Correction of the p-value after multiple tests in a Cox proportional hazard model

We consider a situation which is common in epidemiology, in which several transformations of an explanatory variable are tried in a Cox model and the most significant test is retained. The p-value should then be corrected to take account of the multiplicity of tests. Bonferroni method is often too conservative because the tests may be highly positively correlated. We propose an asymptotically exact correction of the p-value. The method uses the fact that the tests are asymptotically normal to compute numerically the distribution of the maximum of several tests. Counting processes theory is used to derive estimators of the correlations between tests. The method is illustrated by a simulation and an analysis of the relation between concentration of aluminum in drinking water and risk of dementia.