Statistics Colloquium : Dr. Roel Braekers

Abstract: In the analysis of clustered right-censored survival data, frailty and copula models are commonly used to model the influence of covariates on the distribution of a lifetime random variable while taking the association between different lifetimes within a cluster into account. Within the frailty model framework, the conditional hazard function of a lifetime is multiplied by a random effect (a frailty term) common to all lifetimes in the cluster to describe the heterogeneity between the different clusters. In the copula model framework, the viewpoint is different and the joint survival function of all lifetimes in a cluster is modeled by a copula function evaluated in the marginal survival functions of the lifetimes. The association structure between the lifetimes in a cluster is in this way fully described by this copula function and is separated from the marginal behaviour of each lifetime.

In this work, we focus on factor copula functions to model the structure between the life-times. Hereby we assume that the association between the different lifetimes in a clusters depends on a common unknown factor for each cluster. This new methodology allows for clusters to have variable sizes ranging from small to large and allows the intracluster dependence to be flexibly modeled by any parametric family of bivariate copulas. In this way, we encompass a wide range of dependence structures. In the marginal model for the separate lifetime we support the incorporation of covariates (possibly time dependent). 

For this factor copula model, we propose in this work three estimation procedures: both a one- and two-stage parametric method and a two-stage semiparametric method where marginal survival functions are estimated by using a Cox proportional hazards model. For the parameter estimators in the different models we prove that they are consistent and asymptotically normally distributed, and assess their finite sample behavior with simulation studies. Furthermore, we illustrate the proposed methods on a data set containing the time to first insemination after calving in dairy cattle clustered in herds of different sizes.