 l 67.33 208.00 l 68.90 204.86 l 70.48 198.33 l 72.05 196.32 l 73.63 201.31 l 75.20 204.72 l 76.78 213.56 l 78.35 230.82 l 79.93 245.02 l 81.50 223.65 l 83.08 212.45 l 84.66 202.76 l 86.23 199.04 l 87.81 208.22 l 89.38 215.30 l 90.96 220.06 l 92.53 213.78 l 94.11 208.39 l 95.68 203.09 l 97.26 202.92 l 98.84 200.38 l 100.41 202.00 l 101.99 204.09 l 103.56 199.65 l 105.14 196.30 l 106.71 196.67 l 108.29 197.01 l 109.86 202.03 l 111.44 199.93 l 113.01 200.51 l 114.59 202.59 l 116.17 201.71 l 117.74 206.59 l 119.32 205.22 l 120.89 205.69 l S 0.000 0.000 0.000 RG 0.75 w [] 0 d 38.97 194.57 m 124.04 194.57 l S Q q 0.00 144.00 144.00 144.00 re W n BT 0.000 0.000 0.000 rg /F3 1 Tf 10.00 0.00 -0.00 10.00 51.22 264.93 Tm (EraPlus:Surf) Tj ET Q q 182.97 192.47 85.08 56.56 re W n Q q 182.97 192.47 85.08 56.56 re W n 0.000 0.000 0.000 RG 0.75 w [] 0 d 1 J 1 j 10.00 M 187.64 432.00 m 187.69 331.03 l 189.27 286.27 l 190.84 204.84 l 192.42 216.00 l 193.99 211.71 l 195.57 224.58 l 197.15 200.56 l 198.72 207.02 l 200.30 217.19 l 201.87 222.04 l 203.45 220.49 l 205.02 227.43 l 206.60 214.38 l 208.17 203.91 l 209.75 211.95 l 211.33 215.35 l 212.90 216.53 l 214.48 218.58 l 216.05 224.00 l 217.63 218.19 l 219.20 218.79 l 220.78 216.27 l 222.35 211.64 l 223.93 206.87 l 225.50 208.25 l 227.08 204.91 l 228.66 201.90 l 230.23 200.01 l 231.81 198.17 l 233.38 199.62 l 234.96 202.63 l 236.53 204.11 l 238.11 204.59 l 239.68 205.32 l 241.26 209.05 l 242.84 214.38 l 244.41 212.69 l 245.99 211.21 l 247.56 211.36 l 249.14 208.29 l 250.71 204.84 l 252.29 204.88 l 253.86 207.01 l 255.44 205.68 l 257.01 203.04 l 258.59 204.76 l 260.17 204.99 l 261.74 203.58 l 263.32 202.36 l 264.89 200.42 l S Q q 0.000 0.000 0.000 RG 0.75 w [] 0 d 1 J 1 j 10.00 M 186.04 192.47 m 264.89 192.47 l S 186.04 192.47 m 186.04 187.72 l S 201.81 192.47 m 201.81 187.72 l S 217.58 192.47 m 217.58 187.72 l S 233.35 192.47 m 233.35 187.72 l S 249.12 192.47 m 249.12 187.72 l S 264.89 192.47 m 264.89 187.72 l S BT 0.000 0.000 0.000 rg /F2 1 Tf 8.00 0.00 -0.00 8.00 183.81 175.36 Tm (0) Tj /F2 1 Tf 8.00 0.00 -0.00 8.00 206.46 175.36 Tm (20000) Tj /F2 1 Tf 8.00 0.00 -0.00 8.00 238.00 175.36 Tm (40000) Tj ET 182.97 194.57 m 182.97 246.94 l S 182.97 194.57 m 178.21 194.57 l S 182.97 207.66 m 178.21 207.66 l S 182.97 220.75 m 178.21 220.75 l S 182.97 233.85 m 178.21 233.85 l S 182.97 246.94 m 178.21 246.94 l S BT /F2 1 Tf 0.00 8.00 -8.00 0.00 171.56 186.78 Tm (1.00) Tj /F2 1 Tf 0.00 8.00 -8.00 0.00 171.56 212.97 Tm (1.10) Tj /F2 1 Tf 0.00 8.00 -8.00 0.00 171.56 239.15 Tm (1.20) Tj ET 182.97 192.47 m 268.04 192.47 l 268.04 249.03 l 182.97 249.03 l 182.97 192.47 l S Q q 144.00 144.00 144.00 144.00 re W n BT 0.000 0.000 0.000 rg /F2 1 Tf 8.00 0.00 -0.00 8.00 189.04 156.36 Tm (last iteration in chain) Tj /F2 1 Tf 0.00 8.00 -8.00 0.00 152.55 198.91 Tm (shrink factor) Tj ET Q q 182.97 192.47 85.08 56.56 re W n 1.000 0.000 0.000 RG 0.75 w [ 3.00 5.00] 0 d 1 J 1 j 10.00 M 189.56 432.00 m 190.84 223.10 l 192.42 252.94 l 193.99 243.71 l 195.57 282.91 l 197.15 208.59 l 198.72 231.26 l 200.30 257.46 l 201.87 273.04 l 203.45 263.58 l 205.02 289.02 l 206.60 246.76 l 208.17 206.04 l 209.75 239.56 l 211.33 257.93 l 212.90 262.75 l 214.48 270.36 l 216.05 287.95 l 217.63 269.24 l 219.20 271.12 l 220.78 261.45 l 222.35 244.15 l 223.93 228.74 l 225.50 234.12 l 227.08 224.10 l 228.66 215.98 l 230.23 211.16 l 231.81 204.56 l 233.38 210.15 l 234.96 220.60 l 236.53 225.53 l 238.11 227.97 l 239.68 230.64 l 241.26 241.86 l 242.84 258.31 l 244.41 253.44 l 245.99 248.97 l 247.56 249.45 l 249.14 240.13 l 250.71 229.37 l 252.29 229.53 l 253.86 236.17 l 255.44 232.06 l 257.01 223.55 l 258.59 229.01 l 260.17 229.64 l 261.74 225.00 l 263.32 221.01 l 264.89 214.18 l S 0.000 0.000 0.000 RG 0.75 w [] 0 d 182.97 194.57 m 268.04 194.57 l S Q q 144.00 144.00 144.00 144.00 re W n BT 0.000 0.000 0.000 rg /F3 1 Tf 10.00 0.00 -0.00 10.00 202.72 264.93 Tm (Solo:Tide) Tj ET Q q 326.97 192.47 85.08 56.56 re W n Q q 326.97 192.47 85.08 56.56 re W n 0.000 0.000 0.000 RG 0.75 w [] 0 d 1 J 1 j 10.00 M 331.95 432.00 m 333.27 213.55 l 334.84 202.03 l 336.42 207.07 l 337.99 218.27 l 339.57 204.73 l 341.15 203.83 l 342.72 207.19 l 344.30 198.38 l 345.87 199.35 l 347.45 197.07 l 349.02 201.79 l 350.60 198.86 l 352.17 195.85 l 353.75 196.84 l 355.33 197.09 l 356.90 202.65 l 358.48 199.61 l 360.05 200.54 l 361.63 197.93 l 363.20 196.71 l 364.78 195.67 l 366.35 195.43 l 367.93 199.05 l 369.50 198.01 l 371.08 197.38 l 372.66 196.73 l 374.23 195.46 l 375.81 196.67 l 377.38 197.96 l 378.96 196.50 l 380.53 195.76 l 382.11 195.84 l 383.68 195.71 l 385.26 196.11 l 386.84 195.16 l 388.41 195.45 l 389.99 197.06 l 391.56 196.51 l 393.14 196.00 l 394.71 196.02 l 396.29 196.22 l 397.86 198.12 l 399.44 197.36 l 401.01 199.57 l 402.59 199.19 l 404.17 200.03 l 405.74 203.26 l 407.32 202.68 l 408.89 202.14 l S Q q 0.000 0.000 0.000 RG 0.75 w [] 0 d 1 J 1 j 10.00 M 330.04 192.47 m 408.89 192.47 l S 330.04 192.47 m 330.04 187.72 l S 345.81 192.47 m 345.81 187.72 l S 361.58 192.47 m 361.58 187.72 l S 377.35 192.47 m 377.35 187.72 l S 393.12 192.47 m 393.12 187.72 l S 408.89 192.47 m 408.89 187.72 l S BT 0.000 0.000 0.000 rg /F2 1 Tf 8.00 0.00 -0.00 8.00 327.81 175.36 Tm (0) Tj /F2 1 Tf 8.00 0.00 -0.00 8.00 350.46 175.36 Tm (20000) Tj /F2 1 Tf 8.00 0.00 -0.00 8.00 382.00 175.36 Tm (40000) Tj ET 326.97 194.57 m 326.97 246.94 l S 326.97 194.57 m 322.21 194.57 l S 326.97 207.66 m 322.21 207.66 l S 326.97 220.75 m 322.21 220.75 l S 326.97 233.85 m 322.21 233.85 l S 326.97 246.94 m 322.21 246.94 l S BT /F2 1 Tf 0.00 8.00 -8.00 0.00 315.56 186.78 Tm (1.00) Tj /F2 1 Tf 0.00 8.00 -8.00 0.00 315.56 212.97 Tm (1.10) Tj /F2 1 Tf 0.00 8.00 -8.00 0.00 315.56 239.15 Tm (1.20) Tj ET 326.97 192.47 m 412.04 192.47 l 412.04 249.03 l 326.97 249.03 l 326.97 192.47 l S Q q 288.00 144.00 144.00 144.00 re W n BT 0.000 0.000 0.000 rg /F2 1 Tf 8.00 0.00 -0.00 8.00 333.04 156.36 Tm (last iteration in chain) Tj /F2 1 Tf 0.00 8.00 -8.00 0.00 296.55 198.91 Tm (shrink factor) Tj ET Q q 326.97 192.47 85.08 56.56 re W n 1.000 0.000 0.000 RG 0.75 w [ 3.00 5.00] 0 d 1 J 1 j 10.00 M 332.83 432.00 m 333.27 252.28 l 334.84 214.34 l 336.42 227.48 l 337.99 264.07 l 339.57 227.83 l 341.15 223.84 l 342.72 228.29 l 344.30 199.44 l 345.87 202.00 l 347.45 202.68 l 349.02 217.60 l 350.60 208.05 l 352.17 196.14 l 353.75 202.03 l 355.33 203.49 l 356.90 222.02 l 358.48 210.60 l 360.05 212.89 l 361.63 203.86 l 363.20 198.71 l 364.78 198.05 l 366.35 197.65 l 367.93 210.44 l 369.50 206.88 l 371.08 204.70 l 372.66 202.39 l 374.23 197.54 l 375.81 201.63 l 377.38 206.57 l 378.96 201.49 l 380.53 198.74 l 382.11 198.93 l 383.68 198.47 l 385.26 200.13 l 386.84 196.64 l 388.41 196.86 l 389.99 202.75 l 391.56 200.47 l 393.14 199.20 l 394.71 199.29 l 396.29 200.21 l 397.86 206.72 l 399.44 204.43 l 401.01 211.67 l 402.59 210.28 l 404.17 211.35 l 405.74 218.16 l 407.32 217.40 l 408.89 216.43 l S 0.000 0.000 0.000 RG 0.75 w [] 0 d 326.97 194.57 m 412.04 194.57 l S Q q 288.00 144.00 144.00 144.00 re W n BT 0.000 0.000 0.000 rg /F3 1 Tf 10.00 0.00 -0.00 10.00 346.17 264.93 Tm (Surf:Wisk) Tj ET Q q 38.97 48.47 85.08 56.56 re W n Q q 38.97 48.47 85.08 56.56 re W n 0.000 0.000 0.000 RG 0.75 w [] 0 d 1 J 1 j 10.00 M 43.41 432.00 m 43.69 68.75 l 45.27 104.35 l 46.84 62.16 l 48.42 77.64 l 49.99 76.19 l 51.57 73.49 l 53.15 66.33 l 54.72 52.42 l 56.30 55.50 l 57.87 61.73 l 59.45 68.07 l 61.02 76.74 l 62.60 75.04 l 64.17 75.24 l 65.75 92.24 l 67.33 100.38 l 68.90 113.83 l 70.48 103.53 l 72.05 99.78 l 73.63 98.53 l 75.20 95.75 l 76.78 90.21 l 78.35 84.08 l 79.93 77.55 l 81.50 70.90 l 83.08 65.46 l 84.66 62.14 l 86.23 60.02 l 87.81 57.37 l 89.38 57.69 l 90.96 60.13 l 92.53 60.93 l 94.11 56.90 l 95.68 57.48 l 97.26 54.09 l 98.84 57.69 l 100.41 57.37 l 101.99 56.94 l 103.56 55.61 l 105.14 54.33 l 106.71 54.33 l 108.29 54.62 l 109.86 55.65 l 111.44 57.41 l 113.01 56.80 l 114.59 56.66 l 116.17 57.12 l 117.74 55.54 l 119.32 54.81 l 120.89 54.63 l S Q q 0.000 0.000 0.000 RG 0.75 w [] 0 d 1 J 1 j 10.00 M 42.04 48.47 m 120.89 48.47 l S 42.04 48.47 m 42.04 43.72 l S 57.81 48.47 m 57.81 43.72 l S 73.58 48.47 m 73.58 43.72 l S 89.35 48.47 m 89.35 43.72 l S 105.12 48.47 m 105.12 43.72 l S 120.89 48.47 m 120.89 43.72 l S BT 0.000 0.000 0.000 rg /F2 1 Tf 8.00 0.00 -0.00 8.00 39.81 31.36 Tm (0) Tj /F2 1 Tf 8.00 0.00 -0.00 8.00 62.46 31.36 Tm (20000) Tj /F2 1 Tf 8.00 0.00 -0.00 8.00 94.00 31.36 Tm (40000) Tj ET 38.97 50.57 m 38.97 102.94 l S 38.97 50.57 m 34.21 50.57 l S 38.97 63.66 m 34.21 63.66 l S 38.97 76.75 m 34.21 76.75 l S 38.97 89.85 m 34.21 89.85 l S 38.97 102.94 m 34.21 102.94 l S BT /F2 1 Tf 0.00 8.00 -8.00 0.00 27.56 42.78 Tm (1.00) Tj /F2 1 Tf 0.00 8.00 -8.00 0.00 27.56 68.97 Tm (1.10) Tj /F2 1 Tf 0.00 8.00 -8.00 0.00 27.56 95.15 Tm (1.20) Tj ET 38.97 48.47 m 124.04 48.47 l 124.04 105.03 l 38.97 105.03 l 38.97 48.47 l S Q q 0.00 0.00 144.00 144.00 re W n BT 0.000 0.000 0.000 rg /F2 1 Tf 8.00 0.00 -0.00 8.00 45.04 12.36 Tm (last iteration in chain) Tj /F2 1 Tf 0.00 8.00 -8.00 0.00 8.55 54.91 Tm (shrink factor) Tj ET Q q 38.97 48.47 85.08 56.56 re W n 1.000 0.000 0.000 RG 0.75 w [ 3.00 5.00] 0 d 1 J 1 j 10.00 M 43.59 432.00 m 43.69 107.01 l 45.27 205.79 l 46.84 76.82 l 48.42 126.84 l 49.99 129.98 l 51.57 122.93 l 53.15 101.65 l 54.72 54.55 l 56.30 59.54 l 57.87 76.07 l 59.45 92.57 l 61.02 132.00 l 62.60 126.75 l 64.17 128.09 l 65.75 177.89 l 67.33 199.21 l 68.90 238.01 l 70.48 211.01 l 72.05 200.08 l 73.63 195.93 l 75.20 188.16 l 76.78 172.73 l 78.35 154.34 l 79.93 133.96 l 81.50 111.10 l 83.08 91.61 l 84.66 79.18 l 86.23 71.04 l 87.81 63.58 l 89.38 65.83 l 90.96 73.67 l 92.53 76.62 l 94.11 64.59 l 95.68 68.46 l 97.26 62.28 l 98.84 73.94 l 100.41 73.22 l 101.99 71.28 l 103.56 66.29 l 105.14 62.46 l 106.71 62.41 l 108.29 62.88 l 109.86 66.22 l 111.44 71.96 l 113.01 70.15 l 114.59 70.58 l 116.17 72.25 l 117.74 66.47 l 119.32 64.12 l 120.89 63.20 l S 0.000 0.000 0.000 RG 0.75 w [] 0 d 38.97 50.57 m 124.04 50.57 l S Q q 0.00 0.00 144.00 144.00 re W n BT 0.000 0.000 0.000 rg /F3 1 Tf 10.00 0.00 -0.00 10.00 58.17 120.93 Tm (Solo:Solo) Tj ET Q q 182.97 48.47 85.08 56.56 re W n Q q 182.97 48.47 85.08 56.56 re W n 0.000 0.000 0.000 RG 0.75 w [] 0 d 1 J 1 j 10.00 M 187.56 432.00 m 187.69 205.07 l 189.27 69.46 l 190.84 59.46 l 192.42 57.43 l 193.99 62.00 l 195.57 67.62 l 197.15 59.97 l 198.72 52.64 l 200.30 59.00 l 201.87 64.84 l 203.45 63.95 l 205.02 78.53 l 206.60 66.58 l 208.17 62.00 l 209.75 65.08 l 211.33 63.65 l 212.90 69.76 l 214.48 60.69 l 216.05 61.24 l 217.63 58.31 l 219.20 54.30 l 220.78 54.14 l 222.35 57.42 l 223.93 54.29 l 225.50 52.69 l 227.08 52.34 l 228.66 51.98 l 230.23 52.13 l 231.81 52.81 l 233.38 52.94 l 234.96 54.40 l 236.53 54.97 l 238.11 54.68 l 239.68 55.08 l 241.26 54.38 l 242.84 54.91 l 244.41 55.93 l 245.99 54.03 l 247.56 55.61 l 249.14 55.99 l 250.71 55.10 l 252.29 54.04 l 253.86 51.87 l 255.44 51.83 l 257.01 51.04 l 258.59 50.93 l 260.17 51.29 l 261.74 53.49 l 263.32 53.03 l 264.89 52.14 l S Q q 0.000 0.000 0.000 RG 0.75 w [] 0 d 1 J 1 j 10.00 M 186.04 48.47 m 264.89 48.47 l S 186.04 48.47 m 186.04 43.72 l S 201.81 48.47 m 201.81 43.72 l S 217.58 48.47 m 217.58 43.72 l S 233.35 48.47 m 233.35 43.72 l S 249.12 48.47 m 249.12 43.72 l S 264.89 48.47 m 264.89 43.72 l S BT 0.000 0.000 0.000 rg /F2 1 Tf 8.00 0.00 -0.00 8.00 183.81 31.36 Tm (0) Tj /F2 1 Tf 8.00 0.00 -0.00 8.00 206.46 31.36 Tm (20000) Tj /F2 1 Tf 8.00 0.00 -0.00 8.00 238.00 31.36 Tm (40000) Tj ET 182.97 50.57 m 182.97 102.94 l S 182.97 50.57 m 178.21 50.57 l S 182.97 63.66 m 178.21 63.66 l S 182.97 76.75 m 178.21 76.75 l S 182.97 89.85 m 178.21 89.85 l S 182.97 102.94 m 178.21 102.94 l S BT /F2 1 Tf 0.00 8.00 -8.00 0.00 171.56 42.78 Tm (1.00) Tj /F2 1 Tf 0.00 8.00 -8.00 0.00 171.56 68.97 Tm (1.10) Tj /F2 1 Tf 0.00 8.00 -8.00 0.00 171.56 95.15 Tm (1.20) Tj ET 182.97 48.47 m 268.04 48.47 l 268.04 105.03 l 182.97 105.03 l 182.97 48.47 l S Q q 144.00 0.00 144.00 144.00 re W n BT 0.000 0.000 0.000 rg /F2 1 Tf 8.00 0.00 -0.00 8.00 189.04 12.36 Tm (last iteration in chain) Tj /F2 1 Tf 0.00 8.00 -8.00 0.00 152.55 54.91 Tm (shrink factor) Tj ET Q q 182.97 48.47 85.08 56.56 re W n 1.000 0.000 0.000 RG 0.75 w [ 3.00 5.00] 0 d 1 J 1 j 10.00 M 187.94 432.00 m 189.27 76.53 l 190.84 63.16 l 192.42 61.31 l 193.99 83.79 l 195.57 103.52 l 197.15 71.39 l 198.72 57.22 l 200.30 78.54 l 201.87 93.98 l 203.45 93.19 l 205.02 136.48 l 206.60 100.26 l 208.17 85.53 l 209.75 93.89 l 211.33 91.82 l 212.90 111.19 l 214.48 83.51 l 216.05 84.73 l 217.63 73.10 l 219.20 62.23 l 220.78 63.34 l 222.35 73.11 l 223.93 62.09 l 225.50 55.30 l 227.08 53.86 l 228.66 52.86 l 230.23 53.40 l 231.81 55.10 l 233.38 55.11 l 234.96 60.96 l 236.53 62.63 l 238.11 60.02 l 239.68 61.30 l 241.26 59.34 l 242.84 61.82 l 244.41 66.42 l 245.99 60.49 l 247.56 65.22 l 249.14 66.65 l 250.71 63.54 l 252.29 60.57 l 253.86 54.45 l 255.44 53.94 l 257.01 52.21 l 258.59 51.89 l 260.17 52.68 l 261.74 57.79 l 263.32 55.85 l 264.89 53.03 l S 0.000 0.000 0.000 RG 0.75 w [] 0 d 182.97 50.57 m 268.04 50.57 l S Q q 144.00 0.00 144.00 144.00 re W n BT 0.000 0.000 0.000 rg /F3 1 Tf 10.00 0.00 -0.00 10.00 203.84 120.93 Tm (Surf:Surf) Tj ET Q q 326.97 48.47 85.08 56.56 re W n Q q 326.97 48.47 85.08 56.56 re W n 0.000 0.000 0.000 RG 0.75 w [] 0 d 1 J 1 j 10.00 M 331.59 432.00 m 331.69 363.04 l 333.27 103.04 l 334.84 61.68 l 336.42 80.40 l 337.99 75.67 l 339.57 69.38 l 341.15 60.23 l 342.72 73.40 l 344.30 73.51 l 345.87 63.82 l 347.45 57.93 l 349.02 61.64 l 350.60 60.02 l 352.17 57.45 l 353.75 61.51 l 355.33 63.10 l 356.90 68.41 l 358.48 64.23 l 360.05 71.89 l 361.63 64.72 l 363.20 58.42 l 364.78 55.56 l 366.35 59.16 l 367.93 58.80 l 369.50 56.42 l 371.08 55.23 l 372.66 54.57 l 374.23 53.50 l 375.81 52.74 l 377.38 52.55 l 378.96 53.61 l 380.53 54.46 l 382.11 54.55 l 383.68 55.02 l 385.26 53.63 l 386.84 53.60 l 388.41 54.21 l 389.99 56.92 l 391.56 59.05 l 393.14 59.63 l 394.71 59.68 l 396.29 58.71 l 397.86 56.45 l 399.44 54.47 l 401.01 56.19 l 402.59 56.95 l 404.17 59.06 l 405.74 59.56 l 407.32 59.29 l 408.89 56.30 l S Q q 0.000 0.000 0.000 RG 0.75 w [] 0 d 1 J 1 j 10.00 M 330.04 48.47 m 408.89 48.47 l S 330.04 48.47 m 330.04 43.72 l S 345.81 48.47 m 345.81 43.72 l S 361.58 48.47 m 361.58 43.72 l S 377.35 48.47 m 377.35 43.72 l S 393.12 48.47 m 393.12 43.72 l S 408.89 48.47 m 408.89 43.72 l S BT 0.000 0.000 0.000 rg /F2 1 Tf 8.00 0.00 -0.00 8.00 327.81 31.36 Tm (0) Tj /F2 1 Tf 8.00 0.00 -0.00 8.00 350.46 31.36 Tm (20000) Tj /F2 1 Tf 8.00 0.00 -0.00 8.00 382.00 31.36 Tm (40000) Tj ET 326.97 50.57 m 326.97 102.94 l S 326.97 50.57 m 322.21 50.57 l S 326.97 63.66 m 322.21 63.66 l S 326.97 76.75 m 322.21 76.75 l S 326.97 89.85 m 322.21 89.85 l S 326.97 102.94 m 322.21 102.94 l S BT /F2 1 Tf 0.00 8.00 -8.00 0.00 315.56 42.78 Tm (1.00) Tj /F2 1 Tf 0.00 8.00 -8.00 0.00 315.56 68.97 Tm (1.10) Tj /F2 1 Tf 0.00 8.00 -8.00 0.00 315.56 95.15 Tm (1.20) Tj ET 326.97 48.47 m 412.04 48.47 l 412.04 105.03 l 326.97 105.03 l 326.97 48.47 l S Q q 288.00 0.00 144.00 144.00 re W n BT 0.000 0.000 0.000 rg /F2 1 Tf 8.00 0.00 -0.00 8.00 333.04 12.36 Tm (last 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A6I@6AJ4Xv[cxs 6j2,۩S;ðQkcXvP6ΰ6B4JPEm)@59j'5ԃ:b!kMP A00}  aj!YjϣG PSשe'^21QOq,xB5:SHZN>F6I3'1g,zy rn"SI=k` +^{vvk5nԐَYzXXw]bΌ~&1^Kxzcݸ$kA6h| Zj'{q~Ң E.G)?cѓǪOXY$a>eѳL(X|Si2k%) KGU@Ȉ4Ȁ`(@ JA(]ZǺR%*R%WWJ,Rux*'{TkwXG:K})[ϦR-=sMi|WjQfKSjL Vˌ(@%0. mgUk2b(ڎW endstream endobj startxref 883086 %%EOF MNP/inst/doc/MNP.Rnw0000644000176200001440000014200413162676435013566 0ustar liggesusers%\VignetteIndexEntry{MNP} \documentclass[11pt]{article} \usepackage{Rd} %% === margins === \addtolength{\hoffset}{-0.35in} \addtolength{\voffset}{-0.35in} \addtolength{\textwidth}{0.75in} \addtolength{\textheight}{0.75in} %% === basic packages === \usepackage{latexsym} \usepackage{amssymb,amsmath} \usepackage{graphicx} \usepackage{verbatim} %% === bibliography packages === \usepackage{natbib} \bibliographystyle{natbib} %% === hyperref options === \usepackage{color} \usepackage[pdftex, bookmarksopen=true, bookmarksnumbered=true, linkcolor=webred]{hyperref} \definecolor{webgreen}{rgb}{0, 0.5, 0} \definecolor{webblue}{rgb}{0, 0, 0.5} \definecolor{webred}{rgb}{0.5, 0, 0} % == spacing between sections and subsections \usepackage[compact]{titlesec} % === dcolumn package === \usepackage{dcolumn} \newcolumntype{.}{D{.}{.}{-1}} \newcolumntype{d}[1]{D{.}{.}{#1}} \hypersetup{% pdftitle = {MNP: R Package for Fitting Multinomial Probit Model}, pdfauthor = {Kosuke Imai and David A. van Dyk}, } \begin{document} \newcommand\dist{\buildrel\rm d\over\sim} \newcommand\al{\alpha} \newcommand\Y{{\cal Y}} \newcommand\itoN{^N_{i=1}} \newcommand\iton{^n_{i=1}} \newcommand\jtoJ{^J_{j=1}} \newcommand\inv{^{-1}} \newcommand\T{^\top} \newcommand\wt{\widetilde} \newcommand{\tr}{{\rm trace}} \newcommand{\chol}{{\rm Chol}} \newcommand\ind{\stackrel{\rm indep.}{\sim}} \renewcommand\r{\right} \renewcommand\l{\left} \newcommand\Var{{\rm Var}} \newcommand\E{{\rm E}} \newcommand\N{{\rm N}} \newcommand\cur{^{(t)}} \newcommand\pre{^{(t-1)}} \newcommand{\hlink}{\htmladdnormallink} \newcommand\spacingset[1]{\renewcommand{\baselinestretch}% {#1}\small\normalsize} \spacingset{1.5} \newcommand{\mac}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title{{\bf MNP: R Package for Fitting the \\ Multinomial Probit Model}\thanks{An earlier version of this paper appeared in {\it Journal of Statistical Software}, Vol. 14, No. 3 (May 2005), pp.1--32. We thank Jordan Vance for his valuable contribution to this project and Shigeo Hirano for providing the Japanese election dataset. We also thank Doug Bates for helpful advice on Lapack routines and Andrew Martin, Kevin Quinn, users of MNP, and anonymous reviewers and the associate editor for useful suggestions. We gratefully acknowledge funding for this project partially provided by NSF grants DMS-01-04129, DMS-04-38240, and DMS-04-06085, and by the Committee on Research in the Humanities and Social Sciences at Princeton University.}} \author{Kosuke Imai\thanks{Professor, Department of Politics, Princeton University, Princeton, NJ 08544. Phone: 609--258--6610, Fax: 609-258-1110, Email: \href{mailto:kimai@Princeton.Edu}{kimai@Princeton.Edu}, URL: \href{http://imai.princeton.edu}{http://imai.princeton.edu} Department of Politics, Princeton University}\\ David A. van Dyk\thanks{Associate Professor, Department of Statistics, University of California, Irvine, CA 92697-1205. Phone: 949--824--5679, Fax: 949--824--9863, Email: \href{mailto:dvd@uci.edu}{dvd@uci.edu}}} \date{Version 3.1--0} \pdfbookmark[1]{Title Page}{Title Page} \maketitle \begin{abstract} MNP is a publicly available R package that fits the Bayesian multinomial probit model via Markov chain Monte Carlo. The multinomial probit model is often used to analyze the discrete choices made by individuals recorded in survey data. Examples where the multinomial probit model may be useful include the analysis of product choice by consumers in market research and the analysis of candidate or party choice by voters in electoral studies. The MNP software can also fit the model with different choice sets for each individual, and complete or partial individual choice orderings of the available alternatives from the choice set. The estimation is based on the efficient marginal data augmentation algorithm that is developed by \citet{imai:vand:05}. \end{abstract} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \clearpage \section{Introduction} \label{sec:introduction} This paper illustrates how to use MNP, a publicly available R \citep{R:12} package, in order to fit the Bayesian multinomial probit model via Markov chain Monte Carlo. The multinomial probit model is often used to analyze the discrete choices made by individuals recorded in survey data. %Unlike the ordinal probit model, the %multinomial probit model does not assume that the choice set is %inherently ordered. Examples where the multinomial probit model may be useful include the analysis of product choice by consumers in market research and the analysis of candidate or party choice by voters in electoral studies. The MNP software can also fit the model with different choice sets for each individual, and complete or partial individual choice orderings of the available alternatives from the choice set. We use Markov chain Monte Carlo (MCMC) for estimation and computation. In particular, we use the efficient marginal data augmentation MCMC algorithm that is developed by \citet{imai:vand:05}. MNP can be installed in the same way as other R packages via the \texttt{install.packages("MNP")} command. Appendix~\ref{sec:install} gives instructions for obtaining R and installing MNP on Windows, Mac OS X, and Linux/UNIX platforms. Only three commands are necessary to use the MNP software; \texttt{mnp()} fits the multinomial probit model, \texttt{summary()} summarizes the MCMC output, and \texttt{predict()} gives posterior prediction based on the fitted model (In addition, \texttt{coef()} and \texttt{cov.mnp()} allow one to extract the posterior draws of model coefficients and covariance matrix). To run an example script, start R and run the following commands: \spacingset{1}\begin{verbatim} library(MNP) # loads the MNP package example(mnp) # runs the example script \end{verbatim} \spacingset{1.5} Details of the example script are given in Sections~\ref{sec:examp-soap}~and~\ref{sec:examp-japan}. Three appendices describe installation, the commands, and version history. We begin in Section~\ref{sec:mn_probit} with a brief description of the multinomial probit model that MNP is designed to fit. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The Method} \label{sec:mn_probit} MNP implements the marginal data augmentation algorithms for posterior sampling in the multinomial probit model. The MCMC algorithm we implement here is fully described in \citet{imai:vand:05}; we use Scheme 1 of their Algorithm 1. \subsection{The Multinomial Probit Model} \label{sec:mnp} Suppose we have a dataset of size $n$ with $p>2$ choices and $k$ covariates. Here, choices refer to the number of classes in the multinomial model. The word ``choices'' is used because the model is often used to describe how individuals choose among a number of alternatives, e.g., how a voter chooses which candidate to vote for among four candidates running for a particular office. We focus on the case when $p>2$ because when $p=2$, the model reduces to the standard binomial probit model, which can be fit via the \texttt{glm(, family = binomial(probit))} command in R. The multinomial probit model differs from the ordinal probit model in that the former does not assume any inherent ordering on the choices. Thus, although the individuals may have preferences among the available alternatives these ordering are individual specific rather than being characteristics of the alternatives themselves. The ordinal probit model can be fitted via an MCMC algorithm in R by installing a package called MCMCpack \citep{mart:quin:park:09}. Under the multinomial probit model, we assume a multivariate normal distribution on the latent variables, $W_i=(W_{i1},\ldots,W_{i,p-1})$. \begin{equation} W_i = X_i \beta + e_i, \quad e_i \sim \N(0, \Sigma), \ \hbox{for} \ i=1,\ldots,n, \label{eq:mnp} \end{equation} where $X_i$ is a $(p-1) \times k$ matrix of covariates, $\beta$ is $k \times 1$ vector of fixed coefficients, $e_i$ is $(p-1) \times 1$ vector of disturbances, and $\Sigma$ is a $(p-1) \times (p-1)$ positive definite matrix. For the model to be identified, the first diagonal element of $\Sigma$ is constrained, $\sigma_{11}=1$. Please note that starting with version 2.6-1, we use the restriction ${\rm trace}(\Sigma)=p$ as the default identification strategy following the recommendation of \citet{burg:nord:09}. This avoids the arbitrariness of fixing one particular diagonal element. The response variable, $Y_i$, is the index of the choice of individual $i$ among the alternatives in the choice set and is modeled in terms of this latent variable, $W_i$, via \begin{eqnarray} Y_i(W_i) & = & \l\{ \begin{array}{ccl} 0 & {\rm if} & {\rm max}(W_i)<0 \\ j & {\rm if} & {\rm max}(W_i)=W_{ij}>0 \end{array}\r. , \quad {\rm for} \ i=1,\ldots,n, \; {\rm and} \; j=1,\ldots,p-1, \end{eqnarray} where $Y_i$ equal to 0 corresponds to a base category. The matrix $X_i$ may include both choice-specific and individual-specific variables. A choice-specific variable is a variable that has a value for each of the $p$ choices, and these $p$ values may be different for each individual (e.g., the price of a product in a particular region where an individual lives). Choice-specific variables are recorded relative to the baseline choice and thus there are $p-1$ recorded values for each individual. In this way a choice-specific variable is tabulated as a column in $X_i$. Individual-specific variables, on the other hand, take on a value for each individual, but are constant across the choices, e.g., the age or gender of the individual. These variables are tabulated via their interaction with each of the choice indicator variables. Thus, an individual-specific variable corresponds to $p-1$ columns of $X_i$ and $p-1$ components of $\beta$. \subsection{The Multinomial Probit Model with Ordered Preferences} \label{sec:mnpop} In some cases, we observe a complete or partial ordering of $p$ alternatives. For example, we may observe the preferences of each individual among different brands of a product. We denote the outcome variable in such situations by ${\cal Y}_i=\{\Y_{i1}, \ldots, \Y_{ip}\}$ where $i=1,\ldots,n$ indexes individuals and $j=1,\ldots,p$ represent alternatives. If $\Y_{ij} > \Y_{ij'}$ for some $j \ne j'$, we say $j$ is preferred to $j'$. If $\Y_{ij} = \Y_{ij'}$ for some $j \ne j'$, we say individual $i$ is indifferent to the choice between alternatives $j$ and $j'$, but treat the data as if the actual ordering is unknown. In other words, formally we insist on strict inequalities among the preferences, but allow for some inequalities to be unobserved. The preference ordering is assumed to satisfy the usual axioms of preference comparability. Namely, preference is connected: For any $j$ and $j'$, either $\Y_{ij} \le \Y_{ij'}$ or $\Y_{ij} \ge \Y_{ij'}$. Preference also must be transitive: for any $j$,$j'$, and $j''$, $\Y_{ij} \le \Y_{ij'}$ and $\Y_{ij'} \le \Y_{ij''}$ imply $\Y_{ij} \le \Y_{ij''}$. For notational simplicity and without loss of generality, we assume that $\Y_{ij}$ takes an integer value ranging from $0$ to $p-1$. We emphasize that we have not changed the model from Section~\ref{sec:mnp}. Rather, we simply have more observed data: the index of the choice of the individual $i$, $Y_i$, can be computed from $\Y_i$. Thus, we continue to model the preference ordering, $\Y_{i}$, in terms of a latent (multivariate normal) random vector, $W_{i} = (W_{ij},\ldots,W_{i,p-1})$, via \begin{eqnarray} \Y_{ij}(W_i) & = & \#\{W_{ij'}: W_{ij'} < W_{ij} \} \quad {\rm for} \quad i=1,\ldots,n, \quad {\rm and} \quad j = 1,\ldots,p, \label{eq:mop} \end{eqnarray} where $W_{ip}=0$, the distribution of $W_i$ is specified in equation~\ref{eq:mnp}, and $\#\{\cdots\}$ indicates the number of elements in a finite set. This model can be fitted via a slightly modified version of the MCMC algorithm in \citet{imai:vand:05}. In particular, we need only modify the way in which $W_{ij}$ is sampled and use a truncation rule based on Equation~\ref{eq:mop}. \subsection{Prior Specification} Our prior distribution for the multinomial probit model is \begin{equation} \beta \sim \N(0,A\inv) \quad {\rm and} \quad p(\Sigma)\propto |\Sigma|^{-(\nu+p)/2}\l[\tr(S\Sigma\inv)\r]^{-\nu(p-1)/2}, \label{eq:priors} \end{equation} where $A$ is the prior precision matrix of $\beta$, $\nu$ is the prior degrees of freedom parameter for $\Sigma$, and the $(p-1)\times(p-1)$ positive definite matrix $S$ is the prior scale for $\Sigma$; we assume the first diagonal element of $S$ is one. The prior distribution on $\Sigma$ is proper if $\nu\geq p-1$, the prior mean of $\Sigma$ is approximately equal to $S$ if $\nu> p-2$, and the prior variance of $\Sigma$ increase as $\nu$ decreases as long as this variance exists. We also allow for an improper prior on $\beta$, which is $p(\beta) \propto 1$ (i.e., $A=0$).\footnote{Algorithm~2 of \citet{imai:vand:05} allows for a non zero prior mean for $\beta$. Because the update for $\Sigma$ in this sampler is not exactly its complete conditional distribution, however, this algorithm may exhibit undesirable convergence properties in some situations.} Alternate prior specifications were introduced by \citet{mccu:ross:94} and \citet{mccu:pols:ross:00}. The relative advantage of the various prior distributions are discussed by \citet{mccu:pols:ross:00}, \citet{nobi:00}, and \citet{imai:vand:05}. We prefer our choice because it allows us to directly specify the prior distribution on the identifiable model parameters, allows us to specify an improper prior distribution on regression coefficient, and results in a Monte Carlo sampler that is relatively quick to converge. An implementation of of the sampler proposed by \citet{mccu:ross:94} has recently been released in the R package bayesm \citep{ross:mccu:05}. \subsection{Prediction under the Multinomial Probit Model} Predictions of individual preferences given particular values of the covariates can be useful in interpreting the fitted model. Consider a value of the $(p-1)\times k$ matrix of covariates, $X^\star$, that may or may not correspond to the values for one of the observed individuals. We are interested in the distribution of the preferences among the alternatives in the choice set given this value of the covariates. Let $Y^\star$ be the preferred choice and ${\cal Y}^\star=({\cal Y}^\star_1, \ldots, {\cal Y}^\star_p)$ indicate the ordering of the preferences among the available alternatives. As an example, one might be interested in $\Pr(Y^\star = j \mid X^\star)$ for some $j$. By varying $X^\star$, one could explore how preferences are expected to change with covariates. Similarly, one might be interested in how relative preferences such as $\Pr( {\cal Y}_{j}^\star > {\cal Y}_{j^\prime}^\star \mid X^\star)$ are expected to change with the covariates. In the context of a Bayesian analysis, such predictive probabilities are computed via the posterior predictive distribution. This distribution conditions on the observed data, $Y=(Y_1,\ldots, Y_n)$ or ${\cal Y}= ({\cal Y}_1, \ldots, {\cal Y}_n)$, but averages over the uncertainty in the model parameters. For example, \begin{eqnarray} \Pr(Y^\star = j \mid X^\star, Y) & = & \int \Pr(Y^\star = j \mid X^\star, \beta, \Sigma, Y)\, p(\beta, \Sigma \mid Y) \; d(\beta, \Sigma). \end{eqnarray} Thus, the posterior predictive distribution accounts for both variability in the response variable given the model parameters (i.e., the likelihood or sampling distribution) and the uncertainty in the model parameters as quantified in the posterior distribution. Monte Carlo evaluation of the posterior predictive distribution is easy once we obtain a Monte Carlo sample of the model parameters from the posterior distribution: We simply sample according to the likelihood for each Monte Carlo sample from the posterior distribution. This involves sampling the latent variable under the model in (1) and computing the preferred choice using (2) or the ordering of preferences using (3). