On estimation and tests of time-varying effects in the proportional hazards model

Grambsch and Therneau (1994) suggest to use Schoenfeld's Residuals to investigate whether some of the regression coefficients in Cox'(1972) proportional hazards model are time-dependent. Their method is a one-step procedure based on Cox' initial estimate. We suggest an algorithm which in the first step is similar to the Grambsch and Therneau procedure. This algorithm can be shown to converge and the estimates obtained from this procedure have sound properties. The estimation procedure is focussed on the cumulative regression functions that are useful for inferential purposes. The cumulatives can be estmated at $n^{1/2}$-rate. We also present a new general semi-parametric version of Cox' regression model where some of the covariate effects are time-dependent while others have a constant effect. Based on the semi-parametric model we develop an easily implemented procedure for testing if the reression coefficients depend on time. Such tests have not been developed before. The test procedure makes successive testing of time-dependency possible. A small simulation study investigates the finite sample properties of the tests.