Bayesian Mann-Whitney U Test
Also known as: Bayesian rank-sum test, Bayesian Wilcoxon rank-sum test, Bayesian nonparametric two-sample test
The Bayesian Mann-Whitney U test is a nonparametric Bayesian procedure for comparing two independent groups when data are ordinal or non-normal continuous. Instead of a binary reject/fail-to-reject decision, it quantifies the relative evidence for the null and alternative hypotheses through a Bayes factor, allowing researchers to conclude in favour of either hypothesis or express uncertainty.
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When to use it
Use this test when you have two independent groups, data are ordinal or continuous but non-normal (or sample size is too small to verify normality), and you want to quantify evidence rather than just make a binary decision. It is especially valuable when you need to argue in favour of the null hypothesis, when data arrive sequentially and you want to monitor evidence, or when prior knowledge about the effect size is available. Do not use it when data are paired (use the Bayesian Wilcoxon signed-rank test instead), when you have more than two groups (use the Bayesian Kruskal-Wallis test), or when you simply need a familiar p-value output for regulatory compliance.
Strengths & limitations
- Quantifies evidence for both the alternative and the null hypothesis via the Bayes factor.
- Retains the rank-based robustness of the classical Mann-Whitney test — no normality assumption required.
- Supports sequential analysis: evidence can be monitored as data accumulate without inflating the false-positive rate.
- Provides a posterior distribution over the effect size, enabling probabilistic interpretation.
- Allows prior knowledge to be formally incorporated through the Cauchy scale parameter.
- Bayes factor depends on the prior; the Cauchy default may not suit all research contexts.
- Interpretation of Bayes factors requires familiarity with Bayesian inference — less intuitive than p-values for some audiences.
- Computationally more demanding than the classical test, particularly for large samples.
- Does not directly test equality of medians unless additional distributional assumptions are made.
Frequently asked
When should I choose this over the classical Mann-Whitney U test?
Choose the Bayesian version when you want to quantify relative evidence rather than obtain a binary p-value decision, when you need the ability to support the null hypothesis, or when you are conducting sequential analysis. If you only need a simple p-value for a one-time comparison, the classical test is sufficient.
Does this test still avoid the normality assumption?
Yes. Like the classical Mann-Whitney test, the Bayesian version operates on ranks rather than raw values, so it does not require the data to follow a normal distribution. It assumes only that the two samples are independent and that the data are at least ordinal.
How do I choose the prior scale r?
The default r = 1/sqrt(2) encodes a moderate expected effect and is appropriate as a general-purpose prior. For domains where effects are typically small (e.g. cognitive psychology), r = 0.5 may be more appropriate. Always report the chosen r and check robustness by rerunning with a range of plausible values.
Where is this test available in software?
JASP offers a graphical interface for the Bayesian Mann-Whitney test. In R, the BayesFactor package provides the wilcoxonBF() function. Both use the van Doorn et al. (2020) formulation.
Is a Bayes factor of 3 strong evidence?
By the conventional Jeffreys scale, BF10 between 3 and 10 is considered moderate evidence. Values above 10 are strong, and above 30 are very strong. However, these thresholds are guidelines, not hard rules — always contextualise the Bayes factor relative to the research question and prior plausibility.
Sources
- van Doorn, J., Ly, A., Marsman, M., & Wagenmakers, E.-J. (2020). Bayesian rank-based hypothesis testing for the rank sum test, the signed rank test, and Spearman's rho. Journal of Applied Statistics, 47(16), 2984–3006. DOI: 10.1080/02664763.2019.1709053 ↗
- Mann, H. B., & Whitney, D. R. (1947). On a test of whether one of two random variables is stochastically larger than the other. The Annals of Mathematical Statistics, 18(1), 50–60. DOI: 10.1214/aoms/1177730491 ↗
How to cite this page
ScholarGate. (2026, June 3). Bayesian Mann-Whitney U Test. ScholarGate. https://scholargate.app/en/statistics/bayesian-mann-whitney-u-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
- Bayesian Wilcoxon signed-rank testStatistics↔ compare
- Independent samples t-testStatistics↔ compare