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Example 1: Detergent Brand Choice} \label{sec:examp-soap} In this and the next section, we describe the details of two examples of MNP. In this section we use a market research dataset to illustrate the fitting of the multinomial probit model. In Section~\ref{sec:examp-japan} we fit the multinomial probit model with ordered preference to a Japanese election dataset. We also describe how to perform convergence diagnostics of the MCMC sampler and analysis of the Monte Carlo output of MNP using an existing R package. Additional examples of MNP can be found in \citet{imai:vand:05}. \subsection{Preliminaries} \label{subsec:soap-prelim} Our first example analyzes a typical dataset in market research. The dataset contains information about the brand and price of the laundry detergent purchased by 2657 households originally analyzed by \citet{chin:pras:98}. The dataset contains the log prices of six detergent brands -- Tide, Wisk, EraPlus, Surf, Solo, and All -- as well as the brand chosen by each household (Type {\tt help(detergent)} in R for details about the dataset). We fit the multinomial probit model by using \texttt{choice} as the outcome variable and the other six variables as choice-specific covariates. After loading the MNP package, this can be accomplished using the following three commands, \spacingset{1} \begin{verbatim} data(detergent) res <- mnp(choice ~ 1, choiceX = list(Surf=SurfPrice, Tide=TidePrice, Wisk=WiskPrice, EraPlus=EraPlusPrice, Solo=SoloPrice, All=AllPrice), cXnames = c("price"), data = detergent, n.draws = 10000, burnin = 2000, thin = 3, verbose = TRUE) summary(res) \end{verbatim} \spacingset{1.5} The first command loads the example dataset and stores it as the data frame called \texttt{detergent}. The second command fits the multinomial probit model. The default base category in this case is \texttt{All}. (The default base category in MNP is the first factor level of the outcome variable, $Y$.) Each household chooses among the six brands of laundry detergent, i.e., $p=6$. We specify the choice-specific variables, \texttt{choiceX}, using a named list. The elements of the list are the log price of each detergent brand and they are named after the levels of factor variable, \texttt{choice}. We also name the coefficient for this set of choice-specific variables by using \texttt{cXnames}. The argument \texttt{data} allows us to specify the name of the data frame where the data are stored. The model estimates five intercepts and the price coefficient as well as 14 parameters in the covariance matrix, $\Sigma$. We use the default prior distribution; an improper prior distribution for $\beta$ and a diffuse prior distribution for $\Sigma$ with $\nu = p = 6$ and $S=I$. We sample 10,000 replications of the parameter from the resulting posterior distribution, saving every fourth sample after discarding the first 2,000 samples as specified by the arguments, \texttt{n.draws}, \texttt{thin}, and \texttt{burnin}. The argument \texttt{verbose = TRUE} specifies that a progress report and other useful messages be printed while the MCMC sampler is running. The \texttt{summary(res)} command gives a summary of the output including the posterior means and standard deviations of the parameters. The summary is based on the single MCMC chain produced with this call of MNP. Before we can reliably draw conclusions based on these results, we must be sure the chain has converged. Convergence diagnostics are discussed and illustrated in Section~\ref{sec:coda}. The result of the call of \texttt{summary(res)} are as follows. \spacingset{1} \begin{verbatim} Call: mnp(formula = choice ~ 1, data = detergent, choiceX = list(Surf = SurfPrice, Tide = TidePrice, Wisk = WiskPrice, EraPlus = EraPlusPrice, Solo = SoloPrice, All = AllPrice), cXnames = c("price"), n.draws = 10000, burnin = 2000, thin = 3, verbose = TRUE) Coefficients: mean std.dev. 2.5% 97.5% (Intercept):EraPlus 2.567 0.238 2.123 3.03 (Intercept):Solo 1.722 0.247 1.248 2.25 (Intercept):Surf 1.572 0.163 1.259 1.91 (Intercept):Tide 2.716 0.252 2.269 3.22 (Intercept):Wisk 1.620 0.162 1.328 1.96 price -82.102 8.952 -99.896 -66.32 Covariances: mean std.dev. 2.5% 97.5% EraPlus:EraPlus 1.00000 0.00000 1.00000 1.00 EraPlus:Solo 0.82513 0.26942 0.31029 1.36 EraPlus:Surf 0.17021 0.16115 -0.15810 0.48 EraPlus:Tide 0.24872 0.12956 0.00253 0.52 EraPlus:Wisk 0.88170 0.16614 0.54500 1.20 Solo:Solo 2.56481 0.68678 1.53276 4.25 Solo:Surf 0.45246 0.34572 -0.28018 1.13 Solo:Tide 0.50836 0.32706 -0.09069 1.22 Solo:Wisk 1.46997 0.44596 0.65506 2.45 Surf:Surf 1.69005 0.50978 0.92334 2.82 Surf:Tide 0.80762 0.30381 0.34019 1.44 Surf:Wisk 1.01614 0.36503 0.44121 1.85 Tide:Tide 1.32024 0.41669 0.62898 2.25 Tide:Wisk 1.05396 0.30137 0.59323 1.74 Wisk:Wisk 2.58761 0.55076 1.68773 3.82 Base category: All Number of alternatives: 6 Number of observations: 2657 Number of stored MCMC draws: 2000 \end{verbatim} \spacingset{1.5} We emphasize that these results are preliminary because convergence has not yet been assessed. Thus, we delay interpretation of the fit until Section~\ref{sec:examp-soap-final}, after we discuss convergence diagnostics in Section~\ref{sec:coda}. Note that \texttt{coef(res)} and \texttt{cov.mnp(res)} allow one to extract the posterior draws of model coefficients and covariance matrix if desired. Type {\tt help(mnp)} in R for details. \subsection{Using coda for Convergence Diagnostics and Output Analysis} \label{sec:coda} It is possible to use coda \citep*{plum:best:cowl:vine:05}, to perform various convergence diagnostics, as well as to summarize results. The coda package requires a matrix of posterior draws for relevant parameters to be saved as an \texttt{mcmc} object. Here, we illustrate how to use coda to calculate the Gelman-Rubin convergence diagnostic statistic \citep{gelm:rubi:92}. This diagnostic is based on multiple independent Markov chains initiated at over-dispersed starting values. Here, we obtain these chains by independently running the \texttt{mnp()} command three times, specifying different starting values for each time. This can be accomplished by typing the following commands at the R prompt, \spacingset{1} \begin{verbatim} data(detergent) res1 <- mnp(choice ~ 1, choiceX = list(Surf=SurfPrice, Tide=TidePrice, Wisk=WiskPrice, EraPlus=EraPlusPrice, Solo=SoloPrice, All=AllPrice), cXnames = c("price"), data = detergent, n.draws = 50000, verbose = TRUE) res2 <- mnp(choice ~ 1, choiceX = list(Surf=SurfPrice, Tide=TidePrice, Wisk=WiskPrice, EraPlus=EraPlusPrice, Solo=SoloPrice, All=AllPrice), coef.start = c(1, -1, 1, -1, 1, -1)*10, cov.start = matrix(0.5, ncol=5, nrow=5) + diag(0.5, 5), cXnames = c("price"), data = detergent, n.draws = 50000, verbose = TRUE) res3 <- mnp(choice ~ 1, choiceX = list(Surf=SurfPrice, Tide=TidePrice, Wisk=WiskPrice, EraPlus=EraPlusPrice, Solo=SoloPrice, All=AllPrice), coef.start=c(-1, 1, -1, 1, -1, 1)*10, cov.start = matrix(0.9, ncol=5, nrow=5) + diag(0.1, 5), cXnames = c("price"), data = detergent, n.draws = 50000, verbose = TRUE) \end{verbatim} \spacingset{1.5} where we save the output of each chain separately as \texttt{res1}, \texttt{res2}, and \texttt{res3}. The first chain is initiated at the default starting values for all parameters; i.e., a vector of zeros for $\beta$ and an identity matrix for $\Sigma$. The second chain is run starting from a vector of three $10$'s and three $-10$'s for $\beta$ and a matrix with all diagonal elements equal to 1 and all correlations equal to 0.5 for $\Sigma$. Finally, the third chain is run starting from a permutation of the starting value used for $\beta$ in the second chain, and a matrix with all diagonal elements equal to 1 and all correlations equal to 0.9 for $\Sigma$. We again use the default prior specification and obtain 50,000 draws for each chain. We store the output from each of the three chains as an object of class \texttt{mcmc}, and then combine them into a single list using the following commands, \spacingset{1} \begin{verbatim} library(coda) res.coda <- mcmc.list(chain1=mcmc(res1$param[,-7]), chain2=mcmc(res2$param[,-7]), chain3=mcmc(res3$param[,-7])) \end{verbatim} \spacingset{1.5} where the first command loads the coda package\footnote{If you have not used the \texttt{coda} package before, you must install it. At the R prompt, type \texttt{install.packages("coda")}.} and the second command saves the results as an object of class \texttt{mcmc.list}, which is called \texttt{res.coda}. We exclude the 7th column of each chain, because this column corresponds to the first diagonal element of the covariance matrix which is always equal to 1. The following command computes the Gelman-Rubin statistic from these three chains, \spacingset{1} \begin{verbatim} gelman.diag(res.coda, transform = TRUE) \end{verbatim} \spacingset{1.5} where \texttt{transform = TRUE} applies log or logit transformation as appropriate to improve the normality of each of the marginal distributions. \citet{gelm:carl:ster:rubi:04} suggest computing the statistic for each scalar estimate of interest, and to continue to run the chains until the statistics are all less than 1.1. Inference is then based on the Monte Carlo sample obtained by combining the second half of each of the chains. The output of the coda command lists the value and a 97.5\% upper limit of the Gelman-Rubin statistic for each parameter. \spacingset{1}\begin{verbatim} Potential scale reduction factors: Point est. 97.5% quantile (Intercept):EraPlus 1.01 1.02 (Intercept):Solo 1.03 1.08 (Intercept):Surf 1.01 1.05 (Intercept):Tide 1.01 1.02 (Intercept):Wisk 1.01 1.04 price 1.01 1.02 EraPlus:Solo 1.02 1.03 EraPlus:Surf 1.02 1.04 EraPlus:Tide 1.03 1.08 EraPlus:Wisk 1.04 1.13 Solo:Solo 1.01 1.04 Solo:Surf 1.01 1.02 Solo:Tide 1.02 1.07 Solo:Wisk 1.00 1.00 Surf:Surf 1.00 1.00 Surf:Tide 1.01 1.04 Surf:Wisk 1.02 1.06 Tide:Tide 1.02 1.06 Tide:Wisk 1.02 1.08 Wisk:Wisk 1.01 1.04 Multivariate psrf 1.07+0i \end{verbatim} \spacingset{1.5} The Gelman-Rubin statistics are all less than 1.1, suggesting satisfactory convergence has been achieved. (Note that the 97.5% quantile for {\tt EraPlus:Wisk} is greater than 1.1; a more conservative user might want to obtain a set of longer Markov chains and recompute the Gelman-Rubin statistics.) It may also be useful to examine the change in the value of the Gelman-Rubin statistic over the iterations. The following commands produce a graphical summary of the progression of the statistics over iterations. \spacingset{1} \begin{verbatim} gelman.plot(res.coda, transform = TRUE, ylim = c(1,1.2)) \end{verbatim} \spacingset{1.5} where \texttt{ylim = c(1,1.2)} specifies the range of the vertical axis of the plot. The results appear in Figure~\ref{fg:rhat}, as a cumulative evaluation of the Gelman-Rubin statistic over iterations for nine selected parameters. (Three coefficients appear in the first row; three covariance parameters appear in the second row; and three variance parameters appear in the third row.) \begin{figure}[p] \spacingset{1} \includegraphics[scale=1.15]{rhat} \caption{The Gelman-Rubin Statistic Computed with Three Independent Markov Chains for Selected Parameters in the Detergent Example. The first row represents three coefficients, the second row represents three covariances, and the third row represents three variance parameters.} \label{fg:rhat} \end{figure} The coda package can also be used to produce univariate time-series plots of the three chains and univariate density estimate of the posterior distribution. The following commands create these graphs for the price coefficient. %pdf("coda.pdf", width=9, height=4.5) \spacingset{1}\begin{verbatim} res.coda <- mcmc.list(chain1=mcmc(res1$param[25001:50000, "price"], start=25001), chain2=mcmc(res2$param[25001:50000, "price"], start=25001), chain3=mcmc(res3$param[25001:50000, "price"], start=25001)) plot(res.coda, ylab = "price coefficient") \end{verbatim} %dev.off() \spacingset{1.5} Figure \ref{fg:trace} presents the resulting plots. The left panel overlays the time-series plot for each chain with a different color representing each chain. The right panel shows the kernel-smoothed density estimate of the posterior distribution. One can also apply an array of other functions to \texttt{res.coda}. See the coda homepage, \href{http://www-fis.iarc.fr/coda}{http://www-fis.iarc.fr/coda}, for details. \begin{figure} \spacingset{1} \begin{center} \includegraphics[scale=0.75]{coda-small} \end{center} \vspace{-0.25in} \caption{Time-series Plot of Three Independent Markov Chains (Left Panel) and A Density Estimate of the Posterior Distribution of the Price Coefficient (Right Panel). The time-series plot overlays the three chains, each in a different color. A lowess smoothed line is also plotted for each of the three chains. The density estimate is based on all three chains.} \label{fg:trace} \end{figure} \subsection{Final Analysis and Conclusions} \label{sec:examp-soap-final} In the final analysis, we combine the second half of each of the three chains. This is accomplished using the following command that saves the last 25,000 draws from each chain as an \texttt{mcmc} object and combines the \texttt{mcmc} objects into a list, % pdf("rhat.pdf") % par(mex=0.5) % res.coda <- mcmc.list(chain1=mcmc(res1$param[,c(1:2,6,9,14,18,12,16,19,)]), % chain2=mcmc(res2$param[,c(1:2,6,9,14,18,12,16,19,)]), % chain3=mcmc(res3$param[,c(1:2,6,9,14,18,12,16,19,)])) % gelman.plot(res.coda, transform = TRUE, ylim=c(1,1.2)) % dev.off() \spacingset{1}\begin{verbatim} res.coda <- mcmc.list(chain1=mcmc(res1$param[25001:50000,-7], start=25001), chain2=mcmc(res2$param[25001:50000,-7], start=25001), chain3=mcmc(res3$param[25001:50000,-7], start=25001)) summary(res.coda) \end{verbatim} \spacingset{1.5} The second command produces the following summary of the posterior distribution for each parameter based on the combined Monte Carlo sample. \spacingset{1}\begin{verbatim} Iterations = 25001:50000 Thinning interval = 1 Number of chains = 3 Sample size per chain = 25000 1. Empirical mean and standard deviation for each variable, plus standard error of the mean: Mean SD Naive SE Time-series SE (Intercept):EraPlus 2.5398 0.2300 0.0008400 0.014332 (Intercept):Solo 1.7218 0.2227 0.0008131 0.012972 (Intercept):Surf 1.5634 0.1663 0.0006072 0.010462 (Intercept):Tide 2.6971 0.2374 0.0008670 0.015153 (Intercept):Wisk 1.6155 0.1594 0.0005822 0.010221 price -80.9097 8.4292 0.0307791 0.556483 EraPlus:Solo 0.8674 0.2954 0.0010787 0.021698 EraPlus:Surf 0.1226 0.1991 0.0007269 0.014043 EraPlus:Tide 0.2622 0.1525 0.0005568 0.009833 EraPlus:Wisk 0.9062 0.1912 0.0006982 0.012893 Solo:Solo 2.6179 0.7883 0.0028785 0.055837 Solo:Surf 0.5348 0.4307 0.0015728 0.030113 Solo:Tide 0.5570 0.3544 0.0012941 0.024548 Solo:Wisk 1.5442 0.4643 0.0016954 0.031574 Surf:Surf 1.6036 0.4758 0.0017374 0.031269 Surf:Tide 0.7689 0.2992 0.0010926 0.020253 Surf:Wisk 0.9949 0.3548 0.0012955 0.022963 Tide:Tide 1.2841 0.3660 0.0013364 0.024095 Tide:Wisk 1.0658 0.3147 0.0011492 0.020229 Wisk:Wisk 2.5801 0.5523 0.0020167 0.034974 2. Quantiles for each variable: 2.5% 25% 50% 75% 97.5% (Intercept):EraPlus 2.09105 2.38514 2.5321 2.6926 3.0022 (Intercept):Solo 1.28315 1.57316 1.7219 1.8705 2.1639 (Intercept):Surf 1.24132 1.45272 1.5583 1.6701 1.9023 (Intercept):Tide 2.23562 2.53443 2.6886 2.8544 3.1721 (Intercept):Wisk 1.31120 1.50841 1.6104 1.7191 1.9429 price -97.62406 -86.61736 -80.7783 -75.1013 -64.9720 EraPlus:Solo 0.32811 0.65755 0.8480 1.0694 1.4666 EraPlus:Surf -0.24159 -0.01596 0.1131 0.2507 0.5491 EraPlus:Tide -0.02109 0.16081 0.2571 0.3527 0.5957 EraPlus:Wisk 0.54089 0.77643 0.9035 1.0331 1.2917 Solo:Solo 1.37468 2.02720 2.5225 3.0985 4.4160 Solo:Surf -0.31143 0.25493 0.5251 0.8180 1.3880 Solo:Tide -0.05172 0.30518 0.5297 0.7740 1.3106 Solo:Wisk 0.74682 1.21192 1.5131 1.8380 2.5445 Surf:Surf 0.86883 1.25552 1.5377 1.8834 2.6935 Surf:Tide 0.30606 0.55218 0.7285 0.9413 1.4556 Surf:Wisk 0.40231 0.74907 0.9579 1.1957 1.8026 Tide:Tide 0.69841 1.02206 1.2396 1.4999 2.1100 Tide:Wisk 0.51732 0.84824 1.0411 1.2556 1.7579 Wisk:Wisk 1.60662 2.18760 2.5407 2.9238 3.7722 \end{verbatim} \spacingset{1.5} The output shows the mean, standard deviation, and various percentiles of the posterior distributions of the coefficients and the elements of the variance-covariance matrix. The base category is the detergent \texttt{All}. Separate intercepts are estimated for each detergent. The price coefficient is negative and highly statistically significant, agreeing with the standard economic expectation that consumers are less likely to buy more expensive goods. MNP also allows one to calculate the posterior predictive probabilities of each alternative being most preferred given a particular value of the covariates. For example, one can calculate the posterior predictive probabilities using the covariate values of the first two observations by using the \texttt{predict()} command, \spacingset{1} \begin{verbatim} predict(res1, newdata = detergent[1:2,], newdraw = rbind(res1$param[25001:50000,], res2$param[25001:50000,], res3$param[25001:50000,]), type = "prob") \end{verbatim} \spacingset{1.5} where \texttt{res1} is the output object from the \texttt{mnp()} command, and we set \texttt{newdata} to the first two observations of the detergent data set and \texttt{newdraw} to the combined draws from the second half of three chains. Setting \texttt{type = "prob"} causes the function \texttt{predict()} to return the posterior predictive probabilities. Moreover, a new {\tt n.draws} option in \texttt{predict()} command allows one to compute the uncertainty estimates about these predicted probabilities. It is also possible to return a Monte Carlo sample of the the alternative that is most preferred (\texttt{type = "choice"}), a Monte Carlo sample of the latent variables (\texttt{type = "latent"}), or a Monte Carlo sample of the preference-ordered alternatives (\texttt{type = "order"}). (Type \texttt{help(predict.mnp)} in R for more details about the \texttt{predict()} function in MNP.) The above command yields the following output, \spacingset{1} \begin{verbatim} All EraPlus Solo Surf Tide Wisk [1,] 0.01281333 0.1946400 0.12292 0.46208000 0.1401733 0.06737333 [2,] 0.04649333 0.1262133 0.05996 0.03169333 0.3589867 0.37665333 \end{verbatim} \spacingset{1.5} The result indicates that the posterior predictive probability of purchasing \texttt{Surf} is the largest for households with covariates equal to those in the first household in the data set. Under the model, approximately 46\% of such households will purchase \texttt{Surf}. On the other hand, \texttt{All} is the brand least likely to be purchased by these households. The households with covariates equal to the second household are most likely to buy \texttt{Wisk}. Also, they are almost equally likely to purchase \texttt{Tide}. (The posterior predictive probabilities of buying \texttt{Wisk} and \texttt{Tide} are both around 0.35) \section{Example 2: Voters' Preference of Political Parties} \label{sec:examp-japan} Our second example illustrates how to fit the multinomial probit model with ordered preferences (see Section~\ref{sec:mnpop}). \subsection{Preliminaries} We analyze a survey dataset describing the preferences of individual voters in Japan among the political parties. Political scientists may be interested in using the gender, age and education level of voters to predict their party preferences (Type {\tt help(japan)} in R for details about the dataset). The outcome variable is a vector of relative preferences for each of the four parties, i.e., $p=4$. Each of 418 voters is asked to give a score between 0 and 100 to each party. For example, the first voter in the dataset has the following preferences. \begin{verbatim} LDP NFP SKG JCP 80 75 80 0 \end{verbatim} That is, this voter prefers \texttt{LDP} and \texttt{SKG} to \texttt{NFP} and \texttt{JCP}, and between the latter two, she prefers \texttt{NFP} to \texttt{JCP}. Although \texttt{LDP} and \texttt{SKG} have the same preference, we do not constrain the estimated preferences to be the same for these two alternatives. Under the Gaussian random utility model, the probability that the two alternatives having exactly the same preferences is zero. Therefore, inequality constraints are respected, but equality constraints are not. Furthermore, we only preserve the ranking, not the relative numerical values. Therefore, the following coding of the variables, for our purposes, is equivalent to that given above, \spacingset{1} \begin{verbatim} LDP NFP SKG JCP 3 2 3 1 \end{verbatim} \spacingset{1.5} Finally, it is possible to have non-response for one of the categories; e.g., no candidate from a particular party may run in a certain district. If \texttt{NFP = NA}, we have no information about the relative ranking of \texttt{NFP}. \spacingset{1} \begin{verbatim} LDP NFP SKG JCP 3 NA 3 1 \end{verbatim} \spacingset{1.5} In this case, there is no constraint when estimating the preference for this alternative; only the inequality constraint, \texttt{(LDP, SKG) > JCP}, is imposed. All three covariates -- gender, education, and age of voters -- are individual-specific variables rather than choice-specific ones. The model estimates three intercepts and 9 coefficients along with 6 parameters in the covariance matrix. The following commands fit the model, \spacingset{1} \begin{verbatim} data(japan) res <- mnp(cbind(LDP, NFP, SKG, JCP) ~ gender + education + age, data = japan, n.draws = 10000, verbose = TRUE) summary(res) \end{verbatim} \spacingset{1.5} The first command loads the dataset, and the second command fits the model. The base category is \texttt{JCP}, which is the last column of the outcome matrix. The default prior distribution is used as in the previous example: an improper prior distribution for $\beta$ and a diffuse prior distribution for $\Sigma$ with $\nu = p = 4$ and $S=I$. 10,000 draws are obtained with no burnin or thinning. The final command summarizes the Monte Carlo sample and gives the following output, \spacingset{1} \begin{verbatim} Call: mnp(formula = cbind(LDP, NFP, SKG, JCP) ~ gender + education + age, data = japan, n.draws = 10000, verbose = TRUE) Coefficients: mean std.dev. 2.5% 97.5% (Intercept):LDP 0.615184 0.517157 -0.386151 1.61 (Intercept):NFP 0.689753 0.568109 -0.419521 1.79 (Intercept):SKG 0.133961 0.455960 -0.758883 1.02 gendermale:LDP 0.099748 0.152323 -0.194786 0.40 gendermale:NFP 0.216824 0.166103 -0.102108 0.54 gendermale:SKG 0.132661 0.134605 -0.127145 0.40 education:LDP -0.107038 0.074792 -0.253483 0.04 education:NFP -0.107222 0.082324 -0.270127 0.05 education:SKG -0.003728 0.066429 -0.132496 0.13 age:LDP 0.013518 0.006122 0.001492 0.03 age:NFP 0.006948 0.006783 -0.006572 0.02 age:SKG 0.009653 0.005431 -0.000812 0.02 Covariances: mean std.dev. 2.5% 97.5% LDP:LDP 1.0000 0.0000 1.0000 1.00 LDP:NFP 1.0502 0.0585 0.9373 1.16 LDP:SKG 0.7070 0.0622 0.5822 0.82 NFP:NFP 1.4068 0.1359 1.1682 1.70 NFP:SKG 0.7452 0.0864 0.5800 0.91 SKG:SKG 0.6913 0.0874 0.5296 0.87 Base category: JCP Number of alternatives: 4 Number of observations: 418 Number of stored MCMC draws: 10000 \end{verbatim} \spacingset{1.5} \subsection{Convergence Diagnostics, Final Analysis, and Conclusions} In order to evaluate convergence of the MCMC sampler, we again obtain three independent Markov chains by running the \texttt{mnp()} command three times with three sets of different starting values. We use starting values that are relatively dispersed given the preliminary analysis of the previous section. Note that when fitting the multinomial probit model with ordered preferences, the algorithm requires the starting values of the latent variable to respect the order constraints of equation (\ref{eq:mop}). Therefore, the starting values of the parameters cannot be too far away from the posterior mode. The following commands fits the model with the default starting value and two sets of overdispersed starting values, \spacingset{1} \begin{verbatim} res1 <- mnp(cbind(LDP, NFP, SKG, JCP) ~ gender + education + age, data = japan, n.draws = 50000, verbose = TRUE) res2 <- mnp(cbind(LDP, NFP, SKG, JCP) ~ gender + education + age, data = japan, coef.start = c(1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1), cov.start = matrix(0.5, ncol=3, nrow=3) + diag(0.5, 3), n.draws = 50000, verbose = TRUE) res3 <- mnp(cbind(LDP, NFP, SKG, JCP) ~ gender + education + age, data = japan, coef.start = c(-1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1), cov.start = matrix(0.9, ncol=3, nrow=3) + diag(0.1, 3), n.draws = 50000, verbose = TRUE) \end{verbatim} \spacingset{1.5} %res.coda <- mcmc.list(chain1=mcmc(res1$param[,-13]), % chain2=mcmc(res2$param[,-13]), % chain3=mcmc(res3$param[,-13])) %gelman.plot(res.coda, transform = TRUE, ylim = c(1,1.2)) %res.coda <- mcmc.list(chain1=mcmc(res1$param[25001:50000,-13], start=25001), % chain2=mcmc(res2$param[25001:50000,-13], start=25001), % chain3=mcmc(res3$param[25001:50000,-13], start=25001)) %summary(res.coda) We follow the commands used in Section~\ref{sec:coda} and compute the Gelman-Rubin statistic for each parameter. Upon examination of the resulting statistics, we determined that satisfactory convergence has been achieved. For example, the value of the Gelman-Rubin statistic is less than 1.01 for all the parameters. Hence, we base our final analysis on the combined draws from the second half of the three chains (i.e., a total of 75,000 draws using 25,000 draws from each chain). Posterior summaries can be obtained using the coda package as before, \spacingset{1} \begin{verbatim} Iterations = 25001:50000 Thinning interval = 1 Number of chains = 3 Sample size per chain = 25000 1. Empirical mean and standard deviation for each variable, plus standard error of the mean: Mean SD Naive SE Time-series SE (Intercept):LDP 0.60167 0.51421 1.88e-03 8.05e-03 (Intercept):NFP 0.68294 0.56867 2.08e-03 7.95e-03 (Intercept):SKG 0.12480 0.45680 1.67e-03 7.25e-03 gendermale:LDP 0.10668 0.15448 5.64e-04 2.95e-03 gendermale:NFP 0.22240 0.16983 6.20e-04 2.91e-03 gendermale:SKG 0.13897 0.13753 5.02e-04 2.70e-03 education:LDP -0.10517 0.07643 2.79e-04 1.35e-03 education:NFP -0.10634 0.08448 3.08e-04 1.28e-03 education:SKG -0.00258 0.06766 2.47e-04 1.18e-03 age:LDP 0.01361 0.00617 2.25e-05 9.90e-05 age:NFP 0.00698 0.00680 2.48e-05 1.01e-04 age:SKG 0.00972 0.00547 2.00e-05 9.13e-05 LDP:NFP 1.05535 0.05508 2.01e-04 1.15e-03 LDP:SKG 0.71199 0.06125 2.24e-04 1.59e-03 NFP:NFP 1.41860 0.13540 4.94e-04 2.45e-03 NFP:SKG 0.75391 0.08262 3.02e-04 2.12e-03 SKG:SKG 0.70007 0.08488 3.10e-04 2.16e-03 2. Quantiles for each variable: 2.5% 25% 50% 75% 97.5% (Intercept):LDP -0.405757 0.25198 0.60033 0.9476 1.6172 (Intercept):NFP -0.428421 0.30016 0.68058 1.0657 1.7981 (Intercept):SKG -0.769018 -0.18476 0.12335 0.4303 1.0258 gendermale:LDP -0.197199 0.00328 0.10643 0.2105 0.4096 gendermale:NFP -0.110730 0.10856 0.22135 0.3361 0.5566 gendermale:SKG -0.131661 0.04702 0.13905 0.2307 0.4096 education:LDP -0.254631 -0.15718 -0.10519 -0.0533 0.0447 education:NFP -0.271657 -0.16329 -0.10654 -0.0496 0.0595 education:SKG -0.135829 -0.04833 -0.00248 0.0429 0.1306 age:LDP 0.001591 0.00941 0.01361 0.0177 0.0257 age:NFP -0.006336 0.00240 0.00697 0.0115 0.0203 age:SKG -0.000947 0.00603 0.00967 0.0134 0.0206 LDP:NFP 0.944577 1.01919 1.05564 1.0924 1.1623 LDP:SKG 0.587135 0.67120 0.71364 0.7544 0.8266 NFP:NFP 1.181667 1.32454 1.40803 1.5028 1.7125 NFP:SKG 0.590798 0.69778 0.75463 0.8104 0.9125 SKG:SKG 0.538858 0.64219 0.69806 0.7564 0.8711 \end{verbatim} \spacingset{1.5} Here, one of the findings is that older voters tend to prefer \texttt{LDP} as indicated by the statistically significant positive age coefficient for \texttt{LDP}. This is consistent with the conventional wisdom of Japanese politics that the stronghold of \texttt{LDP} is elderly voters. To further investigate the marginal effect of age, we calculate the posterior predictive probabilities of party preference under two scenarios. First, we choose the 10th individual in the survey data and compute the predictive probability that a voter with this set of covariates prefers each of the parties. This can be accomplished by the following commands, \spacingset{1}\begin{verbatim} japan10a <- japan[10,] predict(res1, newdata = japan10a, newdraw = rbind(res1$param[25001:50000,], res2$param[25001:50000,], res3$param[25001:50000,]), type = "prob") \end{verbatim} \spacingset{1.5} where the first command extracts the 10th observation from the Japan data, and the second command computes the predictive probabilities. Note that this individual has the following attributes, \spacingset{1} \begin{verbatim} gender education age male 4 50 \end{verbatim} \spacingset{1.5} The resulting posterior predictive probabilities of being the most preferred party are, \spacingset{1} \begin{verbatim} JCP LDP NFP SKG [1,] 0.107707 0.359267 0.324613 0.208413 \end{verbatim} \spacingset{1.5} The result indicates that under the model, we should expect 36\% of voters with these covariates to prefer \texttt{LDP}, 32\% to prefer \texttt{NFP}, 21\% to prefer \texttt{SKG}, and 11\% to prefer \texttt{JCP}. Next, we change the value of the age variable of this voter from 50 to 75, while holding the other variables constant. We then recompute the posterior predictive probabilities and examine how they change. This can be accomplished using the following commands, \spacingset{1} \begin{verbatim} japan10b <- japan10a japan10b[,"age"] <- 75 predict(res1, newdata = japan10b, newdraw = rbind(res1$param[25001:50000,], res2$param[25001:50000,], res3$param[25001:50000,]), type = "prob") \end{verbatim} \spacingset{1.5} where the first two commands recode the age variable for the voter and the second command makes the prediction. We obtain the following results, \spacingset{1} \begin{verbatim} JCP LDP NFP SKG [1,] 0.06548 0.485467 0.249667 0.199387 \end{verbatim} \spacingset{1.5} The comparison of the two results shows that changing the value of the age variable from 50 to 75 increases the estimated posterior predictive probability of preferring \texttt{LDP} most and by more than 10 percentage points. Interestingly, the predictive probability for \texttt{SKG} changes very little, while that of \texttt{NFP} decreases significantly. This suggests that older voters tend to prefer \texttt{LDP} over \texttt{NFP}. \clearpage \pdfbookmark{References}{References} \bibliography{my,imai} \end{document} MNP/tests/0000755000176200001440000000000013162676245012062 5ustar