REGRESSION USING FRACTIONAL POLYNOMIALS OF CONTINUOUS COVARIATES - PARSIMONIOUS PARAMETRIC MODELING

The relationship between a response variable and one or more continuou s covariates is often curved. Attempts to represent curvature in singl e- or multiple-regression models are usually made by means of polynomi als of the covariates, typically quadratics. However, low order polyno mials offer a limited family of shapes, and high order polynomials may fit poorly at the extreme values of the covariates. We propose an ext ended family of curves, which we call fractional polynomials, whose po wer terms are restricted to a small predefined set of integer and non- integer values. The powers are selected so that conventional polynomia ls are a subset of the family. Regression models using fractional poly nomials of the covariates have appeared in the literature in an ad hoc fashion over a long period; we provide a unified description and a de gree of formalization for them. They are shown to have considerable fl exibility and are straightforward to fit using standard methods. We su ggest an iterative algorithm for covariate selection and model fitting when several covariates are available. We give six examples of the us e of fractional polynomial models in three types of regression analysi s: normal errors, logistic and Cox regression. The examples all relate to medical data: fetal measurements, immunoglobulin concentrations in children, diabetes in children, infertility in women, myelomatosis (a type of leukaemia) and leg ulcers.