On an adaptive test of time-varying effects in Cox regression

Cox’s proportional hazards model is routinely used in many applied fields, especially in bio–medical research. A common phenomenon in medical settings is the presence of a time–dependency in the effect of one or more explanatory variables. It is then crucial to decide whether a covariate effect is constant, as prescribed by the standard Cox regression model, or not. Although several proposal appeared in literature to estimate a time depending effect in Cox model, the problem of testing the null hypothesis of a proportional hazard against different possible alternatives has received less attention. The main point of the present work is to introduce a new test for time–varying effects in the proportional hazards model having power that adapts to the smoothness of the underlying function. Working on the Schoenfeld residuals our procedure is an adaptation to the present setting of a multiple testing technique introduced by Fromont and Laurent in 2006. The results are illustrated with the well-known Mayo liver disease data.