Bayesian Independent Samples t-test
Also known as: Bayesian two-sample t-test, Bayes factor t-test, JZS t-test, Bayesian unpaired t-test
The Bayesian independent samples t-test quantifies evidence for or against a mean difference between two independent groups using a Bayes factor rather than a p-value. Rooted in Jeffreys's probability framework and popularized by Rouder et al. (2009), it places a Cauchy prior on the standardized effect size and returns continuous evidence for both the null and alternative hypotheses.
Key highlights
- Provides continuous, graded evidence for both the null and the alternative hypothesis.
- Allows sequential testing and evidence accumulation without inflating Type I error.
- Yields a posterior distribution over the effect size, enabling richer inference than a point estimate.
- Can support the null hypothesis explicitly, unlike classical p-values.
- The Cauchy prior is theoretically principled and scale-invariant.
Intuition
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How it works
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When to use it
Use the Bayesian independent samples t-test when comparing the means of two independent groups on a continuous outcome and you want to quantify evidence rather than make a binary reject/not-reject decision. It is especially valuable when you need to accumulate evidence across studies, when you expect the effect may be absent, or when sample sizes are too small to trust asymptotic p-values. Avoid it when you have no basis for specifying a prior — in that case, explicitly justify the default Cauchy prior or report results for a range of r values. It is not a substitute for careful data collection; small noisy studies will produce inconclusive Bayes factors rather than clear decisions.
Strengths & limitations
- Provides continuous, graded evidence for both the null and the alternative hypothesis.
- Allows sequential testing and evidence accumulation without inflating Type I error.
- Yields a posterior distribution over the effect size, enabling richer inference than a point estimate.
- Can support the null hypothesis explicitly, unlike classical p-values.
- The Cauchy prior is theoretically principled and scale-invariant.
- The Bayes factor depends on the chosen prior; different priors yield different results.
- Interpretation of Bayes factor thresholds is less standardized than the alpha = 0.05 convention.
- Computationally more involved than the classical t-test, though software (JASP, BayesFactor R package) automates it.
- Still assumes approximate normality of the outcome within each group.
Common pitfalls
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Applications
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Frequently asked
How do I choose the prior scale r?
The default r = sqrt(2)/2 (approximately 0.707) is recommended for a medium prior effect. Use r = 1 for a wider prior expecting larger effects, or r = 0.5 for a narrower prior expecting smaller ones. Always report the chosen r and ideally run a sensitivity analysis across multiple values.
How does BF10 differ from a p-value?
A p-value only quantifies evidence against the null; it cannot support the null. BF10 is symmetric: large values favor H1, small values favor H0. BF10 also does not depend on the sampling plan the way p-values do, making it suitable for sequential or adaptive designs.
Can I use this test without knowing Bayesian statistics?
Yes — software like JASP computes and displays BF10 automatically. However, understanding what the prior represents and how to interpret the Bayes factor scale is important for honest reporting.
What if my data violate normality?
The Bayesian t-test is moderately robust to mild non-normality in large samples, but for clearly non-normal or ordinal data consider a Bayesian Mann-Whitney U test or a robust Bayesian location model.
Is there a Welch correction in the Bayesian version?
Yes. A heteroscedastic (Welch-like) Bayesian t-test exists and is implemented in JASP and the BayesFactor R package. Use it when there is reason to doubt equal variances across the two groups.
Sources
- 1.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.
- 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 Independent Samples t-test. ScholarGate. https://scholargate.app/statistics/bayesian-independent-samples-t-test