A bivariate frailty model for events with a permanent survivor fraction and non-monotonic hazards; with an application to age at first maternity

For certain life cycle events a non-susceptible fraction of subjects will never undergo the event. In demographic applications, examples are provided by marriage and age at first maternity. A model for survival data allowing a permanent survival fraction, non-monotonic failure rates and unobserved frailty is considered here. Regressions are used to explain both the failure time and permanent survival mechanisms and additive correlated errors are included in the general linear models defining these regressions. A hierarchical Bayesian approach is adopted with likelihood conditional on the random frailty effects and a second stage prior defining the bivariate density of those effects. The gain in model fit, and potential effects on inference, from adding frailty is demonstrated in a case study application to age at first maternity in Germany.