On the use of Fractional Polynomials in Dynamic Cox Models

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).