liggesusersMNP/tests/testthat.R0000644000176200001440000000006213162676245014043 0ustar liggesuserslibrary(testthat) library(MNP) test_check("MNP") MNP/tests/testthat/0000755000176200001440000000000013162707044013712 5ustar liggesusersMNP/tests/testthat/test-all.R0000644000176200001440000000456713162676245015606 0ustar liggesusersrm(list=ls()) library(MNP) library(testthat) context("tests MNP") test_that("tests MNP on the detergent data", { # set random seed set.seed(12345) # load the detergent data data(detergent) # run the standard multinomial probit model with intercepts and the price res1 <- mnp(choice ~ 1, choiceX = list(Surf=SurfPrice, Tide=TidePrice, Wisk=WiskPrice, EraPlus=EraPlusPrice, Solo=SoloPrice, All=AllPrice), cXnames = "price", data = detergent, n.draws = 30, burnin = 10, thin = 3, verbose = TRUE) # summarize the results x <- summary(res1) expect_that(length(x), is_equivalent_to(8)) expect_true("coef.table" %in% names(x)) expect_equal(x$coef.table[4, 1], 1.864768, tolerance = 0.001) expect_equal(x$coef.table["(Intercept):Solo", "2.5%"], 1.108233, tolerance = 0.001) expect_equal(x$cov.table[10, 1], 0.9855489, tolerance = 0.001) expect_equal(x$cov.table["Tide:Tide", "mean"], 0.7952514, tolerance = 0.001) # calculate the quantities of interest for the first 3 observations x <- predict(res1, newdata = detergent[1:3,]) expect_that(length(x), is_equivalent_to(4)) expect_true("p" %in% names(x)) expect_that(dim(x$o), is_equivalent_to(c(3, 6, 5))) expect_equal(as.numeric(x$p[1, 3]), 0.2, tolerance = 0.05) expect_equal(as.numeric(x$p[3, "Wisk"]), 0.4, tolerance = 0.05) }) test_that("tests MNP on the Japanese election census", { # set random seed set.seed(12345) # load the Japanese election data data(japan) # run the multinomial probit model with ordered preferences res2 <- mnp(cbind(LDP, NFP, SKG, JCP) ~ gender + education + age, data = japan, verbose = TRUE) # summarize the results x <- summary(res2) expect_that(length(x), is_equivalent_to(8)) expect_true("coef.table" %in% names(x)) expect_equal(x$cov.table[2,1], 1.01725, tolerance = 0.01) expect_equal(x$cov.table["LDP:LDP", "mean"], 0.9667556, tolerance = 0.01) # calculate the predicted probabilities for the 10th observation # averaging over 100 additional Monte Carlo draws given each of MCMC draw. x <- predict(res2, newdata = japan[10,], type = "prob", n.draws = 100, verbose = TRUE) expect_that(length(x), is_equivalent_to(2)) expect_true("p" %in% names(x)) expect_that(dim(x$p), is_equivalent_to(c(1, 4, 5000))) expect_that(x$x[1, "age:LDP"], is_equivalent_to(50)) expect_that(x$x[1, 1], is_equivalent_to(1)) }) MNP/src/0000755000176200001440000000000013162677535011512 5ustar liggesusersMNP/src/Makevars0000644000176200001440000000006313162677535013205 0ustar liggesusers PKG_LIBS = $(LAPACK_LIBS) $(BLAS_LIBS) $(FLIBS) MNP/src/rand.h0000644000176200001440000000041513162677535012607 0ustar liggesusersdouble sTruncNorm(double bd, double mu, double var, int lower); double TruncNorm(double lb, double ub, double mu, double var, int invcdf); void rMVN(double *Sample, double *mean, double **inv_Var, int size); void rWish(double **Sample, double **S, int df, int size); MNP/src/rand.c0000644000176200001440000001225413162677535012606 0ustar liggesusers #include #include #include #include #include #include #include #include "vector.h" #include "subroutines.h" #include "rand.h" double sTruncNorm( double bd, /* bound */ double mu, double var, int lower /* 1 = x > bd, 0 = x < bd */ ) { double z, logb, lambda; double sigma = sqrt(var); double stbd = (bd - mu)/sigma; if (lower == 0) { stbd = (mu - bd)/sigma; } if (stbd > 0) { lambda = 0.5*(stbd + sqrt(stbd*stbd + 4)); logb = 0.5*(lambda*lambda-2*lambda*stbd); do { z = rexp(1/lambda); /* Rprintf("%5g\n", exp(-0.5*(z+stbd)*(z+stbd)+lambda*z-logb)); */ } while (unif_rand() > exp(-0.5*(z+stbd)*(z+stbd)+lambda*z-logb)); } else { do z = norm_rand(); while(z < stbd); } if (lower == 1) { return(z*sigma + mu); } else { return(-z*sigma + mu); } } /* Sample from a univariate truncated Normal distribution (truncated both from above and below): choose either inverse cdf method or rejection sampling method. For rejection sampling, if the range is too far from mu, it uses standard rejection sampling algorithm with exponential envelope function. */ double TruncNorm( double lb, /* lower bound */ double ub, /* upper bound */ double mu, /* mean */ double var, /* variance */ int invcdf /* use inverse cdf method? */ ) { double z; double sigma = sqrt(var); double stlb = (lb-mu)/sigma; /* standardized lower bound */ double stub = (ub-mu)/sigma; /* standardized upper bound */ if(stlb > stub) error("TruncNorm: lower bound is greater than upper bound\n"); if(stlb == stub) { warning("TruncNorm: lower bound is equal to upper bound\n"); return(stlb*sigma + mu); } if (invcdf) { /* inverse cdf method */ z = qnorm(runif(pnorm(stlb, 0, 1, 1, 0), pnorm(stub, 0, 1, 1, 0)), 0, 1, 1, 0); } else { /* rejection sampling method */ double tol=2.0; double temp, M, u, exp_par; int flag=0; /* 1 if stlb, stub <-tol */ if(stub<=-tol){ flag=1; temp=stub; stub=-stlb; stlb=-temp; } if(stlb>=tol){ exp_par=stlb; while(pexp(stub,1/exp_par,1,0) - pexp(stlb,1/exp_par,1,0) < 0.000001) exp_par/=2.0; if(dnorm(stlb,0,1,1) - dexp(stlb,1/exp_par,1) >= dnorm(stub,0,1,1) - dexp(stub,1/exp_par,1)) M=exp(dnorm(stlb,0,1,1) - dexp(stlb,1/exp_par,1)); else M=exp(dnorm(stub,0,1,1) - dexp(stub,1/exp_par,1)); do{ u=unif_rand(); z=-log(1-u*(pexp(stub,1/exp_par,1,0)-pexp(stlb,1/exp_par,1,0)) -pexp(stlb,1/exp_par,1,0))/exp_par; }while(unif_rand() > exp(dnorm(z,0,1,1)-dexp(z,1/exp_par,1))/M ); if(flag==1) z=-z; } else{ do z=norm_rand(); while( zstub ); } } return(z*sigma + mu); } /* Sample from the MVN dist */ void rMVN( double *Sample, /* Vector for the sample */ double *mean, /* The vector of means */ double **Var, /* The matrix Variance */ int size) /* The dimension */ { int j,k; double **Model = doubleMatrix(size+1, size+1); double cond_mean; /* draw from mult. normal using SWP */ for(j=1;j<=size;j++){ for(k=1;k<=size;k++) Model[j][k]=Var[j-1][k-1]; Model[0][j]=mean[j-1]; Model[j][0]=mean[j-1]; } Model[0][0]=-1; Sample[0]=(double)norm_rand()*sqrt(Model[1][1])+Model[0][1]; for(j=2;j<=size;j++){ SWP(Model,j-1,size+1); cond_mean=Model[j][0]; for(k=1;k0) for(k=0;k0) for(k=0;k #include #include #include #include #include #include #include #include "vector.h" #include "rand.h" /* The Sweep operator */ void SWP( double **X, /* The Matrix to work on */ int k, /* The row to sweep */ int size) /* The dim. of X */ { int i,j; if (X[k][k] < 10e-20) error("SWP: singular matrix.\n"); else X[k][k]=-1/X[k][k]; for(i=0;i #include #include #include #include #include int* intArray(int num) { int *iArray = (int *)malloc(num * sizeof(int)); if (!iArray) error("Out of memory error in intArray\n"); return iArray; } void PintArray(int *ivector, int length) { int i; for (i = 0; i < length; i++) Rprintf("%5d\n", ivector[i]); } int** intMatrix(int row, int col) { int i; int **iMatrix = (int **)malloc(row * sizeof(int *)); if (!iMatrix) error("Out of memory error in intMatrix\n"); for (i = 0; i < row; i++) { iMatrix[i] = (int *)malloc(col * sizeof(int)); if (!iMatrix[i]) error("Out of memory error in intMatrix\n"); } return iMatrix; } void PintMatrix(int **imatrix, int row, int col) { int i, j; for (i = 0; i < row; i++) { for (j = 0; j < col; j++) Rprintf("%5d", imatrix[i][j]); Rprintf("\n"); } } double* doubleArray(int num) { double *dArray = (double *)malloc(num * sizeof(double)); if (!dArray) error("Out of memory error in doubleArray\n"); return dArray; } void PdoubleArray(double *dvector, int length) { int i; for (i = 0; i < length; i++) Rprintf("%14g\n", dvector[i]); } double** doubleMatrix(int row, int col) { int i; double **dMatrix = (double **)malloc((size_t)(row * sizeof(double *))); if (!dMatrix) error("Out of memory error in doubleMatrix\n"); for (i = 0; i < row; i++) { dMatrix[i] = (double *)malloc((size_t)(col * sizeof(double))); if (!dMatrix[i]) error("Out of memory error in doubleMatrix\n"); } return dMatrix; } void PdoubleMatrix(double **dmatrix, int row, int col) { int i, j; for (i = 0; i < row; i++) { for (j = 0; j < col; j++) Rprintf("%14g", dmatrix[i][j]); Rprintf("\n"); } } double*** doubleMatrix3D(int x, int y, int z) { int i; double ***dM3 = (double ***)malloc(x * sizeof(double **)); if (!dM3) error("Out of memory error in doubleMatrix3D\n"); for (i = 0; i < x; i++) dM3[i] = doubleMatrix(y, z); return dM3; } void PdoubleMatrix3D(double ***dmatrix3D, int x, int y, int z) { int i, j, k; for (i = 0; i < x; i++) { Rprintf("Fist dimension = %5d\n", i); for (j = 0; j < y; j++) { for (k = 0; k < z; k++) Rprintf("%14g", dmatrix3D[i][j][k]); Rprintf("\n"); } } } long* longArray(int num) { long *lArray = (long *)malloc(num * sizeof(long)); if (!lArray) error("Out of memory error in longArray\n"); return lArray; } void FreeMatrix(double **Matrix, int row) { int i; for (i = 0; i < row; i++) free(Matrix[i]); free(Matrix); } void FreeintMatrix(int **Matrix, int row) { int i; for (i = 0; i < row; i++) free(Matrix[i]); free(Matrix); } void Free3DMatrix(double ***Matrix, int index, int row) { int i; for (i = 0; i < index; i++) FreeMatrix(Matrix[i], row); free(Matrix); } MNP/src/vector.h0000644000176200001440000000121313162677535013162 0ustar liggesusers#include #include int *intArray(int num); void PintArray(int *ivector, int length); int **intMatrix(int row, int col); void PintMatrix(int **imatrix, int row, int col); double *doubleArray(int num); void PdoubleArray(double *dvector, int length); double **doubleMatrix(int row, int col); void PdoubleMatrix(double **dmatrix, int row, int col); double ***doubleMatrix3D(int x, int y, int z); void PdoubleMatrix3D(double ***dmatrix3D, int x, int y, int z); long *longArray(int num); void FreeMatrix(double **Matrix, int row); void FreeintMatrix(int **Matrix, int row); void Free3DMatrix(double ***Matrix, int index, int row); MNP/src/subroutines.h0000644000176200001440000000027313162677535014247 0ustar liggesusers void SWP( double **X, int k, int size); void dinv(double **X, int size, double **X_inv); void dcholdc(double **X, int size, double **L); double ddet(double **X, int size, int give_log); MNP/src/MNP.c0000644000176200001440000004340013162677535012311 0ustar liggesusers/****************************************************************** This file is a part of MNP: R Package for Estimating the Multinomial Probit Models by Kosuke Imai and David A. van Dyk. Copyright: GPL version 2 or later. *******************************************************************/ #include #include #include #include #include #include #include #include "vector.h" #include "subroutines.h" #include "rand.h" void cMNPgibbs(int *piNDim, int *piNCov, int *piNSamp, int *piNGen, double *b0, /* prior mean for beta */ double *pdA0, int *piNu0, double *pdS, double *pdX, int *y, /* response variable: -1 for missing */ double *pdbeta, double *pdSigma, int *piImp, int *invcdf, /* use inverse cdf for TruncNorm? */ int *piBurnin, /* the number of burnin */ int *piKeep, int *itrace, int *verbose, /* 1 if extra print is needed */ int *piMoP, /* 1 if Multinomial ordered Probit */ int *latent, /* 1 if W is stored */ double *pdStore){ /* paramerters from R */ int n_samp = *piNSamp; /* sample size */ int n_gen = *piNGen; /* number of gibbs draws */ int n_cov = *piNCov; /* The number of covariates */ int n_dim = *piNDim; /* The number of indentifiable dimension (p-1) */ int nu0 = *piNu0; /* nu0: degree of freedom, S: scale matrix */ int imp = *piImp; /* 0: improper prior for beta - algorithms 1 & 2 1: proper prior for beta */ int keep = 1; /* counter for keep */ int progress = 1; /* counter for progress report */ double *beta; /* The model parameters */ double **Sigma; /* The unidentifiable variance matrix */ double **X; /* The covariates and outcome var */ double **A0; /* prior precision for beta */ double **S; /* The prior parameters for Sigma */ /* model parameters */ int **Y; /* The response variable */ double **W; /* The missing data! */ double cmean; /* The conditional mean for Wij */ double cvar; /* The conditional variance for Wij */ double maxw=0.0, minw=0.0; /* used for sampling W */ int *maxy; double **Xbeta; double ***PerSig; /* Permuted Sigma for sampling W */ int kindx, lindx; /* Indexes for the permutation of Sigma */ double *mbeta; /* Posterior mean of beta */ double **Vbeta; /* The var of beta, given Yaug */ double **Xstar; /* L*X where L is the Chol factor of SigInv */ double **SigInv; /* The Inverse of Sigma */ double *epsilon; /* The error term */ double **R; /* Sum of squares matrix e'e */ double **SS; /* Sum of squares for sweep operator */ double alpha2; /* alpha^2: Inv-chisq df=nu0 */ double ss; /* scale parameter for alpha^2 */ int i, j, k, l, main_loop; /* used for loops */ /* temporay storages */ int itemp, itempMax, itempMin, *ivtemp, itempS = 0, itempP=ftrunc((double) n_gen/10); double *vtemp; double **mtemp,**mtemp1,**mtemp2; /** get random seed **/ GetRNGstate(); /** defining vectors and matricies **/ Y = intMatrix(n_samp, n_dim+1); W = doubleMatrix(n_samp, n_dim+1); X = doubleMatrix(n_samp*n_dim+n_cov, n_cov+1); Xbeta = doubleMatrix(n_samp, n_dim); SS = doubleMatrix(n_cov+1, n_cov+1); Sigma = doubleMatrix(n_dim, n_dim); SigInv = doubleMatrix(n_dim, n_dim); epsilon = doubleArray(n_samp*n_dim); R = doubleMatrix(n_dim, n_dim); beta = doubleArray(n_cov); mbeta = doubleArray(n_cov); Vbeta = doubleMatrix(n_cov, n_cov); A0 = doubleMatrix(n_cov, n_cov); S = doubleMatrix(n_dim, n_dim); vtemp = doubleArray(n_dim-1); maxy = intArray(n_samp); ivtemp = intArray(n_dim+1); Xstar = doubleMatrix(n_samp*n_dim+n_cov, n_cov+1); mtemp = doubleMatrix(n_cov, n_cov); mtemp1 = doubleMatrix(n_dim, n_dim); mtemp2 = doubleMatrix(n_dim, n_dim); PerSig = doubleMatrix3D(n_dim, n_dim, n_dim); /** Packing Y, X, A0, S, beta, Sigma **/ if(*piMoP) { itemp = 0; for (j = 0; j <= n_dim; j++) for (i = 0; i < n_samp; i++) Y[i][j] = y[itemp++]; for (i = 0; i < n_samp; i++) { /* recording the max of Y[i] */ for (j=0; j<= n_dim; j++) ivtemp[j]=Y[i][j]; R_isort(ivtemp, n_dim+1); maxy[i]=ivtemp[n_dim]; } } itemp = 0; for (k = 0; k < n_cov; k++) for (j = 0; j < n_dim; j++) for (i = 0; i < n_samp; i++) X[i*n_dim+j][k] = pdX[itemp++]; /* PdoubleMatrix(X, n_dim*3, n_cov); */ itemp = 0; for (k = 0; k < n_cov; k++) for (j = 0; j < n_cov; j++) A0[j][k] = pdA0[itemp++]; itemp = 0; for (k = 0; k < n_dim; k++) for (j = 0; j < n_dim; j++) S[j][k] = pdS[itemp++]; itemp = 0; for (j=0; j0) && (maxw > W[i][k])) maxw=W[i][k]; } if(Y[i][k]==(Y[i][j]-1)) { if(itempMin==0) { minw=W[i][k]; itempMin++; } if((itempMin>0) && (minw < W[i][k])) minw=W[i][k]; } } } itemp=0; for (k=0;k *piBurnin) { if(keep==*piKeep) { for(j=0;j 1) { for (j = 0; j <= n_dim; j++) probTemp[j] = 0; } /* compute the mean for each dimension */ for (j = 0; j < n_dim; j++) { Xbeta[j] = 0; for (k = 0; k < n_cov; k++) Xbeta[j] += X[i*n_dim+j][k] * beta[main_loop][k]; } /* PdoubleArray(Xbeta, n_dim); */ /* sample W */ for (j = 0; j < n_extra; j++) { /*dinv(Sigma[main_loop], n_dim, mtemp);*/ rMVN(vtemp, Xbeta, Sigma[main_loop], n_dim); for (k = 0; k < n_dim; k++) W[j][k+1] = vtemp[k]; W[j][0] = 0; } /* which dimension is max for each of n_extra W draws? PdoubleMatrix(W, n_extra, n_dim+1); */ R_max_col2(W, n_extra, n_dim+1, maxdim, 1); /* PintArray(maxdim, n_extra); */ /* order */ for (j = 0; j < n_extra; j++) { for (k = 0; k <= n_dim; k++) { ind[k] = k; sumorder[k] = 0; } revsort(W[j], ind, n_dim+1); /* PintArray(ind, n_dim+1); */ for (k = 0; k <= n_dim; k++) sumorder[ind[k]] += (k+1); if(*verbose) { if(count == itempQ) { Rprintf("%3d percent done.\n", progress*10); itempQ += ftrunc((double) total/10); progress++; R_FlushConsole(); } count++; } } if (n_extra > 1) { /* store probability and mean order */ for (j = 0; j <= n_dim; j++) { itemp = 0; for (k = 0; k < n_extra; k++) if (maxdim[k] == (j+1)) itemp++; prob[itempP++] = ((double) itemp / (double) n_extra); order[itempO++] = ((double) sumorder[j] / (double) n_extra); } } else { /* store choice */ for (j = 0; j <= n_dim; j++) { if (maxdim[0] == (j+1)) { choice[itempC++] = j; probTemp[j]++; } order[itempO++] = sumorder[j]; } } } if (n_extra == 1) for (j = 0; j <= n_dim; j++) prob[itempP++] = ((double) probTemp[j] / (double) n_draw); } /** write out the random seed **/ PutRNGstate(); /* freeing memory */ FreeMatrix(X, n_samp*n_dim); free(vtemp); free(Xbeta); FreeMatrix(W, n_extra); FreeMatrix(beta, n_draw); FreeMatrix(mtemp, n_dim); Free3DMatrix(Sigma, n_draw, n_dim); free(maxdim); free(ind); free(sumorder); free(probTemp); } MNP/src/init.c0000644000176200001440000000135213162677535012622 0ustar liggesusers#include // for NULL #include /* FIXME: Check these declarations against the C/Fortran source code. */ /* .C calls */ extern void cMNPgibbs(void *, void *, void *, void *, void *, void *, void *, void *, void *, void *, void *, void *, void *, void *, void *, void *, void *, void *, void *, void *, void *); extern void predict(void *, void *, void *, void *, void *, void *, void *, void *, void *, void *, void *, void *); static const R_CMethodDef CEntries[] = { {"cMNPgibbs", (DL_FUNC) &cMNPgibbs, 21}, {"predict", (DL_FUNC) &predict, 12}, {NULL, NULL, 0} }; void R_init_MNP(DllInfo *dll) { R_registerRoutines(dll, CEntries, NULL, NULL, NULL); R_useDynamicSymbols(dll, FALSE); } MNP/NAMESPACE0000644000176200001440000000064413103175765012137 0ustar liggesusersuseDynLib(MNP, .registration = TRUE) importFrom("utils", "packageDescription") importFrom("stats", "as.formula", "model.matrix.default", "model.response", "printCoefmat", "quantile", "relevel", "sd") importFrom(MASS, mvrnorm) export(mnp, coef.mnp, cov.mnp, predict.mnp, summary.mnp, print.summary.mnp) S3method(coef, mnp) S3method(summary, mnp) S3method(predict, mnp) S3method(print, mnp) S3method(print, summary.mnp) MNP/data/0000755000176200001440000000000013101760304011610 5ustar liggesusersMNP/data/japan.txt.gz0000644000176200001440000000450413101760304014064 0ustar liggesusers}Zێ-|X/\A B `@H\sĉҗ$v\vx}Mw_|ϏܯL~}ϏWr%u^Qj_"=8c&_3Wp%Չ@^03wp_~=w]fYW/kgߍ+=q1#uʛ?Mz9![*>aM9pڕ8fkzlIdmm~q5\/pc[`鰵~m2]K߭?5Ҧ$e!2s =bCE'T5AύjVA7YhJO^߈qVC ٥\|t RTcYx@`8%Qck# ĎjB ؐҳ-O20> r&!-7:*+fPfEka9L$ɚUYVsmӌg$[tD49osɐd(&9)C{Q6OAa׳Pt%*uoi)wL,+Tm(4JzVTIYwn:s.@Ae9Kߘ ꩸t<2Yl ' :+"Y;9 YIGG=WFO-0`ʣy!JnCT^OdQZ / V=Q(p21US (cv,.pJ*37XHBUҀ)w\9 ՓĚX[@ܑ- ѳy\8T2"]Bټ\y4wh#b*s+dʩ84{Y iH9 Y@I{SU!4-YU5OD)Lszi7Ex FVpGuMausQ:8ȭutjTBlؔIK`!5ܺ]<{BQtA[6vhgl&"t<2L <&.PHuS2f8TY2+ܙEXCBOođT晰3Ty4xkr 6}U?gFr{bgy~'2&ƤGz`\ZO o5^3AViw){Eh߲{qa;5)sHv֛:9b]"xY@*}$/Z4 oY%t0v{Y2)"\gq56a"'y|q)|Kb5K&bzVIk._]z{A}ݢ*<^g=Y(5ۏb/ -[",)Z>y/;C&iW97li Oyt?LV'MNP/data/detergent.txt.gz0000644000176200001440000007176313101760304014767 0ustar liggesusersͽˮ>s%6S~@0HN1b@7CZ%VQ8ۧEu]__ۿ?ۿ?_ma۶sG[أ9;[>|OIh}Up6qUvaU~TOuCʒ;/ҎUF]}yuEro=qSk羝.#;|\φtN['.-mg~E A|^<t10Ϫ^^q;%k,h fV+~|ҡOoޔ;g;LPOG&g8MF%y{Gy<3?͞z]J~^fk+ns'(6M6 oBcg9InTTGۭYrIԑ4D:)˪.% ٣ &fMTW> :%]TDw[t!jc(*M>Jߙ*ADUiuKrRB4Q\+Y}PUr/ž:3 ؟!7UC ùϕwHvQa4ў&)S$ v,{,&EҎDUT<t]q6Iuj- ?Φ@|#c{yiLkXl61ŸGO]l |Oy<+s{S2>vuѤ0Ge0c9 QĨhRtp]'haZ+܇UöǤO}\ncSZ`B]y+%f/ ba>ن2( ܁א6 Jx>w$ %73(Ã'ŭ.cr}]K_T=_X;"E[JV=`ĽPb邤wLS [Řqwǘ&?0ymIwi;z'sOIL7,g8;Fù>)_+`M Mjcad!4y< SX{+AW`ʁMJ8kbZBQK2;JI<!Ԏqu2a X&dt6v~~%6XEziU\VlO H?E̺ ~D.#&J _-ɀV*AIC™B7ŤI@\w yŎ w_8dXVu)>dɼLcEWX4$ A@,II`.I3ρza_{wٍwh^}tm2c+~}PJ$aL >> kF?NOF Mr *%@dN 4 Q/"=,mMiZk;&E^ ғVd8:R~3Mӟ<OYĶ* ,̑Yc|:Z_=t>XPo_۾B6Q+UWc\w_".oBB6;aTc $U%'dϐ?!C.&{ u^EO}(Ut@FV.wk+HsFd^9J0z# }ս~FצQ7][1HGp<FiU$̎(7l 9J~W3P\}:V{ _P;B;zv@9vs:,n!Y~[B* ulha;Ӂ%ݘ{T(v-KPM95Ea}y#YN.5(2KanWEḪW؏5Ѕ}UOjal[ad%ix ;< ړU=l;YbxAT9ƒNEY T},q穉@j*&9eQآ(cFpUxj\NAɠ0/^0β3f|\YR ͅ` eFʯl#t񵿚mg4ԶjcF(Um3o:dlA7Ȓ@CG:3EE1cPXعm ARW?̾5"B 7+p ;VR0Rϥ5 L@9_Jtnkܐ>~\bHf{iG$Gb%Ev_va5m,E *C§'C pil>xԁI[{N9c ],!^4@Q^k\HhGTOy,GާZEb-꥾i# X<]ymfF)wt2]B͖΄^NżE')g!u'gt(kcY8H_c(A4$"g-ohߢ*HO30c)~d\] dguIo]zV򸪅d3>प΃}z/I0QcCs?6XystZ[ei6o.eam6\3IJkA$ "ab^񧳣2YRyBGd"JZc!qh੽vٸ}(69 yd)䙩u? eg. 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The default is the equal tail 95 #' percent credible interval. #' @param ... further arguments passed to or from other methods. #' @return \code{summary.mnp} yields an object of class \code{summary.mnp} #' containing the following elements: \item{call}{The call from \code{mnp}.} #' \item{n.alt}{The total number of alternatives.} \item{base}{The base #' category used for fitting.} \item{n.obs}{The number of observations.} #' \item{n.param}{The number of estimated parameters.} \item{n.draws}{The #' number of Gibbs draws used for the summary.} \item{coef.table}{The summary #' of the posterior distribution of the coefficients. } \item{cov.table}{The #' summary of the posterior distribution of the covariance matrix.} This object #' can be printed by \code{print.summary.mnp} #' @author Kosuke Imai, Department of Politics, Princeton University #' \email{kimai@Princeton.Edu} #' @seealso \code{mnp} #' @keywords methods #' @export summary.mnp summary.mnp <- function(object, CI=c(2.5, 97.5),...){ p <- object$n.alt param <- object$param n.cov <- ncol(param) - p*(p-1)/2 n.draws <- nrow(param) param.table <- cbind(apply(param, 2, mean), apply(param, 2, sd), apply(param, 2, quantile, min(CI)/100), apply(param, 2, quantile, max(CI)/100)) colnames(param.table) <- c("mean", "std.dev.", paste(min(CI), "%", sep=""), paste(max(CI), "%", sep="")) rownames(param.table) <- colnames(param) ans <- list(call = object$call, base = object$base, n.alt=p, n.obs = if(is.matrix(object$y)) nrow(object$y) else length(object$y), n.param = ncol(param)-1, n.draws = n.draws, coef.table= if(n.cov > 1) param.table[1:n.cov,] else matrix(param.table[1,], nrow=1, dimnames=list(rownames(param.table)[1], colnames(param.table))), cov.table=param.table[(n.cov+1):ncol(param),]) class(ans) <- "summary.mnp" return(ans) } MNP/R/onAttach.R0000644000176200001440000000054013101760304012763 0ustar liggesusers".onAttach" <- function(lib, pkg) { mylib <- dirname(system.file(package = pkg)) title <- packageDescription(pkg, lib.loc = mylib)$Title ver <- packageDescription(pkg, lib.loc = mylib)$Version author <- packageDescription(pkg, lib.loc = mylib)$Author packageStartupMessage(pkg, ": ", title, "\nVersion: ", ver, "\nAuthors: ", author, "\n") } MNP/R/cov.mnp.R0000644000176200001440000000364013103226206012606 0ustar liggesusers#' Extract Multinomial Probit Model Covariance Matrix #' #' \code{cov.mnp} is a function which extracts the posterior draws of #' covariance matrix from objects returned by \code{mnp}. #' #' #' @param object An output object from \code{mnp}. #' @param subset A scalar or a numerical vector specifying the row number(s) of #' \code{param} in the output object from \code{mnp}. If specified, the #' posterior draws of covariance matrix for those rows are extracted. The #' default is \code{NULL} where all the posterior draws are extracted. #' @param ... further arguments passed to or from other methods. #' @return When a numerical vector or \code{NULL} is specified for #' \code{subset} argument, \code{cov.mnp} returns a three dimensional array #' where the third dimension indexes posterior draws. When a scalar is #' specified for \code{subset} arugment, \code{cov.mnp} returns a matrix. #' @author Kosuke Imai, Department of Politics, Princeton University #' \email{kimai@Princeton.Edu} #' @seealso \code{mnp}, \code{coef.mnp}; #' @keywords methods #' @export cov.mnp cov.mnp <- function(object, subset = NULL, ...) { if (is.null(subset)) subset <- 1:nrow(object$param) else if (max(subset) > nrow(object$param)) stop(paste("invalid input for `subset.' only", nrow(param), "draws are stored.")) p <- object$n.alt n <- length(subset) Sigma <- array(0, c(p-1, p-1, n)) param <- object$param n.cov <- ncol(param) - p*(p-1)/2 cov <- param[,(n.cov+1):ncol(param)] for (i in 1:n) { count <- 1 for (j in 1:(p-1)) { Sigma[j,j:(p-1),i] <- cov[subset[i],count:(count+p-j-1)] count <- count + p - j } diag(Sigma[,,i]) <- diag(Sigma[,,i]/2) Sigma[,,i] <- Sigma[,,i] + t(Sigma[,,i]) } tmp <- list() tmp[[1]] <- tmp[[2]] <- object$alt[-pmatch(object$base, object$alt)] tmp[[3]] <- as.character(subset) dimnames(Sigma) <- tmp if (n > 1) return(Sigma) else return(Sigma[,,1]) } MNP/R/print.mnp.R0000644000176200001440000000066113101760304013153 0ustar liggesusersprint.mnp <- function (x, digits = max(3, getOption("digits") - 3), ...) { cat("\nCall:\n", deparse(x$call), "\n\n", sep = "") param <- apply(x$param, 2, mean) if (length(param)) { cat("Parameter estimates (posterior means):\n") print.default(format(param, digits = digits), print.gap = 2, quote = FALSE) } else cat("No parameter estimates\n") cat("\n") invisible(x) } MNP/R/coef.mnp.R0000644000176200001440000000275613103226163012744 0ustar liggesusers#' Extract Multinomial Probit Model Coefficients #' #' \code{coef.mnp} is a function which extracts multinomial probit model #' coefficients from ojbects returned by \code{mnp}. \code{coefficients.mnp} is #' an alias for it. \code{coef} method for class \code{mnp}. #' #' #' @aliases coef.mnp coefficients.mnp #' @param object An output object from \code{mnp}. #' @param subset A scalar or a numerical vector specifying the row number(s) of #' \code{param} in the output object from \code{mnp}. If specified, the #' posterior draws of coefficients for those rows are extracted. The default is #' \code{NULL} where all the posterior draws are extracted. #' @param ... further arguments passed to or from other methods. #' @return \code{coef.mnp} returns a matrix (when a numerical vector or #' \code{NULL} is specified for \code{subset} argument) or a vector (when a #' scalar is specified for \code{subset} arugment) of multinomila probit model #' coefficients. #' @author Kosuke Imai, Department of Politics, Princeton University #' \email{kimai@Princeton.Edu} #' @seealso \code{mnp}, \code{cov.mnp}; #' @keywords methods #' @export coef.mnp coef.mnp <- function(object, subset = NULL, ...) { param <- object$param p <- object$n.alt n.cov <- ncol(param) - p*(p-1)/2 n <- nrow(param) if (is.null(subset)) return(param[,1:n.cov]) else if (subset > n) stop(paste("invalid input for `subset.' only", nrow(param), "draws are stored.")) else return(param[subset, 1:n.cov]) } MNP/R/japan.R0000644000176200001440000000226413103175765012335 0ustar liggesusers #' Voters' Preferences of Political Parties in Japan (1995) #' #' This dataset gives voters' preferences of political parties in Japan on the #' 0 (least preferred) - 100 (most preferred) scale. It is based on the 1995 #' survey data of 418 individual voters. The data also include the sex, #' education level, and age of the voters. The survey allowed voters to choose #' among four parties: Liberal Democratic Party (LDP), New Frontier Party #' (NFP), Sakigake (SKG), and Japanese Communist Party (JCP). #' #' #' @name japan #' @docType data #' @format A data frame containing the following 7 variables for 418 #' observations. #' #' \tabular{llll}{ LDP \tab numeric \tab preference for Liberal Democratic #' Party \tab 0 - 100 \cr NFP \tab numeric \tab preference for New Frontier #' Party \tab 0 - 100 \cr SKG \tab numeric \tab preference for Sakigake \tab 0 #' - 100 \cr JCP \tab numeric \tab preference for Japanese Communist Party \tab #' 0 - 100 \cr gender \tab factor \tab gender of each voter \tab \code{male} or #' \code{female} \cr education \tab numeric \tab levels of education for each #' voter \tab \cr age \tab numeric \tab age of each voter \tab } #' @keywords datasets NULL MNP/R/detergent.R0000644000176200001440000000171413103175765013224 0ustar liggesusers #' Detergent Brand Choice #' #' This dataset gives the laundry detergent brand choice by households and the #' price of each brand. #' #' #' @name detergent #' @docType data #' @format A data frame containing the following 7 variables and 2657 #' observations. #' #' \tabular{lll}{ choice \tab factor \tab a brand chosen by each household\cr #' TidePrice \tab numeric \tab log price of Tide\cr WiskPrice \tab numeric \tab #' log price of Wisk\cr EraPlusPrice \tab numeric \tab log price of EraPlus\cr #' SurfPrice \tab numeric \tab log price of Surf\cr SoloPrice \tab numeric \tab #' log price of Solo\cr AllPrice \tab numeric \tab log price of All\cr } #' @references Chintagunta, P. K. and Prasad, A. R. (1998) \dQuote{An Empirical #' Investigation of the `Dynamic McFadden' Model of Purchase Timing and Brand #' Choice: Implications for Market Structure}. \emph{Journal of Business and #' Economic Statistics} vol. 16 no. 1 pp.2-12. #' @keywords datasets NULL MNP/R/print.summary.mnp.R0000644000176200001440000000221313103226315014643 0ustar liggesusers#' Print the summary of the results for the Multinomial Probit Models #' #' \code{summary} print method for class \code{mnp}. #' #' @aliases print.summary.mnp #' @param x An object of class \code{summary.mnp}. #' @param digits the number of significant digits to use when printing. #' @param ... further arguments passed to or from other methods. #' @author Kosuke Imai, Department of Politics, Princeton University #' \email{kimai@Princeton.Edu} #' @seealso \code{mnp} #' @keywords methods #' @export print.summary.mnp print.summary.mnp <- function(x, digits = max(3, getOption("digits") - 3), ...) { cat("\nCall:\n") cat(paste(deparse(x$call), sep = "\n", collapse = "\n"), "\n\n", sep = "") cat("\nCoefficients:\n") printCoefmat(x$coef.table, digits = digits, na.print = "NA", ...) cat("\nCovariances:\n") printCoefmat(x$cov.table, digits = digits, na.print = "NA", ...) cat("\nBase category:", x$base) cat("\nNumber of alternatives:", x$n.alt) cat("\nNumber of observations:", x$n.obs) cat("\nNumber of estimated parameters:", x$n.param) cat("\nNumber of stored MCMC draws:", x$n.draws) cat("\n\n") invisible(x) } MNP/R/mnp.R0000644000176200001440000004114313103236476012032 0ustar liggesusers#' Fitting the Multinomial Probit Model via Markov chain Monte Carlo #' #' \code{mnp} is used to fit (Bayesian) multinomial probit model via Markov #' chain Monte Carlo. \code{mnp} can also fit the model with different choice #' sets for each observation, and complete or partial ordering of all the #' available alternatives. The computation uses the efficient marginal data #' augmentation algorithm that is developed by Imai and van Dyk (2005a). #' #' To fit the multinomial probit model when only the most preferred choice is #' observed, use the syntax for the formula, \code{y ~ x1 + x2}, where \code{y} #' is a factor variable indicating the most preferred choice and \code{x1} and #' \code{x2} are individual-specific covariates. The interactions of #' individual-specific variables with each of the choice indicator variables #' will be fit. #' #' To specify choice-specific covariates, use the syntax, #' \code{choiceX=list(A=cbind(z1, z2), B=cbind(z3, z4), C=cbind(z5, z6))}, #' where \code{A}, \code{B}, and \code{C} represent the choice names of the #' response variable, and \code{z1} and \code{z2} are each vectors of length #' \eqn{n} that record the values of the two choice-specific covariates for #' each individual for choice A, likewise for \code{z3}, \eqn{\ldots}, #' \code{z6}. The corresponding variable names via \code{cXnames=c("price", #' "quantity")} need to be specified, where \code{price} refers to the #' coefficient name for \code{z1}, \code{z3}, and \code{z5}, and #' \code{quantity} refers to that for \code{z2}, \code{z4}, and \code{z6}. #' #' If the choice set varies from one observation to another, use the syntax, #' \code{cbind(y1, y2, y3) ~ x1 + x2}, in the case of a three choice problem, #' and indicate unavailable alternatives by \code{NA}. If only the most #' preferred choice is observed, \code{y1}, \code{y2}, and \code{y3} are #' indicator variables that take on the value one for individuals who prefer #' that choice and zero otherwise. The last column of the response matrix, #' \code{y3} in this particular example syntax, is used as the base category. #' #' To fit the multinomial probit model when the complete or partial ordering of #' the available alternatives is recorded, use the same syntax as when the #' choice set varies (i.e., \code{cbind(y1, y2, y3, y4) ~ x1 + x2}). For each #' observation, all the available alternatives in the response variables should #' be numerically ordered in terms of preferences such as \code{1 2 2 3}. Ties #' are allowed. The missing values in the response variable should be denoted #' by \code{NA}. The software will impute these missing values using the #' specified covariates. The resulting uncertainty estimates of the parameters #' will properly reflect the amount of missing data. For example, we expect the #' standard errors to be larger when there is more missing data. #' #' @aliases mnp MNP #' @param formula A symbolic description of the model to be fit specifying the #' response variable and covariates. The formula should not include the #' choice-specific covariates. Details and specific examples are given below. #' @param data An optional data frame in which to interpret the variables in #' \code{formula} and \code{choiceX}. The default is the environment in which #' \code{mnp} is called. #' @param choiceX An optional list containing a matrix of choice-specific #' covariates for each category. Details and examples are provided below. #' @param cXnames A vector of the names for the choice-specific covariates #' specified in \code{choiceX}. The details and examples are provided below. #' @param base The name of the base category. For the standard multinomial #' probit model, the default is the lowest level of the response variable. For #' the multinomial probit model with ordered preferences, the default base #' category is the last column in the matrix of response variables. #' @param latent logical. If \code{TRUE}, then the latent variable W will be #' returned. See Imai and van Dyk (2005) for the notation. The default is #' \code{FALSE}. #' @param invcdf logical. If \code{TRUE}, then the inverse cdf method is used #' for truncated normal sampling. If \code{FALSE}, then the rejection sampling #' method is used. The default is \code{FALSE}. #' @param trace logical. If \code{TRUE}, then the trace of the variance #' covariance matrix is set to a constant (here, it is equal to \code{n.dim}) #' instead of setting its first diagonal element to 1. The former avoids the #' arbitrariness of fixing one particular diagonal element in order to achieve #' identification (see Burgette and Nordheim, 2009). #' @param n.draws A positive integer. The number of MCMC draws. The default is #' \code{5000}. #' @param p.var A positive definite matrix. The prior variance of the #' coefficients. A scalar input can set the prior variance to the diagonal #' matrix whose diagonal element is equal to that value. The default is #' \code{"Inf"}, which represents an improper noninformative prior distribution #' on the coefficients. #' @param p.df A positive integer greater than \code{n.dim-1}. The prior #' degrees of freedom parameter for the covariance matrix. The default is #' \code{n.dim+1}, which is equal to the total number of alternatives. #' @param p.scale A positive definite matrix. When \code{trace = FALSE}, its #' first diagonal element is set to \code{1} if it is not equal to 1 already. #' The prior scale matrix for the covariance matrix. A scalar input can be used #' to set the scale matrix to a diagonal matrix with diagonal elements equal to #' the scalar input value. The default is \code{1}. #' @param coef.start A vector. The starting values for the coefficients. A #' scalar input sets the starting values for all the coefficients equal to that #' value. The default is \code{0}. #' @param cov.start A positive definite matrix. When \code{trace = FALSE}, its #' first diagonal element is set to \code{1} if it is not equal to 1 already. #' The starting values for the covariance matrix. A scalar input can be used to #' set the starting value to a diagonal matrix with diagonal elements equal to #' the scalar input value. The default is \code{1}. #' @param burnin A positive integer. The burnin interval for the Markov chain; #' i.e., the number of initial Gibbs draws that should not be stored. The #' default is \code{0}. #' @param thin A positive integer. The thinning interval for the Markov chain; #' i.e., the number of Gibbs draws between the recorded values that are #' skipped. The default is \code{0}. #' @param verbose logical. If \code{TRUE}, helpful messages along with a #' progress report of the Gibbs sampling are printed on the screen. The default #' is \code{FALSE}. #' @return An object of class \code{mnp} containing the following elements: #' \item{param}{A matrix of the Gibbs draws for each parameter; i.e., the #' coefficients and covariance matrix. For the covariance matrix, the elements #' on or above the diagonal are returned. } #' \item{call}{The matched call.} #' \item{x}{The matrix of covariates.} #' \item{y}{The vector or matrix of the #' response variable.} #' \item{w}{The three dimensional array of the latent #' variable, W. The first dimension represents the alternatives, and the second #' dimension indexes the observations. The third dimension represents the Gibbs #' draws. Note that the latent variable for the base category is set to 0, and #' therefore omitted from the output.} #' \item{alt}{The names of alternatives.} #' \item{n.alt}{The total number of alternatives.} #' \item{base}{The base #' category used for fitting.} #' \item{invcdf}{The value of #' \code{invcdf}.} #' \item{p.var}{The prior variance for the coefficients.} #' \item{p.df}{The prior #' degrees of freedom parameter for the covariance matrix.} #' \item{p.scale}{The #' prior scale matrix for the covariance matrix.} #' \item{burnin}{The number of #' initial burnin draws.} #' \item{thin}{The thinning interval.} #' @author Kosuke Imai, Department of Politics, Princeton University #' \email{kimai@@Princeton.Edu}, \url{http://imai.princeton.edu}; David A. van #' Dyk, Statistics Section, Department of Mathematics, Imperial College London. #' @seealso \code{coef.mnp}, \code{cov.mnp}, \code{predict.mnp}, #' \code{summary.mnp}; #' @references Imai, Kosuke and David A. van Dyk. (2005a) \dQuote{A Bayesian #' Analysis of the Multinomial Probit Model Using the Marginal Data #' Augmentation,} \emph{Journal of Econometrics}, Vol. 124, No. 2 (February), #' pp.311-334. #' #' Imai, Kosuke and David A. van Dyk. (2005b) \dQuote{MNP: R Package for #' Fitting the Multinomial Probit Models,} \emph{Journal of Statistical #' Software}, Vol. 14, No. 3 (May), pp.1-32. #' #' Burgette, L.F. and E.V. Nordheim. (2009). \dQuote{An alternate identifying #' restriction for the Bayesian multinomial probit model,} \emph{Technical #' report}, Department of Statistics, University of Wisconsin, Madison. #' @keywords models #' @examples #' #' ### #' ### NOTE: this example is not fully analyzed. In particular, the #' ### convergence has not been assessed. A full analysis of these data #' ### sets appear in Imai and van Dyk (2005b). #' ### #' #' ## load the detergent data #' data(detergent) #' ## run the standard multinomial probit model with intercepts and the price #' res1 <- mnp(choice ~ 1, choiceX = list(Surf=SurfPrice, Tide=TidePrice, #' Wisk=WiskPrice, EraPlus=EraPlusPrice, #' Solo=SoloPrice, All=AllPrice), #' cXnames = "price", data = detergent, n.draws = 100, burnin = 10, #' thin = 3, verbose = TRUE) #' ## summarize the results #' summary(res1) #' ## calculate the quantities of interest for the first 3 observations #' pre1 <- predict(res1, newdata = detergent[1:3,]) #' #' ## load the Japanese election data #' data(japan) #' ## run the multinomial probit model with ordered preferences #' res2 <- mnp(cbind(LDP, NFP, SKG, JCP) ~ gender + education + age, data = japan, #' verbose = TRUE) #' ## summarize the results #' summary(res2) #' ## calculate the predicted probabilities for the 10th observation #' ## averaging over 100 additional Monte Carlo draws given each of MCMC draw. #' pre2 <- predict(res2, newdata = japan[10,], type = "prob", n.draws = 100, #' verbose = TRUE) #' #' @export mnp mnp <- function(formula, data = parent.frame(), choiceX = NULL, cXnames = NULL, base = NULL, latent = FALSE, invcdf = FALSE, trace = TRUE, n.draws = 5000, p.var = "Inf", p.df = n.dim+1, p.scale = 1, coef.start = 0, cov.start = 1, burnin = 0, thin = 0, verbose = FALSE) { call <- match.call() mf <- match.call(expand.dots = FALSE) mf$choiceX <- mf$cXnames <- mf$base <- mf$n.draws <- mf$latent <- mf$p.var <- mf$p.df <- mf$p.scale <- mf$coef.start <- mf$invcdf <- mf$trace <- mf$cov.start <- mf$verbose <- mf$burnin <- mf$thin <- NULL mf[[1]] <- as.name("model.frame") mf$na.action <- 'na.pass' mf <- eval.parent(mf) ## fix this parameter p.alpha0 <- 1 ## obtaining Y tmp <- ymatrix.mnp(mf, base=base, extra=TRUE, verbose=verbose) Y <- tmp$Y MoP <- tmp$MoP lev <- tmp$lev base <- tmp$base p <- tmp$p n.dim <- p - 1 if(verbose) cat("\nThe base category is `", base, "'.\n\n", sep="") if (p < 3) stop("The number of alternatives should be at least 3.") if(verbose) cat("The total number of alternatives is ", p, ".\n\n", sep="") if(verbose) { if (trace) cat("The trace restriction is used instead of the diagonal restriction.\n\n") else cat("The diagonal restriction is used instead of the trace restriction.\n\n") } ### obtaining X tmp <- xmatrix.mnp(formula, data=eval.parent(data), choiceX=call$choiceX, cXnames=cXnames, base=base, n.dim=n.dim, lev=lev, MoP=MoP, verbose=verbose, extra=TRUE) X <- tmp$X coefnames <- tmp$coefnames n.cov <- ncol(X) / n.dim ## listwise deletion for X na.ind <- apply(is.na(X), 1, sum) if (ncol(Y) == 1) na.ind <- na.ind + is.na(Y) Y <- Y[na.ind==0,] X <- X[na.ind==0,] n.obs <- nrow(X) if (verbose) { cat("The dimension of beta is ", n.cov, ".\n\n", sep="") cat("The number of observations is ", n.obs, ".\n\n", sep="") if (sum(na.ind>0)>0) { if (sum(na.ind>0)==1) cat("The observation ", (1:length(na.ind))[na.ind>0], " is dropped due to missing values.\n\n", sep="") else { cat("The following ", sum(na.ind>0), " observations are dropped due to missing values:\n", sep="") cat((1:length(na.ind))[na.ind>0], "\n\n") } } } ## checking the prior for beta p.imp <- FALSE if (p.var == Inf) { p.imp <- TRUE p.prec <- diag(0, n.cov) if (verbose) cat("Improper prior will be used for beta.\n\n") } else if (is.matrix(p.var)) { if (ncol(p.var) != n.cov || nrow(p.var) != n.cov) stop("The dimension of `p.var' should be ", n.cov, " x ", n.cov, sep="") if (sum(sign(eigen(p.var)$values) < 1) > 0) stop("`p.var' must be positive definite.") p.prec <- solve(p.var) } else { p.var <- diag(p.var, n.cov) p.prec <- solve(p.var) } p.mean <- rep(0, n.cov) ## checking prior for Sigma p.df <- eval(p.df) if (length(p.df) > 1) stop("`p.df' must be a positive integer.") if (p.df < n.dim) stop(paste("`p.df' must be at least ", n.dim, ".", sep="")) if (abs(as.integer(p.df) - p.df) > 0) stop("`p.df' must be a positive integer.") if (!is.matrix(p.scale)) p.scale <- diag(p.scale, n.dim) if (ncol(p.scale) != n.dim || nrow(p.scale) != n.dim) stop("`p.scale' must be ", n.dim, " x ", n.dim, sep="") if (sum(sign(eigen(p.scale)$values) < 1) > 0) stop("`p.scale' must be positive definite.") else if ((trace == FALSE) & (p.scale[1,1] != 1)) { p.scale[1,1] <- 1 warning("p.scale[1,1] will be set to 1.") } Signames <- NULL for(j in 1:n.dim) for(k in 1:n.dim) if (j<=k) Signames <- c(Signames, paste(if(MoP) lev[j] else lev[j+1], ":", if(MoP) lev[k] else lev[k+1], sep="")) ## checking starting values if (length(coef.start) == 1) coef.start <- rep(coef.start, n.cov) else if (length(coef.start) != n.cov) stop(paste("The dimenstion of `coef.start' must be ", n.cov, ".", sep="")) if (!is.matrix(cov.start)) { cov.start <- diag(n.dim)*cov.start if (!trace) cov.start[1,1] <- 1 } else if (ncol(cov.start) != n.dim || nrow(cov.start) != n.dim) stop("The dimension of `cov.start' must be ", n.dim, " x ", n.dim, sep="") else if (sum(sign(eigen(cov.start)$values) < 1) > 0) stop("`cov.start' must be a positive definite matrix.") else if ((trace == FALSE) & (cov.start[1,1] != 1)) { cov.start[1,1] <- 1 warning("cov.start[1,1] will be set to 1.") } ## checking thinnig and burnin intervals if (burnin < 0) stop("`burnin' should be a non-negative integer.") if (thin < 0) stop("`thin' should be a non-negative integer.") keep <- thin + 1 ## running the algorithm if (latent) n.par <- n.cov + n.dim*(n.dim+1)/2 + n.dim*n.obs else n.par <- n.cov + n.dim*(n.dim+1)/2 if(verbose) cat("Starting Gibbs sampler...\n") # recoding NA into -1 Y[is.na(Y)] <- -1 param <- .C("cMNPgibbs", as.integer(n.dim), as.integer(n.cov), as.integer(n.obs), as.integer(n.draws), as.double(p.mean), as.double(p.prec), as.integer(p.df), as.double(p.scale*p.alpha0), as.double(X), as.integer(Y), as.double(coef.start), as.double(cov.start), as.integer(p.imp), as.integer(invcdf), as.integer(burnin), as.integer(keep), as.integer(trace), as.integer(verbose), as.integer(MoP), as.integer(latent), pdStore = double(n.par*floor((n.draws-burnin)/keep)), PACKAGE="MNP")$pdStore param <- matrix(param, ncol = n.par, nrow = floor((n.draws-burnin)/keep), byrow=TRUE) if (latent) { W <- array(as.vector(t(param[,(n.par-n.dim*n.obs+1):n.par])), dim = c(n.dim, n.obs, floor((n.draws-burnin)/keep)), dimnames = list(lev[!(lev %in% base)], rownames(Y), NULL)) param <- param[,1:(n.par-n.dim*n.obs)] } else W <- NULL colnames(param) <- c(coefnames, Signames) ##recoding -1 back into NA Y[Y==-1] <- NA ## returning the object res <- list(param = param, x = X, y = Y, w = W, call = call, alt = lev, n.alt = p, base = base, invcdf = invcdf, trace = trace, p.mean = if(p.imp) NULL else p.mean, p.var = p.var, p.df = p.df, p.scale = p.scale, burnin = burnin, thin = thin) class(res) <- "mnp" return(res) } MNP/R/predict.mnp.R0000644000176200001440000001650413124567307013471 0ustar liggesusers#' Posterior Prediction under the Bayesian Multinomial Probit Models #' #' Obtains posterior predictions under a fitted (Bayesian) multinomial probit #' model. \code{predict} method for class \code{mnp}. #' #' The posterior predictive values are computed using the Monte Carlo sample #' stored in the \code{mnp} output (or other sample if \code{newdraw} is #' specified). Given each Monte Carlo sample of the parameters and each vector #' of predictor variables, we sample the vector-valued latent variable from the #' appropriate multivariate Normal distribution. Then, using the sampled #' predictive values of the latent variable, we construct the most preferred #' choice as well as the ordered preferences. Averaging over the Monte Carlo #' sample of the preferred choice, we obtain the predictive probabilities of #' each choice being most preferred given the values of the predictor #' variables. Since the predictive values are computed via Monte Carlo #' simulations, each run may produce somewhat different values. The computation #' may be slow if predictions with many values of the predictor variables are #' required and/or if a large Monte Carlo sample of the model parameters is #' used. In either case, setting \code{verbose = TRUE} may be helpful in #' monitoring the progress of the code. #' #' @param object An output object from \code{mnp}. #' @param newdata An optional data frame containing the values of the predictor #' variables. Predictions for multiple values of the predictor variables can be #' made simultaneously if \code{newdata} has multiple rows. The default is the #' original data frame used for fitting the model. #' @param newdraw An optional matrix of MCMC draws to be used for posterior #' predictions. The default is the original MCMC draws stored in \code{object}. #' @param n.draws The number of additional Monte Carlo draws given each MCMC #' draw of coefficients and covariance matrix. The specified number of latent #' variables will be sampled from the multivariate normal distribution, and the #' quantities of interest will be calculated by averaging over these draws. #' This will be particularly useful calculating the uncertainty of predicted #' probabilities. The default is \code{1}. #' @param type The type of posterior predictions required. There are four #' options: \code{type = "prob"} returns the predictive probabilities of being #' the most preferred choice among the choice set. \code{type = "choice"} #' returns the Monte Carlo sample of the most preferred choice, and \code{type #' = "order"} returns the Monte Carlo sample of the ordered preferences, #' @param verbose logical. If \code{TRUE}, helpful messages along with a #' progress report on the Monte Carlo sampling from the posterior predictive #' distributions are printed on the screen. The default is \code{FALSE}. #' @param ... additional arguments passed to other methods. #' @return \code{predict.mnp} yields a list of class #' \code{predict.mnp} containing at least one of the following elements: #' \item{o}{A three dimensional array of the Monte Carlo sample from the posterior predictive #' distribution of the ordered preferences. The first dimension corresponds to #' the rows of \code{newdata} (or the original data set if \code{newdata} is #' left unspecified), the second dimension corresponds to the alternatives in #' the choice set, and the third dimension indexes the Monte Carlo sample. If #' \code{n.draws} is greater than 1, then each entry will be an average over #' these additional draws. } #' \item{p}{A two or three dimensional array of the #' posterior predictive probabilities for each alternative in the choice set #' being most preferred. The first demension corresponds to the rows of #' \code{newdata} (or the original data set if \code{newdata} is left #' unspecified), the second dimension corresponds to the alternatives in the #' choice set, and the third dimension (if it exists) indexes the Monte Carlo #' sample. If \code{n.draws} is greater than 1, then the third diemsion exists #' and indexes the Monte Carlo sample. } #' \item{y}{A matrix of the Monte Carlo #' sample from the posterior predictive distribution of the most preferred #' choice. The first dimension correspond to the rows of \code{newdata} (or the #' original data set if \code{newdata} is left unspecified), and the second #' dimension indexes the Monte Carlo sample. \code{n.draws} will be set to 1 #' when computing this quantity of interest. } #' \item{x}{A matrix of covariates #' used for prediction } #' @author Kosuke Imai, Department of Politics, Princeton University #' \email{kimai@Princeton.Edu} #' @seealso \code{mnp} #' @keywords methods #' @export #' @method predict mnp predict.mnp <- function(object, newdata = NULL, newdraw = NULL, n.draws = 1, type = c("prob", "choice", "order"), verbose = FALSE, ...){ if (NA %in% match(type, c("prob", "choice", "order"))) stop("Invalid input for `type'.") if (n.draws < 1) stop("Invalid input for `n.draws'.") p <- object$n.alt if (is.null(newdraw)) param <- object$param else param <- newdraw n.cov <- ncol(coef(object)) coef <- param[,1:n.cov] n.mcmc <- nrow(coef) cov <- param[,(n.cov+1):ncol(param)] ## get X matrix ready if (is.null(newdata)) x <- object$x else { call <- object$call x <- xmatrix.mnp(as.formula(call$formula), data = newdata, choiceX = call$choiceX, cXnames = eval(call$cXnames), base = object$base, n.dim = p-1, lev = object$alt, MoP = is.matrix(object$y), verbose = FALSE, extra = FALSE) if (nrow(x) > 1) x <- as.matrix(x[apply(is.na(x), 1, sum)==0,]) else if (sum(is.na(x))>0) stop("Invalid input for `newdata'.") } n.obs <- nrow(x) if (verbose) { if (n.obs == 1) cat("There is one observation to predict. Please wait...\n") else cat("There are", n.obs, "observations to predict. Please wait...\n") } alt <- object$alt if (object$base != alt[1]) alt <- c(object$base, alt[1:(length(alt)-1)]) res <- .C("predict", as.double(x), as.integer(n.obs), as.double(coef), as.double(cov), as.integer(p-1), as.integer(n.cov), as.integer(n.mcmc), as.integer(n.draws), as.integer(verbose), prob = if (n.draws > 1) double(n.obs*n.mcmc*p) else double(n.obs*p), choice = double(n.obs*n.mcmc), order = double(n.obs*n.mcmc*p), PACKAGE = "MNP") ans <- list() if ("choice" %in% type) ans$y <- matrix(res$choice, ncol=n.mcmc, nrow = n.obs, byrow = TRUE, dimnames = list(rownames(x), 1:n.mcmc)) else ans$y <- NULL if ("order" %in% type) ans$o <- aperm(array(res$order, dim=c(p, n.mcmc, n.obs), dimnames = list(alt, 1:n.mcmc, rownames(x))), c(3,1,2)) else ans$o <- NULL if ("prob" %in% type) if (n.draws > 1) ans$p <- aperm(array(res$prob, dim = c(p, n.mcmc, n.obs), dimnames = list(alt, 1:n.mcmc, rownames(x))), c(3,1,2)) else ans$p <- matrix(res$prob, nrow = n.obs, ncol = p, byrow = TRUE, dimnames = list(rownames(x), alt)) else ans$p <- NULL ans$x <- x class(ans) <- "predict.mnp" return(ans) } MNP/R/xmatrix.mnp.R0000644000176200001440000000603513101760304013514 0ustar liggesusersxmatrix.mnp <- function(formula, data = parent.frame(), choiceX=NULL, cXnames=NULL, base=NULL, n.dim, lev, MoP=FALSE, verbose=FALSE, extra=FALSE) { call <- match.call() mf <- match.call(expand.dots = FALSE) mf$choiceX <- mf$cXnames <- mf$base <- mf$n.dim <- mf$lev <- mf$MoP <- mf$verbose <- mf$extra <- NULL ## get variables mf[[1]] <- as.name("model.frame.default") mf$na.action <- 'na.pass' mf <- eval.parent(mf) Terms <- attr(mf, "terms") X <- model.matrix.default(Terms, mf) xvars <- as.character(attr(Terms, "variables"))[-1] if ((yvar <- attr(Terms, "response")) > 0) xvars <- xvars[-yvar] xlev <- if (length(xvars) > 0) { xlev <- lapply(mf[xvars], levels) xlev[!sapply(xlev, is.null)] } p <- n.dim + 1 n.obs <- nrow(X) n.cov <- ncol(X) ## expanding X allvnames <- Xnew <- NULL if (ncol(X) > 0) { Xcnames <- colnames(X) for (i in 1:n.cov) { Xv <- X[, Xcnames[i]] Xtmp <- varnames <- NULL for (j in 1:n.dim) { allvnames <- c(allvnames, paste(Xcnames[i], ":", if(MoP) lev[j] else lev[j+1], sep="")) for (k in 1:n.dim) varnames <- c(varnames, paste(Xcnames[i], ":", if(MoP) lev[j] else lev[j+1], sep="")) tmp <- matrix(0, nrow = n.obs, ncol = n.dim) tmp[, j] <- Xv Xtmp <- cbind(Xtmp, tmp) } colnames(Xtmp) <- varnames Xnew <- cbind(Xnew, Xtmp) } } ## checking and adding choice-specific variables if (!is.null(choiceX)) { cX <- eval(choiceX, data) cXn <- unique(names(cX)) if (sum(is.na(pmatch(cXn, lev))) > 0) stop(paste("Error: Invalid input for `choiceX.'\n Some variables do not exist.")) if(MoP) xbase <- as.matrix(cX[[lev[p]]]) else if (is.na(pmatch(base, cXn))) xbase <- NULL else xbase <- as.matrix(cX[[base]]) if (length(cXn) < n.dim) stop(paste("Error: Invalid input for `choiceX.'\n You must specify the choice-specific varaibles at least for all non-base categories.")) if (!is.null(xbase) && length(cXn) != p) stop(paste("Error: Invalid input for `choiceX.'\n You must specify the choice-specific variables at least for all non-base categories.")) if(!is.null(xbase) && verbose) cat("The choice-specific variables of the base category are subtracted from the corresponding variables of the non-base categories.\n\n") for (i in 1:length(cXnames)) for (j in 1:n.dim) { if (length(cXnames) != ncol(as.matrix(cX[[if(MoP) lev[j] else lev[j+1]]]))) stop(paste("Error: The number of variables in `choiceX' and `cXnames' does not match.")) tmp <- matrix(as.matrix(cX[[if(MoP) lev[j] else lev[j+1]]])[,i], ncol=1) if (!is.null(xbase)) tmp <- tmp - xbase[,i] colnames(tmp) <- paste(cXnames[i], ":", if(MoP) lev[j] else lev[j+1], sep="") Xnew <- cbind(Xnew, tmp) } } if(extra) return(list(X=Xnew, coefnames=c(allvnames, cXnames))) else return(Xnew) } MNP/R/ymatrix.mnp.R0000644000176200001440000000226213101760304013513 0ustar liggesusersymatrix.mnp <- function(data, base=NULL, extra=FALSE, verbose=verbose) { ## checking and formatting Y Y <- model.response(data) if (is.matrix(Y)) { # Multinomial ordered Probit model for (i in 1:nrow(Y)) Y[i,] <- match(Y[i,], sort(unique(Y[i,]))) - 1 p <- ncol(Y) lev <- colnames(Y) MoP <- TRUE if(!is.null(base)) stop("Error: The last column of the response matrix must be the base category.\n No need to specify `base.'") base <- lev[p] } else { # standard Multinomial Probit model Y <- as.factor(Y) lev <- levels(Y) if (!is.null(base)) if (base %in% lev) { Y <- relevel(Y, ref = base) lev <- levels(Y) } else { stop(paste("Error: `base' does not exist in the response variable.")) } base <- lev[1] counts <- table(Y) if (any(counts == 0)) { warning(paste("group(s)", paste(lev[counts == 0], collapse = " "), "are empty")) Y <- factor(Y, levels = lev[counts > 0]) lev <- lev[counts > 0] } p <- length(lev) Y <- as.matrix(unclass(Y)) - 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Z&t#aWHJ{.8|iˀ<\8Fu/, EQ2r v#@V@3gZ8j<>~.PnӸ4qzА(r &PLQ>0$ EV,f 8q\-;{z;{n<Arȍ SꖿSF]/$~*\b;0s h+O>3Q 5ןF&6:/6"_0&\ڮKr&qD/ց{t#nFXЫ ԥ.T.B7u.@qb@a%nj Dvqsh{\rt\`(Du3#rHSc +Pu9 >۹T@Hd@ ,q_^2:ЍW~.$Cs8Nش%+`a FW3&0u; 1|j*;(ǿugoR31}x."CkW7x!NJkO]ހO]h endstream endobj 1 0 obj <>stream 2005-03-13T12:14:03 R 2017-05-05T18:31:03-04:00 2017-05-05T18:31:03-04:00 R 2.0.1 application/pdf R Graphics Output uuid:83629924-f7e4-054f-8199-5096877956cb uuid:cedc4e3e-655e-6f41-8b41-ac0b762ffaa4 endstream endobj 2 0 obj <>stream h2U0Pw/+Q0L)T$w endstream endobj 3 0 obj <>stream h2S0Pw.JM,sI,Ip22005064642410҄iiUՙZk委&+jdE %%vv# endstream endobj 4 0 obj <>/Filter/FlateDecode/ID[<22A5B6D098E840A79B898C09A4FDEB7B><455C803382CC45CD88922DC1BEE70B42>]/Info 6 0 R/Length 39/Root 8 0 R/Size 7/Type/XRef/W[1 3 0]>>stream hbb&F xc7#`H9 endstream endobj startxref 116 %%EOF MNP/vignettes/my.bib0000644000176200001440000144675613101760304014041 0ustar liggesusers @manual{R:12, Address = {Vienna, Austria}, Author = {{R Development Core Team}}, Note = {{ISBN} 3-900051-07-0}, Organization = {R Foundation for Statistical Computing}, Title = {R: A Language and Environment for Statistical Computing}, Url = {http://www.R-project.org}, Year = 2012, Bdsk-Url-1 = {http://www.R-project.org}} @article{abad:05, Author = {Abadie, Alberto}, Journal = {Review of Economic Studies}, Pages = {1--19}, Title = {Semiparametric Difference-in-Differences Estimators}, Volume = 72, Year = 2005} @article{abad:angr:imbe:02, Author = {Abadie, Alberto and Angrist, Joshua and Imbens, Guido}, Journal = {Econometrica}, Number = 1, Pages = {91--117}, Title = {Instrumental Variables Estimates of the Effect of Subsidized Training on the Quantiles of Trainee Earnings}, Volume = 70, Year = 2002} @article{abad:diam:hain:10, Author = {Abadie, Alberto and Diamond, Alexis and Hainmueller, Jens}, Journal = {Journal of the American Statistical Association}, Month = {June}, Number = 490, Pages = {493--505}, Title = {Synthetic Control Methods for Comparative Case Studies: Estimating the Effect of California's Tobacco Control Program}, Volume = 105, Year = 2010} @article{abad:imbe:06, Author = {Abadie, Alberto and Imbens, Guido W.}, Journal = {Econometrica}, Number = 1, Pages = {235--267}, Title = {Large Sample Properties of Matching Estimators for Average Treatment Effects}, Volume = 74, Year = 2006} @article{abad:imbe:08, Author = {Abadie, Alberto and Imbens, Guido W.}, Journal = {Econometrica}, Number = {6}, Pages = {1537-1557}, Title = {On the Failure of the Bootstrap for Matching Estimators}, Volume = {76}, Year = {2008}} @article{abad:imbe:11, Author = {Abadie, Alberto and Imbens, Guido W.}, Journal = {Journal of Business and Economic Statistics}, Month = {January}, Number = {1}, Pages = {1--11}, Title = {Bias-Corrected Matching Estimators for Average Treatment Effects}, Volume = {29}, Year = {2011}} @techreport{abad:imbe:11a, Author = {Abadie, Alberto and Imbens, Guido W.}, Institution = {Department of Economics, Harvard University}, Title = {A Martingale Representation for Matching Estimators}, Year = 2011} @Article{abra:83, author = {Abramowitz, Alan I.}, title = {Partisan Redistricting and the 1982 Congressional Elections}, journal = {Journal of Politics}, year = {1983}, OPTkey = {}, volume = {45}, number = {3}, pages = {767--770}, OPTmonth = {}, OPTnote = {}, OPTannote = {} } @article{acem:john:robi:01, Author = {Acemoglu, Daron and Johnson, Simon and Robinson, James A.}, Journal = {American Economic Review}, Number = 5, Pages = {1369--1401}, Title = {The Colonial Origins of Comparative Development}, Volume = 91, Year = 2001} @article{ache:05, Author = {Achen, Christopher H.}, Journal = {Conflict Management and Peace Science}, Number = 4, Pages = {327--339}, Title = {Let's Put Garbage-Can Regressions and Garbage-Can Probits Where They Belong}, Volume = 22, Year = 2005} @article{ache:75, Author = {Achen, Christopher H.}, Journal = {American Political Science Review}, Month = {December}, Number = 4, Pages = {1218--1231}, Title = {Mass Political Attitudes and the Survey Reponse}, Volume = 69, Year = 1975} @book{ache:86, Address = {Berkeley}, Author = {Achen, Christopher H.}, Publisher = {University of California Press}, Title = {The Statistical Analysis of Quasi-experiments}, Year = 1986} @article{ache:bart:04, Author = {Achen, Christopher H. and Bartels, Larry}, Journal = {Working Paper, Princeton University}, Pages = {Presented at the 2004 Annual Meeting of the American Political Science Association}, Title = {Musical Chairs: Pocketbook Voting and the Limits of Democratic Accountability}, Year = 2004} @book{ache:shiv:95, Address = {Chicago}, Author = {Achen, Christopher H. and Shively, W. 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B. and Mealli, F.}, Chapter = {Evaluating the Effects of Job Training Programs on Wages through Principal Stratification}, Publisher = {Elsevier Science Ltd.}, Title = {Advances in Econometrics: Modeling and Evaluating Treatment Effects in Econometrics ({D}. {M}illimet, {J}. {S}mith, and {E}. {V}ytlacil eds.)}, Volume = 21, Year = {Forthcoming}} @article{zhao:etal:11, Author = {Zhao, Yufan and Zeng, Donglin and Socinski, Mark A. and Kosorok, Michael R.}, Journal = {Biometrics}, Month = {December}, Number = {4}, Pages = {1422--1433}, Title = {Reinforcement Learning Strategies for Clinical Trials in Nonsmall Cell Lung Cancer}, Volume = {67}, Year = {2011}} @article{zhao:etal:12, Author = {Zhao, Yingqi and Zeng, Donglin and Rush, John A. and Kosorok, Michael R.}, Journal = {Journal of the American Statistical Association}, Pages = {Forthcoming}, Title = {Estimating Individualized Treatment Rules Using Outcome Weighted Learning}, Year = {2012}} @techreport{zhou:wang:li:05, Author = {Zhou, Xiao-{H}ua and Wang, Richard and Li, Sierra M.