Social Survey Example Revisit
[Data] In the 2000 General Social Survey, 1144 subjects were asked whether, to help the environment, they would be willing to (1) pay higher taxes or (2) accept a cut in living standards.
Social Survey Example
| Pay Higher Taxes |
Yes |
No |
Total |
| Yes |
227 |
132 |
359 |
| No |
107 |
678 |
785 |
| Total |
334 |
810 |
1144 |
[Goal] How can we compare the probabilities of a ``yes" outcome for the two environmental questions?
[Code]
There are several estimation techniques available.
method=quad specifies the maximum likelihood with Gauss-Hermite quadrature approximation
The number of quadrature points is determined automatically. However, it can be specified by method=quad(qpoints=10.
In practice, \(\leq 7\) is generally suffices (Pinheiro and Bates, page 321). One strategy is to check fitting under different quadrature points to see the estimates are stablized.
Choice of the estimation technique is crucial. If the default one (, a pseudo-likelihood method), parameters are as follows:
Effect Estimate Error DF t Value Pr > |t|
Intercept -0.7445 0.1562 1143 -4.77 <.0001
question -0.1167 0.09667 1143 -1.21 0.2275
data survey;
do subject=1 to 227;
response = 1; question = 1; output;
response = 1; question = 2; output;
end;
do subject=227+1 to 227+132;
response = 1; question = 1; output;
response = 0; question = 2; output;
end;
do subject=227+132+1 to 227+132+107;
response = 0; question = 1; output;
response = 1; question = 2; output;
end;
do subject=227+132+107+1 to 227+132+107+678;
response = 0; question = 1; output;
response = 0; question = 2; output;
end;
run;
ods listing;
proc glimmix data=survey method=quad;
class subject;
model response = question / solution dist=binomial link=logit;
random intercept / subject=subject;
run;
ods listing;
proc nlmixed data=survey;
parms alpha = -1, beta = 1, s2 = 10; /* provide initial values */
eta = alpha + beta*question + pair; /* systematic component */
mu = exp(eta)/(1+exp(eta));
model response ~ binary(mu); /* link function */
random pair ~ normal(0, s2) subject=subject; /* random effect */
run;
The GLIMMIX Procedure
Model Information
Data Set WORK.SURVEY
Response Variable response
Response Distribution Binomial
Link Function Logit
Variance Function Default
Variance Matrix Blocked By subject
Estimation Technique Maximum Likelihood
Likelihood Approximation Gauss-Hermite Quadrature
Degrees of Freedom Method Containment
Class Level Information
Class Levels Values
subject 1144 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
49 50 51 52 53 54 55 56 57 58 59 60 61 62 63
64 65 66 67 68 69 70 71 72 73 74 75 76 77 78
79 80 81 82 83 84 85 86 87 88 89 90 91 92 93
94 95 96 97 98 99 100 101 102 103 104 105 106
107 108 109 110 111 112 113 114 115 116 117
118 119 120 121 122 123 124 125 126 127 128
129 130 131 132 133 134 135 136 137 138 139
140 141 142 143 144 145 146 147 148 149 150
151 152 153 154 155 156 157 158 159 160 161
162 163 164 165 166 167 168 169 170 171 172
173 174 175 176 177 178 179 180 181 182 183
184 185 186 187 188 189 190 191 192 193 194
195 196 197 198 199 200 201 202 203 204 205
206 207 208 209 210 211 212 213 214 215 216
217 218 219 220 221 222 223 224 225 226 227
228 229 230 231 232 233 234 235 236 237 238
239 240 241 242 243 244 245 246 247 248 249
250 251 252 253 254 255 256 257 258 259 260
261 262 263 264 265 266 267 268 269 270 271
272 273 274 275 276 277 278 279 280 281 282
283 284 285 286 287 288 289 290 291 292 293
294 295 296 297 298 299 300 301 302 303 304
305 306 307 308 309 310 311 312 313 314 315
316 317 318 319 320 321 322 323 324 325 326
327 328 329 330 331 332 333 334 335 336 337
338 339 340 341 342 343 344 345 346 347 348
349 350 351 352 353 354 355 356 357 358 359
360 361 362 363 364 365 366 367 368 369 370
371 372 373 374 375 376 377 378 379 380 381
382 383 384 385 386 387 388 389 390 391 392
393 394 395 396 397 398 399 400 401 402 403
404 405 406 407 408 409 410 411 412 413 414
