20:
990:
453:
27:
output from calculating pseudo-r-squared values using the "pscl" package by Simon
Jackman. The pseudo-R-squared by McFadden is clearly labelled “McFadden”, which is equal to the pseudo-R-squared by Cohen. Next to this, the pseudo-r-squared by Cox and Snell is labelled “r2ML” and this type of
785:
263:
218:
This is the most analogous index to the squared multiple correlations in linear regression. It represents the proportional reduction in the deviance wherein the deviance is treated as a measure of variation analogous but not identical to the
211:
1058:– the error variances differ for each value of the predicted score. For each value of the predicted score there would be a different value of the proportionate reduction in error. Therefore, it is inappropriate to think of
985:{\displaystyle {\begin{matrix}R_{\text{CS}}^{2}=1-\left({\dfrac {1}{L_{0}}}\right)^{\frac {2(R_{\text{McF}}^{2})}{n}}\\R_{\text{McF}}^{2}=-{\dfrac {n}{2}}\cdot {\dfrac {\ln(1-R_{\text{CS}}^{2})}{\ln L_{0}}}\end{matrix}}}
749:
28:
pseudo-R-squared By Cox and Snell is sometimes simply called “ML”. The last value listed, labelled “r2CU” is the pseudo-r-squared by
Nagelkerke and is the same as the pseudo-r-squared by Cragg and Uhler.
268:
231:
is that it is not monotonically related to the odds ratio, meaning that it does not necessarily increase as the odds ratio increases and does not necessarily decrease as the odds ratio decreases.
571:
448:{\displaystyle {\begin{aligned}R_{\text{CS}}^{2}&=1-\left({\frac {L_{0}}{L_{M}}}\right)^{2/n}\\&=1-\exp \left({\frac {2}{n}}(\ln(L_{0})-\ln(L_{M}))\right)\end{aligned}}}
533:
139:
1175:
Hardin, J. W., Hilbe, J. M. (2007). Generalized linear models and extensions. USA: Taylor & Francis. Page 60,
1176:
658:
609:. Of course, this might not be the case for values exceeding 0.75 as the Cox and Snell index is capped at this value. The likelihood ratio
57:
is used to assess goodness of fit as it represents the proportion of variance in the criterion that is explained by the predictors. In
1160:
1105:
61:
analysis, there is no agreed upon analogous measure, but there are several competing measures each with limitations.
1234:
573:
for a symmetric marginal distribution of events and decreases further for an asymmetric distribution of events.
538:
36:
1054:, that the error variance is the same for all values of the criterion. Logistic regression will always be
1229:
1131:
64:
Four of the most commonly used indices and one less commonly used one are examined in this article:
1019:
For each level of the dependent variable, find the mean of the predicted probabilities of an event.
492:
24:
601:
so that the maximum value is equal to 1. Nevertheless, the Cox and Snell and likelihood ratio
206:{\displaystyle R_{\text{L}}^{2}={\frac {D_{\text{null}}-D_{\text{fitted}}}{D_{\text{null}}}}.}
474:
are the likelihoods for the model being fitted and the null model, respectively. The Cox and
19:
625:
s increase as the proportion of cases increase from 0 to 0.5) and varies between 0 and 1.
8:
58:
43:
1015:
which is a relatively new measure developed by Tjur. It can be calculated in two steps:
1205:
1209:
1156:
1101:
1047:
618:
224:
50:
1197:
1062:
as a proportionate reduction in error in a universal sense in logistic regression.
1051:
645:
1188:
Tjur, Tue (2009). "Coefficients of determination in logistic regression models".
1055:
594:
1093:
597:
in a highly cited
Biometrika paper, provides a correction to the Cox and Snell
605:
s show greater agreement with each other than either does with the
Nagelkerke
35:
values are used when the outcome variable is nominal or ordinal such that the
1223:
1201:
1098:
Applied
Multiple Regression/Correlation Analysis for the Behavioral Sciences
1042:
is that they do not represent the proportionate reduction in error as the
475:
744:{\displaystyle R_{\text{McF}}^{2}=1-{\frac {\ln(L_{M})}{\ln(L_{0})}},}
621:, is independent of the base rate (both Cox and Snell and Nagelkerke
482:
in case of a linear model with normal error. In certain situations,
220:
613:
is often preferred to the alternatives as it is most analogous to
1035:
statistics. The reason these indices of fit are referred to as
1137:. Statistical Horizons LLC and the University of Pennsylvania.
