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

The effects of treatments may differ between persons with different characteristics. Addressing such treatment heterogeneity is crucial to investigate whether patients with specific characteristics are likely to benefit from a new treatment. The current paper presents a novel Bayesian method for superiority 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 a P\'olya-Gamma expansion; b) a transformation procedure to transfer obtained regression coefficients to a more intuitive multivariate probability scale (i.e., success probabilities and the differences between them); and c) a compatible decision procedure for treatment comparison with prespecified decision error rates. 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.