Scaled Fisher consistency for the partial likelihood estimation in various extensions of the Cox model

The Cox proportional hazards model has become the most widely used procedure in survival analysis. The theoretical basis of the original model has been developed in various extensions. In the recent years, vital research has been undertaken involving the incorporation of random effects to survival models. In this setting, the random effect is a variable (frailty) which embraces a variation among individuals or groups of individuals which cannot be explained by observable covariates. The right choice of the frailty distribution is essential for an accurate description of the dependence structure present in the data. In this paper, we aim to investigate the accuracy of inference based on the primer Cox model in the existence of unobserved heterogeneity, that is, when the data generating mechanism is more complex than presumed and described by the kind of an extension of the Cox model with undefined frailty. We show that the conventional partial likelihood estimator under the considered extension is Fisher-consistent up to a scaling factor, provided symmetry-type distributional assumptions on covariates. We also present the results of simulation experiments that reveal an exemplary behaviour of the estimators.