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 problem sounds deceptively simple. In any experience sampling study, a proportion of participants will report exactly the same score on a given construct across every measurement occasion. Someone might rate their daily stress as a three on a scale, every single day, for weeks. The authors call these cases non-varying individuals, meaning that their observed scores on certain state constructs remain constant over the study period. Crucially, the researchers emphasize that this label does not imply such people are inherently incapable of change; it describes an observed pattern in the data, which may reflect genuine stability, restricted circumstances, or response habits such as repeatedly choosing the same scale point. Whatever the cause, the statistical consequences are far from trivial.

Luo and colleagues begin with a motivating example drawn from daily stress research, using measures of daily stressors of the kind captured by instruments such as the Daily Inventory of Stressful Events. When they included non-varying individuals in a univariate autoregressive model, the substantive conclusions changed. Autoregressive parameters, which describe how strongly a person’s stress today depends on their stress yesterday, are the backbone of this modeling tradition. A high autoregressive effect suggests a sluggish system that carries perturbations forward, while a low one suggests rapid recovery. The demonstration showed that the simple act of keeping zero-variability participants in the sample could flip what researchers would report about how stress propagates over time.

To understand why, it helps to consider what a constant time series does to an autoregressive estimate. For an individual whose scores never change, the model sees a perfectly predictable series: yesterday’s value fully determines today’s. Depending on how the model handles such cases, these flat trajectories can pull estimated dynamics toward extreme values. The team’s first simulation study, focused on univariate processes, confirmed the mechanism: as the proportion of non-varying individuals in a sample increased, the autoregressive estimates showed a systematic upward bias. In other words, the population would appear more self-predictive, more inertial, than it truly is, purely because some members contributed flat lines rather than genuine fluctuations.

The second simulation extended the investigation to bivariate processes, where researchers estimate not only how a variable predicts itself over time but also how one variable, say stress, predicts another, say negative affect, at the next occasion. These cross-lagged effects fuel countless claims about emotional reactivity, mutual influence between partners, and feedback loops between symptoms. Here the findings were subtler and arguably more alarming. The average, fixed-effect cross-lagged estimates could look accurate on the surface, giving analysts false confidence. Yet beneath that apparent accuracy, the person-specific cross-lagged effects were systematically distorted, with some individuals’ effects overestimated and others underestimated. A researcher examining individual differences in emotional dynamics, or trying to identify who is most reactive, could be badly misled.

Within the widely used Gaussian autoregressive modeling framework, the authors arrive at a clear practical recommendation: estimate dynamic parameters using a subsample that excludes non-varying individuals. This is not a call to discard participants as defective, but a recognition that a flat observed series carries no information about temporal dynamics and can actively inject bias into estimates pooled across people. The recommendation echoes a broader movement in the methodological literature, which has recently scrutinized related data pathologies in intensive longitudinal designs, including floor effects that compress scores against scale minima and zero inflation that floods datasets with identical values. Non-varying individuals represent a distinct and previously understudied facet of this family of problems.

Importantly, the study does not stop at a blanket rule. Luo, Hu, and Liu delineate when subsample estimates can be interpreted as approximations to the full population, and when they should instead be read as describing only the subsample population. Their simulations indicate that in bivariate processes, once the proportion of non-varying individuals reaches around ten percent or more, the dynamics estimated from the varying subsample can no longer be safely generalized to everyone who was recruited. Below that threshold, the subsample estimates serve as reasonable approximations of the whole; beyond it, researchers must be explicit that their conclusions apply to the subset of people whose states actually fluctuated. This distinction matters for fields such as affective science, where claims about population-level emotional dynamics rest on samples that may contain substantial pockets of stability.

The implications ripple across the many domains that have embraced within-person modeling. Researchers have documented intraindividual variability in pain, sleep, perceived stress, affect in older versus younger adults, state academic self-concept among students, and daily sense of purpose, and meta-analytic work on ecological momentary assessment shows the method now spans virtually every corner of psychology. Some of this literature has already grappled with variability itself as a substantive quantity, asking what predicts within-person variance in applied constructs and whether traits like neuroticism relate to emotional instability. The new findings add a methodological caution to that substantive conversation: before interpreting variability, or dynamics, researchers should first examine each individual’s within-person variability and account for those who show none.

