Using fractional polynomials to model non-linear trends in longitudinal data


Correspondence should be addressed to Dr Jeffrey Long, Quantitative Methods in Education, University of Minnesota, 56 East River Road, Minneapolis, MN 55455, USA (e-mail:


Non-linear growth curves are discussed within the context of the linear mixed model. Non-linearity is modelled with time transformations known as fractional polynomials (FPs) having power terms that can be negative values and fractions with conventional polynomials (CPs) as a special case. Issues of interpretation are discussed with a focus on the instantaneous rate of change in models with and without static correlates. Methods for model selection are presented with emphasis on penalized and non-penalized indices of global fit based on the maximized likelihood and fitted models. Two empirical examples are presented with psychological data in which FPs were fitted along with CPs of equal and next highest order. The results show that the FPs had equal or better fit than the higher-order CPs and had prediction curves with as favourable or more favourable characteristics, such as less extreme behaviour at the edges of the observed time intervals. The results illustrate some of the potential advantages of FPs relative to CPs, which include parsimony, flexibility of curve shape, and the ability to approximate asymptotes. Though FPs are not necessarily suggested as replacements for CPs or other transformations (e.g. piecewise models), they might be useful when the goal is to model non-linear growth trends with smooth curves.