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Berger, Ursula and Gerein, Pia and Ulm, Kurt and Schäfer, Juliane (2000): On the use of Fractional Polynomials in Dynamic Cox Models. Collaborative Research Center 386, Discussion Paper 207
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Abstract

Despite a sophisticated research on modelling of survival data in the last years, the most popular model used in practice is still the proportional hazards regression model proposed by Cox (1972). This is mainly due to its exceptional simplicity. Nevertheless the fundamental assumption of the Cox model is the proportionality of the hazards, which particularly implies that the covariate effects are constant over time. For many applications this assumption is, however, doubtful. Other, more flexible approaches, which are able to cope with non-proportional hazards usually require non-standard estimation techniques, which are often rather complex and thus not favoured in application. Moreover, the selection of an appropriate test-statistic, to examine the improvement of the fit, is not obvious. In this paper we propose a flexible, yet simple method for modelling dynamic effects in survival data within the Cox framework. The method is based on Fractional Polynomials as introduced by Royston and Altman (1994). This allows for a transformation of the dynamic predictor which leads back to the conventional Cox model and hence fitting is straightforward using standard estimation techniques. In addition, it offers the possibility to easily verify the existence of time-variation. We describe a model selection algorithm which enables to include time-varying effects only when evidence is given in the data, in order to construct a model, which is just as complex as needed. We illustrate the properties of the approach in a simulation study and an application to gastric carcinoma data and compare it with other methods (e.g. the residual score test and smoothed Schoenfeld residuals of Grambsch and Therneau, 1994; natural smoothing splines of Hastie and Tibshirani, 1993).