Goodness-of-fit tests and robust statistical inference for the Cox proportional hazards model.

Abstract

The Cox proportional hazards model is a popular statistical tool for analyzing censored failure time data. It assumes the inclusion of all relevant covariates, the log-linear dependence of the hazard function on covariates, and the multiplicative relationship between the baseline hazard function and the regression function of covariates. When these assumptions are violated, the conventional inference procedures based on the partial likelihood function can result in misleading statistical conclusions. However, the current statistical literature lacks convenient goodness-of-fit tests and robust methods for the general Cox method. In this dissertation, we identify two model-based consistent estimators for the inverse of the asymptotic covariance matrix of the maximum partial likelihood estimator of a general Cox model. Under the assumed model, the difference between these two estimators is shown to be asymptotically normal with mean zero and with a covariance matrix which can be consistently estimated. Global goodness-of-fit tests are then constructed based on these results. Extensive Monte Carlo studies indicate that the large-sample approximations to the null distributions of the new test statistics are fairly accurate for moderate-sized samples and that the new tests tend to be more powerful than several existing numerical methods. In addition, we establish the asymptotic normality of the maximum partial likelihood estimator under a possibly misspecified Cox proportional hazards model. A consistent covariance matrix estimator is suggested. For many misspecified Cox models, the asymptotic limit of the maximum partial likelihood estimator, or part of the limit, can be interpreted meaningfully so that the valid statistical inference about the corresponding covariate effects can be drawn based on the new asymptotic theory of the maximum partial likelihood estimator and the related results for the score statistics. Extensive studies demonstrate that the proposed robust tests and interval procedures are adequate for practical use. In contrast, the conventional model-based inference procedures often lead to tests with supranominal size and confidence intervals with poor coverage probability. The proposed robust procedures are finally extended to analyze multivariate incomplete failure time observations.