404:
birth year cohort is insufficiently numerous for statistics, and thus multiple years are often used. Secondly, due to the great ages, accurate records of persons living over 100 years require records dating from the late 19th or early 20th century, when such record-keeping was often not high-quality; further, there is a tendency to exaggerate one's age, which distorts data. Thirdly, granularity is an issue – ideally exact day of birth and death would be used; using only year of birth and death introduces granularity, which adds bias (as discussed below).
115:
414:, using the method of extinct generations, and found that the effect disappeared if various distorting factors were removed. Specifically, they conclude that mortality deceleration is negligible up to the age of 106 in the population studied (beyond this point, reliable data were unavailable) and that the Gompertz law is a good fit, with previous observations of deceleration being spurious, with various causes, discussed below.
314:
25:
183:, concludes that mortality deceleration is negligible up to the age of 106 in the population studied (beyond this point, reliable data were unavailable) and that the Gompertz law is a good fit, with previous observations of deceleration being spurious, with various causes, including bad data and methodological problems – see
403:
Statistical studies of extreme longevity are difficult for a number of factors. Firstly, because few people live to very old ages, a very large population is required for such studies, ideally all born and living in similar conditions (same country, same birth year). In small countries, a single
525:, an organism's redundancy (reserves) are exhausted at extremely old ages, so every random hit results in death. In more detailed accounts, this only holds in relatively simple organisms – in more complex organisms with various subsystems, deceleration is less present or entirely absent.
498:
For example, if hazard rate is high, significantly more deaths should be expected in the first month of the period than the last month (since fewer people survive to the end); assuming that deaths occur evenly throughout the period or in the middle of the period overstates length of
478:
of the logarithm yields an unbiased and maximum likelihood estimator (assuming intervals and change in the hazard rate are small), but other methods, such as the "actuarial estimate", yield bias (especially due to concavity of an exponential function on a linear
518:), is by far the most common explanation of mortality deceleration. Even if individual hazard rate follows a Gompertz law, if the population is heterogeneous, the population hazard rate may exhibit late-life deceleration.
145:
Late-life mortality deceleration is a well-established phenomenon in insects, which often spend much of their lives in a constant hazard rate region, but it is much more controversial in mammals.
277:"the increase of mortality rate with age advances at a slackening rate, that nearly all, perhaps all, methods of graduation of the type of Gompertz’s formula over-state senile mortality"
149:
have found varying conclusions, with some finding short-term periods of mortality deceleration in mice, others not finding such. Baboon studies show no mortality deceleration.
455:
For those of sufficient age, this appears to divide the death rate by their age, because as people age that one year becomes a smaller and smaller fraction of their life.
422:
Given that mortality deceleration in humans had been observed in various studies, but disappeared on the careful analysis (of single-year cohorts in the US) in
552:
The rates of late-life mortality are important for pensions. For example, the mortality rates in late life (after age 85) are of particular interest for the
727:
452:
Use of probability of death (over an discrete interval of time, usually one year), rather than the hazard rate (force of mortality; instantaneous rate)
459:
810:
Economos, A. C. (1979). "A non-Gompertzian paradigm for mortality kinetics of metazoan animals and failure kinetics of manufactured products".
432:
propose several causes; notable, in each instance when such a factor is corrected or diminished, the fit with the
Gompertz law becomes better.
382:
93:
354:
65:
911:"On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies"
361:
72:
245:
This is used synonymously with "mortality leveling-off", or rather to refer to the region where hazard rate is approximately constant.
559:
Late-life mortality rates are of basic importance for understanding aging, both for organisms generally and for humans specifically.
157:
368:
79:
324:
35:
300:, and models have incorporated it. It has been criticized at times, and recently has been very seriously criticized; see below.
800:
350:
61:
225:
towards a limit), and then proceeds at a constant rate (or rather, approaches a constant rate), yielding (slightly sub-)
297:
172:
766:
339:
190:
According to a 2018 paper, statistical errors are the main cause of apparent mortality deceleration in humans.
