Comparing Penalized Splines and Fractional Polynomials for Flexible Modelling of the Effects of Continuous Predictor Variables

Abstract

P(enalized)-splines and fractional polynomials (FPs) have emerged as powerful smoothing techniques with increasing popularity in several fields of applied research. Both approaches provide considerable flexibility, but only limited comparative evaluations of the performance and properties of the two methods have been conducted to date. We thus performed extensive simulations to compare FPs of degree 2 (FP2) and degree 4 (FP4) and P-splines that used generalized cross validation (GCV) and restricted maximum likelihood (REML) for smoothing parameter selection. We evaluated the ability of P-splines and FPs to recover the true functional form of the association between continuous, binary and survival outcomes and exposure for linear, quadratic and more complex, non-linear functions, using different sample sizes and signal to noise ratios. We found that for more curved functions FP2, the current default implementation in standard software, showed considerably bias and consistently higher mean squared error (MSE) compared to spline-based estimators (REML, GCV) and FP4, that performed equally well in most simulation settings. FPs however, are prone to artefacts due to the specific choice of the origin, while P-splines based on GCV reveal sometimes wiggly estimates in particular for small sample sizes. Finally,we highlight the specific features of the approaches in a real dataset