}, Institution = {Department of Biostatistics, University of Washington}, Number = 245, Title = {{ITT} Analysis of Randomized Encouragement Design Studies with Missing-Data}, Type = {Working Paper Series}, Year = 2005} @article{zing:80, Author = {Zinger, A.}, Journal = {Journal of the American Statistical Association}, Month = {March}, Number = 369, Pages = {206--211}, Title = {Variance Estimation in Partially Systematic Sampling}, Volume = 75, Year = 1980} @article{zubi:etal:13, Author = {Zubizarreta, J. R. and Small, D. S. and Goyal, N. K. and Lorch, S. and Rosenbaum, P. R.}, Journal = {Annals of Applied Statistics}, Number = {1}, Pages = {25--50}, Title = {Stronger instruments via integer programming in an observational study of late preterm birth outcomes}, Volume = {7}, Year = {2013}} MNP/vignettes/natbib.bst0000644000176200001440000006375013101760304014673 0ustar liggesusers%% %% This is file `thesis.bst', generated %% on <1995/10/12> with the docstrip utility (2.2i). %% %% The original source files were: %% %% merlin.mbs (with options: `,ay,nat,nm-rev,dt-beg,yr-par,note-yr,vol-bf,vnum-cm,volp-com,jnm-x,ppx,ed,abr,ord,jabr,etal-it,nfss') %% ---------------------------------------- %% *** For Thesis *** %% %------------------------------------------------------------------- % The original source file contains the following version information: % \ProvidesFile{merlin.mbs}[1995/09/04 3.3 (PWD)] % % NOTICE: % This file may be used for non-profit purposes. % It may not be distributed in exchange for money, % other than distribution costs. % % The author provides it `as is' and does not guarantee it in any way. % % Copyright (C) 1994, 1995 Patrick W. Daly %------------------------------------------------------------------- % For use with BibTeX version 0.99a or later %------------------------------------------------------------------- % This bibliography style file is intended for texts in ENGLISH % This is an author-year citation style bibliography. As such, it is % non-standard LaTeX, and requires a special package file to function properly. % Such a package is natbib.sty by Patrick W. Daly % The form of the \bibitem entries is % \bibitem[Jones et al.(1990)]{key}... % \bibitem[Jones et al.(1990)Jones, Baker, and Smith]{key}... % The essential feature is that the label (the part in brackets) consists % of the author names, as they should appear in the citation, with the year % in parentheses following. There must be no space before the opening % parenthesis! % With natbib v5.3, a full list of authors may also follow the year. % In natbib.sty, it is possible to define the type of enclosures that is % really wanted (brackets or parentheses), but in either case, there must % be parentheses in the label. % The \cite command functions as follows: % \cite{key} ==>> Jones et al. (1990) % \cite[]{key} ==>> (Jones et al., 1990) % \cite[chap. 2]{key} ==>> (Jones et al., 1990, chap. 2) % \cite[e.g.][]{key} ==>> (e.g. Jones et al., 1990) % \cite[e.g.][p. 32]{key} ==>> (e.g. Jones et al., p. 32) % \citeauthor{key} Jones et al. % \citefullauthor{key} Jones, Baker, and Smith % \citeyear{key} 1990 %--------------------------------------------------------------------- ENTRY { address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year } {} { label extra.label sort.label short.list } INTEGERS { output.state before.all mid.sentence after.sentence after.block } FUNCTION {init.state.consts} { #0 'before.all := #1 'mid.sentence := #2 'after.sentence := #3 'after.block := } STRINGS { s t } FUNCTION {output.nonnull} { 's := output.state mid.sentence = { ", " * write$ } { output.state after.block = { add.period$ write$ newline$ "\newblock " write$ } { output.state before.all = 'write$ { add.period$ " " * write$ } if$ } if$ mid.sentence 'output.state := } if$ s } FUNCTION {output} { duplicate$ empty$ 'pop$ 'output.nonnull if$ } FUNCTION {output.check} { 't := duplicate$ empty$ { pop$ "empty " t * " in " * cite$ * warning$ } 'output.nonnull if$ } FUNCTION {fin.entry} { add.period$ write$ newline$ } FUNCTION {new.block} { output.state before.all = 'skip$ { after.block 'output.state := } if$ } FUNCTION {new.sentence} { output.state after.block = 'skip$ { output.state before.all = 'skip$ { after.sentence 'output.state := } if$ } if$ } FUNCTION {add.blank} { " " * before.all 'output.state := } FUNCTION {date.block} { new.block } FUNCTION {not} { { #0 } { #1 } if$ } FUNCTION {and} { 'skip$ { pop$ #0 } if$ } FUNCTION {or} { { pop$ #1 } 'skip$ if$ } FUNCTION {non.stop} { duplicate$ "}" * add.period$ #-1 #1 substring$ "." = } FUNCTION {new.block.checkb} { empty$ swap$ empty$ and 'skip$ 'new.block if$ } FUNCTION {field.or.null} { duplicate$ empty$ { pop$ "" } 'skip$ if$ } FUNCTION {emphasize} { duplicate$ empty$ { pop$ "" } { "\emph{" swap$ * "}" * } if$ } FUNCTION {bolden} { duplicate$ empty$ { pop$ "" } { "\textbf{" swap$ * "}" * } if$ } FUNCTION {capitalize} { "u" change.case$ "t" change.case$ } FUNCTION {space.word} { " " swap$ * " " * } % Here are the language-specific definitions for explicit words. % Each function has a name bbl.xxx where xxx is the English word. % The language selected here is ENGLISH FUNCTION {bbl.and} { "and"} FUNCTION {bbl.editors} { "eds." } FUNCTION {bbl.editor} { "ed." } FUNCTION {bbl.edby} { "edited by" } FUNCTION {bbl.edition} { "edn." } FUNCTION {bbl.volume} { "vol." } FUNCTION {bbl.of} { "of" } FUNCTION {bbl.number} { "no." } FUNCTION {bbl.nr} { "no." } FUNCTION {bbl.in} { "in" } FUNCTION {bbl.pages} { "" } FUNCTION {bbl.page} { "" } FUNCTION {bbl.chapter} { "chap." } FUNCTION {bbl.techrep} { "Tech. Rep." } FUNCTION {bbl.mthesis} { "Master's thesis" } FUNCTION {bbl.phdthesis} { "Ph.D. thesis" } FUNCTION {bbl.first} { "1st" } FUNCTION {bbl.second} { "2nd" } FUNCTION {bbl.third} { "3rd" } FUNCTION {bbl.fourth} { "4th" } FUNCTION {bbl.fifth} { "5th" } FUNCTION {bbl.st} { "st" } FUNCTION {bbl.nd} { "nd" } FUNCTION {bbl.rd} { "rd" } FUNCTION {bbl.th} { "th" } MACRO {jan} {"Jan."} MACRO {feb} {"Feb."} MACRO {mar} {"Mar."} MACRO {apr} {"Apr."} MACRO {may} {"May"} MACRO {jun} {"Jun."} MACRO {jul} {"Jul."} MACRO {aug} {"Aug."} MACRO {sep} {"Sep."} MACRO {oct} {"Oct."} MACRO {nov} {"Nov."} MACRO {dec} {"Dec."} FUNCTION {eng.ord} { duplicate$ "1" swap$ * #-2 #1 substring$ "1" = { bbl.th * } { duplicate$ #-1 #1 substring$ duplicate$ "1" = { pop$ bbl.st * } { duplicate$ "2" = { pop$ bbl.nd * } { "3" = { bbl.rd * } { bbl.th * } if$ } if$ } if$ } if$ } MACRO {acmcs} {"ACM Comput. Surv."} MACRO {acta} {"Acta Inf."} MACRO {cacm} {"Commun. ACM"} MACRO {ibmjrd} {"IBM J. Res. Dev."} MACRO {ibmsj} {"IBM Syst.~J."} MACRO {ieeese} {"IEEE Trans. Softw. Eng."} MACRO {ieeetc} {"IEEE Trans. Comput."} MACRO {ieeetcad} {"IEEE Trans. Comput.-Aided Design Integrated Circuits"} MACRO {ipl} {"Inf. Process. Lett."} MACRO {jacm} {"J.~ACM"} MACRO {jcss} {"J.~Comput. Syst. Sci."} MACRO {scp} {"Sci. Comput. Programming"} MACRO {sicomp} {"SIAM J. Comput."} MACRO {tocs} {"ACM Trans. Comput. Syst."} MACRO {tods} {"ACM Trans. Database Syst."} MACRO {tog} {"ACM Trans. Gr."} MACRO {toms} {"ACM Trans. Math. Softw."} MACRO {toois} {"ACM Trans. Office Inf. Syst."} MACRO {toplas} {"ACM Trans. Prog. Lang. Syst."} MACRO {tcs} {"Theoretical Comput. Sci."} INTEGERS { nameptr namesleft numnames } FUNCTION {format.names} { 's := #1 'nameptr := s num.names$ 'numnames := numnames 'namesleft := { namesleft #0 > } { s nameptr "{vv~}{ll}{, jj}{, f.}" format.name$ 't := nameptr #1 > { namesleft #1 > { ", " * t * } { numnames #2 > { "," * } 'skip$ if$ t "others" = { " " * "et~al." emphasize * } { bbl.and space.word * t * } if$ } if$ } 't if$ nameptr #1 + 'nameptr := namesleft #1 - 'namesleft := } while$ } FUNCTION {format.names.ed} { 's := #1 'nameptr := s num.names$ 'numnames := numnames 'namesleft := { namesleft #0 > } { s nameptr "{f.~}{vv~}{ll}{, jj}" format.name$ 't := nameptr #1 > { namesleft #1 > { ", " * t * } { numnames #2 > { "," * } 'skip$ if$ t "others" = { " " * "et~al." emphasize * } { bbl.and space.word * t * } if$ } if$ } 't if$ nameptr #1 + 'nameptr := namesleft #1 - 'namesleft := } while$ } FUNCTION {format.key} { empty$ { key field.or.null } { "" } if$ } FUNCTION {format.authors} { author empty$ { "" } { author format.names } if$ } FUNCTION {format.editors} { editor empty$ { "" } { editor format.names editor num.names$ #1 > { ", " * bbl.editors * } { ", " * bbl.editor * } if$ } if$ } FUNCTION {format.in.editors} { editor empty$ { "" } { editor format.names.ed editor num.names$ #1 > { ", " * bbl.editors * } { ", " * bbl.editor * } if$ } if$ } FUNCTION {format.title} { title empty$ { "" } { title "t" change.case$ } if$ } FUNCTION {format.full.names} {'s := #1 'nameptr := s num.names$ 'numnames := numnames 'namesleft := { namesleft #0 > } { s nameptr "{vv~}{ll}" format.name$ 't := nameptr #1 > { namesleft #1 > { ", " * t * } { numnames #2 > { "," * } 'skip$ if$ t "others" = { " " * "et~al." emphasize * } { bbl.and space.word * t * } if$ } if$ } 't if$ nameptr #1 + 'nameptr := namesleft #1 - 'namesleft := } while$ } FUNCTION {author.editor.key.full} { author empty$ { editor empty$ { key empty$ { cite$ #1 #3 substring$ } 'key if$ } { editor format.full.names } if$ } { author format.full.names } if$ } FUNCTION {author.key.full} { author empty$ { key empty$ { cite$ #1 #3 substring$ } 'key if$ } { author format.full.names } if$ } FUNCTION {editor.key.full} { editor empty$ { key empty$ { cite$ #1 #3 substring$ } 'key if$ } { editor format.full.names } if$ } FUNCTION {make.full.names} { type$ "book" = type$ "inbook" = or 'author.editor.key.full { type$ "proceedings" = 'editor.key.full 'author.key.full if$ } if$ } FUNCTION {output.bibitem} { newline$ "\bibitem[" write$ label write$ ")" make.full.names duplicate$ short.list = { pop$ } { * } if$ "]{" * write$ cite$ write$ "}" write$ newline$ "" before.all 'output.state := } FUNCTION {n.dashify} { 't := "" { t empty$ not } { t #1 #1 substring$ "-" = { t #1 #2 substring$ "--" = not { "--" * t #2 global.max$ substring$ 't := } { { t #1 #1 substring$ "-" = } { "-" * t #2 global.max$ substring$ 't := } while$ } if$ } { t #1 #1 substring$ * t #2 global.max$ substring$ 't := } if$ } while$ } FUNCTION {word.in} { bbl.in capitalize " " * } FUNCTION {format.date} { year duplicate$ empty$ { "empty year in " cite$ * "; set to ????" * warning$ pop$ "????" } 'skip$ if$ before.all 'output.state := " (" swap$ * extra.label * ")" * } FUNCTION {format.btitle} { title emphasize } FUNCTION {tie.or.space.connect} { duplicate$ text.length$ #3 < { "~" } { " " } if$ swap$ * * } FUNCTION {either.or.check} { empty$ 'pop$ { "can't use both " swap$ * " fields in " * cite$ * warning$ } if$ } FUNCTION {format.bvolume} { volume empty$ { "" } { bbl.volume volume tie.or.space.connect series empty$ 'skip$ { bbl.of space.word * series emphasize * } if$ "volume and number" number either.or.check } if$ } FUNCTION {format.number.series} { volume empty$ { number empty$ { series field.or.null } { output.state mid.sentence = { bbl.number } { bbl.number capitalize } if$ number tie.or.space.connect series empty$ { "there's a number but no series in " cite$ * warning$ } { bbl.in space.word * series * } if$ } if$ } { "" } if$ } FUNCTION {is.num} { chr.to.int$ duplicate$ "0" chr.to.int$ < not swap$ "9" chr.to.int$ > not and } FUNCTION {extract.num} { duplicate$ 't := "" 's := { t empty$ not } { t #1 #1 substring$ t #2 global.max$ substring$ 't := duplicate$ is.num { s swap$ * 's := } { pop$ "" 't := } if$ } while$ s empty$ 'skip$ { pop$ s } if$ } FUNCTION {convert.edition} { edition extract.num "l" change.case$ 's := s "first" = s "1" = or { bbl.first 't := } { s "second" = s "2" = or { bbl.second 't := } { s "third" = s "3" = or { bbl.third 't := } { s "fourth" = s "4" = or { bbl.fourth 't := } { s "fifth" = s "5" = or { bbl.fifth 't := } { s #1 #1 substring$ is.num { s eng.ord 't := } { edition 't := } if$ } if$ } if$ } if$ } if$ } if$ t } FUNCTION {format.edition} { edition empty$ { "" } { output.state mid.sentence = { convert.edition "l" change.case$ " " * bbl.edition * } { convert.edition "t" change.case$ " " * bbl.edition * } if$ } if$ } INTEGERS { multiresult } FUNCTION {multi.page.check} { 't := #0 'multiresult := { multiresult not t empty$ not and } { t #1 #1 substring$ duplicate$ "-" = swap$ duplicate$ "," = swap$ "+" = or or { #1 'multiresult := } { t #2 global.max$ substring$ 't := } if$ } while$ multiresult } FUNCTION {format.pages} { pages empty$ { "" } { pages multi.page.check { bbl.pages pages n.dashify tie.or.space.connect } { bbl.page pages tie.or.space.connect } if$ } if$ } FUNCTION {format.vol.num.pages} { volume field.or.null bolden number empty$ 'skip$ { ", " number * * volume empty$ { "there's a number but no volume in " cite$ * warning$ } 'skip$ if$ } if$ pages empty$ 'skip$ { duplicate$ empty$ { pop$ format.pages } { ", " * pages n.dashify * } if$ } if$ } FUNCTION {format.chapter.pages} { chapter empty$ 'format.pages { type empty$ { bbl.chapter } { type "l" change.case$ } if$ chapter tie.or.space.connect pages empty$ 'skip$ { ", " * format.pages * } if$ } if$ } FUNCTION {format.in.ed.booktitle} { booktitle empty$ { "" } { editor empty$ { word.in booktitle emphasize * } { word.in format.in.editors * ", " * booktitle emphasize * } if$ } if$ } FUNCTION {format.thesis.type} { type empty$ 'skip$ { pop$ type "t" change.case$ } if$ } FUNCTION {format.tr.number} { type empty$ { bbl.techrep } 'type if$ number empty$ { "t" change.case$ } { number tie.or.space.connect } if$ } FUNCTION {format.article.crossref} { word.in " \cite{" * crossref * "}" * } FUNCTION {format.book.crossref} { volume empty$ { "empty volume in " cite$ * "'s crossref of " * crossref * warning$ word.in } { bbl.volume capitalize volume tie.or.space.connect bbl.of space.word * } if$ " \cite{" * crossref * "}" * } FUNCTION {format.incoll.inproc.crossref} { word.in " \cite{" * crossref * "}" * } FUNCTION {article} { output.bibitem format.authors "author" output.check author format.key output format.date "year" output.check date.block format.title "title" output.check new.block crossref missing$ { journal emphasize "journal" output.check add.blank format.vol.num.pages output } { format.article.crossref output.nonnull format.pages output } if$ new.block note output fin.entry } FUNCTION {book} { output.bibitem author empty$ { format.editors "author and editor" output.check editor format.key output } { format.authors output.nonnull crossref missing$ { "author and editor" editor either.or.check } 'skip$ if$ } if$ format.date "year" output.check date.block format.btitle "title" output.check crossref missing$ { format.bvolume output new.block format.number.series output new.sentence publisher "publisher" output.check address output } { new.block format.book.crossref output.nonnull } if$ format.edition output new.block note output fin.entry } FUNCTION {booklet} { output.bibitem format.authors output author format.key output format.date "year" output.check date.block format.title "title" output.check new.block howpublished output address output new.block note output fin.entry } FUNCTION {inbook} { output.bibitem author empty$ { format.editors "author and editor" output.check editor format.key output } { format.authors output.nonnull crossref missing$ { "author and editor" editor either.or.check } 'skip$ if$ } if$ format.date "year" output.check date.block format.btitle "title" output.check crossref missing$ { format.bvolume output format.chapter.pages "chapter and pages" output.check new.block format.number.series output new.sentence publisher "publisher" output.check address output } { format.chapter.pages "chapter and pages" output.check new.block format.book.crossref output.nonnull } if$ format.edition output new.block note output fin.entry } FUNCTION {incollection} { output.bibitem format.authors "author" output.check author format.key output format.date "year" output.check date.block format.title "title" output.check new.block crossref missing$ { format.in.ed.booktitle "booktitle" output.check format.bvolume output format.number.series output format.chapter.pages output new.sentence publisher "publisher" output.check address output format.edition output } { format.incoll.inproc.crossref output.nonnull format.chapter.pages output } if$ new.block note output fin.entry } FUNCTION {inproceedings} { output.bibitem format.authors "author" output.check author format.key output format.date "year" output.check date.block format.title "title" output.check new.block crossref missing$ { format.in.ed.booktitle "booktitle" output.check format.bvolume output format.number.series output format.pages output address output new.sentence organization output publisher output } { format.incoll.inproc.crossref output.nonnull format.pages output } if$ new.block note output fin.entry } FUNCTION {conference} { inproceedings } FUNCTION {manual} { output.bibitem format.authors output author format.key output format.date "year" output.check date.block format.btitle "title" output.check organization address new.block.checkb organization output address output format.edition output new.block note output fin.entry } FUNCTION {mastersthesis} { output.bibitem format.authors "author" output.check author format.key output format.date "year" output.check date.block format.btitle "title" output.check new.block bbl.mthesis format.thesis.type output.nonnull school "school" output.check address output new.block note output fin.entry } FUNCTION {misc} { output.bibitem format.authors output author format.key output format.date "year" output.check date.block format.title output new.block howpublished output new.block note output fin.entry } FUNCTION {phdthesis} { output.bibitem format.authors "author" output.check author format.key output format.date "year" output.check date.block format.btitle "title" output.check new.block bbl.phdthesis format.thesis.type output.nonnull school "school" output.check address output new.block note output fin.entry } FUNCTION {proceedings} { output.bibitem format.editors output editor format.key output format.date "year" output.check date.block format.btitle "title" output.check format.bvolume output format.number.series output address output new.sentence organization output publisher output new.block note output fin.entry } FUNCTION {techreport} { output.bibitem format.authors "author" output.check author format.key output format.date "year" output.check date.block format.title "title" output.check new.block format.tr.number output.nonnull institution "institution" output.check address output new.block note output fin.entry } FUNCTION {unpublished} { output.bibitem format.authors "author" output.check author format.key output format.date "year" output.check date.block format.title "title" output.check new.block note "note" output.check fin.entry } FUNCTION {default.type} { misc } READ FUNCTION {sortify} { purify$ "l" change.case$ } INTEGERS { len } FUNCTION {chop.word} { 's := 'len := s #1 len substring$ = { s len #1 + global.max$ substring$ } 's if$ } FUNCTION {format.lab.names} { 's := s #1 "{vv~}{ll}" format.name$ s num.names$ duplicate$ #2 > { pop$ " " * "et~al." emphasize * } { #2 < 'skip$ { s #2 "{ff }{vv }{ll}{ jj}" format.name$ "others" = { " " * "et~al." emphasize * } { bbl.and space.word * s #2 "{vv~}{ll}" format.name$ * } if$ } if$ } if$ } FUNCTION {author.key.label} { author empty$ { key empty$ { cite$ #1 #3 substring$ } 'key if$ } { author format.lab.names } if$ } FUNCTION {author.editor.key.label} { author empty$ { editor empty$ { key empty$ { cite$ #1 #3 substring$ } 'key if$ } { editor format.lab.names } if$ } { author format.lab.names } if$ } FUNCTION {editor.key.label} { editor empty$ { key empty$ { cite$ #1 #3 substring$ } 'key if$ } { editor format.lab.names } if$ } FUNCTION {calc.short.authors} { type$ "book" = type$ "inbook" = or 'author.editor.key.label { type$ "proceedings" = 'editor.key.label 'author.key.label if$ } if$ 'short.list := } FUNCTION {calc.label} { calc.short.authors short.list "(" * year duplicate$ empty$ { pop$ "????" } 'skip$ if$ * 'label := } FUNCTION {sort.format.names} { 's := #1 'nameptr := "" s num.names$ 'numnames := numnames 'namesleft := { namesleft #0 > } { nameptr #1 > { " " * } 'skip$ if$ s nameptr "{vv{ } }{ll{ }}{ f{ }}{ jj{ }}" format.name$ 't := nameptr numnames = t "others" = and { "et al" * } { t sortify * } if$ nameptr #1 + 'nameptr := namesleft #1 - 'namesleft := } while$ } FUNCTION {sort.format.title} { 't := "A " #2 "An " #3 "The " #4 t chop.word chop.word chop.word sortify #1 global.max$ substring$ } FUNCTION {author.sort} { author empty$ { key empty$ { "to sort, need author or key in " cite$ * warning$ "" } { key sortify } if$ } { author sort.format.names } if$ } FUNCTION {author.editor.sort} { author empty$ { editor empty$ { key empty$ { "to sort, need author, editor, or key in " cite$ * warning$ "" } { key sortify } if$ } { editor sort.format.names } if$ } { author sort.format.names } if$ } FUNCTION {editor.sort} { editor empty$ { key empty$ { "to sort, need editor or key in " cite$ * warning$ "" } { key sortify } if$ } { editor sort.format.names } if$ } FUNCTION {presort} { calc.label label sortify " " * type$ "book" = type$ "inbook" = or 'author.editor.sort { type$ "proceedings" = 'editor.sort 'author.sort if$ } if$ #1 entry.max$ substring$ 'sort.label := sort.label * " " * title field.or.null sort.format.title * #1 entry.max$ substring$ 'sort.key$ := } ITERATE {presort} SORT STRINGS { last.label next.extra } INTEGERS { last.extra.num } FUNCTION {initialize.extra.label.stuff} { #0 int.to.chr$ 'last.label := "" 'next.extra := #0 'last.extra.num := } FUNCTION {forward.pass} { last.label label = { last.extra.num #1 + 'last.extra.num := last.extra.num int.to.chr$ 'extra.label := } { "a" chr.to.int$ 'last.extra.num := "" 'extra.label := label 'last.label := } if$ } FUNCTION {reverse.pass} { next.extra "b" = { "a" 'extra.label := } 'skip$ if$ extra.label 'next.extra := extra.label duplicate$ empty$ 'skip$ { "{" swap$ * "}" * } if$ 'extra.label := label extra.label * 'label := } EXECUTE {initialize.extra.label.stuff} ITERATE {forward.pass} REVERSE {reverse.pass} FUNCTION {bib.sort.order} { sort.label " " * year field.or.null sortify * " " * title field.or.null sort.format.title * #1 entry.max$ substring$ 'sort.key$ := } ITERATE {bib.sort.order} SORT FUNCTION {begin.bib} { preamble$ empty$ 'skip$ { preamble$ write$ newline$ } if$ "\begin{thebibliography}{}" write$ newline$ } EXECUTE {begin.bib} EXECUTE {init.state.consts} ITERATE {call.type$} FUNCTION {end.bib} { newline$ "\end{thebibliography}" write$ newline$ } EXECUTE {end.bib} %% End of customized bst file MNP/vignettes/MNP.Rnw0000644000176200001440000014200413162676435014054 0ustar liggesusers%\VignetteIndexEntry{MNP} \documentclass[11pt]{article} \usepackage{Rd} %% === margins === \addtolength{\hoffset}{-0.35in} \addtolength{\voffset}{-0.35in} \addtolength{\textwidth}{0.75in} \addtolength{\textheight}{0.75in} %% === basic packages === \usepackage{latexsym} \usepackage{amssymb,amsmath} \usepackage{graphicx} \usepackage{verbatim} %% === bibliography packages === \usepackage{natbib} \bibliographystyle{natbib} %% === hyperref options === \usepackage{color} \usepackage[pdftex, bookmarksopen=true, bookmarksnumbered=true, linkcolor=webred]{hyperref} \definecolor{webgreen}{rgb}{0, 0.5, 0} \definecolor{webblue}{rgb}{0, 0, 0.5} \definecolor{webred}{rgb}{0.5, 0, 0} % == spacing between sections and subsections \usepackage[compact]{titlesec} % === dcolumn package === \usepackage{dcolumn} \newcolumntype{.}{D{.}{.}{-1}} \newcolumntype{d}[1]{D{.}{.}{#1}} \hypersetup{% pdftitle = {MNP: R Package for Fitting Multinomial Probit Model}, pdfauthor = {Kosuke Imai and David A. van Dyk}, } \begin{document} \newcommand\dist{\buildrel\rm d\over\sim} \newcommand\al{\alpha} \newcommand\Y{{\cal Y}} \newcommand\itoN{^N_{i=1}} \newcommand\iton{^n_{i=1}} \newcommand\jtoJ{^J_{j=1}} \newcommand\inv{^{-1}} \newcommand\T{^\top} \newcommand\wt{\widetilde} \newcommand{\tr}{{\rm trace}} \newcommand{\chol}{{\rm Chol}} \newcommand\ind{\stackrel{\rm indep.}{\sim}} \renewcommand\r{\right} \renewcommand\l{\left} \newcommand\Var{{\rm Var}} \newcommand\E{{\rm E}} \newcommand\N{{\rm N}} \newcommand\cur{^{(t)}} \newcommand\pre{^{(t-1)}} \newcommand{\hlink}{\htmladdnormallink} \newcommand\spacingset[1]{\renewcommand{\baselinestretch}% {#1}\small\normalsize} \spacingset{1.5} \newcommand{\mac}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title{{\bf MNP: R Package for Fitting the \\ Multinomial Probit Model}\thanks{An earlier version of this paper appeared in {\it Journal of Statistical Software}, Vol. 14, No. 3 (May 2005), pp.1--32. We thank Jordan Vance for his valuable contribution to this project and Shigeo Hirano for providing the Japanese election dataset. We also thank Doug Bates for helpful advice on Lapack routines and Andrew Martin, Kevin Quinn, users of MNP, and anonymous reviewers and the associate editor for useful suggestions. We gratefully acknowledge funding for this project partially provided by NSF grants DMS-01-04129, DMS-04-38240, and DMS-04-06085, and by the Committee on Research in the Humanities and Social Sciences at Princeton University.}} \author{Kosuke Imai\thanks{Professor, Department of Politics, Princeton University, Princeton, NJ 08544. Phone: 609--258--6610, Fax: 609-258-1110, Email: \href{mailto:kimai@Princeton.Edu}{kimai@Princeton.Edu}, URL: \href{http://imai.princeton.edu}{http://imai.princeton.edu} Department of Politics, Princeton University}\\ David A. van Dyk\thanks{Associate Professor, Department of Statistics, University of California, Irvine, CA 92697-1205. Phone: 949--824--5679, Fax: 949--824--9863, Email: \href{mailto:dvd@uci.edu}{dvd@uci.edu}}} \date{Version 3.1--0} \pdfbookmark[1]{Title Page}{Title Page} \maketitle \begin{abstract} MNP is a publicly available R package that fits the Bayesian multinomial probit model via Markov chain Monte Carlo. The multinomial probit model is often used to analyze the discrete choices made by individuals recorded in survey data. Examples where the multinomial probit model may be useful include the analysis of product choice by consumers in market research and the analysis of candidate or party choice by voters in electoral studies. The MNP software can also fit the model with different choice sets for each individual, and complete or partial individual choice orderings of the available alternatives from the choice set. The estimation is based on the efficient marginal data augmentation algorithm that is developed by \citet{imai:vand:05}. \end{abstract} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \clearpage \section{Introduction} \label{sec:introduction} This paper illustrates how to use MNP, a publicly available R \citep{R:12} package, in order to fit the Bayesian multinomial probit model via Markov chain Monte Carlo. The multinomial probit model is often used to analyze the discrete choices made by individuals recorded in survey data. %Unlike the ordinal probit model, the %multinomial probit model does not assume that the choice set is %inherently ordered. Examples where the multinomial probit model may be useful include the analysis of product choice by consumers in market research and the analysis of candidate or party choice by voters in electoral studies. The MNP software can also fit the model with different choice sets for each individual, and complete or partial individual choice orderings of the available alternatives from the choice set. We use Markov chain Monte Carlo (MCMC) for estimation and computation. In particular, we use the efficient marginal data augmentation MCMC algorithm that is developed by \citet{imai:vand:05}. MNP can be installed in the same way as other R packages via the \texttt{install.packages("MNP")} command. Appendix~\ref{sec:install} gives instructions for obtaining R and installing MNP on Windows, Mac OS X, and Linux/UNIX platforms. Only three commands are necessary to use the MNP software; \texttt{mnp()} fits the multinomial probit model, \texttt{summary()} summarizes the MCMC output, and \texttt{predict()} gives posterior prediction based on the fitted model (In addition, \texttt{coef()} and \texttt{cov.mnp()} allow one to extract the posterior draws of model coefficients and covariance matrix). To run an example script, start R and run the following commands: \spacingset{1}\begin{verbatim} library(MNP) # loads the MNP package example(mnp) # runs the example script \end{verbatim} \spacingset{1.5} Details of the example script are given in Sections~\ref{sec:examp-soap}~and~\ref{sec:examp-japan}. Three appendices describe installation, the commands, and version history. We begin in Section~\ref{sec:mn_probit} with a brief description of the multinomial probit model that MNP is designed to fit. