Proportionality test in the Cox model with correlated covariates

The correlation effect between covariates on the proportionality test results of a specific covariate in the Cox model is a problem that has already been reported by several authors. The first solution has been proposed for the Kolmogorov-Smirnov (KS) test, the Cramér-von Mises (CvM) test, and the Anderson-Darling (AD) test. It consists of simulating the null distribution of these test statistics, since this is only known if the covariates are uncorrelated. The results of the simulations carried out by the proponents of this solution have not proved its effectiveness in all studied cases. The second solution is based on the fact that the score function used in the tests mentioned above, and in the construction of the score tests, assumes that all other covariates are proportional, which is not always true. The idea is therefore to introduce temporal parameters to these covariates whose meanings match their proportionalities. Such a change in the score function requires estimation of the new parameters introduced for each tested covariate. In this article, we propose a simple technique to eliminate such an effect. The technique involves changing the covariate to be tested by the residual of its linear regression against the other covariates in the model. This change retains the same null hypothesis to be tested with a new covariate that is uncorrelated with the others. A simulation comparison of these techniques is considered.