Stability of multivariable fractional polynomial models with selection of variables and transformations: a bootstrap investigation

Sauerbrei and Royston have recently described an algorithm, based on fractional polynomials, for the simultaneous selection of variables and of suitable transformations for continuous predictors in a multivariable regression setting. They illustrated the approach by analyses of two breast cancer data sets. Here we extend their work by considering how to assess possible instability in such multivariable fractional polynomial models. We first apply the algorithm repeatedly in many bootstrap replicates. We then use log-linear models to investigate dependencies among the inclusion fractions for each predictor and among the simplified classes of fractional polynomial function chosen in the bootstrap samples. To further evaluate the results, we define measures of instability based on a decomposition of the variability of the bootstrap-selected functions in relation to a reference function from the original model. For each data set we are able to identify large, reasonably stable subsets of the bootstrap replications in which the functional forms of the predictors appear fairly stable. Despite the considerable flexibility of the family of fractional polynomials and the consequent risk of overfitting when several variables are considered, we conclude that the multivariable selection algorithm can find stable models.