Bayes Factor

Bayesian approach to hypothesis testing that quantifies evidence for competing hypotheses.

Bayes Factor Overview & Evidence Quantification

What it measures:

The ratio of evidence for one hypothesis relative to another, providing a continuous measure of evidence strength rather than binary accept/reject decisions.

Advantages:

• • Quantifies evidence strength continuously
• • Can provide evidence for null hypothesis
• • No arbitrary significance thresholds
• • Intuitive probabilistic interpretation
• • Incorporates prior knowledge

Considerations:

• • Requires prior specification
• • Computational complexity
• • Different interpretation than p-values
• • Less familiar to many researchers
• • Prior sensitivity issues

Bayes Factor & Marginal Likelihood Ratio

Bayes factor for H1 versus H0

BF10

= Bayes factor favoring the alternative hypothesis

P(Data|H1)

= marginal likelihood of the data under the alternative

P(Data|H0)

= marginal likelihood of the data under the null

BF₀₁ = 1 / BF₁₀ expresses the same evidence in favor of the null.

Posterior odds from prior odds and the Bayes factor

P(H1|Data)P(H0|Data)

= posterior odds after observing the data

P(H1)P(H0)

= prior odds before observing the data

Key Assumptions

• • Priors on model parameters under each hypothesis are correctly specified
• • Marginal likelihoods are computed (or approximated) accurately by integrating over the prior
• • Models are well-defined and mutually exclusive
• • Conclusions should be checked for sensitivity to the chosen prior

Reading the Bayes Factor Evidence Plot

Example Output & Evidence Interpretation

Evidence Categories (BF₁₀):

• BF₁₀ > 100: Extreme evidence for H₁
• 30-100: Very strong evidence for H₁
• 10-30: Strong evidence for H₁
• 3-10: Moderate evidence for H₁
• 1-3: Anecdotal evidence for H₁
• 1/3-1: Anecdotal evidence for H₀
• <1/3: Moderate to strong evidence for H₀

Practical Interpretation:

Evidence for Alternative (BF₁₀ > 3):

Data provides meaningful support for the alternative hypothesis. Consider effect size and practical significance.

Inconclusive Evidence (1/3 < BF₁₀ < 3):

Data does not strongly favor either hypothesis. Consider collecting more data or refining the analysis.

Evidence for Null (BF₁₀ < 1/3):

Data supports the null hypothesis, providing evidence for no effect or difference.

Reporting Guidelines:

• • Report both BF₁₀ and BF₀₁ values
• • Describe prior specification and rationale
• • Include evidence interpretation category
• • Consider sensitivity to prior choices
• • Compare with traditional frequentist results if relevant

sixsigmacenter.com Implementation Guide

Example Data Layout for Bayes Factor Analysis

Two groups for Bayesian hypothesis testing with continuous evidence quantification and prior-informed analysis. Sample Size Recommendation: Minimum 10-20 observations per group for stable Bayes Factor calculations - larger samples provide more reliable evidence quantification and reduced sensitivity to prior specification.

This is how your spreadsheet or CSV file should be structured before you upload it. Match the column names, keep one row per observation, and leave the analysis to the platform.