Bayesian One-Sample t-test
Also known as: Bayesian single-sample t-test, Bayes factor one-sample t-test, JZS one-sample Bayes factor, Bayesian location test
The Bayesian one-sample t-test compares a single group's mean against a fixed reference value using a Bayes factor rather than a p-value. It quantifies the evidence the data provide for the null hypothesis (mean equals the reference) versus the alternative, and yields a full posterior distribution over the effect size — enabling statements about practical magnitude, not just a binary reject-or-retain decision.
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When to use it
Use the Bayesian one-sample t-test when you measure a single continuous variable and want to compare the group mean to a known or theoretically meaningful reference value, and when you want to quantify evidence for both the null and alternative hypotheses rather than simply reject or retain the null. It is especially useful when sample sizes are small and you want to incorporate prior information, or when you need to state the strength of evidence for a null effect. Avoid it when you have no theoretical justification for any prior on effect size and stakeholders require a strict frequentist error-rate guarantee; in that case, the classical one-sample t-test is more appropriate. Also avoid it when your outcome is clearly non-continuous or heavily skewed without transformation.
Strengths & limitations
- Quantifies evidence for the null hypothesis — a capability absent from p-values.
- Provides a full posterior distribution over the effect size, enabling probabilistic statements about magnitude.
- Bayes factor interpretation is continuous: larger values indicate stronger evidence, allowing gradation rather than a binary decision.
- Default Cauchy prior is well-motivated by Rouder et al. (2009) and does not require subjective elicitation in most applied settings.
- Can be updated incrementally as new data arrive, consistent with sequential study designs.
- The Bayes factor depends on the prior — different choices of r yield different BF values, and sensitivity to the prior should always be checked.
- Assumes the sampling distribution of the test statistic follows a t-distribution, which requires approximate normality or sufficiently large n.
- Interpretation of Bayes factor thresholds (e.g. BF > 3 as moderate evidence) is conventional and not universally agreed upon.
- Computational implementation is less familiar to many applied researchers than the classical t-test.
Frequently asked
What does BF10 = 0.2 mean?
A BF10 below 1 means the data are more consistent with the null hypothesis than the alternative. BF10 = 0.2 (equivalently BF01 = 5) indicates moderate evidence in favor of the null — the data are 5 times more likely under H0 than H1. This is a qualitative conclusion the classical t-test cannot provide.
How do I choose the prior scale r?
The default Cauchy scale r = sqrt(2)/2 ≈ 0.707, recommended by Rouder et al. (2009), works well for many social and behavioral science contexts. For expected small effects (e.g. subtle cognitive manipulations) a narrower scale such as r = 0.5 may be preferable. Always report the chosen r and perform a sensitivity analysis by repeating with r = 0.5 and r = 1.0.
Is this the same as the classical one-sample t-test with a Bayesian label?
No. Both use the t statistic as a sufficient statistic, but the inferential output differs fundamentally. The classical test computes a p-value — the probability of a result at least this extreme if H0 is true. The Bayesian test computes a Bayes factor — the relative likelihood of the data under H1 versus H0 — and a posterior credible interval for the effect size.
Can I use it with very small samples?
Yes, and this is one of its advantages over asymptotic frequentist tests. The Bayes factor and posterior are exact given the prior, not asymptotic approximations. However, very small samples (n < 5) will generally produce weak, inconclusive Bayes factors, reflecting genuine uncertainty rather than a false sense of precision.
What software runs this test?
The BayesFactor R package (Morey & Rouder) provides the ttestBF() function. JASP offers a graphical interface with point-and-click Bayesian t-tests. Python users can use the pingouin library, which includes Bayesian t-test functionality with Cauchy priors.
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 ↗
- Kruschke, J. K. (2014). Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan (2nd ed.). Academic Press. ISBN: 978-0124058880
How to cite this page
ScholarGate. (2026, June 3). Bayesian One-Sample t-test. ScholarGate. https://scholargate.app/en/statistics/bayesian-one-sample-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.
- Bayesian Independent Samples t-testStatistics↔ compare
- One-sample t-testStatistics↔ compare