# Willi Sauerbrei: Regression model-building with continuous variables – multivariable fractional polynomials, with extensions for interactions

Willi Sauerbrei (Institute for Medical Biometry and Statistics, University of Freiburg) will give a seminar in the lunch area, 8th floor Niels Henrik Abels hus at 14:15.

**Title**: Regression model-building with continuous variables – multivariable fractional polynomials, with extensions for interactions

**Abstract**: In the analysis of studies in clinical epidemiology, the number of candidate variables for a regression model is often too large and a more parsimonious model is sought. Another key issue is the determination of appropriate dose-response functions for continuous covariates. Often, continuous predictors are either categorized or linearity is assumed. However, both approaches can have major disadvantages and models incorporating non-linear functions may markedly improve the fit. The method of multivariable fractional polynomials (MFP) simultaneously determines suitable functional forms for continuous covariates and eliminates uninfluential covariates (1,2,3). The method also allows categorical and binary covariates.

By analysing data in the framework of linear, logistic and Cox regression models, we discuss model-building issues with an emphasis on MFP. Extensions of MFP have been developed to investigate for interactions between continuous covariates and treatment (MFPI), between two continuous covariates (MFPIgen) and for interactions with time (non-proportional hazards, MFPT) in a Cox model (3,4,5). Using data from a large cohort study, we show that mis-modelling non-linear main effects can introduce spurious interactions between two continuous covariates. In RCTs, we illustrate that our approach has power to identify differential treatment effects, and demonstrate how to estimate and plot a continuous treatment-effect function. In a large simulation studies we could show that MFPI has advantages to several alternative approaches (5).

We conclude that MFP and its extensions for interactions are useful in multivariable model-building with continuous and categorical variables. MFP software for Stata, SAS and R is generally available (6).

Joint work with Patrick Royston (MRC Clinical Trials Unit, London, UK). For more details see http://mfp.imbi.uni-freiburg.de/

References:

- Royston P and Altman DG (1994): Regression using fractional polynomials of continuous covariates: parsimonious parametric modelling (with disc.) Applied Statistics, 43: 429-467
- Sauerbrei W and Royston P (1999): Building multivariable prognostic and diagnostic models: transformation of the predictors using fractional polynomials. Journal of the Royal Statistical Society, Series A, 162: 71-94
- Royston P, Sauerbrei, W (2008): ‘Multivariable Model-Building – A pragmatic approach to regression analysis based on fractional polynomials for modelling continuous variables’. Wiley.
- Sauerbrei W, Royston P, Look M (2007): A new proposal for multivariable modelling of time-varying effects in survival data based on fractional polynomial time-transformation. Biometrical Journal, 49: 453-473
- Royston P., Sauerbrei W. (2014): Interaction of treatment with a continuous variable: simulation study of power for several methods of analysis. Statistics in Medicine, 33: 4695-4708
- Sauerbrei W, Meier-Hirmer C, Benner A, Royston P (2006): Multivariable regression model building by using fractional polynomials: description of SAS, STATA and R programs, Computational Statistics and Data Analysis, 50: 3464-3485