415 416 417 418 419 420 421 422 423 424 425
426 427 428 429 430 431 432 433 434 435 436
437 438 439 440 441 442 443 444 445 446 447
448 449 450 451 452 453 454 455 456 457 458
459 460 461 462 463 464 465 466 467 468 469
470 471 472 473 474 475 476 477 478 479 480
481 482 483 484 485 486 487 488 489 490 491
492 493 494 495 496 497 498 499 500 501 502
503 504 505 506 507 508 509 510 511 512 513
514 515 516 517 518 519 520 521 522 523 524
525 526 527 528 529 530 531 532 533 534 535
536 537 538 539 540 541 542 543 544 545 546
547 548 549 550 551 552 553 554 555 556 557
558 559 560 561 562 563 564 565 566 567 568
569 570 571 572 573 574 575 576 577 578 579
580 581 582 583 584 585 586 587 588 589 590
591 592 593 594 595 596 597 598 599 600 601
602 603 604 605 606 607 608 609 610 611 612
613 614 615 616 617 618 619 620 621 622 623
624 625 626 627 628 629 630 631 632 633 634
635 636 637 638 639 640 641 642 643 644 645
646 647 648 649 650 651 652 653 654 655 656
657 658 659 660 661 662 663 664 665 666 667
668 669 670 671 672 673 674 675 676 677 678
679 680 681 682 683 684 685 686 687 688 689
690 691 692 693 694 695 696 697 698 699 700
701 702 703 704 705 706 707 708 709 710 711
712 713 714 715 716 717 718 719 720 721 722
723 724 725 726 727 728 729 730 731 732 733
734 735 736 737 738 739 740 741 742 743 744
745 746 747 748 749 750 751 752 753 754 755
756 757 758 759 760 761 762 763 764 765 766
767 768 769 770 771 772 773 774 775 776 777
778 779 780 781 782 783 784 785 786 787 788
789 790 791 792 793 794 795 796 797 798 799
800 801 802 803 804 805 806 807 808 809 810
811 812 813 814 815 816 817 818 819 820 821
822 823 824 825 826 827 828 829 830 831 832
833 834 835 836 837 838 839 840 841 842 843
844 845 846 847 848 849 850 851 852 853 854
855 856 857 858 859 860 861 862 863 864 865
866 867 868 869 870 871 872 873 874 875 876
877 878 879 880 881 882 883 884 885 886 887
888 889 890 891 892 893 894 895 896 897 898
899 900 901 902 903 904 905 906 907 908 909
910 911 912 913 914 915 916 917 918 919 920
921 922 923 924 925 926 927 928 929 930 931
932 933 934 935 936 937 938 939 940 941 942
943 944 945 946 947 948 949 950 951 952 953
954 955 956 957 958 959 960 961 962 963 964
965 966 967 968 969 970 971 972 973 974 975
976 977 978 979 980 981 982 983 984 985 986
987 988 989 990 991 992 993 994 995 996 997
998 999 1000 1001 1002 1003 1004 1005 1006
1007 1008 1009 1010 1011 1012 1013 1014 1015
1016 1017 1018 1019 1020 1021 1022 1023 1024
1025 1026 1027 1028 1029 1030 1031 1032 1033
1034 1035 1036 1037 1038 1039 1040 1041 1042
1043 1044 1045 1046 1047 1048 1049 1050 1051
1052 1053 1054 1055 1056 1057 1058 1059 1060
1061 1062 1063 1064 1065 1066 1067 1068 1069
1070 1071 1072 1073 1074 1075 1076 1077 1078
1079 1080 1081 1082 1083 1084 1085 1086 1087
1088 1089 1090 1091 1092 1093 1094 1095 1096
1097 1098 1099 1100 1101 1102 1103 1104 1105
1106 1107 1108 1109 1110 1111 1112 1113 1114
1115 1116 1117 1118 1119 1120 1121 1122 1123
1124 1125 1126 1127 1128 1129 1130 1131 1132
1133 1134 1135 1136 1137 1138 1139 1140 1141
1142 1143 1144
Number of Observations Read 2288
Number of Observations Used 2288
Dimensions
G-side Cov. Parameters 1
Columns in X 2
Columns in Z per Subject 1
Subjects (Blocks in V) 1144
Max Obs per Subject 2
Optimization Information
Optimization Technique Dual Quasi-Newton
Parameters in Optimization 3
Lower Boundaries 1
Upper Boundaries 0
Fixed Effects Not Profiled
Starting From GLM estimates
Quadrature Points 9
Iteration History
Objective Max
Iteration Restarts Evaluations Function Change Gradient
0 0 4 2585.9233051 . 233.5512
1 0 3 2550.1588011 35.76450407 17.2202
2 0 4 2538.0301425 12.12865858 28.23844
3 0 4 2522.7163069 15.31383554 12.69428
4 0 4 2520.5799289 2.13637800 4.421319
5 0 2 2520.5480178 0.03191113 1.665033
6 0 3 2520.5440456 0.00397215 0.069624
7 0 3 2520.5439595 0.00008610 0.037947
8 0 3 2520.5439578 0.00000171 0.000615
Convergence criterion (GCONV=1E-8) satisfied.
