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In this commentary on Pfadt et al. (2026, Psychometrika , 1-35), I first make the case for implementing psychometric methods, such as the conditional standard error of measurement (CSEM), in software that is user-friendly from a practitioner's perspective. Furthermore, I argue that bias and variance in CSEM estimates are still poorly understood and I report a small simulation study comparing the coverage rates of the CSEM estimate recommended by Pfadt et al. with those of the estimated (unconditional) standard error of measurement. The results point to possible directions for future research on CSEM estimation.

A discussion is provided of several issues related to behavioral measurement that arise from Pfadt et al. (2026, Psychometrika, 2026, 1-35). The note may be viewed in part as a complement to their developments regarding precision estimation for individual test scores.

Estimation in exploratory factor analysis often yields estimates on the boundary of the parameter space. Such occurrences, called Heywood cases, are characterized by non-positive variance estimates and can cause numerical instability, convergence failures, and misleading inferences. We derive sufficient conditions on the model and a penalty to the log-likelihood function that guarantee the existence of maximum penalized likelihood estimates in the interior of the parameter space, and that the corresponding estimators possess desirable asymptotic properties expected by the maximum likelihood estimator, namely, consistency and asymptotic normality. Consistency and asymptotic normality follow when penalization is soft enough, in a way that adapts to the information accumulation about the model parameters. We formally show, for the first time, that the penalties of Akaike (1987, Psychometrika, 52, 317-332) and Hirose et al. (2011, Journal of Data Science, 9, 243-259) to the log-likelihood of the normal linear factor model satisfy the conditions for existence, and, hence, deal with Heywood cases. Their vanilla versions, though, can result in questionable finite-sample properties in estimation, inference, and model selection. Our maximum softly-penalized likelihood (MSPL) framework ensures that the resulting estimation and inference procedures are asymptotically optimal. Through comprehensive simulation studies and real data analyses, we illustrate the desirable finite-sample properties of the MSPL estimators.

While the joint modeling of item responses and response times (RTs) has received considerable attention, most existing approaches remain limited to dichotomous items and are not applicable to assessments involving polytomous or mixed-format items. To address this limitation, this article proposes a novel joint modeling framework for graded item responses and RTs. Specifically, we develop a conditional RT model given item responses and integrate it with a marginal response model based on Samejima's graded response model, yielding a conditional joint model for graded item responses and RTs. The model is then embedded within a two-level hierarchical framework to account for the relationship between ability and speed at the population level. A key methodological contribution is the development of a stochastic approximation EM (SAEM) algorithm for estimating the proposed model, which efficiently computes its marginal maximum likelihood estimates. Simulation studies demonstrate the accurate parameter recovery of the SAEM algorithm and indicate that the proposed model outperforms the hierarchical model assuming conditional independence across various testing conditions. Finally, an empirical analysis using data from the 2022 Programme for International Student Assessment illustrates the effectiveness of the graded response-response time model in large-scale assessments.

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Time-varying coefficient modeling (TVCM), which represents regression coefficients as smooth functions of continuous time, provides a flexible framework for uncovering complex patterns of change in levels and associations in intensive longitudinal data. However, conventional TVCM remains limited to investigating directional effects across individuals. By introducing a TVCM formulation of the multivariate normal distribution, the present study extends TVCM to explore change in undirected associations (couplings) and variability, thereby broadening its utility for psychological research. We discuss three versions of this approach: an aggregate-level model and two hierarchical versions capturing interindividual differences in unfolding change, either via person-specific intercepts accounting for onset differences or through fully person-specific coefficient functions smoothed via partial pooling. To illustrate the proposed developments, we apply them to six weeks of intensive longitudinal data from 16 anxiety patients undergoing therapy and examine unfolding changes in the level and volatility of nervousness and threat monitoring, their coupling, as well as between-person heterogeneity in each of these. We further show how inspecting first-order derivatives of the coefficient functions supports identifying periods of stability and change. Finally, we discuss extensions incorporating person-level characteristics to explain heterogeneity in patterns of change and predict outcomes.

