699:
are popular models in social sciences and medical science to assess associations between variables and risk of recurrence, or to predict recurrent event outcomes. Many extensions of survival models based on the Cox proportional hazards approach have been proposed to handle recurrent event data. These
677:
The marginal means/rates model considers all recurrent events of the same subject as a single counting process and does not require time-varying covariates to reflect the past history of the process, which makes it a more flexible model. Instead, the full history of the counting process may influence
30:
The processes which generate events repeatedly over time are referred to as recurrent event processes, which are different from processes analyzed in time-to-event analysis: whereas time-to-event analysis focuses on the time to a single terminal event, individuals may be at risk for subsequent events
668:
is a popular model for recurrent event data, which models the number of recurrences that have occurred. Poisson regression assumes that the number of recurrences has a
Poisson distribution with a fixed rate of recurrence over time. The logarithm of the expected number of recurrences is modeled by a
26:
that analyzes the time until recurrences occur, such as recurrences of traits or diseases. Recurrent events are often analyzed in social sciences and medical studies, for example recurring infections, depressions or cancer recurrences. Recurrent event analysis attempts to answer certain questions,
686:
In multi-state models, the recurrent event processes of individuals are described by different states. The different states may describe the recurrence number, or whether the subject is at risk of recurrence. A change of state is called a transition (or an event) and is central in this framework,
725:
Time to recurrence is often correlated within subjects, as some subjects can be more frail to experiencing recurrences. If the correlated nature of the data is ignored, the confidence intervals (CI) for the estimated rates could be artificially narrow, which may result in false positive results.
651:, which is used to describe the time to a terminal event, recurrent event data can be described using the mean cumulative function, which is the average number of cumulative events experienced by an individual in the study at each point in time since the start of follow-up.
743:
In frailty models, a random effect is included in the recurrent event model which describes the individual excess risk that can not be explained by the included covariates. The frailty term induces dependence among the recurrence times within subjects.
638:
When a heterogeneous group of individuals or processes is considered, the assumption of a common event intensity is no longer plausible. Greater generality can be achieved by incorporating fixed or time-varying covariates in the intensity function.
636:
734:
It is possible to use robust 'sandwich' estimators for the variance of regression coefficients. Robust variance estimators are based on a jackknife estimate, which anticipates correlation within subjects and provides robust standard errors.
687:
which is fully characterized through estimation of transition probabilities between states and transition intensities that are defined as instantaneous hazards of progression to one state, conditional on occupying another state.
312:
505:
137:
500:
234:
388:
164:
717:
Well-known examples of Cox-based recurrent event models are the
Andersen and Gill model, the Prentice, Williams and Petersen model and the Wei–Lin–Weissfeld model
85:
432:
408:
184:
344:
27:
such as: how many recurrences occur on average within a certain time interval? Which factors are associated with a higher or lower risk of recurrence?
779:
349:
Models for recurrent events can be specified by considering the probability distribution for the number of recurrences in short intervals
874:
859:<13::aid-sim279>3.0.co;2-5 "Survival analysis for recurrent event data: an application to childhood infectious diseases"
696:
239:
648:
90:
1014:
437:
944:
R. L. Prentice, B. J. Williams, A. V. Peterson, On the regression analysis of multivariate failure time data,
434:, conditional on the process history, and describes the process mathematically. Define the process history as
52:
Determining the relationship of fixed covariates, treatments, and time-varying factors to event occurrence
631:{\displaystyle \lambda (t|H(t))=\lim _{\Delta t\downarrow 0}{\frac {P(N(t+\Delta t)-N(t)=1)}{\Delta t}}.}
192:
971:"Regression Analysis of Multivariate Incomplete Failure Time Data by Modeling Marginal Distributions"
352:
665:
990:
927:
886:
878:
839:
821:
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23:
982:
949:
917:
870:
829:
811:
767:
858:
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142:
64:
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834:
799:
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169:
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1008:
970:
986:
236:
records the cumulative number of events generated by the process; specifically,
953:
771:
994:
931:
922:
905:
882:
825:
890:
875:
10.1002/(sici)1097-0258(20000115)19:1<13::aid-sim279>3.0.co;2-5
843:
816:
46:
Identifying and characterizing variation across a population of processes
414:
describes the instantaneous probability of an event occurring at time
31:
after the first in recurrent event analysis, until they are censored.
