Survival analysis with delayed entry in selected families with application to human longevity

A tutorial on frailty models

The hazard function plays a central role in survival analysis. In a homogeneous population, the distribution of the time to event, described by the hazard, is the same for each individual. Heterogeneity in the distributions can be accounted for by including covariates in a model for the hazard, for instance a proportional hazards model. In this model, individuals with the same value of the covariates will have the same distribution. It is natural to think that not all covariates that are thought to influence the distribution of the survival outcome are included in the model. This implies that there is unobserved heterogeneity; individuals with the same value of the covariates may have different distributions. One way of accounting for this unobserved heterogeneity is to include random effects in the model. In the context of hazard models for time to event outcomes, such random effects are called frailties, and the resulting models are called frailty models. In this tutorial, we study frailty models for survival outcomes. We illustrate how frailties induce selection of healthier individuals among survivors, and show how shared frailties can be used to model positively dependent survival outcomes in clustered data. The Laplace transform of the frailty distribution plays a central role in relating the hazards, conditional on the frailty, to hazards and survival functions observed in a population. Available software, mainly in R, will be discussed, and the use of frailty models is illustrated in two different applications, one on center effects and the other on recurrent events.

Longevity Around the Turn of the 20th Century: Life-Long Sustained Survival Advantage for Parents of Today's Nonagenarians

Members of longevous families live longer than individuals from similar birth cohorts and delay/escape age-related diseases. Insight into this familial component of longevity can provide important knowledge about mechanisms protecting against age-related diseases. This familial component of longevity was studied in the Leiden Longevity Study which consists of 944 longevous siblings (participants), their parents (N = 842), siblings (N = 2,302), and spouses (N = 809). Family longevity scores were estimated to explore whether human longevity is transmitted preferentially through the maternal or paternal line. Standardized mortality ratios (SMRs) were estimated to investigate whether longevous siblings have a survival advantage compared with longevous singletons and we investigated whether parents of longevous siblings harbor a life-long sustained survival advantage compared with the general Dutch population by estimating lifetime SMRs (L-SMRs). We found that sibships with long-lived mothers and non-long-lived fathers had 0.41 (p = .024) less observed deaths than sibships with long-lived fathers and non-long-lived mothers and 0.48 (p = .008) less observed deaths than sibships with both parents non-long lived. Participants had 18.6 per cent less deaths compared with matched singletons and parents had a life-long sustained survival advantage (L-SMR = 0.510 and 0.688). In conclusion, genetic longevity studies may incorporate the maternal transmission pattern and genes influencing the entire life-course of individuals.