50:
375:
86:
910:
335:
331:
46:
42:
862:"Mortality Measurement at Advanced Ages: A Study of the Social Security Administration Death Master File"
522:
205:
Hazard rate increasing at a decreasing rate (rather than increasing log-linearly as in the
Gompertz law).
1012:
442:
There is at times a tendency for old people to exaggerate their ages; this reduces apparent mortality.
540:
Late-life mortality deceleration can be modeled via modifications of the
Gompertz law, using various
296:
Following these studies, late-life mortality deceleration became one of the pillars of the theory of
146:
258:, 2. Mortality at Advanced Ages: A Historical Review (pp. 433–435)); a detailed survey is given in
746:
556:
generation, which will reach this age starting in 2030, and for pensions funding calculations.
426:, it is natural to ask what causes this discrepancy – why was mortality deceleration observed?
788:
280:"the possibility that with advancing age the rate of mortality asymptotes to a finite value"
861:
138:
increases at a decreasing rate in late life rather than increasing exponentially as in the
8:
1017:
475:
474:
Since the
Gompertz law is log-linear, working on a semi-log scale and taking the central
990:
961:
932:
889:
845:
758:
652:
617:
953:
936:
906:
894:
837:
829:
796:
750:
657:
639:
541:
411:
230:
226:
849:
762:
265:
Late-life mortality deceleration was first proposed as occurring in human aging, in
982:
922:
884:
880:
876:
821:
742:
647:
629:
163:
Late-life mortality deceleration was first proposed as occurring in human aging in
179:; here "late life" is typically "after 85 years of age". However, a recent paper,
857:
634:
167:(which also introduced the Gompertz law), and observed as occurring in humans in
1006:
957:
833:
812:
643:
618:"Errors as a primary cause of late-life mortality deceleration and plateaus"
269:, which also introduced the Gompertz law. It was observed and quantified in
927:
898:
754:
661:
153:
139:
973:
Olshansky, S. J. (1998). "On the
Biodemography of Aging: A Review Essay".
841:
135:
127:
965:
994:
825:
119:
944:
Greenwood, M.; Irwin, J. O. (1939). "The biostatistics of senility".
553:
489:
Mixing together several birth cohorts with different mortality levels
468:
222:
986:
726:
Shahrestani, Parvin; Mueller, Laurence D.; Rose, Michael R. (2009).
528:
Less risky behavior and more sheltered environment for older people.
495:
Crude assumptions, such as uniformity of death over a time interval
313:
122:
for 2003, showing considerable acceleration rather than deceleration
24:
585:
273:, and reproduced in many later studies. Greenwood and Irwin wrote:
575:
573:
571:
114:
668:
568:
685:
683:
156:
of manufactured products; this analogy is elaborated in the
695:
680:
417:
410:
examined single birth-year cohorts from the United States
915:
707:
793:
The
Lifespan of Animals, Volume 5: Colloquia on Ageing
725:
458:
Probabilities of death can be reconstructed using the
510:
Several causes are proposed for late-life mortality:
791:. In Wolstenholme, G.E.W.; O'Connor, M. O. (eds.).
597:
856:
674:
591:
579:
429:
423:
407:
255:
180:
171:, and has since become one of the pillars of the
1004:
943:
713:
701:
689:
270:
198:Three related terms are used in this context:
168:
789:"Note on Some Mathematical Mortality Models"
514:Population heterogeneity (first proposed in
492:Using cross-sectional instead of cohort data
340:introducing citations to additional sources
51:introducing citations to additional sources
795:. Boston: Little, Brown. pp. 302–311.
972:
952:(1). Wayne State University Press: 1–23.
926:
888:
747:10.1093/acprof:oso/9780199754229.001.0001
651:
633:
259:
158:reliability theory of aging and longevity
905:
809:
603:
418:Why was mortality deceleration observed?
330:Relevant discussion may be found on the
266:
164:
152:An analogous deceleration occurs in the
113:
41:Relevant discussion may be found on the
1005:
615:
292:are 0.439 for women and 0.544 for men"
254:A brief historical review is given in
217:More strongly, hazard rate eventually
786:
515:
307:
18:
13:
531:Various evolutionary explanations.
