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

Area of Science:

• Statistics
• Bayesian Inference

Background:

• Student's t-test, a fundamental statistical tool, has been used for over a century.
• Objective Bayesian procedures are crucial for routine statistical practice.

Purpose of the Study:

• To propose an objective and robust Bayesian approach for one-sample and two-sample mean comparisons.
• To introduce novel Bayes factors and a corrected Bayesian Information Criterion (BIC-TESS) for hypothesis testing.

Main Methods:

• Utilized intrinsic and Berger robust priors for Bayes factor calculation.
• Developed BIC-TESS based on effective sample size (TESS) for comparing population means.
• Conducted simulation experiments to evaluate the performance of the proposed methods.

Main Results:

• The proposed Bayesian methodologies consistently favored the null hypothesis when means and variances were equal.
• Demonstrated finite sample consistency and stable qualitative behavior across simulations.
• Applied to Gosset sleep data, finding significant evidence for differing average sleep hours between treatments.

Conclusions:

• The developed Bayesian approach offers a reliable and objective alternative for hypothesis testing in mean comparisons.
• The methods are practical, especially for moderate to large sample sizes, providing results close to conventional approaches.
• The study validates the utility of robust Bayesian methods in real-world data analysis.