Effects of Different Hazard Ratios on Asymptotic Relative Efficiency of Estimates from Cox's Model

Abstract

Jeong and Oakes (2003) investigated the asymptotic relative efficiencies (ARE) of the partial likelihood estimator and the semiparametric cumulative hazard estimator from the proportional hazards model under the assumption of gamma hazard ratio. The results showed that the loss of efficiency of the estimators relative to an exponential parametric model can be substantial, especially for the estimates of survival probability, when the regression parameter is not close to 0. A natural question arises whether the efficiency loss holds for different distributional assumptions for the hazard ratio in the proportional hazards model. In this article, we derive some general formulas for the variances of the parametric and semiparametric estimators from the Cox's model, which can be applied to other types of hazard ratio models than gamma. The ARE of the semiparametric estimators under the positive stable and inverse Gaussian hazard ratio models are compared to ones under the gamma hazard ratio model. Special types of censorship models and some of their interesting results are also presented.

Key Words:

• Conditional proportional censoring
• Efficiency
• Frailty model
• Inverse Gaussian
• Laplace transform
• Marginal proportional censoring
• Positive stable
• Proportional hazards model