Estimating average regression effect under non-proportional hazards

We present an estimator of average regression effect under a non-proportional hazards model, where the regression effect of the covariates on the log hazard ratio changes with time. In the absence of censoring, the new estimate coincides with the usual partial likelihood estimate, both estimates being consistent for a parameter having an interpretation as an average population regression effect. In the presence of an independent censorship, the new estimate is still consistent for this same population parameter, whereas the partial likelihood estimate will converge to a different quantity that depends on censoring. We give an approximation of the population average effect as \({\int}\ {\beta}(t)dF(t)\) ⁠. The new estimate is easy to compute, requiring only minor modifications to existing softwares. We illustrate the use of the average effect estimate on a breast cancer dataset from Institut Curie. The behavior of the estimator, its comparison with the partial likelihood estimate, as well as the approximation by \({\int}\ {\beta}(t)dF(t)\) are studied via simulation.