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

The research team, led by Rosa Carmen Sotelo-Casas of the University of Guadalajara, monitored benthic communities in three bay systems: Chamela Bay, the Tenacatita-Barra de Navidad bays, and the Santiago-Manzanillo bays. These locations were chosen because they harbor some of the best-preserved coral ecosystems in the region, each maintaining an average live coral cover of at least 29 percent. Between February 2010 and October 2023, divers combined visual surveys along belt transects with underwater video analysis, extracting forty frames from each recording and overlaying a grid of fifty random points per frame, which yielded two thousand scored points per video. This dual technique allowed the researchers to quantify the cover of reef-building corals, algal turf, fleshy macroalgae, encrusting calcareous algae, and inert substrate with enough precision to track change across thirteen turbulent years.

The reefs themselves are dominated by a single remarkable genus. Pocillopora, the branching coral that forms the structural backbone of these systems, accounted for an average of 30 percent of total benthic cover, dwarfing the contributions of Porites, Pavona, and Psammocora. In total, the team recorded eighteen species of scleractinian coral across the region, with the Santiago-Manzanillo bays supporting the highest richness at sixteen species, followed by Tenacatita-Barra de Navidad with fourteen and Chamela Bay with eight. The rest of the seascape was a dynamic mosaic: algal turf averaged 29 percent, inert substrates 25 percent, encrusting calcareous algae 9 percent, and fleshy macroalgae just 4.5 percent, with sponges, octocorals, bryozoans, and other sessile invertebrates each contributing less than 2 percent.

What makes this region scientifically fascinating is its position at a biogeographic crossroads. The Central Mexican Pacific sits in a transitional zone between the Warm Temperate Northeast Pacific and the Tropical Eastern Pacific ecoregions, bounded to the north by the entrance to the Gulf of California and to the south by the Isthmus of Tehuantepec. The region experiences two sharply contrasting hydroclimatic seasons. From February to June, the equatorward branch of the California Current drives strong wind-powered coastal upwelling, pushing cold, nutrient-rich deep water to the surface and fueling blooms of phytoplankton. From July to January, the poleward Mexican Coastal Current takes over, promoting warm, nutrient-poor conditions. Sea surface temperatures swing from an average of 22.5 degrees Celsius in the dry season to 28.3 degrees in the rainy season, a gradient that can widen by up to 10 degrees when El Niño or La Niña events superimpose their thermal anomalies on the local cycle.

To untangle how these forces shape the reefs, the researchers paired their biological monitoring with satellite-derived environmental data, including sea surface temperature, chlorophyll-a concentration, and the diffuse attenuation coefficient, a measure of water clarity. They then applied a battery of statistical tools rarely seen together in reef ecology: Granger causality tests to determine whether large-scale ENSO indices actually drive local thermal anomalies, periodogram-based dissimilarity measures to compare the rhythms of environmental time series, and beta regression models to link each benthic group’s cover to specific environmental variables with quarterly and semiannual time lags. The results were striking. Of the seven ENSO indices tested, only the Niño 4 index showed consistent and significant causality with local sea surface temperature anomalies at all three sites, indicating that only certain flavors of El Niño genuinely reach these coastal waters.

The interplay between global climate oscillations and local upwelling proved to be the decisive factor in reef survival. When the 2010 to 2012 La Niña event overlapped with the spring 2012 upwelling season, sea surface temperatures plunged below 22 degrees Celsius, thermal anomalies reached roughly minus 4 degrees, chlorophyll-a concentrations climbed above 10 milligrams per cubic meter, and water turbidity hit record values. Conversely, during the 2015 to 2016 El Niño nicknamed Godzilla, the upwelling season kept temperatures below 30 degrees throughout the spring and delayed anomalies exceeding 2 degrees until autumn, effectively buffering the reefs against the worst of the marine heatwave. The timing, the authors conclude, is everything: when an ENSO event opposes the local seasonal cycle, upwelling acts as a stress mitigator, but when the two align, thermal extremes are amplified.

