The use of fractional polynomials to model interactions between treatment and continuous covariates in clinical trials

We consider modelling and testing for `interaction' between a continuous covariate X and a categorical covariate C in a regression model. Here C represents two treatment arms in a parallel-group clinical trial and X is a prognostic factor which may influence response to treatment. Usually X is categorised into groups according to cut-point(s) and the interaction is analysed in a model with main effects and multiplicative terms. A trend test of the effect of C over the ordered categories from X may be performed and is likely to have better power. The cut-point approach raises several well-known and difficult issues for the analyst, including dependency of the results on the choice of cut-point, loss of power due to categorisation, and the danger of `over-fitting' if several cut-points are considered in a search for `optimality' (Altman et al., 1994). We will describe an approach to avoid such problems based on fractional polynomial (FP) modelling of X, without categorisation, overall and at each level of C (Royston and Sauerbrei, 2002). The first step is to construct a multivariable adjustment model which may contain binary covariates and FP transformations of continuous covariates other than X. The second step involves FP modelling of X within the adjustment model. Stata software to fit the models will be demonstrated using example datasets, mainly from cancer studies. The examples show the power of the approach in detecting and displaying interactions in real data from randomised controlled trials with a survival-time outcome.