Effects of model misspecification on tests of no randomized treatment effect arising from Cox's proportional hazards model

We examine the asymptotic and small sample properties of model-based and ro bust tests of the null hypothesis of no randomized treatment affect based o n the partial likelihood arising from an arbitrarily misspecified Cox propo rtional hazards model. When the distribution of the censoring variable is e ither conditionally independent of the treatment group given covariates or conditionally independent of covariates given the treatment group, the nume rators of the partial likelihood treatment score and Wald tests have asympt otic mean equal to 0 under the null hypothesis, regardless of whether or ho w the Cox model is misspecified. We show that the model-based variance esti mators used in the calculation of the model-based tests are not, in general , consistent under model misspecification, yet using analytic consideration s and simulations we show that their true sizes can be as close to the nomi nal value as tests calculated with robust variance estimators. As a special case, we show that the model-based log-rank test is asymptotically valid. When the Cox model is misspecified and the distribution of censoring depend s on both treatment group and covariates, the asymptotic distributions of t he resulting partial likelihood treatment score statistic and maximum parti al likelihood estimator do not, in general, have a zero mean under the null hypothesis. Here neither the fully model-based tests, including the log-ra nk test, nor the robust tests will be asymptotically valid, and we show thr ough simulations that the distortion to test size can be substantial.