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

The effects of treatments may differ between persons with different characteristics. Addressing such treatment heterogeneity is crucial to identify who benefits from a new treatment, but can be complex in the context of multiple correlated outcomes. The current paper presents a novel Bayesian method for superiority and inferiority decision-making in the context of randomized controlled trials with multivariate binary responses and heterogeneous treatment effects. The framework is based on three elements: a) Bayesian multivariate logistic regression analysis with P\'olya-Gamma expansion; b) a transformation procedure to tranfer obtained regression coefficients to the more intuitive multivariate probability scale (i.e. success probabilities and differences between them); and c) a compatible decision procedure for treatment comparison. Procedures for a priori sample size estimation under a non-informative prior distribution are included. A numerical evaluation demonstrated that decisions based on a priori sample size estimation resulted in anticipated error rates among the trial population as well as subpopulations. Further, average and conditional treatment effect parameters could be estimated unbiasedly when the sample was large enough. Illustration with the International Stroke Trial dataset revealed a trend towards heterogeneous effects among stroke patients: Something that would have remained undetected when analyses were limited to average treatment effects.