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The Method} \label{sec:mn_probit} MNP implements the marginal data augmentation algorithms for posterior sampling in the multinomial probit model. The MCMC algorithm we implement here is fully described in \citet{imai:vand:05}; we use Scheme 1 of their Algorithm 1. \subsection{The Multinomial Probit Model} \label{sec:mnp} Suppose we have a dataset of size $n$ with $p>2$ choices and $k$ covariates. Here, choices refer to the number of classes in the multinomial model. The word ``choices'' is used because the model is often used to describe how individuals choose among a number of alternatives, e.g., how a voter chooses which candidate to vote for among four candidates running for a particular office. We focus on the case when $p>2$ because when $p=2$, the model reduces to the standard binomial probit model, which can be fit via the \texttt{glm(, family = binomial(probit))} command in R. The multinomial probit model differs from the ordinal probit model in that the former does not assume any inherent ordering on the choices. Thus, although the individuals may have preferences among the available alternatives these ordering are individual specific rather than being characteristics of the alternatives themselves. The ordinal probit model can be fitted via an MCMC algorithm in R by installing a package called MCMCpack \citep{mart:quin:park:09}. Under the multinomial probit model, we assume a multivariate normal distribution on the latent variables, $W_i=(W_{i1},\ldots,W_{i,p-1})$. \begin{equation} W_i = X_i \beta + e_i, \quad e_i \sim \N(0, \Sigma), \ \hbox{for} \ i=1,\ldots,n, \label{eq:mnp} \end{equation} where $X_i$ is a $(p-1) \times k$ matrix of covariates, $\beta$ is $k \times 1$ vector of fixed coefficients, $e_i$ is $(p-1) \times 1$ vector of disturbances, and $\Sigma$ is a $(p-1) \times (p-1)$ positive definite matrix. For the model to be identified, the first diagonal element of $\Sigma$ is constrained, $\sigma_{11}=1$. Please note that starting with version 2.6-1, we use the restriction ${\rm trace}(\Sigma)=p$ as the default identification strategy following the recommendation of \citet{burg:nord:09}. This avoids the arbitrariness of fixing one particular diagonal element. The response variable, $Y_i$, is the index of the choice of individual $i$ among the alternatives in the choice set and is modeled in terms of this latent variable, $W_i$, via \begin{eqnarray} Y_i(W_i) & = & \l\{ \begin{array}{ccl} 0 & {\rm if} & {\rm max}(W_i)<0 \\ j & {\rm if} & {\rm max}(W_i)=W_{ij}>0 \end{array}\r. , \quad {\rm for} \ i=1,\ldots,n, \; {\rm and} \; j=1,\ldots,p-1, \end{eqnarray} where $Y_i$ equal to 0 corresponds to a base category. The matrix $X_i$ may include both choice-specific and individual-specific variables. A choice-specific variable is a variable that has a value for each of the $p$ choices, and these $p$ values may be different for each individual (e.g., the price of a product in a particular region where an individual lives). Choice-specific variables are recorded relative to the baseline choice and thus there are $p-1$ recorded values for each individual. In this way a choice-specific variable is tabulated as a column in $X_i$. Individual-specific variables, on the other hand, take on a value for each individual, but are constant across the choices, e.g., the age or gender of the individual. These variables are tabulated via their interaction with each of the choice indicator variables. Thus, an individual-specific variable corresponds to $p-1$ columns of $X_i$ and $p-1$ components of $\beta$. \subsection{The Multinomial Probit Model with Ordered Preferences} \label{sec:mnpop} In some cases, we observe a complete or partial ordering of $p$ alternatives. For example, we may observe the preferences of each individual among different brands of a product. We denote the outcome variable in such situations by ${\cal Y}_i=\{\Y_{i1}, \ldots, \Y_{ip}\}$ where $i=1,\ldots,n$ indexes individuals and $j=1,\ldots,p$ represent alternatives. If $\Y_{ij} > \Y_{ij'}$ for some $j \ne j'$, we say $j$ is preferred to $j'$. If $\Y_{ij} = \Y_{ij'}$ for some $j \ne j'$, we say individual $i$ is indifferent to the choice between alternatives $j$ and $j'$, but treat the data as if the actual ordering is unknown. In other words, formally we insist on strict inequalities among the preferences, but allow for some inequalities to be unobserved. The preference ordering is assumed to satisfy the usual axioms of preference comparability. Namely, preference is connected: For any $j$ and $j'$, either $\Y_{ij} \le \Y_{ij'}$ or $\Y_{ij} \ge \Y_{ij'}$. Preference also must be transitive: for any $j$,$j'$, and $j''$, $\Y_{ij} \le \Y_{ij'}$ and $\Y_{ij'} \le \Y_{ij''}$ imply $\Y_{ij} \le \Y_{ij''}$. For notational simplicity and without loss of generality, we assume that $\Y_{ij}$ takes an integer value ranging from $0$ to $p-1$. We emphasize that we have not changed the model from Section~\ref{sec:mnp}. Rather, we simply have more observed data: the index of the choice of the individual $i$, $Y_i$, can be computed from $\Y_i$. Thus, we continue to model the preference ordering, $\Y_{i}$, in terms of a latent (multivariate normal) random vector, $W_{i} = (W_{ij},\ldots,W_{i,p-1})$, via \begin{eqnarray} \Y_{ij}(W_i) & = & \#\{W_{ij'}: W_{ij'} < W_{ij} \} \quad {\rm for} \quad i=1,\ldots,n, \quad {\rm and} \quad j = 1,\ldots,p, \label{eq:mop} \end{eqnarray} where $W_{ip}=0$, the distribution of $W_i$ is specified in equation~\ref{eq:mnp}, and $\#\{\cdots\}$ indicates the number of elements in a finite set. This model can be fitted via a slightly modified version of the MCMC algorithm in \citet{imai:vand:05}. In particular, we need only modify the way in which $W_{ij}$ is sampled and use a truncation rule based on Equation~\ref{eq:mop}. \subsection{Prior Specification} Our prior distribution for the multinomial probit model is \begin{equation} \beta \sim \N(0,A\inv) \quad {\rm and} \quad p(\Sigma)\propto |\Sigma|^{-(\nu+p)/2}\l[\tr(S\Sigma\inv)\r]^{-\nu(p-1)/2}, \label{eq:priors} \end{equation} where $A$ is the prior precision matrix of $\beta$, $\nu$ is the prior degrees of freedom parameter for $\Sigma$, and the $(p-1)\times(p-1)$ positive definite matrix $S$ is the prior scale for $\Sigma$; we assume the first diagonal element of $S$ is one. The prior distribution on $\Sigma$ is proper if $\nu\geq p-1$, the prior mean of $\Sigma$ is approximately equal to $S$ if $\nu> p-2$, and the prior variance of $\Sigma$ increase as $\nu$ decreases as long as this variance exists. We also allow for an improper prior on $\beta$, which is $p(\beta) \propto 1$ (i.e., $A=0$).\footnote{Algorithm~2 of \citet{imai:vand:05} allows for a non zero prior mean for $\beta$. Because the update for $\Sigma$ in this sampler is not exactly its complete conditional distribution, however, this algorithm may exhibit undesirable convergence properties in some situations.} Alternate prior specifications were introduced by \citet{mccu:ross:94} and \citet{mccu:pols:ross:00}. The relative advantage of the various prior distributions are discussed by \citet{mccu:pols:ross:00}, \citet{nobi:00}, and \citet{imai:vand:05}. We prefer our choice because it allows us to directly specify the prior distribution on the identifiable model parameters, allows us to specify an improper prior distribution on regression coefficient, and results in a Monte Carlo sampler that is relatively quick to converge. An implementation of of the sampler proposed by \citet{mccu:ross:94} has recently been released in the R package bayesm \citep{ross:mccu:05}. \subsection{Prediction under the Multinomial Probit Model} Predictions of individual preferences given particular values of the covariates can be useful in interpreting the fitted model. Consider a value of the $(p-1)\times k$ matrix of covariates, $X^\star$, that may or may not correspond to the values for one of the observed individuals. We are interested in the distribution of the preferences among the alternatives in the choice set given this value of the covariates. Let $Y^\star$ be the preferred choice and ${\cal Y}^\star=({\cal Y}^\star_1, \ldots, {\cal Y}^\star_p)$ indicate the ordering of the preferences among the available alternatives. As an example, one might be interested in $\Pr(Y^\star = j \mid X^\star)$ for some $j$. By varying $X^\star$, one could explore how preferences are expected to change with covariates. Similarly, one might be interested in how relative preferences such as $\Pr( {\cal Y}_{j}^\star > {\cal Y}_{j^\prime}^\star \mid X^\star)$ are expected to change with the covariates. In the context of a Bayesian analysis, such predictive probabilities are computed via the posterior predictive distribution. This distribution conditions on the observed data, $Y=(Y_1,\ldots, Y_n)$ or ${\cal Y}= ({\cal Y}_1, \ldots, {\cal Y}_n)$, but averages over the uncertainty in the model parameters. For example, \begin{eqnarray} \Pr(Y^\star = j \mid X^\star, Y) & = & \int \Pr(Y^\star = j \mid X^\star, \beta, \Sigma, Y)\, p(\beta, \Sigma \mid Y) \; d(\beta, \Sigma). \end{eqnarray} Thus, the posterior predictive distribution accounts for both variability in the response variable given the model parameters (i.e., the likelihood or sampling distribution) and the uncertainty in the model parameters as quantified in the posterior distribution. Monte Carlo evaluation of the posterior predictive distribution is easy once we obtain a Monte Carlo sample of the model parameters from the posterior distribution: We simply sample according to the likelihood for each Monte Carlo sample from the posterior distribution. This involves sampling the latent variable under the model in (1) and computing the preferred choice using (2) or the ordering of preferences using (3). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Example 1: Detergent Brand Choice} \label{sec:examp-soap} In this and the next section, we describe the details of two examples of MNP. In this section we use a market research dataset to illustrate the fitting of the multinomial probit model. In Section~\ref{sec:examp-japan} we fit the multinomial probit model with ordered preference to a Japanese election dataset. We also describe how to perform convergence diagnostics of the MCMC sampler and analysis of the Monte Carlo output of MNP using an existing R package. Additional examples of MNP can be found in \citet{imai:vand:05}. \subsection{Preliminaries} \label{subsec:soap-prelim} Our first example analyzes a typical dataset in market research. The dataset contains information about the brand and price of the laundry detergent purchased by 2657 households originally analyzed by \citet{chin:pras:98}. The dataset contains the log prices of six detergent brands -- Tide, Wisk, EraPlus, Surf, Solo, and All -- as well as the brand chosen by each household (Type {\tt help(detergent)} in R for details about the dataset). We fit the multinomial probit model by using \texttt{choice} as the outcome variable and the other six variables as choice-specific covariates. After loading the MNP package, this can be accomplished using the following three commands, \spacingset{1} \begin{verbatim} data(detergent) res <- mnp(choice ~ 1, choiceX = list(Surf=SurfPrice, Tide=TidePrice, Wisk=WiskPrice, EraPlus=EraPlusPrice, Solo=SoloPrice, All=AllPrice), cXnames = c("price"), data = detergent, n.draws = 10000, burnin = 2000, thin = 3, verbose = TRUE) summary(res) \end{verbatim} \spacingset{1.5} The first command loads the example dataset and stores it as the data frame called \texttt{detergent}. The second command fits the multinomial probit model. The default base category in this case is \texttt{All}. (The default base category in MNP is the first factor level of the outcome variable, $Y$.) Each household chooses among the six brands of laundry detergent, i.e., $p=6$. We specify the choice-specific variables, \texttt{choiceX}, using a named list. The elements of the list are the log price of each detergent brand and they are named after the levels of factor variable, \texttt{choice}. We also name the coefficient for this set of choice-specific variables by using \texttt{cXnames}. The argument \texttt{data} allows us to specify the name of the data frame where the data are stored. The model estimates five intercepts and the price coefficient as well as 14 parameters in the covariance matrix, $\Sigma$. We use the default prior distribution; an improper prior distribution for $\beta$ and a diffuse prior distribution for $\Sigma$ with $\nu = p = 6$ and $S=I$. We sample 10,000 replications of the parameter from the resulting posterior distribution, saving every fourth sample after discarding the first 2,000 samples as specified by the arguments, \texttt{n.draws}, \texttt{thin}, and \texttt{burnin}. The argument \texttt{verbose = TRUE} specifies that a progress report and other useful messages be printed while the MCMC sampler is running. The \texttt{summary(res)} command gives a summary of the output including the posterior means and standard deviations of the parameters. The summary is based on the single MCMC chain produced with this call of MNP. Before we can reliably draw conclusions based on these results, we must be sure the chain has converged. Convergence diagnostics are discussed and illustrated in Section~\ref{sec:coda}. The result of the call of \texttt{summary(res)} are as follows. \spacingset{1} \begin{verbatim} Call: mnp(formula = choice ~ 1, data = detergent, choiceX = list(Surf = SurfPrice, Tide = TidePrice, Wisk = WiskPrice, EraPlus = EraPlusPrice, Solo = SoloPrice, All = AllPrice), cXnames = c("price"), n.draws = 10000, burnin = 2000, thin = 3, verbose = TRUE) Coefficients: mean std.dev. 2.5% 97.5% (Intercept):EraPlus 2.567 0.238 2.123 3.03 (Intercept):Solo 1.722 0.247 1.248 2.25 (Intercept):Surf 1.572 0.163 1.259 1.91 (Intercept):Tide 2.716 0.252 2.269 3.22 (Intercept):Wisk 1.620 0.162 1.328 1.96 price -82.102 8.952 -99.896 -66.32 Covariances: mean std.dev. 2.5% 97.5% EraPlus:EraPlus 1.00000 0.00000 1.00000 1.00 EraPlus:Solo 0.82513 0.26942 0.31029 1.36 EraPlus:Surf 0.17021 0.16115 -0.15810 0.48 EraPlus:Tide 0.24872 0.12956 0.00253 0.52 EraPlus:Wisk 0.88170 0.16614 0.54500 1.20 Solo:Solo 2.56481 0.68678 1.53276 4.25 Solo:Surf 0.45246 0.34572 -0.28018 1.13 Solo:Tide 0.50836 0.32706 -0.09069 1.22 Solo:Wisk 1.46997 0.44596 0.65506 2.45 Surf:Surf 1.69005 0.50978 0.92334 2.82 Surf:Tide 0.80762 0.30381 0.34019 1.44 Surf:Wisk 1.01614 0.36503 0.44121 1.85 Tide:Tide 1.32024 0.41669 0.62898 2.25 Tide:Wisk 1.05396 0.30137 0.59323 1.74 Wisk:Wisk 2.58761 0.55076 1.68773 3.82 Base category: All Number of alternatives: 6 Number of observations: 2657 Number of stored MCMC draws: 2000 \end{verbatim} \spacingset{1.5} We emphasize that these results are preliminary because convergence has not yet been assessed. Thus, we delay interpretation of the fit until Section~\ref{sec:examp-soap-final}, after we discuss convergence diagnostics in Section~\ref{sec:coda}. Note that \texttt{coef(res)} and \texttt{cov.mnp(res)} allow one to extract the posterior draws of model coefficients and covariance matrix if desired. Type {\tt help(mnp)} in R for details. \subsection{Using coda for Convergence Diagnostics and Output Analysis} \label{sec:coda} It is possible to use coda \citep*{plum:best:cowl:vine:05}, to perform various convergence diagnostics, as well as to summarize results. The coda package requires a matrix of posterior draws for relevant parameters to be saved as an \texttt{mcmc} object. Here, we illustrate how to use coda to calculate the Gelman-Rubin convergence diagnostic statistic \citep{gelm:rubi:92}. This diagnostic is based on multiple independent Markov chains initiated at over-dispersed starting values. Here, we obtain these chains by independently running the \texttt{mnp()} command three times, specifying different starting values for each time. This can be accomplished by typing the following commands at the R prompt, \spacingset{1} \begin{verbatim} data(detergent) res1 <- mnp(choice ~ 1, choiceX = list(Surf=SurfPrice, Tide=TidePrice, Wisk=WiskPrice, EraPlus=EraPlusPrice, Solo=SoloPrice, All=AllPrice), cXnames = c("price"), data = detergent, n.draws = 50000, verbose = TRUE) res2 <- mnp(choice ~ 1, choiceX = list(Surf=SurfPrice, Tide=TidePrice, Wisk=WiskPrice, EraPlus=EraPlusPrice, Solo=SoloPrice, All=AllPrice), coef.start = c(1, -1, 1, -1, 1, -1)*10, cov.start = matrix(0.5, ncol=5, nrow=5) + diag(0.5, 5), cXnames = c("price"), data = detergent, n.draws = 50000, verbose = TRUE) res3 <- mnp(choice ~ 1, choiceX = list(Surf=SurfPrice, Tide=TidePrice, Wisk=WiskPrice, EraPlus=EraPlusPrice, Solo=SoloPrice, All=AllPrice), coef.start=c(-1, 1, -1, 1, -1, 1)*10, cov.start = matrix(0.9, ncol=5, nrow=5) + diag(0.1, 5), cXnames = c("price"), data = detergent, n.draws = 50000, verbose = TRUE) \end{verbatim} \spacingset{1.5} where we save the output of each chain separately as \texttt{res1}, \texttt{res2}, and \texttt{res3}. The first chain is initiated at the default starting values for all parameters; i.e., a vector of zeros for $\beta$ and an identity matrix for $\Sigma$. The second chain is run starting from a vector of three $10$'s and three $-10$'s for $\beta$ and a matrix with all diagonal elements equal to 1 and all correlations equal to 0.5 for $\Sigma$. Finally, the third chain is run starting from a permutation of the starting value used for $\beta$ in the second chain, and a matrix with all diagonal elements equal to 1 and all correlations equal to 0.9 for $\Sigma$. We again use the default prior specification and obtain 50,000 draws for each chain. We store the output from each of the three chains as an object of class \texttt{mcmc}, and then combine them into a single list using the following commands, \spacingset{1} \begin{verbatim} library(coda) res.coda <- mcmc.list(chain1=mcmc(res1$param[,-7]), chain2=mcmc(res2$param[,-7]), chain3=mcmc(res3$param[,-7])) \end{verbatim} \spacingset{1.5} where the first command loads the coda package\footnote{If you have not used the \texttt{coda} package before, you must install it. At the R prompt, type \texttt{install.packages("coda")}.} and the second command saves the results as an object of class \texttt{mcmc.list}, which is called \texttt{res.coda}. We exclude the 7th column of each chain, because this column corresponds to the first diagonal element of the covariance matrix which is always equal to 1. The following command computes the Gelman-Rubin statistic from these three chains, \spacingset{1} \begin{verbatim} gelman.diag(res.coda, transform = TRUE) \end{verbatim} \spacingset{1.5} where \texttt{transform = TRUE} applies log or logit transformation as appropriate to improve the normality of each of the marginal distributions. \citet{gelm:carl:ster:rubi:04} suggest computing the statistic for each scalar estimate of interest, and to continue to run the chains until the statistics are all less than 1.1. Inference is then based on the Monte Carlo sample obtained by combining the second half of each of the chains. The output of the coda command lists the value and a 97.5\% upper limit of the Gelman-Rubin statistic for each parameter. \spacingset{1}\begin{verbatim} Potential scale reduction factors: Point est. 97.5% quantile (Intercept):EraPlus 1.01 1.02 (Intercept):Solo 1.03 1.08 (Intercept):Surf 1.01 1.05 (Intercept):Tide 1.01 1.02 (Intercept):Wisk 1.01 1.04 price 1.01 1.02 EraPlus:Solo 1.02 1.03 EraPlus:Surf 1.02 1.04 EraPlus:Tide 1.03 1.08 EraPlus:Wisk 1.04 1.13 Solo:Solo 1.01 1.04 Solo:Surf 1.01 1.02 Solo:Tide 1.02 1.07 Solo:Wisk 1.00 1.00 Surf:Surf 1.00 1.00 Surf:Tide 1.01 1.04 Surf:Wisk 1.02 1.06 Tide:Tide 1.02 1.06 Tide:Wisk 1.02 1.08 Wisk:Wisk 1.01 1.04 Multivariate psrf 1.07+0i \end{verbatim} \spacingset{1.5} The Gelman-Rubin statistics are all less than 1.1, suggesting satisfactory convergence has been achieved. (Note that the 97.5% quantile for {\tt EraPlus:Wisk} is greater than 1.1; a more conservative user might want to obtain a set of longer Markov chains and recompute the Gelman-Rubin statistics.) It may also be useful to examine the change in the value of the Gelman-Rubin statistic over the iterations. The following commands produce a graphical summary of the progression of the statistics over iterations. \spacingset{1} \begin{verbatim} gelman.plot(res.coda, transform = TRUE, ylim = c(1,1.2)) \end{verbatim} \spacingset{1.5} where \texttt{ylim = c(1,1.2)} specifies the range of the vertical axis of the plot. The results appear in Figure~\ref{fg:rhat}, as a cumulative evaluation of the Gelman-Rubin statistic over iterations for nine selected parameters. (Three coefficients appear in the first row; three covariance parameters appear in the second row; and three variance parameters appear in the third row.) \begin{figure}[p] \spacingset{1} \includegraphics[scale=1.15]{rhat} \caption{The Gelman-Rubin Statistic Computed with Three Independent Markov Chains for Selected Parameters in the Detergent Example. The first row represents three coefficients, the second row represents three covariances, and the third row represents three variance parameters.} \label{fg:rhat} \end{figure} The coda package can also be used to produce univariate time-series plots of the three chains and univariate density estimate of the posterior distribution. The following commands create these graphs for the price coefficient. %pdf("coda.pdf", width=9, height=4.5) \spacingset{1}\begin{verbatim} res.coda <- mcmc.list(chain1=mcmc(res1$param[25001:50000, "price"], start=25001), chain2=mcmc(res2$param[25001:50000, "price"], start=25001), chain3=mcmc(res3$param[25001:50000, "price"], start=25001)) plot(res.coda, ylab = "price coefficient") \end{verbatim} %dev.off() \spacingset{1.5} Figure \ref{fg:trace} presents the resulting plots. The left panel overlays the time-series plot for each chain with a different color representing each chain. The right panel shows the kernel-smoothed density estimate of the posterior distribution. One can also apply an array of other functions to \texttt{res.coda}. See the coda homepage, \href{http://www-fis.iarc.fr/coda}{http://www-fis.iarc.fr/coda}, for details. \begin{figure} \spacingset{1} \begin{center} \includegraphics[scale=0.75]{coda-small} \end{center} \vspace{-0.25in} \caption{Time-series Plot of Three Independent Markov Chains (Left Panel) and A Density Estimate of the Posterior Distribution of the Price Coefficient (Right Panel). The time-series plot overlays the three chains, each in a different color. A lowess smoothed line is also plotted for each of the three chains. The density estimate is based on all three chains.} \label{fg:trace} \end{figure} \subsection{Final Analysis and Conclusions} \label{sec:examp-soap-final} In the final analysis, we combine the second half of each of the three chains. This is accomplished using the following command that saves the last 25,000 draws from each chain as an \texttt{mcmc} object and combines the \texttt{mcmc} objects into a list, % pdf("rhat.pdf") % par(mex=0.5) % res.coda <- mcmc.list(chain1=mcmc(res1$param[,c(1:2,6,9,14,18,12,16,19,)]), % chain2=mcmc(res2$param[,c(1:2,6,9,14,18,12,16,19,)]), % chain3=mcmc(res3$param[,c(1:2,6,9,14,18,12,16,19,)])) % gelman.plot(res.coda, transform = TRUE, ylim=c(1,1.2)) % dev.off() \spacingset{1}\begin{verbatim} res.coda <- mcmc.list(chain1=mcmc(res1$param[25001:50000,-7], start=25001), chain2=mcmc(res2$param[25001:50000,-7], start=25001), chain3=mcmc(res3$param[25001:50000,-7], start=25001)) summary(res.coda) \end{verbatim} \spacingset{1.5} The second command produces the following summary of the posterior distribution for each parameter based on the combined Monte Carlo sample. \spacingset{1}\begin{verbatim} Iterations = 25001:50000 Thinning interval = 1 Number of chains = 3 Sample size per chain = 25000 1. Empirical mean and standard deviation for each variable, plus standard error of the mean: Mean SD Naive SE Time-series SE (Intercept):EraPlus 2.5398 0.2300 0.0008400 0.014332 (Intercept):Solo 1.7218 0.2227 0.0008131 0.012972 (Intercept):Surf 1.5634 0.1663 0.0006072 0.010462 (Intercept):Tide 2.6971 0.2374 0.0008670 0.015153 (Intercept):Wisk 1.6155 0.1594 0.0005822 0.010221 price -80.9097 8.4292 0.0307791 0.556483 EraPlus:Solo 0.8674 0.2954 0.0010787 0.021698 EraPlus:Surf 0.1226 0.1991 0.0007269 0.014043 EraPlus:Tide 0.2622 0.1525 0.0005568 0.009833 EraPlus:Wisk 0.9062 0.1912 0.0006982 0.012893 Solo:Solo 2.6179 0.7883 0.0028785 0.055837 Solo:Surf 0.5348 0.4307 0.0015728 0.030113 Solo:Tide 0.5570 0.3544 0.0012941 0.024548 Solo:Wisk 1.5442 0.4643 0.0016954 0.031574 Surf:Surf 1.6036 0.4758 0.0017374 0.031269 Surf:Tide 0.7689 0.2992 0.0010926 0.020253 Surf:Wisk 0.9949 0.3548 0.0012955 0.022963 Tide:Tide 1.2841 0.3660 0.0013364 0.024095 Tide:Wisk 1.0658 0.3147 0.0011492 0.020229 Wisk:Wisk 2.5801 0.5523 0.0020167 0.034974 2. Quantiles for each variable: 2.5% 25% 50% 75% 97.5% (Intercept):EraPlus 2.09105 2.38514 2.5321 2.6926 3.0022 (Intercept):Solo 1.28315 1.57316 1.7219 1.8705 2.1639 (Intercept):Surf 1.24132 1.45272 1.5583 1.6701 1.9023 (Intercept):Tide 2.23562 2.53443 2.6886 2.8544 3.1721 (Intercept):Wisk 1.31120 1.50841 1.6104 1.7191 1.9429 price -97.62406 -86.61736 -80.7783 -75.1013 -64.9720 EraPlus:Solo 0.32811 0.65755 0.8480 1.0694 1.4666 EraPlus:Surf -0.24159 -0.01596 0.1131 0.2507 0.5491 EraPlus:Tide -0.02109 0.16081 0.2571 0.3527 0.5957 EraPlus:Wisk 0.54089 0.77643 0.9035 1.0331 1.2917 Solo:Solo 1.37468 2.02720 2.5225 3.0985 4.4160 Solo:Surf -0.31143 0.25493 0.5251 0.8180 1.3880 Solo:Tide -0.05172 0.30518 0.5297 0.7740 1.3106 Solo:Wisk 0.74682 1.21192 1.5131 1.8380 2.5445 Surf:Surf 0.86883 1.25552 1.5377 1.8834 2.6935 Surf:Tide 0.30606 0.55218 0.7285 0.9413 1.4556 Surf:Wisk 0.40231 0.74907 0.9579 1.1957 1.8026 Tide:Tide 0.69841 1.02206 1.2396 1.4999 2.1100 Tide:Wisk 0.51732 0.84824 1.0411 1.2556 1.7579 Wisk:Wisk 1.60662 2.18760 2.5407 2.9238 3.7722 \end{verbatim} \spacingset{1.5} The output shows the mean, standard deviation, and various percentiles of the posterior distributions of the coefficients and the elements of the variance-covariance matrix. The base category is the detergent \texttt{All}. Separate intercepts are estimated for each detergent. The price coefficient is negative and highly statistically significant, agreeing with the standard economic expectation that consumers are less likely to buy more expensive goods. MNP also allows one to calculate the posterior predictive probabilities of each alternative being most preferred given a particular value of the covariates. For example, one can calculate the posterior predictive probabilities using the covariate values of the first two observations by using the \texttt{predict()} command, \spacingset{1} \begin{verbatim} predict(res1, newdata = detergent[1:2,], newdraw = rbind(res1$param[25001:50000,], res2$param[25001:50000,], res3$param[25001:50000,]), type = "prob") \end{verbatim} \spacingset{1.5} where \texttt{res1} is the output object from the \texttt{mnp()} command, and we set \texttt{newdata} to the first two observations of the detergent data set and \texttt{newdraw} to the combined draws from the second half of three chains. Setting \texttt{type = "prob"} causes the function \texttt{predict()} to return the posterior predictive probabilities. Moreover, a new {\tt n.draws} option in \texttt{predict()} command allows one to compute the uncertainty estimates about these predicted probabilities. It is also possible to return a Monte Carlo sample of the the alternative that is most preferred (\texttt{type = "choice"}), a Monte Carlo sample of the latent variables (\texttt{type = "latent"}), or a Monte Carlo sample of the preference-ordered alternatives (\texttt{type = "order"}). (Type \texttt{help(predict.mnp)} in R for more details about the \texttt{predict()} function in MNP.) The above command yields the following output, \spacingset{1} \begin{verbatim} All EraPlus Solo Surf Tide Wisk [1,] 0.01281333 0.1946400 0.12292 0.46208000 0.1401733 0.06737333 [2,] 0.04649333 0.1262133 0.05996 0.03169333 0.3589867 0.37665333 \end{verbatim} \spacingset{1.5} The result indicates that the posterior predictive probability of purchasing \texttt{Surf} is the largest for households with covariates equal to those in the first household in the data set. Under the model, approximately 46\% of such households will purchase \texttt{Surf}. On the other hand, \texttt{All} is the brand least likely to be purchased by these households. The households with covariates equal to the second household are most likely to buy \texttt{Wisk}. Also, they are almost equally likely to purchase \texttt{Tide}. (The posterior predictive probabilities of buying \texttt{Wisk} and \texttt{Tide} are both around 0.35) \section{Example 2: Voters' Preference of Political Parties} \label{sec:examp-japan} Our second example illustrates how to fit the multinomial probit model with ordered preferences (see Section~\ref{sec:mnpop}). \subsection{Preliminaries} We analyze a survey dataset describing the preferences of individual voters in Japan among the political parties. Political scientists may be interested in using the gender, age and education level of voters to predict their party preferences (Type {\tt help(japan)} in R for details about the dataset). The outcome variable is a vector of relative preferences for each of the four parties, i.e., $p=4$. Each of 418 voters is asked to give a score between 0 and 100 to each party. For example, the first voter in the dataset has the following preferences. \begin{verbatim} LDP NFP SKG JCP 80 75 80 0 \end{verbatim} That is, this voter prefers \texttt{LDP} and \texttt{SKG} to \texttt{NFP} and \texttt{JCP}, and between the latter two, she prefers \texttt{NFP} to \texttt{JCP}. Although \texttt{LDP} and \texttt{SKG} have the same preference, we do not constrain the estimated preferences to be the same for these two alternatives. Under the Gaussian random utility model, the probability that the two alternatives having exactly the same preferences is zero. Therefore, inequality constraints are respected, but equality constraints are not. Furthermore, we only preserve the ranking, not the relative numerical values. Therefore, the following coding of the variables, for our purposes, is equivalent to that given above, \spacingset{1} \begin{verbatim} LDP NFP SKG JCP 3 2 3 1 \end{verbatim} \spacingset{1.5} Finally, it is possible to have non-response for one of the categories; e.g., no candidate from a particular party may run in a certain district. If \texttt{NFP = NA}, we have no information about the relative ranking of \texttt{NFP}. \spacingset{1} \begin{verbatim} LDP NFP SKG JCP 3 NA 3 1 \end{verbatim} \spacingset{1.5} In this case, there is no constraint when estimating the preference for this alternative; only the inequality constraint, \texttt{(LDP, SKG) > JCP}, is imposed. All three covariates -- gender, education, and age of voters -- are individual-specific variables rather than choice-specific ones. The model estimates three intercepts and 9 coefficients along with 6 parameters in the covariance matrix. The following commands fit the model, \spacingset{1} \begin{verbatim} data(japan) res <- mnp(cbind(LDP, NFP, SKG, JCP) ~ gender + education + age, data = japan, n.draws = 10000, verbose = TRUE) summary(res) \end{verbatim} \spacingset{1.5} The first command loads the dataset, and the second command fits the model. The base category is \texttt{JCP}, which is the last column of the outcome matrix. The default prior distribution is used as in the previous example: an improper prior distribution for $\beta$ and a diffuse prior distribution for $\Sigma$ with $\nu = p = 4$ and $S=I$. 