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

• Published in: Journal of the Royal Statistical Society. Series B, Statistical methodology Vol. 79; no. 2; pp. 483 - 505
• Main Authors: Prenen, Leen, Braekers, Roel, Duchateau, Luc
• Format: Journal Article
• Language: English
• Published: Oxford John Wiley & Sons Ltd 01.03.2017; Oxford University Press
• Subjects: Archimedean copula; Clusters; Consistency; Estimators; Frailty; Mathematical models; Methodology; Multivariate survival data; Parameter estimation; Parameters; Statistical analysis; Statistics; Studies; Survival; Survival analysis; Varying cluster size
• ISSN: 1369-7412; 1467-9868
• DOI: 10.1111/rssb.12174

For the analysis of clustered survival data, two different types of model that take the association into account are commonly used: frailty models and copula models. Frailty models assume that, conditionally 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 in contrast 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 to set up manageable estimation procedures for the different model parameters. We describe 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 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.