A new Bayesian two-sample t-test for effect size estimation under uncertainty based on a two-component Gaussian mixture with known allocations and the region of practical equivalence

TL;DR

This paper proposes a Bayesian two-sample t-test based on a Gaussian mixture model that estimates effect size under uncertainty, emphasizing estimation over hypothesis testing, and is implemented in an R package.

Contribution

It introduces a novel Bayesian approach for effect size estimation using a Gaussian mixture model, focusing on uncertainty quantification and practical equivalence testing.

Findings

The method provides reliable effect size estimates under uncertainty.

Simulation studies demonstrate the method's effectiveness.

The approach is implemented in the R package bayest.

Abstract

Testing differences between a treatment and control group is common practice in biomedical research like randomized controlled trials (RCT). The standard two-sample t-test relies on null hypothesis significance testing (NHST) via p-values, which has several drawbacks. Bayesian alternatives were recently introduced using the Bayes factor, which has its own limitations. This paper introduces an alternative to current Bayesian two-sample t-tests by interpreting the underlying model as a two-component Gaussian mixture in which the effect size is the quantity of interest, which is most relevant in clinical research. Unlike p-values or the Bayes factor, the proposed method focusses on estimation under uncertainty instead of explicit hypothesis testing. Therefore, via a Gibbs sampler the posterior of the effect size is produced, which is used subsequently for either estimation under uncertainty or explicit hypothesis testing based on the region of practical equivalence (ROPE). An illustrative example, theoretical results and a simulation study show the usefulness of the proposed method, and the test is made available in the R package bayest.