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 lifetimes.
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