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

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 assumed model, the normalized difference between the maximum partial likelihood estimator and a weighted parameter estimator is shown to converge weakly to a multivariate normal with mean zero and with a covariance matrix for which a consistent estimator is proposed. The asymptotic properties of the weighted parameter estimators and those of the related goodness-of-fit tests under misspecified Cox models are also investigated. In particular, it is demonstrated that a goodness-of-fit test with a monotone weight function is consistent against monotone departures from the proportional hazards assumption. Versatile testing procedures with broad sensitivities can be developed based on simultaneous use of several weight functions. Three examples using real data are presented.