Bayesian Moderation Analysis
Also known as: Bayesian interaction analysis, Bayesian moderated regression, Bayesian moderator testing, BMA
Bayesian moderation analysis tests whether the relationship between a predictor and an outcome changes depending on the value of a third variable (the moderator). By placing prior distributions on all model parameters and updating them with observed data, it yields full posterior distributions for the interaction effect — enabling direct probability statements about the moderation rather than binary significance decisions.
Read the full method
Sign in with a free account to read this section.
Method map
The neighbourhood of related methods — select a node to explore.
When to use it
Use Bayesian moderation analysis when you want to test whether a third variable changes the strength or direction of a predictor-outcome relationship and you need more than a binary significant/non-significant answer — particularly when sample sizes are small to moderate and frequentist power is limited, when you can specify meaningful prior information, or when the research question concerns the magnitude of the interaction rather than its mere existence. It is well-suited to confirmatory studies where prior findings can be formalised as informative priors. Do not use it as a substitute for careful model specification: an interaction term requires that both main effects are also in the model. Avoid it when the moderation hypothesis is entirely data-driven and no defensible priors exist, in which case classical moderated regression with bootstrap confidence intervals may be more transparent to reviewers.
Strengths & limitations
- Provides full posterior distributions for the interaction coefficient, enabling direct probability statements about the moderation.
- Handles small and moderate samples better than frequentist tests by borrowing strength from priors.
- Allows incorporation of prior evidence from past studies or expert knowledge via informative priors.
- Credible intervals have a direct probability interpretation that does not require the repeated-sampling logic of frequentist confidence intervals.
- Can naturally accommodate multilevel structures, non-normal outcomes, and complex designs within the same Bayesian framework.
- Choice of prior distributions is subjective and can influence results, especially in small samples; sensitivity analyses are required.
- MCMC sampling is computationally more demanding than ordinary least squares and requires convergence checking.
- Posterior summaries and credible intervals are less familiar to many applied researchers and reviewers than standard regression output.
- Does not automatically eliminate the problems of multiple testing when many moderators are examined.
Frequently asked
How does Bayesian moderation differ from standard moderated regression?
Both estimate the same linear model with an interaction term. The difference is inferential: standard moderated regression uses p-values and frequentist confidence intervals, while Bayesian moderation yields posterior distributions and credible intervals. The Bayesian approach allows direct probability statements about the interaction coefficient and incorporates prior information.
What prior should I use for the interaction coefficient?
A weakly informative prior — for example, Normal(0, 1) on standardised coefficients — is a safe default that regularises estimation without strongly constraining the result. If past studies report a specific range for the interaction, an informative prior can encode that knowledge. Always report the prior chosen and assess sensitivity by re-running with alternative prior widths.
Should I centre the predictor and moderator before the analysis?
Yes. Mean-centring X and W before forming the product term XW reduces multicollinearity and makes the main effect coefficients interpretable as effects at the mean of the other variable. This is standard practice in both frequentist and Bayesian moderation analysis.
How do I know whether the moderation is practically meaningful?
Beyond checking whether the credible interval for the interaction excludes zero, compute simple slopes at substantively meaningful values of W and examine their posterior distributions. You can also define a region of practical equivalence (ROPE) around zero and check whether the posterior for the interaction falls inside it, indicating a negligible effect.
Can I use Bayesian moderation with non-continuous outcomes?
Yes. The Bayesian framework generalises naturally to logistic, Poisson, ordinal, and other regression models; you simply specify the appropriate likelihood. Software such as brms in R makes it straightforward to fit Bayesian generalised linear models with interaction terms.
Sources
- Hayes, A. F. (2018). Introduction to Mediation, Moderation, and Conditional Process Analysis: A Regression-Based Approach (2nd ed.). Guilford Press. ISBN: 978-1462534654
- Kruschke, J. K. (2015). 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 Moderation Analysis. ScholarGate. https://scholargate.app/en/statistics/bayesian-moderation-analysis
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 Moderated MediationStatistics↔ compare
- Moderated MediationStatistics↔ compare