Asymptotic Behavior of Cox's Partial Likelihood and its Application to Variable Selection

Area of Science:

• Statistics
• Biostatistics
• Survival Analysis

Background:

• Cox's proportional hazards model is widely used for survival data analysis.
• Variable selection is crucial for building reliable Cox models.
• The asymptotic properties of partial likelihood are not fully understood, impacting model selection procedures.

Purpose of the Study:

• To investigate the asymptotic behavior of partial likelihood for Cox's model.
• To analyze the model selection consistency of penalized partial likelihood methods.
• To evaluate the performance of generalized cross-validation (GCV) for tuning parameter selection.

Main Methods:

• Theoretical analysis of the asymptotic behavior of partial likelihood.
• Derivation of convergence rates for the sample average of partial likelihood.
• Application of asymptotic results to penalized partial likelihood with GCV tuning.
• Empirical validation using Monte Carlo simulations and a real data example.

Main Results:

• Partial likelihood does not converge to a finite value like ordinary likelihood; its sample average diverges logarithmically with sample size.
• Penalized partial likelihood with GCV tuning demonstrates model selection consistency for Cox models.
• GCV, AIC, and BIC are not universally model selection consistent for Cox models, unlike in linear regression.

Conclusions:

• The unique asymptotic behavior of partial likelihood necessitates specialized approaches for variable selection in Cox models.
• GCV provides a theoretically sound method for tuning parameter selection, ensuring model consistency.
• Findings are supported by simulation studies and a practical data application, enhancing confidence in the proposed methods.