Effects of Model Misspecification on Tests of No Randomized Treatment Effect Arising From Cox’s Proportional Hazards Model

Abstract

We examine the asymptotic and small sample properties of model-based and robust tests of the null hypothesis of no randomized treatment effect based on the partial likelihood arising from an arbitrarily misspeci®ed Cox proportional hazards model. When the distribution of the censoring variable is either conditionally independent of the treatment group given covariates or conditionally independent of covariates given the treatment group, the numerators of the partial likelihood treatment score and Wald tests have asymptotic mean equal to 0 under the null hypothesis, regardless of whether or how the Cox model is misspeci®ed. We show that the modelbased variance estimators used in the calculation of the model-based tests are not, in general, consistent under model misspeci®cation, yet using analytic considerations and simulations we show that their true sizes can be as close to the nominal 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 misspeci®ed and the distribution of censoring depends on both treatment group and covariates, the asymptotic distributions of the resulting partial likelihood treatment score statistic and maximum partial likelihood estimator do not, in general, have a zero mean under the null hypothesis. Here neither the fully model-based tests, including the log-rank test, nor the robust tests will be asymptotically valid, and we show through simulations that the distortion to test size can be substantial.

The Effects of Misspecifying Cox's Regression Model on Randomized Treatment Group Comparisons

“…Then the test statistic n −1/2 U n has an asymptotic normal distribution with mean 0 under H 0 , regardless of whether or not the model (2) is misspecified (DiRienzo and Lagakos, 2001a). Furthermore, when either of these conditions hold, consistent estimates of the variance of n −1/2 U n are easily derived, yielding asymptotically valid inference whether or not the relationship between T and (X, W ) is properly specified.…”

“…To provide some insight into why either of these conditions are necessary for valid inference, note that at baseline (that is, when t = 0), the distribution of W is independent of X because of randomization; when either C ⊥ X | W or C ⊥ W | X holds and H 0 is true, it is implied that X ⊥ W | Y (t) = 1, t > 0, which is necessary for n −1/2 U n to have mean 0 asymptotically. For a proof of these results, see Appendix A of DiRienzo and Lagakos (2001a) or Kong and Slud (1997).…”

“…We note that for the special case of the log-rank test, use of the model-based variance estimator of n −1/2 U n also results in a valid asymptotic test, and appears to usually provide nominal finite-sample type I errors (see DiRienzo and Lagakos, 2001a), so that use of a robust variance estimator is not needed. Lagakos and Schoenfeld (1984) investigated the effects of various types of misspecification of the working model (2) on the power of n −1/2 U n .…”

“…Recent work has shown that the impact of model misspecification on the validity of resulting tests hinges on whether the distribution of the potential censoring time either (i) is conditionally independent of treatment group given covariates or conditionally independent of covariates given treatment group, or (ii) depends on both treatment group and covariates. In the first case, resulting test statistics have an asymptotic normal distribution with mean zero under the null hypothesis and that consistent variance estimates are readily obtainable (see Kong andSlud, 1997 andLagakos, 2001a). In the second case, the asymptotic mean of the test statistic is not necessarily equal to zero under the null hypothesis when the proportional hazards working model is misspecified.…”

Bayesian Inference and Testing of Group Differences in Brain Networks

“…Hence, these procedures are expected to fail in scenarios where the changes in the network-valued random variable are due to variations in more complex functionals. Model misspecification can have a major effect on the quality of inference (Deegan, 1976;Begg and Lagakos, 1990;DiRienzo and Lagakos, 2001), providing biased and inaccurate conclusions.…”