351:"Late-life mortality deceleration"
62:"Late-life mortality deceleration"
14:
1029:
975:Population and Development Review
869:North American Actuarial Journal
860:; Gavrilova, Natalia S. (2011).
323:relies largely or entirely on a
312:
298:biodemography of human longevity
202:Late-life mortality deceleration
173:biodemography of human longevity
132:late-life mortality deceleration
34:relies largely or entirely on a
23:
521:Redundancy exhaustion – in the
430:Gavrilov & Gavrilova (2011)
424:Gavrilov & Gavrilova (2011)
408:Gavrilov & Gavrilova (2011)
181:Gavrilov & Gavrilova (2011)
881:10.1080/10920277.2011.10597629
719:
609:
256:Gavrilov & Gavrilova (2011
184:
1:
779:
675:Gavrilov & Gavrilova 2011
592:Gavrilov & Gavrilova 2011
580:Gavrilov & Gavrilova 2011
635:10.1371/journal.pbio.2006776
616:Newman, Saul Justin (2018).
562:
547:
303:
271:Greenwood & Irwin (1939)
193:
176:
169:Greenwood & Irwin (1939)
134:is the disputed theory that
7:
535:
523:reliability theory of aging
471:of theoretical hazard rate
240:Late-life mortality plateau
10:
1034:
714:Greenwood & Irwin 1939
702:Greenwood & Irwin 1939
690:Greenwood & Irwin 1939
249:
505:
283:"the limiting values of
221:increasing (or rather,
928:10.1098/rstl.1825.0026
820:(3). Springer: 74–76.
212:Mortality leveling-off
123:
787:Beard, R. E. (1959).
735:Current Aging Science
117:
16:Theory in gerontology
336:improve this article
47:improve this article
858:Gavrilov, Leonid A.
594:, pp. 433–434.
476:difference quotient
826:10.1007/bf02432250
728:"Does Aging Stop?"
124:
1013:Actuarial science
802:978-0-470-71494-2
439:Age exaggeration
412:Death Master File
401:
400:
386:
231:radioactive decay
227:exponential decay
112:
111:
97:
1025:
998:
969:
940:
930:
902:
892:
866:
853:
806:
774:
773:
771:
765:. Archived from
732:
723:
717:
711:
705:
699:
693:
687:
678:
672:
666:
665:
655:
637:
628:(12): e2006776.
613:
607:
601:
595:
589:
583:
577:
396:
393:
387:
385:
344:
316:
308:
260:Olshansky (1998)
242:
241:
214:
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107:
104:
98:
96:
55:
27:
19:
1033:
1032:
1028:
1027:
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1024:
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987:10.2307/2807981
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803:
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688:
681:
673:
669:
614:
610:
602:
598:
590:
586:
578:
569:
565:
550:
542:logistic models
538:
508:
462:, if necessary.
420:
397:
391:
388:
345:
343:
329:
317:
306:
291:
267:Gompertz (1825)
252:
239:
238:
211:
210:
196:
165:Gompertz (1825)
108:
102:
99:
56:
54:
40:
28:
17:
12:
11:
5:
1031:
1021:
1020:
1015:
1000:
999:
981:(2): 381–393.
970:
941:
903:
875:(3): 432–447.
854:
807:
801:
783:
781:
778:
776:
775:
772:on 2019-02-25.
718:
706:
694:
679:
677:, p. 433.
667:
608:
596:
584:
582:, p. 434.