906:"Cox's Regression Model for Counting Processes: A Large Sample Study"
800:"Modelling recurrent events: a tutorial for analysis in epidemiology"
314:
is the number of events occurring over the time interval
43:
Understanding and describing individual event processes
700:
models can be characterized by four model components:
242:
508:
440:
420:
396:
355:
320:
307:{\textstyle N(t)=\sum _{k=1}^{\infty }I(T_{k}\leq t)}
195:
172:
145:
93:
67:
390:, given the history of event occurrence before time
857:Kelly, Patrick J.; Lim, Lynette L-Y. (2000-01-15).
948:, Volume 68, Issue 2, August 1981, Pages 373–379,
630:
494:
426:
402:
382:
338:
306:
228:
178:
158:
131:
79:
61:For a single recurrent event process starting at
542:
39:Objectives of recurrent event analysis include:
975:Journal of the American Statistical Association
798:Amorim, Leila DAF; Cai, Jianwen (2014-12-09).
132:{\displaystyle 0\leq T_{1}<T_{2}<\dots }
691:Extended Cox proportional hazards (PH) models
669:linear combination of explanatory variables.
8:
763:The Statistical Analysis of Recurrent Events
489:
456:
223:
196:
904:Andersen, P. K.; Gill, R. D. (1982-12-01).
766:. Statistics for Biology and Health. 2007.
655:Statistical models for recurrent event data
502:, then the intensity is formally defined as
495:{\displaystyle H(t)=\{N(s):0\leq s<t\}}
921:
833:
815:
713:Correction for within-subject correlation
560:
545:
518:
507:
439:
419:
395:
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319:
289:
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262:
241:
194:
171:
150:
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117:
104:
92:
66:
753:
678:the mean function of recurrent events.
721:Correlated event times within subjects
804:International Journal of Epidemiology
7:
793:
791:
643:Description of recurrent event data
616:
581:
546:
371:
274:
14:
229:{\displaystyle \{N(t),0\leq t\}}
987:10.1080/01621459.1989.10478873
697:Cox proportional hazard models
611:
602:
596:
587:
572:
566:
552:
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532:
526:
519:
512:
468:
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450:
444:
383:{\displaystyle [t,t+\Delta t)}
377:
356:
333:
321:
301:
282:
252:
246:
208:
202:
139:denote the event times, where
1:
49:Comparing groups of processes
16:Branch of survival analysis
1031:
673:Marginal means/rates model
772:10.1007/978-0-387-69810-6
186:th event. The associated
965:Wei, L. J.; Lin, D. Y.;
910:The Annals of Statistics
647:As a counterpart of the
20:Recurrent event analysis
954:10.1093/biomet/68.2.373
57:Notation and frameworks
923:10.1214/aos/1176345976
863:Statistics in Medicine
632:
496:
428:
404:
384:
340:
308:
278:
230:
180:
160:
133:
81:
633:
497:
429:
405:
385:
341:
309:
258:
231:
181:
161:
159:{\displaystyle T_{k}}
134:
82:
506:
438:
418:
394:
353:
318:
240:
193:
170:
143:
91:
65:
166:is the time of the
80:{\displaystyle t=0}
981:(408): 1065–1073.
