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 survivalanalysis. The theoretical basis of the original model has been developed in various exten-sions. In the recent years, vital research has been undertaken involving the incorporation ofrandom 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 ex-plained by observable covariates. The right choice of the frailty distribution is essential foran accurate description of the dependence structure present in the data. In this paper, weaim to investigate the accuracy of inference based on the primer Cox model in the existenceof unobserved heterogeneity, that is, when the data generating mechanism is more complexthan presumed and described by the kind of an extension of the Cox model with undefinedfrailty. We show that the conventional partial likelihood estimator under the considered ex-tension is Fisher-consistent up to a scaling factor, provided symmetry-type distributionalassumptions on covariates. We also present the results of simulation experiments that revealan exemplary behaviour of the estimators.