Commentary on Andersen and Gill's "Cox's Regression Model for Counting Processes: A Large Sample Study"

In this issue Andersen and Gill (hereafter AG) present a stimulating development of asymptotic distribution theory for the Cox regression model with time-dependent covariates. They use a counting process formulation for the failure time data and martingale covergence results. This approach involves such conditions as $\sigma$ -algebra right continuity and predictable, locally bounded, covariate processes. In this commentary we consider the implications of such assumptions for likelihood factorization and covariate modeling. In particular, it is noted that the partial likelihood function modeled by AG cannot, in general, involve covariate measurements at the random failure times. Some related work by the authors on a partial likelihood function that may involve covariate values at the random failure times is briefly discussed. Assumptions under which the intensity process modeled by AG has a standard "hazard" function interpretation are described and some generalizations of the AG results are mentioned.In this issue Andersen and Gill (hereafter AG) present a stimulating development of asymptotic distribution theory for the Cox regression model with time-dependent covariates. They use a counting process formulation for the failure time data and martingale covergence results. This approach involves such conditions as $\sigma$-algebra right continuity and predictable, locally bounded, covariate processes. In this commentary we consider the implications of such assumptions for likelihood factorization and covariate modeling. In particular, it is noted that the partial likelihood function modeled by AG cannot, in general, involve covariate measurements at the random failure times. Some related work by the authors on a partial likelihood function that may involve covariate values at the random failure times is briefly discussed. Assumptions under which the intensity process modeled by AG has a standard "hazard" function interpretation are described and some generalizations of the AG results are mentioned.

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