A strategy for modelling the effect of a continuous covariate in medicine and epidemiology

Low-dimensional parametric models are well understood, straightforward to communicate to other workers, have very smooth curves and may easily be checked for consistency with background scientific knowledge or understanding. They should therefore be ideal tools with which to represent smooth relationships between a continuous predictor and an outcome variable in medicine and epidemiology. Unfortunately, a seriously restricted set of such models is used routinely in practical data analysis - typically, linear, quadratic or occasionally cubic polynomials, or sometimes a power or logarithmic transformation of a covariate. Since their flexibility is limited, it is not surprising that the fit of such models is often poor. Royston and Altman's recent work on fractional polynomials has extended the range of available functions. It is clearly crucial that the chosen final model fits the data well. Achieving a good fit with minimal restriction on the functional form has been the motivation behind the major recent research effort on non-parametric curve-fitting techniques. Here I propose that one such model, a (possibly over-fitted) cubic smoothing spline, may be used to define a suitable reference curve against which the fit of a parametric model may be checked. I suggest a significance test for the purpose and examine its type I error and power in a small simulation study. Several families of parametric models, including some with sigmoid curves, are considered. Their suitability in fitting regression relationships found in several real data sets is investigated. With all the example data sets, a simple parametric model can be found which fits the data approximately as well as a cubic smoothing spline, but without the latter's tendency towards artefacts in the fitted curve.