The authors also situate their work within a rapidly maturing methodological ecosystem. Dynamic structural equation models, Bayesian multilevel approaches, and related frameworks have made person-specific dynamics routinely estimable, and open resources such as databases of openly available experience sampling datasets are flooding the field with intensive longitudinal data. As data volume grows, so does the chance that non-varying individuals slip through automated pipelines unnoticed. The study’s data and code have been made openly available on the Open Science Framework, allowing applied researchers to reproduce the simulations and apply the recommended screening procedures to their own projects. The work was supported by the National Natural Science Foundation of China, and the authors report no conflicts of interest.

For the practicing scientist, the takeaway is a short checklist with long consequences. Screen every participant’s time series for zero observed variability before fitting dynamic models. Report how many individuals were excluded and why. Interpret autoregressive and cross-lagged estimates from the varying subsample as population approximations only when the excluded fraction is small, and as subsample descriptions when it approaches or exceeds roughly ten percent in bivariate settings. And resist the temptation to treat flat responders as noise to be quietly absorbed; as this study demonstrates, their inclusion can inflate autoregressive estimates and silently scramble person-specific cross-lagged effects even while group averages look pristine. In a research culture increasingly built on the promise of capturing life as it actually unfolds, the people whose numbers never move turn out to be among the most consequential figures in the dataset, precisely because of the stillness they bring to it.

Subject of Research: Statistical handling of non-varying individuals in within-person dynamic modeling of intensive longitudinal data

Article Title: Examining within-person variability of each individual: How should we deal with non-varying individuals?

Article References: Luo, X., Hu, Y., & Liu, H. (2026). Examining within-person variability of each individual: How should we deal with non-varying individuals?. Behavior Research Methods, 58(10), Article 285. https://doi.org/10.3758/s13428-026-03171-1

Image Credits: AI Generated

DOI: 10.3758/s13428-026-03171-1

Keywords: within-person variability, intensive longitudinal data, autoregressive models, cross-lagged effects, dynamic structural equation modeling, ecological momentary assessment, non-varying individuals, simulation study, daily stress, parameter bias, Behavior Research Methods, psychological methods

The study, published in the journal Behavior Research Methods, was led by Milla Pihlajamäki together with Ginette Lafit, Olivia J. Kirtley, Inez Myin-Germeys, and Gudrun Eisele. The team recruited 192 students and randomly assigned 64 of them to each of three incentive conditions. The first group received a fixed payment for taking part, the standard arrangement in most experience sampling research. The second group received the same fixed payment plus personalized feedback, a summary of the data they had contributed, on the theory that seeing one’s own psychological patterns might motivate more careful responding. The third group was paid per beep, earning money incrementally with every completed prompt, a structure that mirrors the piece-rate logic of behavioral economics rather than a flat salary model.

The protocol itself was demanding by design. For fourteen consecutive days, participants received nine prompts — or beeps — per day on their smartphones, producing up to 126 measurement moments per person and roughly 24,000 potential data points across the sample. This intensity is precisely what makes experience sampling so scientifically valuable and so operationally fragile. Unlike a one-off questionnaire, an intensive longitudinal design asks people to interrupt whatever they are doing — in a lecture, at dinner, with friends — to report on their inner states. Compliance, in this context, is not a given; it is an achievement that must be engineered through careful protocol design, and incentives are one of the most powerful levers available.

What makes the study methodologically notable is that the researchers did not stop at counting completed surveys. They systematically distinguished between data quantity and data quality, treating the two as separable outcomes that incentives might influence in different directions. Quantity was straightforward: compliance rates, or the proportion of beeps answered, and how that proportion changed over the two-week study period. Quality required more nuance. The team screened for careless responding — the phenomenon where participants click through items without genuine attention — and examined whether the temporal dynamics of such responses shifted across the study, alongside retrospective measures of participant burden and experience.

