Bayesian methodsBayesianModel

Bayesian ANOVA

Q: How is BF₁₀ different from a p-value?

A p-value measures how surprising the data are under the null hypothesis alone; it cannot quantify evidence for the null. BF₁₀ is the ratio of the probability of the data under the alternative to the probability under the null, so it simultaneously measures evidence for both hypotheses. A BF₁₀ of 0.1 is clear evidence for the null; a p-value of 0.5 is not.

Q: What does the Cauchy prior with r = 0.707 mean?

The Cauchy prior places a probability distribution over the standardised effect sizes of the group differences before seeing the data. The scale parameter r = 0.707 (approximately √2/2) is the Rouder et al. default: it expects effect sizes of moderate magnitude and is deliberately non-committal about direction. It is called a 'default' because no domain-specific prior knowledge is required to use it, and it has become the community standard for Bayesian ANOVA.

Q: When should I use Bayesian ANOVA instead of classical one-way ANOVA?

Bayesian ANOVA is the better choice when (a) you want to be able to conclude that groups do not differ, not just fail to reject the null; (b) you want full posterior distributions and credible intervals rather than point estimates and confidence intervals; or (c) you want to communicate the strength of evidence on a continuous, interpretable scale. Classical ANOVA remains practical when sample sizes are large, prior specification is of no interest, and a binary reject/do-not-reject decision is sufficient.

Q: What convergence checks are required after MCMC sampling?

At minimum: (1) R-hat values below 1.01 for all parameters, indicating the chains have mixed; (2) an effective sample size large enough for stable posterior summaries (typically several hundred or more per parameter); and (3) visual inspection of trace plots to confirm the chains are not stuck. If any of these diagnostics fail, the MCMC output should not be used to draw conclusions — increase the number of iterations or re-examine the model.

Also known as: bayesian analysis of variance, bayes factor ANOVA, JZS ANOVA, Bayesçi ANOVA — Bayes Faktörü ile Grup Karşılaştırması

OriginatorRouder, Morey, Speckman & ProvinceYear2012Sources1Related methods6

Bayesian ANOVA, formalised by Rouder, Morey, Speckman and Province (2012), tests whether group means differ by quantifying the evidence for the alternative hypothesis relative to the null using the Bayes Factor (BF₁₀). Unlike classical ANOVA, it can also measure evidence in favour of the null hypothesis, making it equally informative when groups do not differ.

Key highlights

Quantifies evidence for both the alternative and the null hypothesis, unlike a p-value which can only reject the null.
Provides a continuous, interpretable scale of evidence (BF₁₀ > 3 moderate, > 10 strong, > 30 very strong).
Uses a principled, literature-standard Cauchy prior (r = 0.707) that does not require subjective elicitation.
Results are reported as posterior distributions with credible intervals, conveying full parameter uncertainty.

Intuition

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How it works

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When to use it

Use Bayesian ANOVA when comparing continuous outcomes across two or more independent groups and when you need to quantify evidence for both the alternative and the null hypothesis. It is particularly valuable when you expect group differences may be absent and want more than 'p > 0.05' to say so, when you wish to incorporate a principled default prior on effect sizes, or when the sample size is moderate and a full uncertainty picture is needed. A minimum of about 10 observations total is advisable; with fewer, the Bayes Factor is dominated by the Cauchy prior and interpretation is unreliable. Normality violations should prompt a prior-sensitivity analysis, as the Cauchy prior assumption can affect the BF.

Strengths & limitations

Strengths

Quantifies evidence for both the alternative and the null hypothesis, unlike a p-value which can only reject the null.
Provides a continuous, interpretable scale of evidence (BF₁₀ > 3 moderate, > 10 strong, > 30 very strong).
Uses a principled, literature-standard Cauchy prior (r = 0.707) that does not require subjective elicitation.
Results are reported as posterior distributions with credible intervals, conveying full parameter uncertainty.

Limitations

With n < 10, the Bayes Factor is overly sensitive to the prior and yields unreliable evidence.
The Cauchy prior (r = 0.707) is a default that may not reflect the true prior expectations in every domain; prior-sensitivity analysis is mandatory.
Does not test specific contrasts between groups without additional post-hoc Bayesian comparisons.
Posterior sampling requires MCMC and convergence checks, adding computational and interpretive overhead compared to classical ANOVA.

Common pitfalls

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Applications

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Frequently asked

How is BF₁₀ different from a p-value?

A p-value measures how surprising the data are under the null hypothesis alone; it cannot quantify evidence for the null. BF₁₀ is the ratio of the probability of the data under the alternative to the probability under the null, so it simultaneously measures evidence for both hypotheses. A BF₁₀ of 0.1 is clear evidence for the null; a p-value of 0.5 is not.

What does the Cauchy prior with r = 0.707 mean?

The Cauchy prior places a probability distribution over the standardised effect sizes of the group differences before seeing the data. The scale parameter r = 0.707 (approximately √2/2) is the Rouder et al. default: it expects effect sizes of moderate magnitude and is deliberately non-committal about direction. It is called a 'default' because no domain-specific prior knowledge is required to use it, and it has become the community standard for Bayesian ANOVA.

When should I use Bayesian ANOVA instead of classical one-way ANOVA?

Bayesian ANOVA is the better choice when (a) you want to be able to conclude that groups do not differ, not just fail to reject the null; (b) you want full posterior distributions and credible intervals rather than point estimates and confidence intervals; or (c) you want to communicate the strength of evidence on a continuous, interpretable scale. Classical ANOVA remains practical when sample sizes are large, prior specification is of no interest, and a binary reject/do-not-reject decision is sufficient.

What convergence checks are required after MCMC sampling?

At minimum: (1) R-hat values below 1.01 for all parameters, indicating the chains have mixed; (2) an effective sample size large enough for stable posterior summaries (typically several hundred or more per parameter); and (3) visual inspection of trace plots to confirm the chains are not stuck. If any of these diagnostics fail, the MCMC output should not be used to draw conclusions — increase the number of iterations or re-examine the model.

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.
DOI 10.1016/j.jmp.2012.08.001

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ScholarGate. (2026, June 1). Bayesian ANOVA. ScholarGate. https://scholargate.app/bayesian/bayesian-anova