Fit Statistics
-2 Log Likelihood 2520.54
AIC (smaller is better) 2526.54
AICC (smaller is better) 2526.55
BIC (smaller is better) 2541.67
CAIC (smaller is better) 2544.67
HQIC (smaller is better) 2532.26
Fit Statistics for Conditional Distribution
-2 log L(response | r. effects) 1041.77
Pearson Chi-Square 702.92
Pearson Chi-Square / DF 0.31
Covariance Parameter Estimates
Standard
Cov Parm Subject Estimate Error
Intercept subject 8.1121 1.2028
Solutions for Fixed Effects
Standard
Effect Estimate Error DF t Value Pr > |t|
Intercept -1.4172 0.2358 1143 -6.01 <.0001
question -0.2094 0.1299 1143 -1.61 0.1072
Type III Tests of Fixed Effects
Num Den
Effect DF DF F Value Pr > F
question 1 1143 2.60 0.1072
The NLMIXED Procedure
Specifications
Data Set WORK.SURVEY
Dependent Variable response
Distribution for Dependent Variable Binary
Random Effects pair
Distribution for Random Effects Normal
Subject Variable subject
Optimization Technique Dual Quasi-Newton
Integration Method Adaptive Gaussian
Quadrature
Dimensions
Observations Used 2288
Observations Not Used 0
Total Observations 2288
Subjects 1144
Max Obs per Subject 2
Parameters 3
Quadrature Points 11
Initial Parameters
Negative
Log
alpha beta s2 Likelihood
-1 1 10 1496.76272
Iteration History
Negative
Log Maximum
Iteration Calls Likelihood Difference Gradient Slope
1 6 1261.1651 235.5976 9.49199 -2037.78
2 8 1260.9312 0.233967 0.91885 -0.41606
3 12 1260.8860 0.045185 2.68291 -0.03103
4 16 1260.8210 0.064942 0.97461 -0.03836
5 22 1260.1873 0.633706 2.02640 -0.02159
6 25 1260.1478 0.039503 0.80245 -0.12789
7 28 1260.1378 0.010028 0.064909 -0.01874
8 31 1260.1378 0.000053 0.002600 -0.00011
9 34 1260.1378 5.278E-8 0.000086 -1.06E-7
NOTE: GCONV convergence criterion satisfied.
Fit Statistics
-2 Log Likelihood 2520.3
AIC (smaller is better) 2526.3
AICC (smaller is better) 2526.3
BIC (smaller is better) 2541.4
Parameter Estimates
Standard 95% Confidence
Parameter Estimate Error DF t Value Pr > |t| Limits
alpha -1.4268 0.2381 1143 -5.99 <.0001 -1.8939 -0.9596
beta -0.2102 0.1302 1143 -1.62 0.1066 -0.4656 0.04515
s2 8.2743 1.2756 1143 6.49 <.0001 5.7714 10.7771
Parameter Estimates
Parameter Gradient
alpha 0.000057
beta 0.000086
s2 1.537E-6
Social Survey Example Revisit
[Data] In the 2000 General Social Survey, 1144 subjects were asked whether, to help the environment, they would be willing to (1) pay higher taxes or (2) accept a cut in living standards.
[Goal] How can we compare the probabilities of a ``yes" outcome for the two environmental questions?
[Code]
There are several estimation techniques available.
method=quadspecifies the maximum likelihood with Gauss-Hermite quadrature approximationThe number of quadrature points is determined automatically. However, it can be specified by
method=quad(qpoints=10.In practice, \(\leq 7\) is generally suffices (Pinheiro and Bates, page 321). One strategy is to check fitting under different quadrature points to see the estimates are stablized.
Choice of the estimation technique is crucial. If the default one (, a pseudo-likelihood method), parameters are as follows: