Bayesian t-Test
Bayesian t-Test for Two-Group Comparison · Also known as: bayesian two-sample t-test, bayes factor t-test, Bayesçi t-Testi
The Bayesian t-test, formalised by Rouder and colleagues in 2009, is a two-group comparison method that works within a Bayesian framework. Instead of a p-value, it produces a Bayes Factor (BF₁₀) that quantifies the evidence the data provide for the alternative hypothesis relative to the null, and it reports the full posterior distribution of the standardised effect size δ with a highest-density interval.
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When to use it
Use the Bayesian t-test when you are comparing two groups on a continuous outcome and you need more than a binary significant-or-not conclusion. It is especially valuable when the sample is small — where frequentist p-values are highly variable — because a sensible prior on effect size stabilises inference. It is also the right choice when you need to quantify evidence for the null hypothesis (frequentist tests cannot do this), or when you want to report effect size with its full posterior uncertainty rather than just a point estimate and a standard error. The method requires a minimum sample of about ten observations per group; below that, the posterior is dominated almost entirely by the prior and the Bayes Factor becomes unreliable.
Strengths & limitations
- Produces a Bayes Factor that quantifies evidence for both H₁ and H₀, unlike a p-value which cannot support the null.
- Reports the full posterior distribution of the effect size δ with a 95% HDI, giving honest uncertainty rather than a single number.
- Performs better than the frequentist t-test in small samples by incorporating a principled prior on effect size.
- Results are directly interpretable in probability terms without reference to sampling distributions or long-run frequency.
- When sample size is below about ten, the posterior is dominated by the prior and the Bayes Factor does not reliably reflect the data.
- The Cauchy prior scale r is an assumption; different defensible choices can shift BF₁₀ noticeably, so prior-sensitivity analysis is mandatory.
- If the distributional assumption is violated and a sensitivity analysis is skipped, the Cauchy prior scale can dominate the Bayes Factor.
Frequently asked
What does BF₁₀ = 10 actually mean?
It means the observed data are ten times more probable under the hypothesis that a difference exists (H₁) than under the hypothesis of no difference (H₀). Following Jeffreys' (1961) scale, BF₁₀ > 10 is conventionally labelled strong evidence for H₁, but this label is a guideline, not a sharp threshold.
How do I choose the Cauchy scale parameter r?
The default r = 0.707 recommended by Rouder et al. (2009) is a reasonable starting point for most social-science contexts. If you have genuine prior information about effect size — for instance from a meta-analysis — you can set r accordingly. Whatever you choose must be stated in the report, and you should test at least one alternative value to assess how sensitive BF₁₀ is to that choice.
Can the Bayesian t-test confirm the null hypothesis?
Yes — this is one of its key advantages over frequentist testing. A BF₁₀ < 1/10 (equivalently BF₀₁ > 10) provides strong evidence that H₀ is more compatible with the data than H₁. A frequentist p-value greater than 0.05 cannot make this claim; it only fails to reject the null.
What is a Highest Density Interval and how does it differ from a confidence interval?
The 95% HDI is the shortest posterior interval that contains 95% of the probability mass. It is a direct statement: given the data and prior, there is a 95% probability that δ lies in that range. A frequentist 95% confidence interval is a statement about the long-run coverage of the procedure across hypothetical replications, not about any single computed interval. The two intervals often overlap numerically but carry fundamentally different interpretations.
Sources
- Rouder, J. N., Speckman, P. L., Sun, D., Morey, R. D. & Iverson, G. (2009). Bayesian t Tests for Accepting and Rejecting the Null Hypothesis. Psychonomic Bulletin & Review, 16(2), 225–237. DOI: 10.3758/PBR.16.2.225 ↗
How to cite this page
ScholarGate. (2026, June 1). Bayesian t-Test for Two-Group Comparison. ScholarGate. https://scholargate.app/en/bayesian/bayesian-t-test
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
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