Choice of scale for cubic smoothing spline models in medical applications

The determination of the functional form of the relationship between an outcome variable and one or more continuous covariates is an important aspect of the modelling of medical data. For correct interpretation of the data it is essential that the functional form be specified at least approximately correctly. I show that for given model complexity, logarithmic transformation of a covariate can greatly improve the fit of one of the most useful and convenient non-parametric regression models, the cubic smoothing spline. A mathematical rationale for the idea is given. I propose a diagnostic for deciding initially whether a log transformation is needed. The method is illustrated using several medical data sets. No special software other than that used for fitting the spline models is needed.