10,000 draws are obtained with no burnin or thinning. The final command summarizes the Monte Carlo sample and gives the following output, \spacingset{1} \begin{verbatim} Call: mnp(formula = cbind(LDP, NFP, SKG, JCP) ~ gender + education + age, data = japan, n.draws = 10000, verbose = TRUE) Coefficients: mean std.dev. 2.5% 97.5% (Intercept):LDP 0.615184 0.517157 -0.386151 1.61 (Intercept):NFP 0.689753 0.568109 -0.419521 1.79 (Intercept):SKG 0.133961 0.455960 -0.758883 1.02 gendermale:LDP 0.099748 0.152323 -0.194786 0.40 gendermale:NFP 0.216824 0.166103 -0.102108 0.54 gendermale:SKG 0.132661 0.134605 -0.127145 0.40 education:LDP -0.107038 0.074792 -0.253483 0.04 education:NFP -0.107222 0.082324 -0.270127 0.05 education:SKG -0.003728 0.066429 -0.132496 0.13 age:LDP 0.013518 0.006122 0.001492 0.03 age:NFP 0.006948 0.006783 -0.006572 0.02 age:SKG 0.009653 0.005431 -0.000812 0.02 Covariances: mean std.dev. 2.5% 97.5% LDP:LDP 1.0000 0.0000 1.0000 1.00 LDP:NFP 1.0502 0.0585 0.9373 1.16 LDP:SKG 0.7070 0.0622 0.5822 0.82 NFP:NFP 1.4068 0.1359 1.1682 1.70 NFP:SKG 0.7452 0.0864 0.5800 0.91 SKG:SKG 0.6913 0.0874 0.5296 0.87 Base category: JCP Number of alternatives: 4 Number of observations: 418 Number of stored MCMC draws: 10000 \end{verbatim} \spacingset{1.5} \subsection{Convergence Diagnostics, Final Analysis, and Conclusions} In order to evaluate convergence of the MCMC sampler, we again obtain three independent Markov chains by running the \texttt{mnp()} command three times with three sets of different starting values. We use starting values that are relatively dispersed given the preliminary analysis of the previous section. Note that when fitting the multinomial probit model with ordered preferences, the algorithm requires the starting values of the latent variable to respect the order constraints of equation (\ref{eq:mop}). Therefore, the starting values of the parameters cannot be too far away from the posterior mode. The following commands fits the model with the default starting value and two sets of overdispersed starting values, \spacingset{1} \begin{verbatim} res1 <- mnp(cbind(LDP, NFP, SKG, JCP) ~ gender + education + age, data = japan, n.draws = 50000, verbose = TRUE) res2 <- mnp(cbind(LDP, NFP, SKG, JCP) ~ gender + education + age, data = japan, coef.start = c(1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1), cov.start = matrix(0.5, ncol=3, nrow=3) + diag(0.5, 3), n.draws = 50000, verbose = TRUE) res3 <- mnp(cbind(LDP, NFP, SKG, JCP) ~ gender + education + age, data = japan, coef.start = c(-1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1), cov.start = matrix(0.9, ncol=3, nrow=3) + diag(0.1, 3), n.draws = 50000, verbose = TRUE) \end{verbatim} \spacingset{1.5} %res.coda <- mcmc.list(chain1=mcmc(res1$param[,-13]), % chain2=mcmc(res2$param[,-13]), % chain3=mcmc(res3$param[,-13])) %gelman.plot(res.coda, transform = TRUE, ylim = c(1,1.2)) %res.coda <- mcmc.list(chain1=mcmc(res1$param[25001:50000,-13], start=25001), % chain2=mcmc(res2$param[25001:50000,-13], start=25001), % chain3=mcmc(res3$param[25001:50000,-13], start=25001)) %summary(res.coda) We follow the commands used in Section~\ref{sec:coda} and compute the Gelman-Rubin statistic for each parameter. Upon examination of the resulting statistics, we determined that satisfactory convergence has been achieved. For example, the value of the Gelman-Rubin statistic is less than 1.01 for all the parameters. Hence, we base our final analysis on the combined draws from the second half of the three chains (i.e., a total of 75,000 draws using 25,000 draws from each chain). Posterior summaries can be obtained using the coda package as before, \spacingset{1} \begin{verbatim} Iterations = 25001:50000 Thinning interval = 1 Number of chains = 3 Sample size per chain = 25000 1. Empirical mean and standard deviation for each variable, plus standard error of the mean: Mean SD Naive SE Time-series SE (Intercept):LDP 0.60167 0.51421 1.88e-03 8.05e-03 (Intercept):NFP 0.68294 0.56867 2.08e-03 7.95e-03 (Intercept):SKG 0.12480 0.45680 1.67e-03 7.25e-03 gendermale:LDP 0.10668 0.15448 5.64e-04 2.95e-03 gendermale:NFP 0.22240 0.16983 6.20e-04 2.91e-03 gendermale:SKG 0.13897 0.13753 5.02e-04 2.70e-03 education:LDP -0.10517 0.07643 2.79e-04 1.35e-03 education:NFP -0.10634 0.08448 3.08e-04 1.28e-03 education:SKG -0.00258 0.06766 2.47e-04 1.18e-03 age:LDP 0.01361 0.00617 2.25e-05 9.90e-05 age:NFP 0.00698 0.00680 2.48e-05 1.01e-04 age:SKG 0.00972 0.00547 2.00e-05 9.13e-05 LDP:NFP 1.05535 0.05508 2.01e-04 1.15e-03 LDP:SKG 0.71199 0.06125 2.24e-04 1.59e-03 NFP:NFP 1.41860 0.13540 4.94e-04 2.45e-03 NFP:SKG 0.75391 0.08262 3.02e-04 2.12e-03 SKG:SKG 0.70007 0.08488 3.10e-04 2.16e-03 2. Quantiles for each variable: 2.5% 25% 50% 75% 97.5% (Intercept):LDP -0.405757 0.25198 0.60033 0.9476 1.6172 (Intercept):NFP -0.428421 0.30016 0.68058 1.0657 1.7981 (Intercept):SKG -0.769018 -0.18476 0.12335 0.4303 1.0258 gendermale:LDP -0.197199 0.00328 0.10643 0.2105 0.4096 gendermale:NFP -0.110730 0.10856 0.22135 0.3361 0.5566 gendermale:SKG -0.131661 0.04702 0.13905 0.2307 0.4096 education:LDP -0.254631 -0.15718 -0.10519 -0.0533 0.0447 education:NFP -0.271657 -0.16329 -0.10654 -0.0496 0.0595 education:SKG -0.135829 -0.04833 -0.00248 0.0429 0.1306 age:LDP 0.001591 0.00941 0.01361 0.0177 0.0257 age:NFP -0.006336 0.00240 0.00697 0.0115 0.0203 age:SKG -0.000947 0.00603 0.00967 0.0134 0.0206 LDP:NFP 0.944577 1.01919 1.05564 1.0924 1.1623 LDP:SKG 0.587135 0.67120 0.71364 0.7544 0.8266 NFP:NFP 1.181667 1.32454 1.40803 1.5028 1.7125 NFP:SKG 0.590798 0.69778 0.75463 0.8104 0.9125 SKG:SKG 0.538858 0.64219 0.69806 0.7564 0.8711 \end{verbatim} \spacingset{1.5} Here, one of the findings is that older voters tend to prefer \texttt{LDP} as indicated by the statistically significant positive age coefficient for \texttt{LDP}. This is consistent with the conventional wisdom of Japanese politics that the stronghold of \texttt{LDP} is elderly voters. To further investigate the marginal effect of age, we calculate the posterior predictive probabilities of party preference under two scenarios. First, we choose the 10th individual in the survey data and compute the predictive probability that a voter with this set of covariates prefers each of the parties. This can be accomplished by the following commands, \spacingset{1}\begin{verbatim} japan10a <- japan[10,] predict(res1, newdata = japan10a, newdraw = rbind(res1$param[25001:50000,], res2$param[25001:50000,], res3$param[25001:50000,]), type = "prob") \end{verbatim} \spacingset{1.5} where the first command extracts the 10th observation from the Japan data, and the second command computes the predictive probabilities. Note that this individual has the following attributes, \spacingset{1} \begin{verbatim} gender education age male 4 50 \end{verbatim} \spacingset{1.5} The resulting posterior predictive probabilities of being the most preferred party are, \spacingset{1} \begin{verbatim} JCP LDP NFP SKG [1,] 0.107707 0.359267 0.324613 0.208413 \end{verbatim} \spacingset{1.5} The result indicates that under the model, we should expect 36\% of voters with these covariates to prefer \texttt{LDP}, 32\% to prefer \texttt{NFP}, 21\% to prefer \texttt{SKG}, and 11\% to prefer \texttt{JCP}. Next, we change the value of the age variable of this voter from 50 to 75, while holding the other variables constant. We then recompute the posterior predictive probabilities and examine how they change. This can be accomplished using the following commands, \spacingset{1} \begin{verbatim} japan10b <- japan10a japan10b[,"age"] <- 75 predict(res1, newdata = japan10b, newdraw = rbind(res1$param[25001:50000,], res2$param[25001:50000,], res3$param[25001:50000,]), type = "prob") \end{verbatim} \spacingset{1.5} where the first two commands recode the age variable for the voter and the second command makes the prediction. We obtain the following results, \spacingset{1} \begin{verbatim} JCP LDP NFP SKG [1,] 0.06548 0.485467 0.249667 0.199387 \end{verbatim} \spacingset{1.5} The comparison of the two results shows that changing the value of the age variable from 50 to 75 increases the estimated posterior predictive probability of preferring \texttt{LDP} most and by more than 10 percentage points. Interestingly, the predictive probability for \texttt{SKG} changes very little, while that of \texttt{NFP} decreases significantly. This suggests that older voters tend to prefer \texttt{LDP} over \texttt{NFP}. \clearpage \pdfbookmark{References}{References} \bibliography{my,imai} \end{document} MNP/vignettes/rhat.pdf0000644000176200001440000007217313101760304014352 0ustar liggesusers%PDF-1.1 %ρ 1 0 obj << /CreationDate (D:20050317135032) /ModDate (D:20050317135032) /Title (R Graphics Output) /Producer (R 2.0.1) /Creator (R) >> endobj 2 0 obj << /Type /Catalog /Pages 3 0 R >> endobj 5 0 obj << /Type /Font /Subtype /Type1 /Name /F1 /BaseFont /ZapfDingbats >> endobj 6 0 obj << /Type /Page /Parent 3 0 R /Contents 7 0 R /Resources 4 0 R >> endobj 7 0 obj << /Length 8 0 R >> stream q Q q 38.97 336.47 85.08 56.56 re W n 0.000 0.000 0.000 RG 0.75 w [] 0 d 1 J 1 j 10.00 M 44.52 432.00 m 45.27 357.67 l 46.84 342.27 l 48.42 343.36 l 49.99 356.48 l 51.57 347.96 l 53.15 341.85 l 54.72 350.78 l 56.30 348.18 l 57.87 344.46 l 59.45 340.69 l 61.02 349.31 l 62.60 349.07 l 64.17 344.48 l 65.75 356.41 l 67.33 356.81 l 68.90 365.35 l 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0000029253 00000 n 0000029362 00000 n trailer << /Size 15 /Info 1 0 R /Root 2 0 R >> startxref 29440 %%EOF MNP/vignettes/imai.bib0000644000176200001440000006061713101760304014316 0ustar liggesusers @Unpublished{barb:imai:13, author = {Barber, Michael and Imai, Kosuke}, title = {Estimating Neighborhood Effects on Turnout from Geocoded Voter Registration Records}, note = {Working Paper available at \url{http://imai.princeton.edu/research/neighbor.html}}, OPTkey = {}, OPTmonth = {}, year = {2013}, OPTannote = {} } @misc{blai:imai:11, Author = {Blair, Graeme and Imai, Kosuke}, Howpublished = {hdl:1902.1/17040}, Note = {The Dataverse Network}, Title = {Replication data for: Statistical Analysis of List Experiments}, Year = {2011}} @article{blai:imai:12, Author = {Blair, Graeme and Imai, Kosuke}, Journal = {Political Analysis}, Month = {Winter}, Number = {1}, Pages = {47--77}, Title = {Statistical Analysis of List Experiments}, Volume = {20}, Year = {2012}} @misc{blai:imai:14, Author = {Blair, Graeme and Imai, Kosuke}, Howpublished = {available at the {Comprehensive R Archive Network (CRAN)}}, Note = {\url{http://CRAN.R-project.org/package=list}}, Title = {{list}: Statistical Methods for the Item Count Technique and List Experiment}, Year = {2014}} @article{blai:imai:lyal:14, author = {Blair, Graeme and Imai, Kosuke and Lyall, Jason}, title = {Comparing and Combining List and Endorsement Experiments: Evidence from {A}fghanistan}, Journal = {American Journal of Political Science}, year = {2014}, volume = {58}, number = {4}, month = {October}, pages = {1043--1063}, } @article{blai:imai:zhou:15, author = {Blair, Graeme and Imai, Kosuke and Zhou, Yang-Yang}, title = {Design and Analysis of Randomized Response Technique}, year = {2015}, journal = {Journal of the American Statistical Association}, pages = {Forthcoming}, volume = {110}, number = {511}, pages = {1304--1319}, month = {September}, } @misc{blai:imai:zhou:15b, Author = {Graeme Blair and Kosuke Imai and Yang-Yang Zhou}, Howpublished = {http://dx.doi.org/10.7910/DVN/AIO5BR}, Note = {The Dataverse Network}, Title = {Replication data for: Design and Analysis of the Randomized Response Technique}, Year = {2015}} @misc{blai:zhou:imai:15, Author = {Blair, Graeme and Zhou, Yang-Yang and Imai, Kosuke}, Howpublished = {available at the {Comprehensive R Archive Network (CRAN)}}, Note = {\url{http://CRAN.R-project.org/package=rr}}, Title = {{rr}: Statistical Methods for the Randomized Response}, Year = {2015}} @article{bull:imai:shap:11, Author = {Bullock, Will and Imai, Kosuke and Shapiro, Jacob N.}, Journal = {Political Analysis}, Month = {Autumn}, Number = {4}, Pages = {363--384}, Title = {Statistical Analysis of Endorsement Experiments: Measuring Support for Militant Groups in {P}akistan}, Volume = {19}, Year = {2011}} @misc{bull:imai:shap:11a, Author = {Bullock, Will and Imai, Kosuke and Shapiro, Jacob N.}, Howpublished = {hdl:1902.1/14840}, Note = {The Dataverse Network}, Title = {Replication data for: Statistical Analysis of Endorsement Experiments: Measuring Support for Militant Groups in {P}akistan}, Year = {2011}} @unpublished{clin:imai:pems:09, Author = {Clinton, Joshua and Imai, Kosuke and Pemstein, Daniel}, Note = {Working paper available at \url{http://imai.princeton.edu/research/}}, Title = {Statistical Tests of Agenda Control Using Roll Call Data}, Year = {2009}} @misc{egam:ratk:imai:15, Author = {Egami, Naoki and Ratkovic, Marc and Imai, Kosuke}, Howpublished = {available at the {Comprehensive R Archive Network (CRAN)}}, Note = {\url{http://CRAN.R-project.org/package=FindIt}}, Title = {{FindIt}: Finding Heterogeneous Treatment Effects}, Year = {2015}} @misc{fifi:tarr:imai:15, Author = {Fifield, Benjamin and Tarr, Alexander and Imai, Kosuke}, Howpublished = {available as an {R} package at the {GitHub}}, Note = {\url{https://github.com/redistricting/redist}}, Title = {redist: Markov Chain Monte Carlo Methods for Redistricting Simulation}, Year = {2015}} @misc{fong:etal:15, Author = {Fong, Christian and Ratkovic, Marc and Hazlett, Chad and Imai, Kosuke}, Howpublished = {available at the {Comprehensive R Archive Network (CRAN)}}, Note = {\url{http://CRAN.R-project.org/package=CBPS}}, Title = {{CBPS}: R Package for Covariate Balancing Propensity Score}, Year = {2015}} @TechReport{fong:hazl:imai:15, author = {Fong, Christian and Hazlett, Chad and Imai, Kosuke}, title = {Covariate Balancing Propensity Score for General Treatment Regimes}, institution = {Department of Politics, Princeton University}, year = {2015}, OPTkey = {}, OPTtype = {}, OPTnumber = {}, OPTaddress = {}, OPTmonth = {}, OPTnote = {}, OPTannote = {} } @unpublished{gold:etal:08, Author = {Goldstein, Daniel G. and Imai, Kosuke and G{\"o}ritz, Anja S. and Gollwitzer, Peter M.}, Note = {Working paper available at SSRN \url{http://ssrn.com/abstract=977000}}, Title = {Nudging Turnout: Mere Measurement and Implementation Planning of Intentions to Vote}, Year = {2008}} @Unpublished{hira:etal:11, author = {Hirano, Shigeo and Imai, Kosuke and Shiraito, Yuki and Taniguchi, Masaaki}, title = {Policy Positions in Mixed Member Electoral Systems:Evidence from {Japan}}, note = {Working Paper available at \url{http://imai.princeton.edu/research/japan.html}}, OPTkey = {}, OPTmonth = {}, year = {2011}, OPTannote = {} } @unpublished{ho:imai:04, Author = {Ho, Daniel E. and Kosuke Imai}, Note = {Princeton Law \& Public Affairs Paper No. 04-001; Harvard Public Law Working Paper No. 89; available at SSRN \url{http://ssrn.com/abstract=496863}}, Title = {The Impact of Partisan Electoral Regulation: Ballot Effects from the California Alphabet Lottery, 1978--2002}, Year = {2004}} @article{ho:imai:06, Author = {Ho, Daniel E. and Imai, Kosuke}, Journal = {Journal of the American Statistical Association}, Month = {September}, Number = {475}, Pages = {888--900}, Title = {Randomization Inference with Natural Experiments: An Analysis of Ballot Effects in the 2003 {C}alifornia Recall Election}, Volume = {101}, Year = {2006}} @article{ho:imai:08, Author = {Ho, Daniel E. and Imai, Kosuke}, Journal = {Public Opinion Quarterly}, Month = {Summer}, Number = {2}, Pages = {216--240}, Title = {Estimating Causal Effects of Ballot Order from a Randomized Natural Experiment: {C}alifornia Alphabet Lottery, 1978--2002}, Volume = {72}, Year = {2008}} @article{ho:imai:king:stua:07, Author = {Ho, Daniel E. and Imai, Kosuke and King, Gary and Stuart, Elizabeth A.}, Journal = {Political Analysis}, Month = {Summer}, Number = {3}, Pages = {199--236}, Title = {Matching as Nonparametric Preprocessing for Reducing Model Dependence in Parametric Causal Inference}, Volume = {15}, Year = {2007}} @article{ho:imai:king:stua:11, Author = {Ho, Daniel E. and Imai, Kosuke and King, Gary and Stuart, Elizabeth A.}, Journal = {Journal of Statistical Software}, Month = {June}, Note = {Software available at the {C}omprehensive {R} {A}rchive {N}etwork, \url{http://CRAN.R-project.org/package=MatchIt}}, Number = {8}, Pages = {1--28}, Title = {Match{I}t: Nonparametric Preprocessing for Parametric Causal Inference}, Volume = {42}, Year = {2011}} @article{hori:imai:tani:07, Author = {Horiuchi, Yusaku and Imai, Kosuke and Taniguchi, Naoko}, Journal = {American Journal of Political Science}, Month = {July}, Number = {3}, Pages = {669--687}, Title = {Designing and Analyzing Randomized Experiments: Application to a {J}apanese Election Survey Experiment}, Volume = {51}, Year = {2007}} @phdthesis{imai:03, Author = {Imai, Kosuke}, School = {Department of Government, Harvard University}, Title = {Essays on Political Methodology}, Year = {2003}} @article{imai:05, Author = {Imai, Kosuke}, Journal = {American Political Science Review}, Month = {May}, Number = {2}, Pages = {283--300}, Title = {Do Get-Out-The-Vote Calls Reduce Turnout?: The Importance of Statistical Methods for Field Experiments}, Volume = {99}, Year = {2005}} @article{imai:08, Author = {Imai, Kosuke}, Journal = {Statistics \& Probability Letters}, Month = {February}, Number = {2}, Pages = {144--149}, Title = {Sharp Bounds on the Causal Effects in Randomized Experiments with ``Truncation-by-Death''}, Volume = {78}, Year = {2008}} @article{imai:08a, Author = {Imai, Kosuke}, Journal = {Statistics in Medicine}, Month = {October}, Number = {24}, Pages = {4857--4873}, Title = {Variance Identification and Efficiency Analysis in Randomized Experiments under the Matched-Pair Design}, Volume = {27}, Year = {2008}} @misc{imai:08b, Author = {Imai, Kosuke}, Howpublished = {available at the Comprehensive {R} Archive Network (CRAN)}, Note = {\url{http://CRAN.R-project.org/package=experiment}}, Title = {experiment: R package for designing and analyzing randomized experiments}, Year = {2008}} @article{imai:09, Author = {Imai, Kosuke}, Journal = {Journal of the Royal Statistical Society, {Series C} (Applied Statistics)}, Month = {February}, Number = {1}, Pages = {83--104}, Title = {Statistical Analysis of Randomized Experiments with Nonignorable Missing Binary Outcomes: An Application to a Voting Experiment}, Volume = {58}, Year = {2009}} @article{imai:11, Author = {Imai, Kosuke}, Journal = {Journal of the American Statistical Association}, Month = {June}, Number = {494}, Pages = {407--416}, Title = {Multivariate Regression Analysis for the Item Count Technique}, Volume = {106}, Year = {2011}} @Article{imai:12, author = {Imai, Kosuke}, title = {Comments: Improving Weighting Methods for Causal Mediation Analysis}, journal = {Journal of Research on Educational Effectiveness}, year = {2012}, OPTkey = {}, volume = {5}, number = {3}, pages = {293--295}, OPTmonth = {}, OPTnote = {}, OPTannote = {} } @Book{imai:16, author = {Imai, Kosuke}, ALTeditor = {}, title = {A First Course in Quantitative Social Science}, publisher = {A book manuscript}, year = {2016}, OPTkey = {}, OPTvolume = {}, OPTnumber = {}, OPTseries = {}, OPTaddress = {}, OPTedition = {}, OPTmonth = {}, OPTnote = {}, OPTannote = {} } @inbook{imai:etal:10, Address = {New York}, Author = {Imai, Kosuke and Keele, Luke and Tingley, Dustin and Yamamoto, Teppei}, Chapter = {Causal Mediation Analysis Using {R}}, Pages = {129--154}, Publisher = {Springer}, Series = {Lecture Notes in Statistics}, Title = {Advances in Social Science Research Using {R} (ed. H. D. Vinod)}, Year = {2010}} @article{imai:etal:11, Author = {Imai, Kosuke and Keele, Luke and Tingley, Dustin and Yamamoto, Teppei}, Journal = {American Political Science Review}, Month = {November}, Number = {4}, Pages = {765--789}, Title = {Unpacking the Black Box of Causality: Learning about Causal Mechanisms from Experimental and Observational Studies}, Volume = {105}, Year = {2011}} @misc{imai:etal:11a, Author = {Imai, Kosuke and Keele, Luke and Tingley, Dustin and Yamamoto, Teppei}, Howpublished = {hdl:1902.1/16467}, Note = {The Dataverse Network}, Title = {Replication data for: Unpacking the Black Box of Causality: Learning about Causal Mechanisms from Experimental and Observational Studies}, Year = {2011}} @Article{imai:etal:14, author = {Imai, Kosuke and Keele, Luke and Tingley, Dustin and Yamamoto, Teppei}, title = {Comment on {Pearl}: Practical Implications of Theoretical Results for Causal Mediation Analysis}, journal = {Psychological Methods}, year = {2014}, volume = {19}, number = {4}, pages = {482--487}, month = {December}, } @Article{imai:jo:stua:11, author = {Imai, Kosuke and Jo, Booil and Stuart, Elizabeth A.}, title = {Commentary: Using Potential Outcomes to Understand Causal Mediation Analysis}, journal = {Multivariate Behavioral Research}, year = {2011}, OPTkey = {}, volume = {46}, number = {5}, pages = {842--854}, OPTmonth = {}, OPTnote = {}, OPTannote = {} } @article{imai:keel:ting:10, Author = {Imai, Kosuke and Keele, Luke and Tingley, Dustin}, Journal = {Psychological Methods}, Month = {December}, Number = {4}, Pages = {309--334}, Title = {A General Approach to Causal Mediation Analysis}, Volume = {15}, Year = {2010}} @misc{imai:keel:ting:10a, Author = {Imai, Kosuke and Keele, Luke and Tingley, Dustin}, Howpublished = {hdl:1902.1/14801}, Note = {The Dataverse Network}, Title = {Replication data for: A General Approach to Causal Mediation Analysis}, Year = {2010}} @article{imai:keel:yama:10, Author = {Imai, Kosuke and Keele, Luke and Yamamoto, Teppei}, Journal = {Statistical Science}, Month = {February}, Number = {1}, Pages = {51--71}, Title = {Identification, Inference, and Sensitivity Analysis for Causal Mediation Effects}, Volume = {25}, Year = {2010}} @misc{imai:keel:yama:10a, Author = {Imai, Kosuke and Keele, Luke and Yamamoto, Teppei}, Howpublished = {hdl:1902.1/14412}, Note = {The Dataverse Network}, Title = {Replication data for: Identification, Inference, and Sensitivity Analysis for Causal Mediation Effects}, Year = {2010}} @unpublished{imai:kim:12, Author = {Imai, Kosuke and Kim, In Song}, Note = {Working paper available at \url{http://imai.princeton.edu/research/FEmatch.html}}, Title = {On the Use of Linear Fixed Effects Regression Estimators for Causal Inference}, Year = {2012}} @article{imai:king:04, Author = {Imai, Kosuke and King, Gary}, Journal = {Perspectives on Politics}, Month = {September}, Number = {3}, Pages = {537--549}, Title = {Did Illegal Overseas Absentee Ballots Decide the 2000 {U.S.} Presidential Election?}, Volume = {2}, Year = {2004}} @article{imai:king:lau:08, Author = {Imai, Kosuke and King, Gary and Lau, Olivia}, Journal = {Journal of Computational and Graphical Statistics}, Month = {December}, Number = {4}, Pages = {892--913}, Title = {Toward A Common Framework of Statistical Analysis and Development}, Volume = {17}, Year = {2008}} @misc{imai:king:lau:08S, Author = {Imai, Kosuke and King, Gary and Lau, Olivia}, Howpublished = {available at the {Comprehensive R Archive Network (CRAN)}}, Note = {\url{http://CRAN.R-project.org/package=Zelig}}, Title = {Zelig: Everyone's Statistical Software}, Year = {2008}} @article{imai:king:nall:09, Author = {Imai, Kosuke and King, Gary and Nall, Clayton}, Journal = {Statistical Science}, Month = {February}, Number = {1}, Pages = {29--53}, Title = {The Essential Role of Pair Matching in Cluster-Randomized Experiments, with Application to the {M}exican Universal Health Insurance Evaluation (with discussions)}, Volume = {24}, Year = {2009}} @article{imai:king:nall:09a, Author = {Imai, Kosuke and King, Gary and Nall, Clayton}, Journal = {Statistical Science}, Month = {February}, Number = {1}, Pages = {65--72}, Title = {Rejoinder: Matched Pairs and the Future of Cluster-Randomized Experiments}, Volume = {24}, Year = {2009}} @article{imai:king:stua:08, Author = {Imai, Kosuke and King, Gary and Stuart, Elizabeth A.}, Journal = {Journal of the Royal Statistical Society, {Series A} (Statistics in Society)}, Month = {April}, Number = {2}, Pages = {481--502}, Title = {Misunderstandings among Experimentalists and Observationalists about Causal Inference}, Volume = {171}, Year = {2008}} @misc{imai:lo:olms:15, Author = {Imai, Kosuke and Lo, James and Olmsted, Jonathan}, Howpublished = {available at the {Comprehensive R Archive Network (CRAN)}}, Note = {\url{http://CRAN.R-project.org/package=emIRT}}, Title = {emIRT: EM Algorithms for Estimating Item Response Theory Models}, Year = {2015}} @article{imai:lu:stra:08, Author = {Imai, Kosuke and Lu, Ying and Strauss, Aaron}, Journal = {Political Analysis}, Month = {Winter}, Number = {1}, Pages = {41--69}, Title = {Bayesian and Likelihood Inference for $2 \times 2$ Ecological Tables: An Incomplete Data Approach}, Volume = {16}, Year = {2008}} @article{imai:lu:stra:11, Author = {Imai, Kosuke and Lu, Ying and Strauss, Aaron}, Journal = {Journal of Statistical Software}, Month = {June}, Note = {Software available at {T}he {C}omprehensive {R} {A}rchive {N}etwork, \url{http://CRAN.R-project.org/package=eco}}, Number = {5}, Pages = {1--23}, Title = {{eco}: {R} package for Ecological Inference in $2 \times 2$ Tables}, Volume = {42}, Year = {2011}} @Misc{imai:park:gree:14a, OPTkey = {}, author = {Imai, Kosuke and Park, Bethany and Greene, Kenneth F.}, title = {Replication data for: Using the Predicted Responses from List Experiments as Explanatory Variables in Regression Models}, howpublished = {\url{http://dx.doi.org/10.7910/DVN/27083}}, OPTmonth = {}, year = {2014}, note = {The Dataverse Network}, OPTannote = {} } @article{imai:park:gree:15, author = {Imai, Kosuke and Park, Bethany and Greene, Kenneth F.}, title = {Using the Predicted Responses from List Experiments as Explanatory Variables in Regression Models}, journal = {Political Analysis}, year = {2015}, volume = {23}, number = {2}, pages = {180--196}, month = {Spring}, OPTannote = {} } @article{imai:ratk:13, Author = {Imai, Kosuke and Ratkovic, Marc}, Journal = {Annals of Applied Statistics}, Title = {Estimating Treatment Effect Heterogeneity in Randomized Program Evaluation}, Year = {2013}, Volume = {7}, Number = {1}, Month = {March}, Pages = {443--470}} @article{imai:ratk:14, Author = {Imai, Kosuke and Ratkovic, Marc}, Title = {Covariate Balancing Propensity Score}, Year = {2014}, Journal = {Journal of the Royal Statistical Society, {Series B} (Statistical Methodology)}, Volume = {76}, Number = {1}, Month = {January}, Pages = {243--263} } @article{imai:ratk:15, author = {Imai, Kosuke and Ratkovic, Marc}, title = {Robust Estimation of Inverse Probability Weights for Marginal Structural Models}, year = {2015}, journal = {Journal of the American Statistical Association}, pages = {1013--1023}, month = {September}, Volume = {110}, Number = {511} } @article{imai:sone:07, Author = {Imai, Kosuke and Soneji, Samir}, Journal = {Journal of the American Statistical Association}, Month = {December}, Number = {480}, Pages = {1199--1211}, Title = {On the Estimation of Disability-Free Life Expectancy: {S}ullivan's Method and Its Extension}, Volume = {102}, Year = {2007}} @article{imai:stra:11, Author = {Imai, Kosuke and Strauss, Aaron}, Journal = {Political Analysis}, Month = {Winter}, Number = {1}, Pages = {1--19}, Title = {Estimation of Heterogeneous Treatment Effects from Randomized Experiments, with Application to the Optimal Planning of the Get-out-the-vote Campaign}, Volume = {19}, Year = {2011}} @misc{imai:ting:11a, Author = {Imai, Kosuke and Tingley, Dustin}, Howpublished = {hdl:1902.1/16378}, Note = {The Dataverse Network}, Title = {Replication data for: A Statistical Method for Empirical Testing of Competing Theories}, Year = {2011}} @article{imai:ting:12, Author = {Imai, Kosuke and Tingley, Dustin}, Journal = {American Journal of Political Science}, Month = {January}, Number = {1}, Pages = {218--236}, Title = {A Statistical Method for Empirical Testing of Competing Theories}, Volume = {56}, Year = {2012}} @misc{imai:ting:yama:11a, Author = {Imai, Kosuke and Tingley, Dustin and Yamamoto, Teppei}, Howpublished = {hdl:1902.1/16416}, Note = {The Dataverse Network}, Title = {Replication data for: Experimental Designs for Identifying Causal Mechanisms}, Year = {2011}} @article{imai:ting:yama:13, Author = {Imai, Kosuke and Tingley, Dustin and Yamamoto, Teppei}, Journal = {Journal of the Royal Statistical Society, {Series A} (Statistics in Society)}, Number = {1}, Pages = {5--51}, Title = {Experimental Designs for Identifying Causal Mechanisms (with discussions)}, Volume = {176}, Month = {January}, Year = {2013}} @article{imai:vand:04, Author = {Imai, Kosuke and van Dyk, David A.}, Journal = {Journal of the American Statistical Association}, Month = {September}, Number = {467}, Pages = {854--866}, Title = {Causal Inference with General Treatment Regimes: Generalizing the Propensity Score}, Volume = {99}, Year = {2004}} @article{imai:vand:05, Author = {Imai, Kosuke and van Dyk, David A.}, Journal = {Journal of Econometrics}, Month = {February}, Number = {2}, Pages = {311--334}, Title = {A {B}ayesian Analysis of the Multinomial Probit Model Using Marginal Data Augmentation}, Volume = {124}, Year = {2005}} @article{imai:vand:05a, Author = {Imai, Kosuke and van Dyk, David A.