790:
42:
cannot be applied as a measure for goodness of fit and when a
1022:
Take the absolute value of the difference between these means
535:. For example, for logistic regression, the upper bound is
251:
is an alternative index of goodness of fit related to the
1031:
A word of caution is in order when interpreting pseudo-
920:
905:
823:
788:
661:
541:
495:
266:
142:
1091:
984:
743:
565:
527:
447:
205:
227:analysis. One limitation of the likelihood ratio
1221:
1092:Cohen, Jacob; Cohen, Patricia; West, Steven G.;
255:value from linear regression. It is given by:
489:may be problematic as its maximum value is
566:{\displaystyle R_{\text{CS}}^{2}\leq 0.75}
1132:"Measures of fit for logistic regression"
1146:
1144:
1100:(3rd ed.). Routledge. p. 502.
234:
18:
1087:
1085:
1083:
1081:
1079:
1077:
1075:
1222:
1150:
1141:
1187:
1125:
1123:
1121:
1119:
1117:
1072:
628:
576:
53:, the squared multiple correlation,
1129:
13:
777:are then related respectively by,
478:index corresponds to the standard
14:
1246:
1114:
1026:
1050:does. Linear regression assumes
763:by Allison. The two expressions
114:
1181:
1169:
997:
953:
929:
869:
851:
732:
719:
708:
695:
643:by McFadden (sometimes called
433:
430:
417:
405:
392:
383:
1:
1065:
528:{\displaystyle 1-L_{0}^{2/n}}
37:coefficient of determination
7:
1153:Applied Logistic Regression
10:
1251:
16:Statistical measure of fit
1151:Menard, Scott W. (2002).
46:is used to fit a model.
1202:10.1198/tast.2009.08210
1235:Regression diagnostics
1155:(2nd ed.). SAGE.
986:
756:and is preferred over
745:
567:
529:
449:
207:
29:
1190:American Statistician
987:
746:
568:
530:
450:
208:
22:
786:
659:
539:
493:
264:
140:
952:
897:
868:
807:
676:
556:
524:
285:
157:
131:is given by Cohen:
59:logistic regression
44:likelihood function
1230:Statistical ratios
982:
980:
976:
938:
914:
883:
854:
839:
793:
741:
662:
563:
542:
525:
502:
445:
443:
271:
203:
143:
30:
1162:978-0-7619-2208-7
1130:Allison, Paul D.
1107:978-0-8058-2223-6
1048:linear regression
975:
945:
913:
890:
876:
861:
838:
800:
736:
669:
619:linear regression
549:
381:
326:
278:
225:linear regression
198:
195:
184:
171:
150:
68:Likelihood ratio
51:linear regression
1242:
1214:
1213:
1185:
1179:
1173:
1167:
1166:
1148:
1139:
1138:
1136:
1127:
1112:
1111:
1089:
1061:
1052:homoscedasticity
1045:
1041:
1034:
1011:
1008:Allison prefers
1000:
991:
989:
988:
983:
981:
977:
974:
973:
972:
956:
951:
946:
943:
921:
915:
906:
896:
891:
888:
878:
877:
872:
867:
862:
859:
846:
844:
840:
837:
836:
824:
806:
801:
798:
773:
766:
759:
750:
748:
747:
742:
737:
735:
731:
730:
711:
707:
706:
687:
675:
670:
667:
650:) is defined as
646:likelihood ratio
642:
631:
624:
616:
612:
608:
604:
600:
589:
579:
572:
570:
569:
564:
555:
550:
547:
534:
532:
531:
526:
523:
519:
510:
485:
481:
473:
466:
454:
452:
451:
446:
444:
440:
436:
429:
428:
404:
403:
382:
374:
349:
345:
344:
340:
331:
327:
325:
324:
315:
314:
305:
284:
279:
276:
254:
247:
241:by Cox and Snell
237:
230:
212:
210:
209:
204:
199:
197:
196:
193:
187:
186:
185:
182:
173:
172:
169:
162:
156:
151:
148:
127:
117:
107:
98:
89:
80:
71:
56:
41:
33:Pseudo-R-squared
1250:
1249:
1245:
1244:
1243:
1241:
1240:
1239:
1220:
1219:
1218:
1217:
1186:
1182:
1174:
1170:
1163:
1149:
1142:
1134:
1128:
1115:
1108:
1094:Aiken, Leona S.
