An Objective and Robust Bayes Factor for the Hypothesis Test One Sample and Two Population Means.

It has been over 100 years since the discovery of one of the most fundamental statistical tests: the Student's t test. However, reliable conventional and objective Bayesian procedures are still essential for routine practice. In this work, we proposed an objective and robust Bayesian approach for hypothesis testing for one-sample and two-sample mean comparisons when the assumption of equal variances holds. The newly proposed Bayes factors are based on the intrinsic and Berger robust prior. Additionally, we introduced a corrected version of the Bayesian Information Criterion (BIC), denoted BIC-TESS, which is based on the effective sample size (TESS), for comparing two population means. We studied our developed Bayes factors in several simulation experiments for hypothesis testing. Our methodologies consistently provided strong evidence in favor of the null hypothesis in the case of equal means and variances. Finally, we applied the methodology to the original Gosset sleep data, concluding strong evidence favoring the hypothesis that the average sleep hours differed between the two treatments. These methodologies exhibit finite sample consistency and demonstrate consistent qualitative behavior, proving reasonably close to each other in practice, particularly for moderate to large sample sizes.

From p-Values to Posterior Probabilities of Null Hypotheses.

Minimum Bayes factors are commonly used to transform two-sided p-values to lower bounds on the posterior probability of the null hypothesis, in particular the bound -e·p·log(p). This bound is easy to compute and explain; however, it does not behave as a Bayes factor. For example, it does not change with the sample size. This is a very serious defect, particularly for moderate to large sample sizes, which is precisely the situation in which p-values are the most problematic. In this article, we propose adjusting this minimum Bayes factor with the information to approximate an exact Bayes factor, not only when p is a p-value but also when p is a pseudo-p-value. Additionally, we develop a version of the adjustment for linear models using the recent refinement of the Prior-Based BIC.