817:10.1093/ije/dyu222
695:Extensions of the
649:Kaplan–Meier curve
628:
559:
492:
424:
412:intensity function
400:
380:
336:
304:
226:
176:
156:
129:
77:
1015:Survival analysis
781:978-0-387-69809-0
682:Multi-state model
623:
541:
427:{\displaystyle t}
403:{\displaystyle t}
179:{\displaystyle k}
24:survival analysis
1022:
999:
998:
962:
956:
942:
936:
935:
925:
901:
895:
894:
854:
848:
847:
837:
819:
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637:
635:
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629:
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614:
561:
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522:
501:
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431:
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389:
387:
386:
381:
345:
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342:
339:{\displaystyle }
337:
313:
311:
310:
305:
294:
293:
277:
272:
235:
233:
232:
227:
188:counting process
185:
183:
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177:
165:
163:
162:
157:
155:
154:
138:
136:
135:
130:
122:
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109:
108:
86:
84:
83:
78:
1030:
1029:
1025:
1024:
1023:
1021:
1020:
1019:
1005:
1004:
1003:
1002:
964:
963:
959:
943:
939:
903:
902:
898:
856:
855:
851:
797:
796:
789:
782:
760:
759:
755:
750:
741:
732:
730:Robust variance
723:
707:Baseline hazard
693:
684:
675:
662:
657:
645:
615:
562:
504:
503:
436:
435:
416:
415:
392:
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351:
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316:
315:
285:
238:
237:
191:
190:
168:
167:
146:
141:
140:
113:
100:
89:
88:
63:
62:
59:
37:
22:is a branch of
17:
12:
11:
5:
1028:
1026:
1018:
1017:
1007:
1006:
1001:
1000:
957:
937:
896:
849:
810:(1): 324–333.
787:
780:
752:
751:
749:
746:
740:
739:Frailty models
737:
731:
728:
722:
719:
715:
714:
711:
708:
705:
704:Risk intervals
692:
689:
683:
680:
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621:
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534:
531:
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521:
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491:
488:
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479:
476:
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467:
464:
461:
458:
455:
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449:
446:
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423:
399:
379:
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373:
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367:
364:
361:
358:
335:
332:
329:
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323:
303:
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297:
292:
288:
284:
281:
276:
271:
268:
265:
261:
257:
254:
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222:
219:
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213:
210:
207:
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201:
198:
175:
153:
149:
128:
125:
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116:
112:
107:
103:
99:
96:
76:
73:
70:
58:
55:
54:
53:
50:
47:
44:
36:
33:
15:
13:
10:
9:
6:
4:
3:
2:
1027:
1016:
1013:
1012:
1010:
996:
992:
988:
984:
980:
976:
972:
968:
967:Weissfeld, L.
961:
958:
955:
951:
947:
941:
938:
933:
929:
924:
919:
915:
911:
907:
900:
897:
892:
888:
884:
880:
876:
872:
868:
864:
860:
853:
850:
845:
841:
836:
831:
827:
823:
818:
813:
809:
805:
801:
794:
792:
788:
783:
777:
773:
769:
765:
764:
757:
754:
747:
745:
738:
736:
729:
727:
720:
718:
712:
709:
706:
703:
702:
701:
698:
690:
688:
681:
679:
672:
670:
667:
666:Poisson model
660:Poisson model
659:
654:
652:
650:
642:
640:
625:
619:
608:
605:
599:
593:
590:
584:
578:
575:
569:
563:
555:
549:
538:
529:
523:
515:
509:
486:
483:
480:
477:
474:
471:
465:
459:
453:
447:
441:
421:
413:
397:
374:
368:
365:
362:
359:
347:
330:
327:
324:
298:
295:
290:
286:
279:
269:
266:
263:
259:
255:
249:
243:
220:
217:
214:
211:
205:
199:
189:
173:
151:
147:
126:
123:
118:
114:
110:
105:
101:
97:
94:
74:
71:
68:
56:
51:
48:
45:
42:
41:
40:
34:
32:
28:
25:
21:
978:
974:
960:
945:
940:
913:
909:
899:
869:(1): 13–33.
866:
862:
852:
807:
803:
762:
756:
742:
733:
724:
716:
694:
685:
676:
663:
646:
411:
348:
187:
60:
38:
35:Introduction
29:
19:
18:
946:Biometrika
748:References
995:0162-1459
932:0090-5364
883:0277-6715
826:1464-3685
617:Δ
591:−
582:Δ
553:↓
547:Δ
510:λ
478:≤
372:Δ
296:≤
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260:∑
218:≤
127:…
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1009:Category
969:(1989).
891:10623910
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710:Risk set
835:4339761
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776:ISBN
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