A simulation study of the estimation quality in the double-Cox model with shared frailty for non-proportional hazards survival analysis

The Cox regression, a semi-parametric method of survival analysis, is extremely popular in biomedical applications. The proportional hazards assumption is a key requirement in the Cox model. To accommodate non-proportional hazards, we propose to parameterise the shape parameter of the baseline hazard function using the additional, separate Cox-regression term which depends on the vector of the covariates. We call this model the double-Cox model. The R programs for fitting the double-Cox model are available on Github. We formally introduce the double-Cox model with shared frailty and investigate, by simulation, the estimation bias and the coverage of the proposed point and interval estimation methods for the Gompertz and the Weibull baseline hazards. In applications with low frailty variance and a large number of clusters, the marginal likelihood estimation is almost unbiased and the profile likelihood-based confidence intervals provide good coverage for all model parameters. We also compare the results from the over-fitted double-Cox model to those from the standard Cox model with frailty in the case of the scale-only proportional hazards. Results of our simulations on the bias and coverage of the model parameters are provided in 12 Tables and in 145 A4 Figures, 178 pages in total.