566:
564:
561:
549:
546:
537:
534:
533:
532:
529:
526:
519:
507:
504:
503:
502:
501:
500:
493:
490:
483:
482:
481:
480:
467:Use of biased
465:
464:
463:
460:Sacher formula
456:
446:
445:
444:
443:
435:Data quality:
419:
416:
399:
398:
334:. Please help
320:
318:
311:
305:
302:
294:
293:
287:
281:
278:
251:
248:
247:
246:
243:
235:
234:
215:
207:
206:
203:
195:
192:
147:Rodent studies
110:
109:
45:. Please help
31:
29:
22:
15:
9:
6:
4:
3:
2:
1030:
1019:
1016:
1014:
1011:
1010:
1008:
996:
992:
988:
984:
980:
976:
971:
967:
963:
959:
955:
951:
947:
946:Human Biology
942:
938:
934:
929:
924:
920:
916:
912:
908:
904:
900:
896:
891:
886:
882:
878:
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870:
863:
859:
855:
851:
847:
843:
839:
835:
831:
827:
823:
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814:
808:
804:
798:
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785:
784:
768:
764:
760:
756:
752:
748:
744:
740:
736:
729:
722:
715:
710:
704:, p. 21.
703:
698:
692:, p. 14.
691:
686:
684:
676:
671:
663:
659:
654:
649:
645:
641:
636:
631:
627:
623:
619:
612:
605:
604:Economos 1979
600:
593:
588:
581:
576:
574:
572:
567:
560:
557:
555:
545:
543:
530:
527:
524:
520:
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511:
497:
496:
494:
491:
488:
487:
486:
485:Methodology:
477:
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466:
461:
457:
454:
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451:
450:
449:
441:
440:
438:
437:
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431:
427:
425:
415:
413:
409:
405:
395:
384:
381:
377:
374:
370:
367:
363:
360:
356:
353: –
352:
348:
347:Find sources:
341:
337:
333:
327:
326:
325:single source
321:This section
319:
315:
310:
309:
301:
299:
290:
286:
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268:
263:
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121:
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95:
92:
88:
85:
81:
78:
74:
71:
67:
64: –
63:
59:
58:Find sources:
52:
48:
44:
38:
37:
36:single source
32:This article
30:
26:
21:
20:
978:
974:
949:
945:
918:
914:
907:Gompertz, B.
872:
868:
817:
811:
792:
767:the original
738:
734:
721:
709:
697:
670:
625:
622:PLOS Biology
621:
611:
599:
587:
558:
551:
539:
516:Beard (1959)
509:
484:
447:
434:
428:
421:
406:
402:
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379:
372:
365:
358:
346:
322:
295:
288:
284:
264:
253:
218:
197:
189:
162:
154:failure rate
151:
144:
140:Gompertz law
131:
125:
100:
90:
83:
76:
69:
57:
33:
921:: 513–585.
741:(1): 3–11.
448:Technical:
136:hazard rate
128:gerontology
1018:Senescence
1007:Categories
780:References
469:estimators
392:March 2020
362:newspapers
223:asymptotes
120:life table
118:US period
73:newspapers
958:0018-7143
937:145157003
834:0161-9152
644:1545-7885
563:Citations
554:baby boom
548:Relevance
332:talk page
304:Criticism
194:Phenomena
185:criticism
43:talk page
966:41447403
909:(1825).
899:22308064
850:11362971
763:19036844
755:20021395
662:30571676
536:Modeling
229:, as in
103:May 2021
995:2807981
890:3269912
653:6301557
479:scale).
376:scholar
250:History
177:history
87:scholar
993:
964:
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935:
897:
887:
848:
842:486056
840:
832:
799:
761:
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506:Causes
378:
371:
364:
357:
349:
175:– see
89:
82:
75:
68:
60:
991:JSTOR
962:JSTOR
933:S2CID
865:(PDF)
846:S2CID
770:(PDF)
759:S2CID
731:(PDF)
499:life.
383:JSTOR
369:books
219:stops
94:JSTOR
80:books
954:ISSN
895:PMID
838:PMID
830:ISSN
797:ISBN
751:PMID
658:PMID
640:ISSN
355:news
66:news
983:doi
923:doi
919:115
885:PMC
877:doi
822:doi
813:AGE
743:doi
648:PMC
630:doi
338:by
126:In
49:by
1009::
989:.
979:24
977:.
960:.
950:11
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931:.
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913:.
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883:.
873:15
871:.
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828:.
816:.
757:.
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737:.
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626:16
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620:.
570:^
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262:.
187:.
160:.
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130:,
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901:.
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342:.
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