Evaluating predictive biomarkers for a binary outcome with linear versus logistic regression – Practical recommendations for the choice of the model

A predictive biomarker can forecast whether a patient benefits from a specific treatment under study. To establish predictiveness of a biomarker, a statistical interaction between the biomarker status and the treatment group concerning the clinical outcome needs to be shown. In clinical trials looking at a binary outcome, linear or logistic regression models may be used to evaluate the interaction, but the effects in the two models are different and differently interpreted. Specifically, the effects are estimated as absolute risk reductions (ARRs) and odds ratios (ORs) in the linear and logistic model, thus measuring the effect on an additive and multiplicative scale, respectively. We derived the relationship between the effects of the linear and the logistic regression model allowing for translations between the effect estimates between both models. In addition, we performed a comprehensive simulation study to compare the power of the two models under a variety of scenarios in different study designs. In general, the differences in power to detect interaction were minor, and visible differences were detected in rather unrealistic scenarios of effect size combinations and were usually in favor of the logistic model. Based on our results and theoretical considerations, we recommend to 1) estimate logistic regression models because of their statistical properties, 2) test for interaction effects and 3) calculate and report both ARRs and ORs from these using the formulae provided.