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

Cox’s proportional hazards model is routinely used in many ap- plied fields, especially in bio–medical research. A common phe- nomenon 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 pro- posal appeared in literature to estimate a time depending effect in Cox model, the problem of testing the null hypothesis of a propor- tional 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 hav- ing 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.