Each reef told its own story of loss and recovery. Chamela Bay, the northernmost and deepest site, kept live coral cover above 19 percent throughout the study, dipping to its lowest point in 2017 before steadily recovering to roughly 29 percent by 2023. The Tenacatita-Barra de Navidad bays suffered the harshest blow, losing 28 percent of coral cover between 2013 and 2014, when cover fell to about 14 percent from a starting point above 46 percent in 2010, yet the system rebounded to approximately 38 percent by 2023. The Santiago-Manzanillo bays, which consistently held the highest coral cover, peaked near 52 percent in 2019 before declining 13 percent in 2023, a drop the researchers attribute to the combined effect of the prolonged 2023 El Niño and heated water discharged from a thermoelectric power plant in the Cuyutlán lagoon near the port of Manzanillo.

The beta regression models revealed that every benthic group responds to a distinct set of environmental levers, and that these levers differ from site to site. Coral cover at Chamela Bay was best explained by the previous semiannual chlorophyll-a value, with a negative relationship, while at Tenacatita-Barra de Navidad the key drivers were local sea surface temperature and thermal anomalies, both negative, and at Santiago-Manzanillo it was the previous quarterly thermal anomaly. Algal turf, the great opportunist of reef surfaces, responded to thermal anomalies with site-specific lags, expanding as anomalies rose at the two northern sites but shrinking at Santiago-Manzanillo, a pattern suggesting that turf dynamics are governed less by temperature itself than by the availability of bare space created when corals die. Encrusting calcareous algae peaked at all three sites in 2017, shortly after the Godzilla El Niño, apparently exploiting the combination of moderate irradiance, elevated nutrients, and fresh settlement space left behind by coral mortality.

These findings carry implications that extend far beyond the Mexican coast. Globally, roughly one-third of the planet’s live coral cover has been lost over the past five decades, and during the 2023 El Niño event, reef ecosystems elsewhere in the Mexican Pacific suffered catastrophic losses of up to 90 percent of coral cover. Yet the Central Mexican Pacific reefs persisted, and the study points to two explanations. First, pronounced seasonal upwelling appears to shield corals from the most severe thermal anomalies, preventing local temperatures from exceeding critical thresholds and boosting productivity that buffers heat stress. Second, the historical gauntlet of thermal events these populations have endured may have selected for strains and colonies that are genetically and physiologically more resistant, a process reinforced by recent evidence that corals exposed to repeated bleaching develop metabolomic and epigenetic adaptations in their tissues, symbiotic algae, and bacterial communities.

The study also delivers a sobering caveat. Resilience is not immunity. The 2023 decline at Santiago-Manzanillo demonstrates that when a powerful El Niño synchronizes with local anthropogenic warming, even the region’s most robust reefs can be pushed past their tolerance limits. The researchers caution that long-term genetic monitoring will be needed to determine whether these coral assemblages are undergoing population turnover after each thermal crisis, and they note that classifying algae into functional groups, while useful for detecting broad patterns, may mask species-level differences driving site-specific behavior. Still, in an era when coral reef news is dominated by collapse, the Central Mexican Pacific offers something increasingly precious: a documented case of reefs that bend under environmental pressure, recover with remarkable speed, and may hold the secrets to how corals everywhere might weather the century ahead.

Subject of Research: Long-term benthic changes and resilience of coral reef ecosystems in the Central Mexican Pacific in response to ENSO events and upwelling

Article Title: Ten years of coral reef benthic changes in the Central Mexican Pacific

Article References: Sotelo-Casas, R. C., Rodríguez-Zaragoza, F. A., Rodríguez-Troncoso, A. P., Cupul-Magaña, A. L., & Godínez-Domínguez, E. (2025). Ten years of coral reef benthic changes in the Central Mexican Pacific. Discover Ecology, 1(1), Article 14. https://doi.org/10.1007/s44396-025-00017-6

Image Credits: AI Generated

DOI: 10.1007/s44396-025-00017-6

Keywords: coral reefs, Central Mexican Pacific, ENSO, El Niño, upwelling, Pocillopora, algal turf, coral bleaching, benthic cover, sea surface temperature, reef resilience, beta regression