Goodness-of-Fit Analysis for the Cox Regression Model Based on a Class of Parameter Estimators

Abstract

Abstract In this article we propose a class of estimation functions for the vector of regression parameters in the Cox proportional hazards model with possibly time-dependent covariates by incorporating the weight functions commonly used in weighted log-rank tests into the partial likelihood score function. The resulting estimators behave much like the conventional maximum partial likelihood estimator in that they are consistent and asymptotically normal. When the Cox model is inappropriate, however, the estimators with different weight functions generally converge to nonidentical constant vectors. For example, the magnitude of the parameter estimator using the Kaplan–Meier survival estimator as the weight function will be stochastically larger than that of the maximum partial likelihood estimator if covariate effects diminish over time. Such facts motivate us to develop goodness-of-fit methods for the Cox regression model by comparing parameter estimators with different weight functions. Under the assume...