Abstract
Multivariate survival data are frequently encountered in biomedical applications in the form of clustered failures (or recurrent events data). A popular way of analyzing such data is by using shared frailty models, which assume that the proportional hazards assumption holds conditional on an unobserved cluster‐specific random effect. Such models are often incorporated in more complicated joint models in survival analysis.If the random effect distribution has finite expectation, then the conditional proportional hazards assumption does not carry over to the marginal models. It has been shown that, for univariate data, this makes it impossible to distinguish between the presence of unobserved heterogeneity (eg, due to missing covariates) and marginal nonproportional hazards. We show that time‐dependent covariate effects may falsely appear as evidence in favor of a frailty model also in the case of clustered failures or recurrent events data, when the cluster size or number of recurrent events is small. When true unobserved heterogeneity is present, the presence of nonproportional hazards leads to overestimating the frailty effect. We show that this phenomenon is somewhat mitigated as the cluster size grows.We carry out a simulation study to assess the behavior of test statistics and estimators for frailty models in such contexts. The gamma, inverse Gaussian, and positive stable shared frailty models are contrasted using a novel software implementation for estimating semiparametric shared frailty models. Two main questions are addressed in the contexts of clustered failures and recurrent events: whether covariates with a time‐dependent effect may appear as indication of unobserved heterogeneity and whether the additional presence of unobserved heterogeneity can be detected in this case. Finally, the practical implications are illustrated in a real‐world data analysis example.
Highlights
The Cox proportional hazards model[1] is widely used for analyzing survival data
We address the following question: how “multivariate” must the data be so that shared frailty models reflect the strength of the effects of unobserved heterogeneity, rather than a possible time-dependent effect of the observed covariates? This question is addressed through a simulation study, where we simulate time-dependent hazard ratios and attempt to estimate different frailty models
It is well known that a proportional hazards frailty model and a nonproportional hazards model cannot be distinguished on the basis of the data alone
Summary
The Cox proportional hazards model[1] is widely used for analyzing survival data. More generally, this model may be used for multivariate survival data, in the form of clustered failures or recurrent events.[2]. This additional information means that, in theory, the time-dependent covariate effects and the unobserved heterogeneity scenarios can be distinguished This is not the case in univariate survival data, which can be seen as a scenario where clusters have size 1 and where individuals experience at most one event. In twin studies, the lifetime of the second twin is always censored after birth In this case, there is virtually no information on the correlation of the event times, but a shared frailty model is still identifiable if covariates are present.[6] When the cluster size is small, there is a confounding between the regression parameters and the dependence structure.7,Ch 7.2.7.
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