Bayesian One-Way ANOVA
Also known as: Bayesian ANOVA, BF ANOVA, Bayes factor one-way ANOVA, Bayesian F-test
Bayesian one-way ANOVA tests whether the means of three or more independent groups differ by computing a Bayes factor — a ratio that quantifies how much more likely the data are under a model that allows group differences than under the null model that assumes equal means. Unlike the classical F-test, it provides direct evidence for or against the null hypothesis rather than merely rejecting or retaining it.
Key highlights
- Quantifies evidence for the null hypothesis, not just against it — a unique advantage over p-values.
- Produces a single interpretable number (Bayes factor) that is continuous and communicates strength of evidence.
- Coherently incorporates prior information and updates it with data, supporting cumulative science.
- Allows evidence monitoring and sequential updating as data accumulate without inflating error rates.
- Performed in user-friendly software (JASP) without requiring deep Bayesian programming.
Intuition
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How it works
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When to use it
Use Bayesian one-way ANOVA when you have a continuous outcome measured in three or more independent groups and you want to quantify evidence rather than simply make a binary reject/retain decision. It is especially valuable when sample sizes are small (where classical tests are underpowered), when you want to support the null hypothesis as a genuine scientific claim, or when you want to incorporate prior knowledge from previous studies. Assumptions are broadly the same as classical one-way ANOVA: independence of observations, approximate normality within groups, and homogeneity of variance. Do not use it when groups are not independent (use Bayesian repeated-measures ANOVA instead), when the outcome is ordinal or severely non-normal in small samples, or when you have no reasonable basis for specifying or accepting a default prior.
Strengths & limitations
- Quantifies evidence for the null hypothesis, not just against it — a unique advantage over p-values.
- Produces a single interpretable number (Bayes factor) that is continuous and communicates strength of evidence.
- Coherently incorporates prior information and updates it with data, supporting cumulative science.
- Allows evidence monitoring and sequential updating as data accumulate without inflating error rates.
- Performed in user-friendly software (JASP) without requiring deep Bayesian programming.
- Results depend on the choice of prior; different priors can yield different Bayes factors, especially with small samples.
- Computationally more demanding than classical ANOVA; marginal likelihoods may require MCMC approximation.
- Threshold labels (e.g. 'moderate', 'strong') are heuristic conventions, not hard boundaries.
- Less familiar to reviewers in some disciplines; may require additional explanation in manuscripts.
- Post-hoc pairwise comparisons are less standardised than in classical ANOVA.
Common pitfalls
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Applications
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Frequently asked
How do I choose the prior scale r?
The default r = 0.5 (medium) from Rouder et al. (2012) is appropriate when you have no strong prior information. Use r = 0.354 (small) if effects in your domain are consistently modest, or r = 0.707 (large) if large effects are plausible. Conduct a sensitivity analysis — if conclusions change dramatically across reasonable r values, report this and interpret cautiously.
What software runs Bayesian one-way ANOVA easily?
JASP (free, GUI-based) provides Bayesian one-way ANOVA with default priors, Bayes factor tables, and posterior plots out of the box. In R, the BayesFactor package (anovaBF function) and brms (for full Bayesian modelling) are the main options. SPSS does not natively support it.
Can I use this when variances are unequal across groups?
The standard formulation assumes homogeneity of variance. If Levene's test is significant, the classical approach switches to Welch's ANOVA. In Bayesian terms, you can extend the model to allow group-specific variance parameters, but this is not available in point-and-click software; it requires custom brms or Stan code.
How is Bayesian ANOVA different from classical one-way ANOVA?
Classical ANOVA gives an F statistic and p-value indicating whether data are unlikely under the null. Bayesian ANOVA gives a Bayes factor indicating how much more (or less) likely the data are under the alternative than the null. The Bayesian approach can support the null positively, and it avoids the binary threshold logic of significance testing.
Do I still need post-hoc tests?
Yes, if the Bayes factor favours M1 you will want to identify which groups differ. In JASP you can request Bayesian pairwise comparisons with automatic multiplicity correction. These produce Bayes factors for each pair, analogous to Tukey post-hoc tests but in a Bayesian framework.
Sources
- 1.Rouder, J. N., Morey, R. D., Speckman, P. L., & Province, J. M. (2012). Default Bayes factors for ANOVA designs. Journal of Mathematical Psychology, 56(5), 356–374.
- 2.Jeffreys, H. (1961). Theory of Probability (3rd ed.). Oxford University Press.ISBN 978-0198503682
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Cite this page
ScholarGate. (2026, June 3). Bayesian one-way ANOVA. ScholarGate. https://scholargate.app/statistics/bayesian-one-way-anova