Assessing Frailty Survival Models in Describing Variations Caused by Unobserved Covariates

Abstract

The classical Cox proportional hazards model is commonly used to assess the effects of risk factors in a homogeneous population. Quite often, variation among individuals towards risk beyond known important risk factors may lead to unobserved covariates causing heterogeneity in a population. Under these circumstances, a frailty model is introduced to include random effects, the so-called frailty term, in survival analysis to take unobserved heterogeneity or correlation into account. In this study, we investigated the performance of a classical Cox model with and without gamma frailty as well as a parametric Weibull model with gamma frailty using its bias and its mean square error of prediction of the estimated fixed effect and the estimated heterogeneity parameter in model comparison. Via simulation studies, we found that the classical Cox model with some level of unobserved heterogeneity generated bias and effected variances of the parameter estimates. Moreover, these variances by either the classical Cox or the Weibull model with the gamma frailty term were less than those of the classical Cox model in all situations. These results suggest that the both models with gamma frailty are more robust than the classical Cox model in terms of the censoring rate, group size and the number of groups, and the magnitude of the variance parameter effects.

We also applied the models to real-life data from bone marrow transplantation in the treatment of acute leukemia, the results from which showed that risk factors under the classical Cox model were different from those of the two models with the frailty term, which might be the consequence of bias. Therefore, Cox model does not always provide an adequate fit to the data. From our results, we found that incorporating the frailty term into a survival model is a good choice either with or without variations caused by unobserved covariates.