Information criteria for Firth's penalized partial likelihood approach in Cox regression models

In the estimation of Cox regression models, maximum partial likelihood estimates might be infinite in a monotone likelihood setting, where partial likelihood converges to a finite value and parameter estimates converge to infinite values. To address monotone likelihood, previous studies have applied Firth's bias correction method to Cox regression models. However, while the model selection criteria for Firth's penalized partial likelihood approach have not yet been studied, a heuristic AIC-type information criterion can be used in a statistical package. Application of the heuristic information criterion to data obtained from a prospective observational study of patients with multiple brain metastases indicated that the heuristic information criterion selects models with many parameters and ignores the adequacy of the model. Moreover, we showed that the heuristic information criterion tends to select models with many regression parameters as the sample size increases. Thereby, in the present study, we propose an alternative AIC-type information criterion based on the risk function. A Bayesian information criterion type was also evaluated. Further, the presented simulation results confirm that the proposed criteria performed well in a monotone likelihood setting. The proposed AIC-type criterion was applied to prospective observational study data.