Prevalent cohort studies and unobserved heterogeneity.

*(English)*Zbl 1436.62580Summary: Consider lifetimes originating at a series of calendar times \(t_1 ,t_2 , \dots \). At a certain time \(t_0\) a cross-sectional sample is taken, generating a sample of current durations (backward recurrence times) of survivors until \(t_0\) and a prevalent cohort study consisting of survival times left-truncated at the current durations. A Lexis diagram is helpful in visualizing this situation. Survival analysis based on current durations and prevalent cohort studies is now well-established as long as all covariates are observed. The general problems with unobserved covariates have been well understood for ordinary prospective follow-up studies, with the good help of hazard rate models incorporating frailties: as for ordinary regression models, the added noise generates attenuation in the regression parameter estimates. For prevalent cohort studies this attenuation remains, but in addition one needs to take account of the differential selection of the survivors from initiation \(t_i\) to cross-sectional sampling at \(t_0\). This paper intends to survey the recent development of these matters and the consequences for routine use of hazard rate models or accelerated failure time models in the many cases where unobserved heterogeneity may be an issue. The study was inspired by concrete problems in the study of time-to-pregnancy, and we present various simulation results inspired by this particular application.

Reviewer: Reviewer (Berlin)

##### MSC:

62P10 | Applications of statistics to biology and medical sciences; meta analysis |

62N02 | Estimation in survival analysis and censored data |

62N01 | Censored data models |

62N05 | Reliability and life testing |

62G05 | Nonparametric estimation |

62P25 | Applications of statistics to social sciences |

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

##### Keywords:

current duration; survivor selection; attenuation; survival analysis; unobserved heterogeneity
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\textit{N. Keiding} et al., Lifetime Data Anal. 25, No. 4, 712--738 (2019; Zbl 1436.62580)

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##### References:

[1] | Aalen OO (1994) Effects of frailty in survival analysis. Stat Methods Med Res 3:227-243 |

[2] | Aalen OO, Borgan Ø, Gjessing HK (2008) Survival and event history analysis: a process point of view. Springer, New York · Zbl 1204.62165 |

[3] | Aalen OO, Cook RJ, Røysland K (2015a) Does Cox analysis of a randomized survival study yield a causal treatment effect? Lifetime Data Anal 21:579-593 · Zbl 1333.62228 |

[4] | Aalen OO, Valberg M, Grotmol T, Tretli S (2015b) Understanding variation in disease risk: the elusive concept of frailty (with discussion). Int J Epidemiol 44:1408-1421 |

[5] | Ali MM, Marshall T, Babiker AG (2001) Analysis of incomplete durations with application to contraceptive use. J R Stat Soc A 164:549-563 · Zbl 1002.62537 |

[6] | Andersen PK, Borgan Ø, Gill RD, Keiding N (1993) Statistical models based on counting processes. Springer, New York, p 767 · Zbl 0769.62061 |

[7] | Anderson JE, Louis TA (1995) Survival analysis using a scale change random effects model. J Am Stat Assoc 90:669-679 · Zbl 0826.62084 |

[8] | Anonymous [W. Dale] (1777). A Supplement to Calculations of the Value of Annuities, Published for the Use of Societies Instituted for Benefit of Age Containing Various Illustration of the Doctrine of Annuities, and Compleat Tables of the Value of 1£. Immediate Annuity. (Being the Only Ones Extant by Half-Yearly Interest and Payments). Together with Investigations of the State of the Laudable Society of Annuitants; Showing What Annuity Each Member Hath Purchased, and Real Mortality Therein, from its Institution Compared with Dr. Halley’s Table. Also Several publications, Letters, and Anecdotes Relative to that Society. And Explanatory of Proceedings to the Present year. London: Ridley |

[9] | Cox DR (1972) Regression models and life tables (with discussion). J R Stat Soc B 34:187-220 · Zbl 0243.62041 |

[10] | Duron S, Slama R, Ducot B, Bohet A, Sørensen DN, Keiding N, Moreau C, Bouyer J (2013) Cumulative incidence rate of medical consultation for fertility problems—analysis of a prevalent cohort using competing risks. Hum Reprod 28:2872-2879 |

[11] | Eijkemans MJC, Leridon H, Keiding N, Slama R (2019) A systematic comparison of designs to study human fecundity. Epidemiology 30:120-129 |

[12] | Farewell VT, Prentice RL (1977) A study of distributional shape in life testing. Technometrics 19:69-75 · Zbl 0352.62018 |

[13] | Hald A (1949) Maximum likelihood estimation of the parameters of a normal distribution which is truncated at a known point. Skandinavisk Aktuarietidskrift 32:119-134 · Zbl 0041.46502 |

[14] | Henderson R, Oman P (1999) Effect of frailty on marginal regression estimates in survival analysis. J R Stat Soc B 61:367-379 · Zbl 0913.62097 |

[15] | Hernan MA (2010) The hazards of hazard ratios. Epidemiology 21:13-15 |

[16] | Hougaard P (2000) Analysis of multivariate survival data. Springer, New York · Zbl 0962.62096 |

[17] | Hougaard P, Myglegaard P, Borch-Johnsen K (1994) Heterogeneity models of disease susceptibility, with application to diabetic nephropathy. Biometrics 50:1178-1188 |

