From p-Values to Posterior Probabilities of Null Hypotheses

Area of Science:

• Statistics
• Hypothesis Testing
• Bayesian Inference

Background:

• Minimum Bayes factors, like -e·p·log(p), are used to convert p-values to posterior probabilities of the null hypothesis.
• Existing methods fail to account for sample size, a critical flaw for moderate to large datasets where p-values are most problematic.

Purpose of the Study:

• To propose an adjusted minimum Bayes factor that approximates an exact Bayes factor.
• To ensure the adjusted Bayes factor is sensitive to sample size variations.
• To extend the adjustment method for both p-values and pseudo-p-values, including applications in linear models.

Main Methods:

• Adjusting the minimum Bayes factor using information to approximate an exact Bayes factor.
• Developing a specific adjustment for linear models utilizing the Prior-Based BIC refinement.

Main Results:

• The proposed adjustment corrects the sample size defect of the minimum Bayes factor.
• The adjusted Bayes factor provides a more accurate lower bound for the posterior probability of the null hypothesis.
• The method is applicable to both p-values and pseudo-p-values.

Conclusions:

• The adjusted minimum Bayes factor offers a more reliable measure for hypothesis testing, especially with larger sample sizes.
• This approach enhances the interpretability and validity of Bayes factors derived from p-values.
• The extension to linear models provides a practical tool for statistical modeling.