}, Journal = {Journal of Statistical Software}, Month = {May}, Note = {Software available at the {C}omprehensive {R} {A}rchive {N}etwork, \url{http://cran.r-project.org/package=MNP}}, Number = {3}, Pages = {1--32}, Title = {{MNP}: {R} Package for Fitting the Multinomial Probit Model}, Volume = {14}, Year = {2005}} @article{imai:yama:10, Author = {Imai, Kosuke and Yamamoto, Teppei}, Journal = {American Journal of Political Science}, Month = {April}, Number = {2}, Pages = {543--560}, Title = {Causal Inference with Differential Measurement Error: Nonparametric Identification and Sensitivity Analysis}, Volume = {54}, Year = {2010}} @misc{imai:yama:12a, Author = {Imai, Kosuke and Yamamoto, Teppei}, Howpublished = {hdl:1902.1/19036}, Note = {The Dataverse Network}, Title = {Replication data for: Identification and Sensitivity Analysis for Multiple Causal Mechanisms: Revisiting Evidence from Framing Experiments}, Year = {2012}} @article{imai:yama:13, Author = {Imai, Kosuke and Yamamoto, Teppei}, Journal = {Political Analysis}, Title = {Identification and Sensitivity Analysis for Multiple Causal Mechanisms: Revisiting Evidence from Framing Experiments}, Year = {2013}, Volume = {21}, Number = {2}, Month = {Spring}, Pages = {141--171}} @misc{kim:imai:12, Author = {Kim, In Song and Imai, Kosuke}, Howpublished = {available at the {Comprehensive R Archive Network (CRAN)}}, Note = {\url{http://CRAN.R-project.org/package=wfe}}, Title = {wfe: Weighted Linear Fixed Effects Regression Models for Causal Inference}, Year = {2012}} @article{king:etal:09, Author = {King, Gary and Gakidou, Emmanuela and Imai, Kosuke and Lakin, Jason and Moore, Ryan T. and Ravishankar, Nirmala and Vargas, Manett and T{\'e}llez-Rojo, Martha Mar{\'\i}a and Hern{\'a}ndez {\'A}vila, Juan Eugenio and Hern{\'a}ndez {\'A}vila, Mauricio and Hern{\'a}ndez Llamas, H{\'e}ctor}, Journal = {The Lancet}, Month = {April}, Number = {9673}, Pages = {1447--1454}, Title = {Public Policy for the Poor? A Randomised Assessment of the {M}exican Universal Health Insurance Programme}, Volume = {373}, Year = {2009}} @article{lyal:blai:imai:13, Author = {Lyall, Jason and Blair, Graeme and Imai, Kosuke}, Journal = {American Political Science Review}, Title = {Explaining Support for Combatants during Wartime: A Survey Experiment in {A}fghanistan}, Year = {2013}, Volume = {107}, Number = {4}, Month = {November}, Pages = {679--705}} @article{lyal:imai:shir:15, author = {Lyall, Jason and Imai, Kosuke and Shiraito, Yuki}, title = {Coethnic Bias and Wartime Informing}, journal = {Journal of Politics}, volume = {77}, month = {July}, year = {2015}, number = {3}, pages = {833--848} } @article{rose:imai:shap:15, author = {Rosenfeld, Bryn and Imai, Kosuke and Shapiro, Jacob}, title = {An Empirical Validation Study of Popular Survey Methodologies for Sensitive Questions}, journal = {American Journal of Political Science}, OPTkey = {}, OPTmonth = {}, year = {2015}, pages = {Forthcoming} } @misc{rose:imai:shap:15a, Author = {Rosenfeld, Bryn and Imai, Kosuke and Shapiro, Jacob}, Howpublished = {\url{http://dx.doi.org/10.7910/DVN/29911}}, Note = {The Dataverse Network}, Title = {Replication data for: An Empirical Validation Study of Popular Survey Methodologies for Sensitive Questions}, Year = {2015}} @misc{shir:imai:12, Author = {Shiraito, Yuki and Imai, Kosuke}, Howpublished = {available at the {Comprehensive R Archive Network (CRAN)}}, Note = {\url{http://CRAN.R-project.org/package=endorse}}, Title = {endorse: R Package for Analyzing Endorsement Experiments}, Year = {2012}} @article{ting:etal:14, Author = {Tingley, Dustin and Yamamoto, Teppei and Hirose, Kentaro and Keele, Luke and Imai, Kosuke}, Journal = {Journal of Statistical Software}, Month = {August}, Note = {Software available at {T}he {C}omprehensive {R} {A}rchive {N}etwork, \url{http://CRAN.R-project.org/package=mediation}}, Number = {5}, Pages = {1--38}, Title = {mediation: {R} Package for Causal Mediation Analysis}, Volume = {59}, Year = {2014}} MNP/MD50000644000176200001440000000430013162707044011215 0ustar liggesusers3ee63bcf5243c6396e0592e3d124939e *ChangeLog dcf686b1e76d12012e8591bae3c4b324 *DESCRIPTION f8f774d9d0b6b798b2158bd77c7b126a *NAMESPACE 0fb08f470802dd710f15dfc2cc3e91ab *R/coef.mnp.R 28797d8b4e633d8bc6a64244c59b2ee7 *R/cov.mnp.R b3458f250f294938c1a33c3c99b03600 *R/detergent.R 1c6fd60d1858ed12d7a0844a81cb4c8e *R/japan.R a397bf827fe34252e292eb0f5125cda3 *R/mnp.R 5e40e20eb515081275508f808be52c3e *R/onAttach.R 3b525cf682b5da4474f52733c44cd891 *R/predict.mnp.R e822d2e847f5daf2922a019fa2a4c491 *R/print.mnp.R 7bb6ce567e9bae8a05147424af1ce021 *R/print.summary.mnp.R 3aad35e5779bc160ee891c3e279109be *R/summary.mnp.R cfdc76deb4c1e71bdb9422bf9f2cecd9 *R/xmatrix.mnp.R f594a4e2db80f456c9dee49d5f9efa6c *R/ymatrix.mnp.R ac3b415e9cbd5c0af266d7de0711d9d6 *build/vignette.rds c342e6c52a792ec58aac9efa0be703ae *data/detergent.txt.gz bb1181c0a078518436a2a7c9ee5594a9 *data/japan.txt.gz 9ca37d86e3f3aaea7e63f7e9df31b561 *inst/doc/MNP.Rnw 80fb33f1bb588e10968cc8e5b1f6ebc7 *inst/doc/MNP.pdf 037ee471d43097393839f831aef8f4d7 *man/coef.mnp.Rd 34ff270e9f6c9d48dfa0b39b0a0dbfa7 *man/cov.mnp.Rd 2545b41aa3684b8422d3409eb4564ab0 *man/detergent.Rd 066b8b5b232441ce0438b65ab8585497 *man/japan.Rd 4c86e8668e9128a5c47374864134c3f4 *man/mnp.Rd 36a2126cac4bec3da085cbb702d74cc5 *man/predict.mnp.Rd 6c0ab6e2649092561bf1cd71b7b57630 *man/print.summary.mnp.Rd 59f5eafed02261018ca55766955ab259 *man/summary.mnp.Rd d9088c8b8546c15516c135dbf430f572 *src/MNP.c f009e46fcf131d28ea4ead122961b7bd *src/Makevars 9ead8c15c72782f0d75f78322643ca34 *src/init.c 595360fa13934613523dee91b89a9358 *src/rand.c e79ea40e8f33e593ba24b0fff4cdbba7 *src/rand.h c4520f8ee86dbf71c0f6225b34764975 *src/subroutines.c d7c8fe15ffb32a769555c331cfb6021b *src/subroutines.h 9e39d6679ad4e0a3dabbe9f9056268c1 *src/vector.c 87ab6c76b848c81968161e883914b2ce *src/vector.h d98a81a351b4b9d7e5719183c29bf86a *tests/testthat.R 47e5fb1ebfc6f1024106151db9b9de83 *tests/testthat/test-all.R 9ca37d86e3f3aaea7e63f7e9df31b561 *vignettes/MNP.Rnw c22fa2eb5a78e4ca63fcf217dbc20612 *vignettes/coda-small.pdf bb2e14064fb13a0bbbac088622fddeed *vignettes/imai.bib 9e8e8a987bf60c74055949b108f3808c *vignettes/my.bib 283b8deb4463167133f19cb178da5d67 *vignettes/natbib.bst f953c93983eafa93f4ccc1e8d26a7c36 *vignettes/rhat.pdf MNP/build/0000755000176200001440000000000013162677535012022 5ustar liggesusersMNP/build/vignette.rds0000644000176200001440000000026713162677535014366 0ustar liggesusersb```b`fad`b2 1# 'f +Gf cSY& Ċ  Depends: R (>= 2.1), MASS, utils Suggests: testthat Description: Fits the Bayesian multinomial probit model via Markov chain Monte Carlo. The multinomial probit model is often used to analyze the discrete choices made by individuals recorded in survey data. Examples where the multinomial probit model may be useful include the analysis of product choice by consumers in market research and the analysis of candidate or party choice by voters in electoral studies. The MNP package can also fit the model with different choice sets for each individual, and complete or partial individual choice orderings of the available alternatives from the choice set. The estimation is based on the efficient marginal data augmentation algorithm that is developed by Imai and van Dyk (2005). ``A Bayesian Analysis of the Multinomial Probit Model Using the Data Augmentation,'' Journal of Econometrics, Vol. 124, No. 2 (February), pp. 311-334. Detailed examples are given in Imai and van Dyk (2005). ``MNP: R Package for Fitting the Multinomial Probit Model.'' Journal of Statistical Software, Vol. 14, No. 3 (May), pp. 1-32. . LazyLoad: yes LazyData: yes License: GPL (>= 2) URL: http://imai.princeton.edu/software/MNP.html BugReports: https://github.com/kosukeimai/MNP/issues NeedsCompilation: yes RoxygenNote: 6.0.1 Packaged: 2017-09-27 10:37:17 UTC; kimai Author: Kosuke Imai [aut, cre], David van Dyk [aut], Hubert Jin [ctb] Repository: CRAN Date/Publication: 2017-09-27 11:40:20 UTC MNP/ChangeLog0000644000176200001440000000734613162676351012502 0ustar liggesusersVersion Date Description 3.1-0 09.27.17 testthat added 3.0-2 06.28.17 better registration of C functions 3.0-1 05.05.17 Documentation generation via roxygen2, registering C functions 2.6-4 06.09.13 a label error for latent variables (thanks to Sky Xue) 2.6-3 12.06.11 a minor change to eliminate warnings for R2.14-0 2.6-2 10.27.10 a bug fixed in predict() (thanks to Lane Burgette) 2.6-1 09.23.09 trace restriction used as the default (thanks to Lane Burgette) 2.5-6 03.24.08 an important bug fix for the treatment of base categories (thanks to Ryan Black and Taeyoung Park) 2.5-5 08.01.07 returning X matrix from predict() 2.5-4 05.24.07 another bug fix in predict() (thanks to Andrew Owen) 2.5-3 12.04.06 a minor change 2.5-2 11.30.06 a bug fix in predict() 2.5-1 11.21.06 rewrite predict() in C for speedup. Changed moredraw to n.draws 2.4-2 10.17.06 bug fix in {\tt moredraw} option (thanks to Ken Benoit) 2.4-1 10.05.06 added an option, moredraw, to predict.mnp() (thanks to Ken Benoit) 2.3-9 09.21.06 minor changes to be consistent with R version 2.4-0 2.3-8 04.26.06 removing C warnings for Windows platform 2.3-7 04.24.06 some very minor fixes to the C code 2.3-6 01.11.06 print the number of estimated parameters in summary() (thanks to S.C. Wang) 2.3-5 12.27.05 -1 is allowed in formula (thanks to Daeyoung Koh) 2.3-4 09.06.05 added inverse CDF method option for truncated normal sampling 2.3-3 06.23.05 made Gibbs sampler slightly more efficient 2.3-2 06.02.05 minor changes to NAMESPACE and DESCRIPTOIN files 2.3-1 05.27.05 added coef.mnp() and cov.mnp() (thanks to Natasha Zharinova) 2.2-3 05.12.05 minor changes to the documentation; version published in Journal of Statistical Software 2.2-2 05.09.05 minor changes to the documentation 2.2-1 05.01.05 stable release for R 2.1.0; The observations with missing values in X will be deleted in mnp() and predict() (thanks to Natasha Zharinova) 2.1-2 03.22.05 added an option, newdraw, for predict() method 2.1-1 02.25.05 improved predict() method; documentation enhanced and edited 2.0-1 02.12.05 added predict() method (thanks to Xavier Gerard and Saleem Shaik) 1.4-1 12.16.04 improved error handling (thanks to Kjetil Halvorsen) 1.3-2 11.17.04 stable release for R 2.0.1; minor updates of the documentation 1.3-1 10.09.04 stable release for R 2.0.0; updating vector.c 1.2-1 09.26.04 optionally stores the latent variable (thanks to Colin McCulloch) 1.1-2 09.14.04 minor fix in mnp() (thanks to Ken Shultz) 1.1-1 08.28.04 major and minor changes: namespace implemented 1.0-4 07.14.04 users can interrupt the C process within R (thanks to Kevin Quinn) 1.0-3 06.30.04 bug fix in xmatrix.mnp() (thanks to Andrew Martin) 1.0-2 06.29.04 removed p.alpha0 parameter 1.0-1 06.23.04 official release 0.9-13 05.28.04 bug fix in ymatrix.mnp() 0.9-12 05.23.04 updating the documentation and help files 0.9-11 05.08.04 bug fix in cXnames() (thanks to Liming Wang) 0.9-10 05.03.04 first stable version; bug fix in labeling 0.9-9 05.02.04 bug fix in sampling of W; added summary.mnp() and print.summary.mnp() 0.9-8 04.29.04 improving sampling of W, replace printf() with Rprintf() 0.9-7 04.27.04 missing data allowed for all models, varying choice sets allowed. 0.9-6 04.26.04 missing data allowed in the response variable for standard MNP; a major bug fixed for MoP (thanks to Shigeo Hirano) 0.9-5 04.25.04 improper prior handled by algorithm 1 0.9-4 04.21.04 bug fix in MoP, R 1.9.0 compatible, changes in mprobit.R 0.9-3 04.10.04 rWish() modified with an improved algorithm 0.9-2 03.22.04 first public beta version 0.9-1 03.20.04 first beta version MNP/man/0000755000176200001440000000000013103175765011467 5ustar liggesusersMNP/man/japan.Rd0000644000176200001440000000234113103175765013047 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/japan.R \docType{data} \name{japan} \alias{japan} \title{Voters' Preferences of Political Parties in Japan (1995)} \format{A data frame containing the following 7 variables for 418 observations. \tabular{llll}{ LDP \tab numeric \tab preference for Liberal Democratic Party \tab 0 - 100 \cr NFP \tab numeric \tab preference for New Frontier Party \tab 0 - 100 \cr SKG \tab numeric \tab preference for Sakigake \tab 0 - 100 \cr JCP \tab numeric \tab preference for Japanese Communist Party \tab 0 - 100 \cr gender \tab factor \tab gender of each voter \tab \code{male} or \code{female} \cr education \tab numeric \tab levels of education for each voter \tab \cr age \tab numeric \tab age of each voter \tab }} \description{ This dataset gives voters' preferences of political parties in Japan on the 0 (least preferred) - 100 (most preferred) scale. It is based on the 1995 survey data of 418 individual voters. The data also include the sex, education level, and age of the voters. The survey allowed voters to choose among four parties: Liberal Democratic Party (LDP), New Frontier Party (NFP), Sakigake (SKG), and Japanese Communist Party (JCP). } \keyword{datasets} MNP/man/summary.mnp.Rd0000644000176200001440000000256013103226434014235 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/summary.mnp.R \name{summary.mnp} \alias{summary.mnp} \title{Summarizing the results for the Multinomial Probit Models} \usage{ \method{summary}{mnp}(object, CI = c(2.5, 97.5), ...) } \arguments{ \item{object}{An output object from \code{mnp}.} \item{CI}{A 2 dimensional vector of lower and upper bounds for the credible intervals used to summarize the results. The default is the equal tail 95 percent credible interval.} \item{...}{further arguments passed to or from other methods.} } \value{ \code{summary.mnp} yields an object of class \code{summary.mnp} containing the following elements: \item{call}{The call from \code{mnp}.} \item{n.alt}{The total number of alternatives.} \item{base}{The base category used for fitting.} \item{n.obs}{The number of observations.} \item{n.param}{The number of estimated parameters.} \item{n.draws}{The number of Gibbs draws used for the summary.} \item{coef.table}{The summary of the posterior distribution of the coefficients. } \item{cov.table}{The summary of the posterior distribution of the covariance matrix.} This object can be printed by \code{print.summary.mnp} } \description{ \code{summary} method for class \code{mnp}. } \seealso{ \code{mnp} } \author{ Kosuke Imai, Department of Politics, Princeton University \email{kimai@Princeton.Edu} } \keyword{methods} MNP/man/detergent.Rd0000644000176200001440000000201213103175765013732 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/detergent.R \docType{data} \name{detergent} \alias{detergent} \title{Detergent Brand Choice} \format{A data frame containing the following 7 variables and 2657 observations. \tabular{lll}{ choice \tab factor \tab a brand chosen by each household\cr TidePrice \tab numeric \tab log price of Tide\cr WiskPrice \tab numeric \tab log price of Wisk\cr EraPlusPrice \tab numeric \tab log price of EraPlus\cr SurfPrice \tab numeric \tab log price of Surf\cr SoloPrice \tab numeric \tab log price of Solo\cr AllPrice \tab numeric \tab log price of All\cr }} \description{ This dataset gives the laundry detergent brand choice by households and the price of each brand. } \references{ Chintagunta, P. K. and Prasad, A. R. (1998) \dQuote{An Empirical Investigation of the `Dynamic McFadden' Model of Purchase Timing and Brand Choice: Implications for Market Structure}. \emph{Journal of Business and Economic Statistics} vol. 16 no. 1 pp.2-12. } \keyword{datasets} MNP/man/coef.mnp.Rd0000644000176200001440000000237513103226434013460 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/coef.mnp.R \name{coef.mnp} \alias{coef.mnp} \alias{coefficients.mnp} \title{Extract Multinomial Probit Model Coefficients} \usage{ \method{coef}{mnp}(object, subset = NULL, ...) } \arguments{ \item{object}{An output object from \code{mnp}.} \item{subset}{A scalar or a numerical vector specifying the row number(s) of \code{param} in the output object from \code{mnp}. If specified, the posterior draws of coefficients for those rows are extracted. The default is \code{NULL} where all the posterior draws are extracted.} \item{...}{further arguments passed to or from other methods.} } \value{ \code{coef.mnp} returns a matrix (when a numerical vector or \code{NULL} is specified for \code{subset} argument) or a vector (when a scalar is specified for \code{subset} arugment) of multinomila probit model coefficients. } \description{ \code{coef.mnp} is a function which extracts multinomial probit model coefficients from ojbects returned by \code{mnp}. \code{coefficients.mnp} is an alias for it. \code{coef} method for class \code{mnp}. } \seealso{ \code{mnp}, \code{cov.mnp}; } \author{ Kosuke Imai, Department of Politics, Princeton University \email{kimai@Princeton.Edu} } \keyword{methods} MNP/man/predict.mnp.Rd0000644000176200001440000001133213103226434014167 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/predict.mnp.R \name{predict.mnp} \alias{predict.mnp} \title{Posterior Prediction under the Bayesian Multinomial Probit Models} \usage{ \method{predict}{mnp}(object, newdata = NULL, newdraw = NULL, n.draws = 1, type = c("prob", "choice", "order"), verbose = FALSE, ...) } \arguments{ \item{object}{An output object from \code{mnp}.} \item{newdata}{An optional data frame containing the values of the predictor variables. Predictions for multiple values of the predictor variables can be made simultaneously if \code{newdata} has multiple rows. The default is the original data frame used for fitting the model.} \item{newdraw}{An optional matrix of MCMC draws to be used for posterior predictions. The default is the original MCMC draws stored in \code{object}.} \item{n.draws}{The number of additional Monte Carlo draws given each MCMC draw of coefficients and covariance matrix. The specified number of latent variables will be sampled from the multivariate normal distribution, and the quantities of interest will be calculated by averaging over these draws. This will be particularly useful calculating the uncertainty of predicted probabilities. The default is \code{1}.} \item{type}{The type of posterior predictions required. There are four options: \code{type = "prob"} returns the predictive probabilities of being the most preferred choice among the choice set. \code{type = "choice"} returns the Monte Carlo sample of the most preferred choice, and \code{type = "order"} returns the Monte Carlo sample of the ordered preferences,} \item{verbose}{logical. If \code{TRUE}, helpful messages along with a progress report on the Monte Carlo sampling from the posterior predictive distributions are printed on the screen. The default is \code{FALSE}.} \item{...}{additional arguments passed to other methods.} } \value{ \code{predict.mnp} yields a list of class \code{predict.mnp} containing at least one of the following elements: \item{o}{A three dimensional array of the Monte Carlo sample from the posterior predictive distribution of the ordered preferences. The first dimension corresponds to the rows of \code{newdata} (or the original data set if \code{newdata} is left unspecified), the second dimension corresponds to the alternatives in the choice set, and the third dimension indexes the Monte Carlo sample. If \code{n.draws} is greater than 1, then each entry will be an average over these additional draws. } \item{p}{A two or three dimensional array of the posterior predictive probabilities for each alternative in the choice set being most preferred. The first demension corresponds to the rows of \code{newdata} (or the original data set if \code{newdata} is left unspecified), the second dimension corresponds to the alternatives in the choice set, and the third dimension (if it exists) indexes the Monte Carlo sample. If \code{n.draws} is greater than 1, then the third diemsion exists and indexes the Monte Carlo sample. } \item{y}{A matrix of the Monte Carlo sample from the posterior predictive distribution of the most preferred choice. The first dimension correspond to the rows of \code{newdata} (or the original data set if \code{newdata} is left unspecified), and the second dimension indexes the Monte Carlo sample. \code{n.draws} will be set to 1 when computing this quantity of interest. } \item{x}{A matrix of covariates used for prediction } } \description{ Obtains posterior predictions under a fitted (Bayesian) multinomial probit model. \code{predict} method for class \code{mnp}. } \details{ The posterior predictive values are computed using the Monte Carlo sample stored in the \code{mnp} output (or other sample if \code{newdraw} is specified). Given each Monte Carlo sample of the parameters and each vector of predictor variables, we sample the vector-valued latent variable from the appropriate multivariate Normal distribution. Then, using the sampled predictive values of the latent variable, we construct the most preferred choice as well as the ordered preferences. Averaging over the Monte Carlo sample of the preferred choice, we obtain the predictive probabilities of each choice being most preferred given the values of the predictor variables. Since the predictive values are computed via Monte Carlo simulations, each run may produce somewhat different values. The computation may be slow if predictions with many values of the predictor variables are required and/or if a large Monte Carlo sample of the model parameters is used. In either case, setting \code{verbose = TRUE} may be helpful in monitoring the progress of the code. } \seealso{ \code{mnp} } \author{ Kosuke Imai, Department of Politics, Princeton University \email{kimai@Princeton.Edu} } \keyword{methods} MNP/man/mnp.Rd0000644000176200001440000002415213103236607012545 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/mnp.R \name{mnp} \alias{mnp} \alias{MNP} \title{Fitting the Multinomial Probit Model via Markov chain Monte Carlo} \usage{ mnp(formula, data = parent.frame(), choiceX = NULL, cXnames = NULL, base = NULL, latent = FALSE, invcdf = FALSE, trace = TRUE, n.draws = 5000, p.var = "Inf", p.df = n.dim + 1, p.scale = 1, coef.start = 0, cov.start = 1, burnin = 0, thin = 0, verbose = FALSE) } \arguments{ \item{formula}{A symbolic description of the model to be fit specifying the response variable and covariates. The formula should not include the choice-specific covariates. Details and specific examples are given below.} \item{data}{An optional data frame in which to interpret the variables in \code{formula} and \code{choiceX}. The default is the environment in which \code{mnp} is called.} \item{choiceX}{An optional list containing a matrix of choice-specific covariates for each category. Details and examples are provided below.} \item{cXnames}{A vector of the names for the choice-specific covariates specified in \code{choiceX}. The details and examples are provided below.} \item{base}{The name of the base category. For the standard multinomial probit model, the default is the lowest level of the response variable. For the multinomial probit model with ordered preferences, the default base category is the last column in the matrix of response variables.} \item{latent}{logical. If \code{TRUE}, then the latent variable W will be returned. See Imai and van Dyk (2005) for the notation. The default is \code{FALSE}.} \item{invcdf}{logical. If \code{TRUE}, then the inverse cdf method is used for truncated normal sampling. If \code{FALSE}, then the rejection sampling method is used. The default is \code{FALSE}.} \item{trace}{logical. If \code{TRUE}, then the trace of the variance covariance matrix is set to a constant (here, it is equal to \code{n.dim}) instead of setting its first diagonal element to 1. The former avoids the arbitrariness of fixing one particular diagonal element in order to achieve identification (see Burgette and Nordheim, 2009).} \item{n.draws}{A positive integer. The number of MCMC draws. The default is \code{5000}.} \item{p.var}{A positive definite matrix. The prior variance of the coefficients. A scalar input can set the prior variance to the diagonal matrix whose diagonal element is equal to that value. The default is \code{"Inf"}, which represents an improper noninformative prior distribution on the coefficients.} \item{p.df}{A positive integer greater than \code{n.dim-1}. The prior degrees of freedom parameter for the covariance matrix. The default is \code{n.dim+1}, which is equal to the total number of alternatives.} \item{p.scale}{A positive definite matrix. When \code{trace = FALSE}, its first diagonal element is set to \code{1} if it is not equal to 1 already. The prior scale matrix for the covariance matrix. A scalar input can be used to set the scale matrix to a diagonal matrix with diagonal elements equal to the scalar input value. The default is \code{1}.} \item{coef.start}{A vector. The starting values for the coefficients. A scalar input sets the starting values for all the coefficients equal to that value. The default is \code{0}.} \item{cov.start}{A positive definite matrix. When \code{trace = FALSE}, its first diagonal element is set to \code{1} if it is not equal to 1 already. The starting values for the covariance matrix. A scalar input can be used to set the starting value to a diagonal matrix with diagonal elements equal to the scalar input value. The default is \code{1}.} \item{burnin}{A positive integer. The burnin interval for the Markov chain; i.e., the number of initial Gibbs draws that should not be stored. The default is \code{0}.} \item{thin}{A positive integer. The thinning interval for the Markov chain; i.e., the number of Gibbs draws between the recorded values that are skipped. The default is \code{0}.} \item{verbose}{logical. If \code{TRUE}, helpful messages along with a progress report of the Gibbs sampling are printed on the screen. The default is \code{FALSE}.} } \value{ An object of class \code{mnp} containing the following elements: \item{param}{A matrix of the Gibbs draws for each parameter; i.e., the coefficients and covariance matrix. For the covariance matrix, the elements on or above the diagonal are returned. } \item{call}{The matched call.} \item{x}{The matrix of covariates.} \item{y}{The vector or matrix of the response variable.} \item{w}{The three dimensional array of the latent variable, W. The first dimension represents the alternatives, and the second dimension indexes the observations. The third dimension represents the Gibbs draws. Note that the latent variable for the base category is set to 0, and therefore omitted from the output.} \item{alt}{The names of alternatives.} \item{n.alt}{The total number of alternatives.} \item{base}{The base category used for fitting.} \item{invcdf}{The value of \code{invcdf}.} \item{p.var}{The prior variance for the coefficients.} \item{p.df}{The prior degrees of freedom parameter for the covariance matrix.} \item{p.scale}{The prior scale matrix for the covariance matrix.} \item{burnin}{The number of initial burnin draws.} \item{thin}{The thinning interval.} } \description{ \code{mnp} is used to fit (Bayesian) multinomial probit model via Markov chain Monte Carlo. \code{mnp} can also fit the model with different choice sets for each observation, and complete or partial ordering of all the available alternatives. The computation uses the efficient marginal data augmentation algorithm that is developed by Imai and van Dyk (2005a). } \details{ To fit the multinomial probit model when only the most preferred choice is observed, use the syntax for the formula, \code{y ~ x1 + x2}, where \code{y} is a factor variable indicating the most preferred choice and \code{x1} and \code{x2} are individual-specific covariates. The interactions of individual-specific variables with each of the choice indicator variables will be fit. To specify choice-specific covariates, use the syntax, \code{choiceX=list(A=cbind(z1, z2), B=cbind(z3, z4), C=cbind(z5, z6))}, where \code{A}, \code{B}, and \code{C} represent the choice names of the response variable, and \code{z1} and \code{z2} are each vectors of length \eqn{n} that record the values of the two choice-specific covariates for each individual for choice A, likewise for \code{z3}, \eqn{\ldots}, \code{z6}. The corresponding variable names via \code{cXnames=c("price", "quantity")} need to be specified, where \code{price} refers to the coefficient name for \code{z1}, \code{z3}, and \code{z5}, and \code{quantity} refers to that for \code{z2}, \code{z4}, and \code{z6}. If the choice set varies from one observation to another, use the syntax, \code{cbind(y1, y2, y3) ~ x1 + x2}, in the case of a three choice problem, and indicate unavailable alternatives by \code{NA}. If only the most preferred choice is observed, \code{y1}, \code{y2}, and \code{y3} are indicator variables that take on the value one for individuals who prefer that choice and zero otherwise. The last column of the response matrix, \code{y3} in this particular example syntax, is used as the base category. To fit the multinomial probit model when the complete or partial ordering of the available alternatives is recorded, use the same syntax as when the choice set varies (i.e., \code{cbind(y1, y2, y3, y4) ~ x1 + x2}). For each observation, all the available alternatives in the response variables should be numerically ordered in terms of preferences such as \code{1 2 2 3}. Ties are allowed. The missing values in the response variable should be denoted by \code{NA}. The software will impute these missing values using the specified covariates. The resulting uncertainty estimates of the parameters will properly reflect the amount of missing data. For example, we expect the standard errors to be larger when there is more missing data. } \examples{ ### ### NOTE: this example is not fully analyzed. In particular, the ### convergence has not been assessed. A full analysis of these data ### sets appear in Imai and van Dyk (2005b). ### ## load the detergent data data(detergent) ## run the standard multinomial probit model with intercepts and the price res1 <- mnp(choice ~ 1, choiceX = list(Surf=SurfPrice, Tide=TidePrice, Wisk=WiskPrice, EraPlus=EraPlusPrice, Solo=SoloPrice, All=AllPrice), cXnames = "price", data = detergent, n.draws = 100, burnin = 10, thin = 3, verbose = TRUE) ## summarize the results summary(res1) ## calculate the quantities of interest for the first 3 observations pre1 <- predict(res1, newdata = detergent[1:3,]) ## load the Japanese election data data(japan) ## run the multinomial probit model with ordered preferences res2 <- mnp(cbind(LDP, NFP, SKG, JCP) ~ gender + education + age, data = japan, verbose = TRUE) ## summarize the results summary(res2) ## calculate the predicted probabilities for the 10th observation ## averaging over 100 additional Monte Carlo draws given each of MCMC draw. pre2 <- predict(res2, newdata = japan[10,], type = "prob", n.draws = 100, verbose = TRUE) } \references{ Imai, Kosuke and David A. van Dyk. (2005a) \dQuote{A Bayesian Analysis of the Multinomial Probit Model Using the Marginal Data Augmentation,} \emph{Journal of Econometrics}, Vol. 124, No. 2 (February), pp.311-334. Imai, Kosuke and David A. van Dyk. (2005b) \dQuote{MNP: R Package for Fitting the Multinomial Probit Models,} \emph{Journal of Statistical Software}, Vol. 14, No. 3 (May), pp.1-32. Burgette, L.F. and E.V. Nordheim. (2009). \dQuote{An alternate identifying restriction for the Bayesian multinomial probit model,} \emph{Technical report}, Department of Statistics, University of Wisconsin, Madison. } \seealso{ \code{coef.mnp}, \code{cov.mnp}, \code{predict.mnp}, \code{summary.mnp}; } \author{ Kosuke Imai, Department of Politics, Princeton University \email{kimai@Princeton.Edu}, \url{http://imai.princeton.edu}; David A. van Dyk, Statistics Section, Department of Mathematics, Imperial College London. } \keyword{models} MNP/man/cov.mnp.Rd0000644000176200001440000000227213103226434013327 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/cov.mnp.R \name{cov.mnp} \alias{cov.mnp} \title{Extract Multinomial Probit Model Covariance Matrix} \usage{ cov.mnp(object, subset = NULL, ...) } \arguments{ \item{object}{An output object from \code{mnp}.} \item{subset}{A scalar or a numerical vector specifying the row number(s) of \code{param} in the output object from \code{mnp}. If specified, the posterior draws of covariance matrix for those rows are extracted. The default is \code{NULL} where all the posterior draws are extracted.} \item{...}{further arguments passed to or from other methods.} } \value{ When a numerical vector or \code{NULL} is specified for \code{subset} argument, \code{cov.mnp} returns a three dimensional array where the third dimension indexes posterior draws. When a scalar is specified for \code{subset} arugment, \code{cov.mnp} returns a matrix. } \description{ \code{cov.mnp} is a function which extracts the posterior draws of covariance matrix from objects returned by \code{mnp}. } \seealso{ \code{mnp}, \code{coef.mnp}; } \author{ Kosuke Imai, Department of Politics, Princeton University \email{kimai@Princeton.Edu} } \keyword{methods} MNP/man/print.summary.mnp.Rd0000644000176200001440000000131213103226434015362 0ustar liggesusers% Generated by roxygen2: do not edit by hand % Please edit documentation in R/print.summary.mnp.R \name{print.summary.mnp} \alias{print.summary.mnp} \title{Print the summary of the results for the Multinomial Probit Models} \usage{ \method{print}{summary.mnp}(x, digits = max(3, getOption("digits") - 3), ...) } \arguments{ \item{x}{An object of class \code{summary.mnp}.} \item{digits}{the number of significant digits to use when printing.} \item{...}{further arguments passed to or from other methods.} } \description{ \code{summary} print method for class \code{mnp}. } \seealso{ \code{mnp} } \author{ Kosuke Imai, Department of Politics, Princeton University \email{kimai@Princeton.Edu} } \keyword{methods}