Increasing the flexibility of mixed models by using fractional polynomials

The class of regression models incorporating Fractional Polynomials (FPs), proposed by Royston and colleagues in the 1990’s, has been extensively studied and shown to be fruitful in the presence of nonlinearity between the response variable and continuous covariates. FP functions provide an alternative to higher-order polynomials and splines for dealing with lack-of-fit. Mixed models may also benefit from this class of curves in the presence of non-linearity. The inclusion of FP functions into the structure oflinear mixed models has been previously explored, though for simple layouts, e.g. a single covariate in the random intercept model. This paper proposes a general strategy for model-building and variable selection that takes advantage of the FPs within the framework of linear mixed models. Application of the methodto three data sets from the literature, known for violating the linearity assumption, illustrates that it is possible to solve the problem of lack-of-fit by using fewer terms in the model than the usual approach of fitting higher-order polynomials.