Bayesian multivariate logistic regression for superiority and inferiority decision-making under treatment heterogeneity

The effects of a treatment may differ between patients with different characteristics. Addressing such treatment heterogeneity is crucial to identify which patients benefit from a treatment, but can be complex in the context of multiple correlated binary outcomes. The current paper presents a novel Bayesian method for estimation and inference for heterogeneous treatment effects in a multivariate binary setting. The framework is suitable for prediction of heterogeneous treatment effects and superiority/inferiority decision-making within subpopulations, while taking advantage of the size of the entire study sample. We introduce a decision-making framework based on Bayesian multivariate logistic regression analysis with a P\'olya-Gamma expansion. The obtained regression coefficients are transformed into differences between success probabilities of treatments to allow for treatment comparison in terms of point estimation and superiority and/or inferiority decisions for different (sub)populations. Procedures for a priori sample size estimation under a non-informative prior distribution are included in the framework. A numerical evaluation demonstrated that a) average and conditional treatment effect parameters could be estimated unbiasedly when the sample is large enough; b) decisions based on a priori sample size estimation resulted in anticipated error rates. Application to the International Stroke Trial dataset revealed a heterogeneous treatment effect: The model showed conditional treatment effects in opposite directions for patients with different levels of blood pressure, while the average treatment effect among the trial population was close to zero.