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Salanti, G. and Ulm, Kurt (2003): Modelling time-varying effects in Cox model under order restrictions. Collaborative Research Center 386, Discussion Paper 319
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Abstract

The violation of the proportional hazards assumption in Cox model occurs quite often in studies concerning solid tumours or leukaemia. Then the time varying coefficients model is its most popular extension used. The function f(t) that measures the time variation of a covariate, can be assessed through several smoothing techniques, such as cubic splines. However, for practical propose, it is more convenient to assess f(t) by a step function. The main drawback of this approach is the lack of stability since there is no standard method of defining the cutpoints of the underlined step function. The variation in the effect of a predictor can be assumed to be monotonic during the observational period. In these cases, we propose a method to estimate f(t) based on the isotonic regression framework. Applying the idea of Grambsch and Therneau, where smoothing the Schoenfeld residuals plotted against time reveal the shape of the underlined f(t) function, we use the Pooled Adjacent Violators Algorithm as smoother. As a result a set of cutpoints is returned without any a priori information about their location. Subsequently, the corresponding step function is introduced in the model and the standard likelihood-based method is applied to estimate it while adjusting for other covariates. This approach presents the advantage that additional decisions that can effect the result, as the number of knots in cubic splines, do not need to be taken. The performance of the provided PH test and the stability of the method are explored in a simulation study.