Interaction of treatment with a continuous variable: simulation study of significance level for several methods of analysis

Interactions between treatments and covariates in RCTs are a key topic. Standard methods for modelling treatment-covariate interactions with continuous covariates are categorisation or linear functions. Both approaches are easily criticised, but for different reasons. Multivariable fractional polynomial interactions, an approach based on fractional polynomials with the linear interaction model as the simplest special case, was proposed. Four variants of multivariable fractional polynomial interaction (FLEX1-FLEX4), allowing varying flexibility in functional form, were suggested. However, their properties are unknown, and comparisons with other procedures are unavailable. Additionally, we consider various methods based on categorisation and on cubic regression splines. We present the results of a simulation study to determine the significance level (probability of a type 1 error) of various tests for interaction between a binary covariate ('treatment effect') and a continuous covariate in univariate analysis. We consider a simplified setting in which the response variable is conditionally normally distributed, given the continuous covariate. We consider two main cases with the covariate distribution well behaved (approximately symmetric) or badly behaved (positively skewed). We construct nine scenarios with different functional forms for the main effect. In the well-behaved case, significance levels are in general acceptably close to nominal and are slightly better for the larger sample size (n = 250 and 500 were investigated). In the badly behaved case, departures from nominal are more pronounced for several approaches. For a final assessment of these results and recommendations for practice, a study of power is needed.