In psychometric sciences, such as social or behavioral sciences, and, similarly, in medical sciences, it is increasingly common to deal with longitudinal data organized as high-dimensional multidimensional arrays, also known as tensors. Within this framework, the time-continuous property of longitudinal data often implies a smooth functional structure on one of the tensor modes. To help researchers investigate such data, we introduce a new tensor decomposition approach based on the PARAFAC decomposition. Our approach allows researchers to represent a high-dimensional functional tensor as a low-dimensional set of functions and feature matrices. Furthermore, to capture the underlying randomness of the statistical setting more efficiently, we introduce a probabilistic latent model in the decomposition. A covariance-based block-relaxation algorithm is derived to obtain estimates of model parameters. Thanks to the covariance formulation of the solving procedure and thanks to the probabilistic modeling, the method can be used in sparse and irregular sampling schemes, making it applicable in numerous settings. Our approach is applied in the psychometric setting to help characterize multiple neurocognitive scores observed over time in the Alzheimer's Disease Neuroimaging Initiative study. Finally, intensive simulations show a notable advantage of our method in reconstructing tensors.

This study investigates the relationship between daily interpersonal stress (binary, time-varying) and suicidal behavior (binary, time-varying) using 90 days of daily diary data from 106 adolescents assessed immediately after discharge from acute psychiatric treatment. It addresses two key complexities: the rarity of suicidal events and non-monotone, non-ignorable missingness in both the outcome and the predictor. Because existing methods often fail to accommodate these complexities, leading to biased estimates, a Bayesian selection model is specified. The model integrates a mixed-effects complementary log-log regression for rare events with a missingness model that accounts for non-monotone, non-ignorable missingness in the outcome. A probit mixed-effects model is used for the time-varying predictor, along with a corresponding missingness model for its non-monotone, non-ignorable missingness. Empirical results support the applicability of the specified model to longitudinal studies involving rare events and complex missing-data structures. Furthermore, a simulation study demonstrates parameter recovery and highlights bias in focal parameters when sensitivity parameters in the outcome and missingness models are ignored.

Behavioral models are instrumental for studying human cognition, yet many inferences derived from such models fail to generalize. We argue that this is driven in part by the increasing complexity of behavioral models, where non-linearities and discontinuities create dynamic parameter interactions that limit the generalizability of inferences across different contexts, experiments, and datasets. We first demonstrate the problems that arise from parameter dependency. We then propose a new methodological framework for understanding the generalizability of behavioral modeling results using multivariate sampling distributions for the model parameters. We derive and validate novel sampling distributions for complex non-linear behavioral models by transforming the mimicry between different parameter values into the chances of one set of parameters being inferred from data generated by another set of parameters. Our approach is computationally scalable to evaluate how model estimates change across the parameter space and different experiments, which can limit the generalizability of experimental results. We then apply our approach to current behavioral models, revealing new theoretical insights. Using our approach, we reinterpret results from recent modeling work in decision-making and category learning. We conclude by discussing the implications of our proposed framework for building stronger, more generalizable psychological research and theory through behavioral modeling.

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Psychological research has long centered around questionnaire assessments, but now digital devices, especially smartphones, enable the collection of real-world behavioral data through mobile sensing. While this data collection method offers unique opportunities, it also introduces new methodological challenges, as mobile-sensing data are highly complex and high in dimensionality (i.e., timestamped events with millisecond resolution), requiring advanced preprocessing to derive psychologically meaningful variables. This article highlights these challenges by reviewing the current state of data preprocessing based on app usage logs from smartphones. Afterward, it presents three preprocessing cases that vary in complexity across the dimensions of data enrichment-which involves adding context to raw data by integrating information from external and internal sources (including ecological momentary assessments)-and data aggregation-which entails summarizing data in different ways, from basic descriptive statistics to sophisticated machine-learning models. For each case, potential pitfalls are identified, and extensions are discussed to refine our preprocessing pipelines and accommodate different data types and research questions. By outlining these preprocessing strategies, this manuscript demonstrates the rich potential of mobile-sensing data for extracting nuanced behavioral variables beyond simple person-level summaries and aims to inspire the development of more advanced research questions based on sensing data.