1090:
1073:
1068:
1059:
1056:heteroscedastic
1043:
1039:
1032:
1029:
1014:
1009:
1006:
1003:
998:
979:
978:
968:
964:
957:
947:
942:
922:
919:
904:
892:
887:
880:
879:
863:
858:
847:
845:
832:
828:
822:
818:
817:
802:
797:
789:
787:
784:
783:
776:
771:
769:
764:
762:
757:
726:
722:
712:
702:
698:
688:
686:
671:
666:
660:
657:
656:
640:
637:
634:
629:
622:
614:
610:
606:
602:
598:
595:Nico Nagelkerke
592:
587:
585:
582:
577:
551:
546:
540:
537:
536:
515:
511:
506:
494:
491:
490:
488:
483:
479:
472:
468:
465:
461:
442:
441:
424:
420:
399:
395:
373:
372:
368:
347:
346:
336:
332:
320:
316:
310:
306:
304:
300:
299:
286:
280:
275:
267:
265:
262:
261:
252:
250:
245:
243:
240:
235:
228:
192:
188:
181:
177:
168:
164:
163:
161:
152:
147:
141:
138:
137:
130:
125:
123:
120:
115:
110:
105:
101:
96:
92:
87:
83:
78:
74:
69:
54:
39:
17:
12:
11:
5:
1248:
1238:
1237:
1232:
1216:
1215:
1196:(4): 366–372.
1180:
1168:
1161:
1140:
1113:
1106:
1070:
1069:
1067:
1064:
1028:
1027:Interpretation
1025:
1024:
1023:
1020:
1012:
1005:
1001:
996:
995:
994:
993:
992:
971:
967:
963:
960:
955:
950:
941:
937:
934:
931:
928:
925:
918:
912:
909:
903:
900:
895:
886:
882:
881:
875:
871:
866:
857:
853:
850:
843:
835:
831:
827:
821:
816:
813:
810:
805:
796:
792:
791:
774:
767:
760:
754:
753:
752:
751:
740:
734:
729:
725:
721:
718:
715:
710:
705:
701:
697:
694:
691:
685:
682:
679:
674:
665:
636:
632:
627:
593:, proposed by
590:
584:
580:
575:
562:
559:
554:
545:
522:
518:
514:
509:
505:
501:
498:
486:
470:
463:
458:
457:
456:
455:
439:
435:
432:
427:
423:
419:
416:
413:
410:
407:
402:
398:
394:
391:
388:
385:
380:
377:
371:
367:
364:
361:
358:
355:
352:
350:
348:
343:
339:
335:
330:
323:
319:
313:
309:
303:
298:
295:
292:
289:
287:
283:
274:
270:
269:
248:
242:
238:
233:
216:
215:
214:
213:
202:
191:
180:
176:
167:
160:
155:
146:
128:
122:
118:
113:
112:
111:
108:
102:
99:
93:
90:
84:
81:
77:Cox and Snell
75:
72:
23:This displays
15:
9:
6:
4:
3:
2:
1247:
1236:
1233:
1231:
1228:
1227:
1225:
1211:
1207:
1203:
1199:
1195:
1191:
1184:
1178:
1172:
1164:
1158:
1154:
1147:
1145:
1133:
1126:
1124:
1122:
1120:
1118:
1109:
1103:
1099:
1095:
1088:
1086:
1084:
1082:
1080:
1078:
1076:
1071:
1063:
1057:
1053:
1049:
1038:
1021:
1018:
1017:
1016:
969:
965:
961:
958:
948:
939:
935:
932:
926:
923:
916:
910:
907:
901:
898:
893:
884:
873:
864:
855:
848:
841:
833:
829:
825:
819:
814:
811:
808:
803:
794:
782:
781:
780:
779:
778:
738:
727:
723:
716:
713:
703:
699:
692:
689:
683:
680:
677:
672:
663:
655:
654:
653:
652:
651:
649:
647:
626:
620:
596:
583:by Nagelkerke
574:
560:
557:
552:
543:
520:
516:
512:
507:
503:
499:
496:
477:
437:
425:
421:
414:
411:
408:
400:
396:
389:
386:
378:
375:
369:
365:
362:
359:
356:
353:
351:
341:
337:
333:
328:
321:
317:
311:
307:
301:
296:
293:
290:
288:
281:
272:
260:
259:
258:
257:
256:
232:
226:
222:
200:
189:
178:
174:
165:
158:
153:
144:
136:
135:
134:
133:
132:
103:
94:
85:
76:
67:
66:
65:
62:
60:
52:
47:
45:
38:
34:
26:
21:
1193:
1189:
1183:
1177:Google Books
1171:
1152:
1097:
1036:
1030:
1007:
755:
644:
638:
586:
459:
244:
217:
124:
63:
48:
32:
31:
639:The pseudo
635:by McFadden
86:Nagelkerke
1224:Categories
1066:References
1210:121927418
962:
936:−
927:
917:⋅
902:−
815:−
717:
693:
684:−
558:≤
500:−
415:
409:−
390:
366:
360:−
297:−
175:−
95:McFadden
1096:(2002).
221:variance
121:by Cohen
1004:by Tjur
1208:
1159:
1104:
1037:pseudo
460:where
183:fitted
1206:S2CID
1135:(PDF)
648:index
476:Snell
104:Tjur
1157:ISBN
1102:ISBN
770:and
561:0.75
467:and
194:null
170:null
1198:doi
1046:in
889:McF
860:McF
768:McF
668:McF
633:McF
617:in
363:exp
223:in
100:McF
49:In
1226::
1204:.
1194:63
1192:.
1143:^
1116:^
1074:^
959:ln
944:CS
924:ln
799:CS
775:CS
761:CS
714:ln
690:ln
548:CS
487:CS
412:ln
387:ln
277:CS
249:CS
239:CS
82:CS
1212:.
1200::
1165:.
1110:.
1060:R
1044:R
1040:R
1033:R
1013:T
1010:R
1002:T
999:R
970:0
966:L
954:)
949:2
940:R
933:1
930:(
911:2
908:n
899:=
894:2
885:R
874:n
870:)
865:2
856:R
852:(
849:2
842:)
834:0
830:L
826:1
820:(
812:1
809:=
804:2
795:R
772:R
765:R
758:R
739:,
733:)
728:0
724:L
720:(
709:)
704:M
700:L
696:(
681:1
678:=
673:2
664:R
641:R
630:R
623:R
615:R
611:R
607:R
603:R
599:R
591:N
588:R
581:N
578:R
553:2
544:R
521:n
517:/
513:2
508:0
504:L
497:1
484:R
480:R
471:0
469:L
464:M
462:L
438:)
434:)
431:)
426:M
422:L
418:(
406:)
401:0
397:L
393:(
384:(
379:n
376:2
370:(
357:1
354:=
342:n
338:/
334:2
329:)
322:M
318:L
312:0
308:L
302:(
294:1
291:=
282:2
273:R
253:R
246:R
236:R
229:R
201:.
190:D
179:D
166:D
159:=
154:2
149:L
145:R
129:L
126:R
119:L
116:R
109:T
106:R
97:R
91:N
88:R
79:R
73:L
70:R
55:R
40:R
25:R
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
