Extending the Archimedean copula methodology to model multivariate survival data grouped in clusters of variable size

TL;DR

This paper extends Archimedean copula models to handle multivariate survival data with variable-sized clusters, providing new estimators and demonstrating their properties through simulations and real data application.

Contribution

It introduces a copula model for clustered survival data with variable cluster sizes using Archimedean copulas, along with consistent and asymptotically normal estimators.

Findings

Estimators are consistent and asymptotically normal.

Simulation studies show good finite sample properties.

Application to cow insemination data demonstrates practical utility.

Abstract

For the analysis of clustered survival data, two different types of models that take the association into account, are commonly used: frailty models and copula models. Frailty models assume that conditional on a frailty term for each cluster, the hazard functions of individuals within that cluster are independent. These unknown frailty terms with their imposed distribution are used to express the association between the different individuals in a cluster. Copula models on the other hand assume that the joint survival function of the individuals within a cluster is given by a copula function, evaluated in the marginal survival function of each individual. It is the copula function which describes the association between the lifetimes within a cluster. A major disadvantage of the present copula models over the frailty models is that the size of the different clusters must be small and equal in order to set up manageable estimation procedures for the different model parameters. We describe in this manuscript a copula model for clustered survival data where the clusters are allowed to be moderate to large and varying in size by considering the class of Archimedean copulas with completely monotone generator. We develop both one- and two-stage estimators for the different copula parameters. Furthermore we show the consistency and asymptotic normality of these estimators. Finally, we perform a simulation study to investigate the finite sample properties of the estimators. We illustrate the method on a data set containing the time to first insemination in cows, with cows clustered in herds.