Differential equation models have become increasingly popular for investigating dynamic processes. However, commonly used two-stage estimation methods, such as the generalized local linear approximation (GLLA), often produce biased parameter estimates. This study proposes a bias-correction method for GLLA estimates in second-order differential equation models. The method solves a bias-correction equation (derived from the relation between true parameter values and their biased estimates) via stochastic approximation, producing asymptotically unbiased estimates even with large initial bias. We first demonstrate the application of the bias-correction method by correcting the bias of a single parameter in a simple second-order differential equation model (i.e., the linear oscillator model with time-independent measurement error). We then extend the method to a more commonly used second-order differential equation model (i.e., the damped linear oscillator model), examining its performance in simultaneously addressing multiple parameters and incorporating time-dependent dynamic error. A simulation study shows that the bias-correction method substantially reduces bias in GLLA estimates, yielding highly accurate and precise parameter estimates. An empirical illustration further compares the results of GLLA and the bias-correction method. Our findings highlight the effectiveness of the proposed method in improving parameter estimation for differential equation models, offering an enhanced approach for analyzing dynamic processes.

Multiple response (MR) items-such as multiple true-false, multiple-select, and select-N items-are increasingly used in assessments to identify partial knowledge and differentiate latent abilities more accurately. Allowing multiple selections, MR items provide richer information and reduce guessing effects compared to single-answer multiple-choice items. However, traditional scoring methods (e.g., Dichotomous, Ripkey, Partial scoring) compress response combination (RC) data, losing valuable information and ignoring issues like local dependence and incompatibility across item types. To address these challenges, we introduce a novel psychometric model framework: the Multiple Response Model with Inter-option Local Dependencies (MRM-LD), and its simplified version, the Multiple Response Model (MRM). These models preserve RC data across MR item types, offering a more comprehensive understanding for MR assessment. Parameters for MRM-LD and MRM were estimated using Markov chain Monte Carlo algorithms in Stan and R. Empirical data from an eighth-grade physics test showed that MRM-LD and MRM outperform Graded Response Model and Nominal Response Model combined with three scoring methods, by retaining more test information, improving reliability and validity, and providing more detailed analysis of item characteristics. Simulation studies confirmed the proposed models perform robustly under various conditions, including small samples and few items, demonstrating their applicability across diverse testing scenarios.

Test speededness, caused by time constraints, can impact examinees' performance, leading to decreased response accuracy, particularly toward the end of the test. Most existing methods for detecting test speededness rely on specific distributional assumptions for response times (RTs), such as the lognormal distribution, which may lead to incorrect statistical inference if the true data distribution deviates from these assumptions. This article proposes a novel Bootstrap-CUSUM method for detecting test speededness, which is robust to non-normality in log-RTs. By constructing a cumulative sum (CUSUM) person-fit statistic for log-RTs and using the multiplier bootstrap to estimate its empirical distribution, our method facilitates individual-level detection and changepoint estimation. We prove the theoretical consistency of the method under both null and alternative hypotheses. Simulation studies show that the Bootstrap-CUSUM method outperforms the likelihood ratio test, Wald test, and score test in terms of correct classification rate, true detection rate, and false positive rate, demonstrating superior robustness and adaptability across different data distributions. The real data analysis further demonstrates the practical utility of the proposed method for detecting test speededness.

Over 45 years ago, William Revelle proposed a reliability measure based on the worst split-half of a test or scale, commonly known as Revelle's beta, to assess the general factor saturation. However, to this day, there is no reliable method for computing this measure, as existing approaches are either computationally infeasible or insufficiently accurate in identifying the worst split-half. This difficulty arises because the number of candidate splits increases exponentially with the number of items. In this article, we show that computing Revelle's beta is conceptually equivalent to divisive ("top-down") hierarchical clustering. This insight allows us to reduce the number of candidate splits to a quadratic problem, making the computation feasible. We specify theoretical conditions under which this approach is guaranteed to recover the worst split-half. To validate the efficiency of our approach, we conduct simulation studies and analyze real-world data. Code implementations accompanying this work are available online, together with Supplementary Material.

