Adaptive Fractional Polynomial Modeling

Abstract

Regression analyses reported in the applied research literature commonly assume that relationships are linear in predictors without assessing this assumption. Fractional polynomials provide a general approach for addressing nonlinearity through power transforms of predictors using real valued powers. An adaptive approach for generating fractional polynomial models is presented based on heuristic search through alternative power transforms of predictors guided by k -fold likelihood cross-validation (LCV) scores and controlled by tolerance parameters indicating how much a reduction in the LCV score can be tolerated at given stages of the search. The search optionally can generate geometric combinations, that is, products of power transforms of multiple predictors, thereby supporting nonlinear moderation analyses. Positive valued continuous outcomes can be power transformed as well as predictors. These methods are demonstrated using data from a study of family management for mothers of children with chronic physical conditions. The example analyses demonstrate that power transformation of a predictor may be required to identify that a relationship holds between that predictor and an outcome (dependent or response) variable. Consideration of geometric combinations can identify moderation effects not identifiable using linear relationships or power transforms of interactions. Power transformation of positive valued continuous out comes along with their primary predictors can resolve model assumption problems.