Careless responding matters because its consequences are surprisingly severe. As the methodological literature has repeatedly shown, even a modest proportion of inattentive respondents can distort correlations, inflate or deflate reliability estimates, and lead analysts astray in ways that are hard to detect after the fact. In experience sampling data, the problem is compounded by the sheer number of measurement moments and the fatigue that accumulates as the days wear on. A participant might respond attentively on day one and slide into patterned, mechanical answers by day ten. The Leuven team, drawing on their own prior work on the temporal dynamics of careless responding, built this dimension directly into the analysis, using analyses of variance and multilevel linear and logistic regression models to test whether incentive structure affected not only average behavior but its trajectory over time.

The headline finding is refreshingly clean: payment per beep increased compliance rates compared with the other two conditions, and that was essentially where the differences ended. The incremental payment scheme did not make participants answer more carelessly, did not change their reported burden, and did not alter their retrospective experience of the study in measurable ways. Personalized feedback, meanwhile — despite its intuitive appeal and growing popularity in mobile health and self-tracking applications — produced no measurable advantage over a plain fixed payment on any of the outcomes examined. The feedback condition neither boosted compliance nor improved the quality of responses, a null result that carries real practical weight for researchers weighing the added complexity of generating individualized reports against any assumed motivational payoff.

The finding that pay-per-beep worked without degrading quality is worth unpacking, because incentives in psychology have a complicated reputation. Classic motivation research has documented cases where external rewards crowd out intrinsic motivation or encourage gaming of the reward structure. In an experience sampling context, one might have worried that paying per beep would incentivize rapid, low-effort completion — quantity at the expense of quality. Recent related work on game-based rewards in experience sampling found exactly that pattern: gamification increased data quantity but reduced data quality. The Leuven results suggest that a straightforward monetary increment per completed beep does not carry the same risk, at least not in a motivated student population completing a two-week protocol.

The authors’ recommendation follows directly from the evidence: for motivated samples such as students, use payment per beep to boost data quantity. This is not a trivial prescription. Compliance in intensive longitudinal research varies enormously across populations and study designs, with meta-analyses reporting widely different completion rates depending on the sample, the burden of the protocol, and the population under study. Clinical populations, adolescents, and people experiencing acute distress often show lower compliance, and every percentage point of missing data constrains the statistical models that researchers use to extract meaning from these datasets. Multilevel time-series analyses, network models of psychological dynamics, and idiographic prediction approaches all benefit directly from denser, more complete measurement streams.

There are, of course, boundaries to the conclusion. The study was conducted in a student sample, a population that is generally well-motivated, digitally fluent, and accustomed to participating in research. Whether payment per beep would deliver the same benefit — and remain free of quality costs — in harder-to-reach populations, longer protocols, or studies measuring stigmatized behaviors remains an open question. The reward magnitudes involved were also specific to this context; the psychology of incremental incentives plausibly depends on the size of each increment relative to the total payment. And burden, while measured, was assessed retrospectively and through momentary reports within a demanding 14-day design; incentive effects on the lived experience of participation might differ in studies with different rhythms and demands.

Still, the study exemplifies a broader and welcome shift in psychological methods research: treating study design choices — payment schemes, feedback provision, sampling frequency, questionnaire length — as empirical questions rather than conventions inherited from previous studies. The team post-registered their study, documented every deviation transparently, and made all materials and analysis code openly available on the Open Science Framework, with data accessible through a secure checkout system. In a field increasingly aware that methodological decisions ripple through every downstream finding, knowing that a simple per-beep payment can fill in more of the data grid without compromising what fills those cells is exactly the kind of actionable, evidence-based guidance that turns methodological debate into better science.

Subject of Research: Effects of incentive type on data quantity and quality in experience sampling research

Article Title: The effect of incentive type on data quality and quantity in an experience sampling study in a student population

Article References: Pihlajamäki, M., Lafit, G., Kirtley, O. J., Myin-Germeys, I., & Eisele, G. (2026). The effect of incentive type on data quality and quantity in an experience sampling study in a student population. Behavior Research Methods, 58(11), Article 300. https://doi.org/10.3758/s13428-026-03164-0

Image Credits: AI Generated

DOI: 10.3758/s13428-026-03164-0

Keywords: experience sampling, ecological momentary assessment, incentives, compliance, data quality, careless responding, survey methodology, participant payment, personalized feedback, intensive longitudinal data, burden, psychological methods