The problem begins with the data themselves. Psychology is full of outcomes that are naturally constrained. Accuracy on a memory test is a proportion of correct responses. Confidence ratings on a slider scale fall between two endpoints. The share of time a participant spends looking at a stimulus, the fraction of items recalled, the percentage of conspiracy beliefs reduced after an intervention — all of these quantities are bounded. Yet the workhorse tools of the field, from ordinary least squares regression to the t test and analysis of variance, assume that outcomes are drawn from a normal distribution, which stretches infinitely in both directions. When researchers feed bounded data into these models, the model can, in principle, predict values below zero or above one, and the resulting estimates of means, variances, and effects can become warped, especially when data pile up near the boundaries.

Historically, researchers have tried to work around this with transformations. Square root and arcsine transformations, some dating back to the 1930s, were devised to make proportions behave more like normally distributed variables. But the tutorial’s authors, following a long line of statistical literature, argue that these fixes are clumsy and increasingly unnecessary. The beta distribution, a flexible family of probability distributions defined precisely on the interval from zero to one, offers a principled alternative. Depending on its two shape parameters, the beta distribution can be symmetric, skewed toward either boundary, U-shaped, or nearly uniform, which makes it remarkably well suited to the messy reality of proportion data in the behavioral sciences.

Beta regression, first formalized for rates and proportions in 2004 by Ferrari and Cribari-Neto, extends this idea into a full modeling framework analogous to the generalized linear models that revolutionized the analysis of counts and binary outcomes. Instead of modeling the mean of a normal distribution, beta regression models the mean of a beta-distributed outcome, typically through a link function such as the logit, which maps the bounded interval onto the unbroken real line. A second parameter governs the precision, or how tightly the data cluster around the mean, and that precision itself can depend on predictors. This means the model can capture not only how the average outcome changes with an experimental manipulation but also how the spread of outcomes changes, a capability that classical models simply lack.

The tutorial’s centerpiece is a worked example drawn from the psychological literature on learning and memory, specifically studies of how instructor fluency shapes perceptions of learning. In these experiments, participants watch lectures delivered either fluently or disfluently and then judge how much they learned, alongside measures of actual learning. The outcome of interest is a proportion, making it a textbook case for beta regression. The authors walk readers through fitting a Bayesian beta regression model in R using the brms package, which provides a high-level interface to the Stan probabilistic programming language while retaining the familiar formula syntax of standard R functions such as lm. For researchers trained in conventional regression, the transition is deliberately gentle: the model is specified with the same kind of formula, but the underlying machinery samples from posterior distributions rather than computing single point estimates.

The Bayesian framing is not incidental. The authors explain that with sufficiently weak, uniform priors, Bayesian estimation of a beta regression reproduces the maximum likelihood estimates one would obtain in a frequentist analysis, so the choice of framework is less about producing different numbers and more about the inferential machinery surrounding them. Bayesian posterior distributions yield direct statements about plausible parameter values, uncertainty intervals that behave intuitively, and a coherent way to incorporate prior knowledge. The authors also address a common objection head-on: yes, Bayesian computation is slower, but modern engines are efficient, brms supports parallelization for large datasets, and in explanatory research the priority is specifying the right model, not shaving seconds off the computation.

Interpretation is where many applied researchers stumble, and the tutorial devotes careful attention to it. Coefficients in a beta regression live on the logit scale, which is notoriously opaque. The authors therefore demonstrate how to translate results back onto the response scale — the familiar territory of predicted proportions — using the marginaleffects and easystats packages. Rather than reporting a raw coefficient, researchers can compute predicted proportions for each condition, differences between them, and uncertainty around those predictions, producing quantities that read like plain English: participants in one condition recalled, on average, a proportion of X, while those in another recalled Y. The authors frame models as prediction machines, echoing a broader movement in quantitative psychology to convert confusing coefficients into clear, interpretable quantities.

Real data rarely cooperate perfectly, and the tutorial confronts the messiest feature of bounded outcomes: exact zeros and ones. The beta distribution assigns zero probability to the boundary points themselves, so observations sitting precisely at zero or one require special handling. The authors introduce zero-inflated and zero-one-inflated beta models, which add a separate process generating boundary observations, and they note that some methodologists prefer the term hurdle model for the same reason. In their example data, a single observation equals exactly one, and they show how a zero-one-inflated beta model can accommodate it. They also present the ordered beta regression model developed by Kubinec, a parsimonious alternative that handles continuous data with both lower and upper bounds, including boundary values, within a single framework, and they point readers to newer options such as the beta-gate model and extended-support beta regression for continuous processes near the boundaries.

Model comparison and checking receive their due as well. The tutorial demonstrates leave-one-out cross-validation, implemented through the loo function in brms, which lets researchers compare competing beta models on out-of-sample predictive accuracy. Convergence of the Markov chain Monte Carlo sampler is assessed with modern diagnostics, and the authors emphasize robust summaries of posterior distributions. For researchers who prefer a frequentist framework, an appendix shows how to fit equivalent beta models with the glmmTMB package, making the tutorial accessible regardless of statistical philosophy. The entire workflow was written in R with Quarto, and the authors used the rix package with the Nix ecosystem to snapshot not just the R packages but the full system dependencies, so that anyone can reproduce the exact computational environment from the archived materials.

The stakes extend well beyond one subfield. Proportions and bounded scales appear across the sciences, from ecological studies quantifying ecosystem vulnerability to phonetics research on categorical speech perception to ecological momentary assessments of substance use. As Bayesian methods continue their rapid rise in psychology, propelled by the open science movement, tutorials like this one lower the barrier to adopting tools that match the data rather than forcing data to match the tools. All code, data, and materials are openly archived on Zenodo and GitHub, and the authors assume only basic familiarity with regression and R. For a discipline increasingly alert to the fragility of its statistical practices, the message is simple: when your outcome lives between zero and one, model it that way, and the beta distribution — flexible, principled, and now thoroughly documented — is waiting to do the job properly.

Subject of Research: Bayesian beta regression for analyzing bounded proportion outcomes in psychological research

Article Title: A beta way: A tutorial on Bayesian beta regression for psychological research

Article References: Geller, J., Kubinec, R., Pelleriti, C. M. P., & Vuorre, M. (2026). A beta way: A tutorial on Bayesian beta regression for psychological research. Behavior Research Methods, 58(11), Article 309. https://doi.org/10.3758/s13428-026-03106-w

Image Credits: AI Generated

DOI: 10.3758/s13428-026-03106-w

Keywords: beta regression, Bayesian inference, psychological methods, proportion data, brms, Stan, R tutorial, Behavior Research Methods, statistics, learning and memory, zero-inflated models, quantitative psychology