This article develops an analysis pipeline for quantifying and relating mouth shape variation to the emotions perceived from facial expressions. We use open-source data that contains ratings from 802 fairgoers on 27 smile-like expressions. Each rater was given a list of seven emotions (happy, sad, anger, contempt, fear, surprise, and disgust) and asked to select all of the words that best described the facial expression. To develop a generalizable method for quantifying mouth shape variation, we leverage statistical shape analysis techniques to parameterize each mouth's shape in terms of 30 systematically placed landmarks that outline the upper and lower lips. Furthermore, we demonstrate that a three-dimensional representation of these landmark coordinates produces an interpretable feature set that outperforms the original and full-dimensional feature sets in terms of predictive performance. To connect the mouth shape features to the emotion ratings, we develop a nonparametric multinomial regression model that is capable of shrinkage and selection with high-dimensional predictors. Our results demonstrate that the proposed method can produce easily interpretable model predictions that enhance our understanding of the nature in which subtle variations in mouth shape affect the perception of a facial expression.

Several textbooks give the incorrect formula to calculate the standard error estimates for standardized regression coefficients. As a remedy, two analytic methods have been developed: (1) the delta method and (2) the covariance structure modeling method. However, neither method is applicable to compute the standard error estimates for unstandardized regression coefficients of products of Z-scores. In the literature, a nonparametric bootstrap procedure is advocated to test the significance of unstandardized regression coefficients of products of Z-scores. In this article, we propose a simple analytic approach that can produce the standard error estimates not only for standardized regression coefficients when interaction terms are not included, but also for unstandardized regression coefficients when interaction terms of Z-scores are included. Two numeric examples are used to compare our analytic approach with the existing methods, and simulation studies are conducted to further evaluate the performances of regular regression, our analytic approach, and the nonparametric bootstrap procedure at finite sample sizes. It is found that (1) regular regression performs well only when the variances of predictor variables are small, (2) our analytic approach performs well at the sample size of 200 or larger, and (3) the nonparametric bootstrap procedure performs (almost) perfectly in all conditions.

Violations of the assumption of local independence are a fundamental issue in item response theory as they threaten model validity and bias the parameter estimates. For such a reason, a plethora of tests and approaches has been devised in the last 40 years to detect or to model such violations. Nonetheless, local dependence (LD) remains an open problem, with somewhat blurred boundaries due to the lack of a general framework for dealing with the different notions of dependence that have been suggested in the literature. The present contribution has a two-fold aim: On the one hand, to review and collect some of the approaches available in the literature; on the other hand, by following a unified perspective on assessment models introduced by Noventa et al. (2024, Journal of Mathematical Psychology 122, 102872) to suggest a possible systematization of some existing and some new approaches to LD. As a result, deterministic and probabilistic modeling mechanisms of LD are formalized and discussed.

Cognitive diagnostic models (CDMs) provide fine-grained diagnostic feedback by modeling the relationship between latent attributes and item responses. Two key components required for CDM implementation are the Q-matrix, which links items to attributes, and the attribute hierarchy, which defines prerequisite relationships among attributes. In many practical settings, both structures are specified by experts based on cognitive theory. In this article, we propose a novel Bayesian estimation method that simultaneously learns the Q-matrix, the attribute hierarchy, and the parameters of the deterministic inputs, noisy "and" gate (DINA) model. We develop a Metropolis-Hastings within Gibbs algorithm and integrate a mini-batch strategy to improve computational efficiency. We conducted a series of simulation studies to evaluate the performance of the proposed algorithm under varying conditions, including sample size, test length, hierarchy structure, mini-batch size, and threshold settings. The results demonstrate strong recovery rates for both latent structures and item parameters, confirming the accuracy and robustness of our method. A real data application further illustrates the utility of the proposed framework in uncovering interpretable diagnostic structures. Our findings offer practical guidance for researchers seeking to implement CDMs when both the Q-matrix and attribute hierarchy are unknown.