[18] | Kaplan EL, Meier P (1958) Non-parametric estimation from incomplete observations. J Am Stat Assoc 53:457-481 · Zbl 0089.14801 |

[19] | Keiding N (1987) The method of expected number of deaths 1786-1886-1986. Int Stat Rev 55:1-20 · Zbl 0616.62001 |

[20] | Keiding, N.; Klein, JP (ed.); Goel, PK (ed.), Independent delayed entry (with discussion), 309-326 (1992), Dordrecht · Zbl 0761.62156 |

[21] | Keiding N (2006) Event history analysis and the cross-section. Stat Med 25:2343-2364 |

[22] | Keiding N, Sørensen DN, Ducot B, Bouyer J, Slama R. The current duration approach to analysing time-to-pregnancy: direct validation using an imbedded prevalent cohort study (in preparation) |

[23] | Keiding N, Andersen PK, Klein JP (1997) The role of frailty models and accelerated failure time models in describing heterogeneity due to omitted covariates. Stat Med 16:215-224 |

[24] | Keiding N, Kvist K, Hartvig H, Tvede M, Juul S (2002) Estimating time to pregnancy from current durations in a cross-sectional sample. Biostatistics 3:565-578 · Zbl 1138.62353 |

[25] | Keiding N, Fine JP, Hansen OH, Slama R (2011) Accelerated failure time regression for backward recurrence times and current durations. Stat Prob Lett 81:724-729 · Zbl 1217.62152 |

[26] | Keiding N, Hansen OHH, Sørensen DN, Slama R (2012) The current duration approach to estimating time to pregnancy (with discussion). Scand J Stat 39:185-213 · Zbl 1246.62211 |

[27] | Leridon H (2004) Can assisted reproduction technology compensate for the natural decline in fertility with age? A model assessment. Hum Reprod 19:1548-1553 |

[28] | Louis JF, Thoma ME, Sørensen DN, McLain AC, King RB, Sundaram R, Keiding N, Louis GMB (2013) The prevalence of couple infertility in the United States from a male perspective: evidence from a nationally representative sample. Andrology 1:741-748 |

[29] | Martinussen T, Vansteelandt S (2013) On collapsibility and confounding bias in Cox and Aalen regression models. Lifetime Data Anal 19:279-296 · Zbl 1322.62253 |

[30] | Martinussen T, Vansteelandt S, Andersen PK (2018) Subtleties in the interpretation of hazard ratios. arXiv:1810.09192 [math.ST] |

[31] | McLain AC, Sundaram R, Thoma M, Louis GMB (2018) Cautionary note on “Semiparametric modeling of grouped current duration data with preferential reporting”. arXiv:1801.00775 [stat.AP] |

[32] | McLain AC, Sundaram R, Thoma M, Louis GMB (2014) Semiparametric modeling of grouped current duration data with preferential reporting. Stat Med 33:3961-3972 |

[33] | McLaughlin KA, Green JG, Gruber MJ, Sampson NA, Zaslavsky AM, Kessler RC (2010) Childhood adversities and adult psychiatric disorders in the National Comorbidity Survey Replication II. Arch Gen Psychiatry 67:124-132 |

[34] | Munda M, Rotolo F, Legrand C (2012) Parfm: parametric frailty models in R. J Stat Softw 51:11 |

[35] | Polis CB, Cox CM, Tuncalp Ö, McLain AC, Thoma ME (2017) Estimating infertility prevalence in low-to-middle-income countries: an application of a current duration approach to Demographic and Health Survey data. Hum Reprod 32:1064-1074 |

[36] | Scheike T, Keiding N (2006) Design and analysis of time to pregnancy. Stat Methods Med Res 15:127-140 · Zbl 1122.62374 |

[37] | Slama R, Ducot B, Carstensen L, Lorente C, de La Rochebrochard E, Leridon H, Keiding N, Bouyer J (2006) Feasibility of the current duration approach to study human fecundity. Epidemiology 17:440-449 |

[38] | Slama R, Hansen O, Ducot B, Bohet A, Sorensen D, Giorgis Allemand L, Eijkemans MJC, Rosetta L, Thalabard JC, Keiding N, Bouyer J (2012) Estimation of the frequency of involuntary infertility on a nation-wide basis. Hum Reprod 27:1489-1498 |

[39] | Therneau T (2012) survival: survival analysis, including penalised likelihood. R package version 2.36-14. http://CRAN.R-project.org/package=survival |

[40] | Thoma ME, McLain AC, Louis JF, King RB, Trumble AC, Sundaram R, Louis GMB (2013) Prevalence of infertility in the United States as estimated by the current duration approach and a traditional constructed approach. Fertil Steril 99:1324-1331 |

[41] | van Es B, Klaassen CAJ, Oudshoorn K (2000) Survival analysis under cross sectional sampling: length bias and multiplicative censoring. J Stat Plan Inf 91:295-312 · Zbl 0969.62062 |

[42] | Vaupel JW, Manton KG, Stallard E (1979) The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography 16:439-454 |

[43] | Weinberg CS, Gladen BC (1986) The beta-geometric distribution applied to comparative fecundity studies. Biometrics 42:547-560 |

[44] | Yamaguchi K (2003) Accelerated failure-time mover-stayer regression models for the analysis of last episode data. Sociol Methodol 33:81-110 |

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