Building multivariable prognostic and diagnostic models: transformation of the predictors by using fractional polynomials

To be useful to clinicians, prognostic and diagnostic indices must be derived from accurate models developed by using appropriate data sets. We show that fractional polynomials, which extend ordinary polynomials by including non-positive and fractional powers, may be used as the basis of such models. We describe how to fit fractional polynomials in several continuous covariates simultaneously, and we propose ways of ensuring that the resulting models are parsimonious and consistent with basic medical knowledge. The methods are applied to two breast cancer data sets, one from a prognostic factors study in patients with positive lymph nodes and the other from a study to diagnose malignant or benign tumours by using colour Doppler blood flow mapping. We investigate the problems of biased parameter estimates in the final model and overfitting using cross-validation calibration to estimate shrinkage factors. We adopt bootstrap resampling to assess model stability. We compare our new approach with conventional modelling methods which apply stepwise variables selection to categorized covariates. We conclude that fractional polynomial methodology can be very successful in generating simple and appropriate models.