PCA is a popular tool for exploring and summarizing multivariate data, especially those consisting of many variables. PCA, however, is often not simple to interpret, as the components are a linear combination of the variables. To address this issue, numerous methods have been proposed to sparsify the nonzero coefficients in the components, including rotation-thresholding methods and, more recently, PCA methods subject to sparsity inducing penalties or constraints. Here, we offer guidelines on how to choose among the different sparse PCA methods. Current literature misses clear guidance on the properties and performance of the different sparse PCA methods, often relying on the misconception that the equivalence of the formulations for ordinary PCA also holds for sparse PCA. To guide potential users of sparse PCA methods, we first discuss several popular sparse PCA methods in terms of where the sparseness is imposed on the loadings or on the weights, assumed model, and optimization criterion used to impose sparseness. Second, using an extensive simulation study, we assess each of these methods by means of performance measures such as squared relative error, misidentification rate, and percentage of explained variance for several data generating models and conditions for the population model. Finally, two examples using empirical data are considered.

Cognitive diagnostic models (CDMs) have arisen as advanced psychometric models in the past few decades for assessments that intend to measure students’ mastery of a set of attributes. Recently, quite a few studies attempted to extend CDMs to longitudinal versions, and they all tended to model transition probabilities from non-mastery to mastery or vice versa for each attribute separately, with an exception of a few studies (e.g., Chen et al. 2018; Madison & Bradshaw 2018). However, these pioneering works have not taken into consideration the attribute relationships and the ever-changing attributes in a learning period. In this paper, we consider a profile-level latent transition CDM (TCDM hereafter), which can not only identify transition probabilities across the same attributes over time, but also the transition pathways across different attributes. Two versions of the penalized expectation-maximization (PEM) algorithms are proposed to shrink the probabilities associated with impermissible transition pathways to 0 and, thereby, help explore attribute relationships in a longitudinal setting. Simulation results reveal that PEM with group penalty holds great promise for identifying learning trajectories.

In this article, we analyse the usefulness of multidimensional scaling in relation to performing K-means clustering on a dissimilarity matrix, when the dimensionality of the objects is unknown. In this situation, traditional algorithms cannot be used, and so K-means clustering procedures are being performed directly on the basis of the observed dissimilarity matrix. Furthermore, the application of criteria originally formulated for two-mode data sets to determine the number of clusters depends on their possible reformulation in a one-mode situation. The linear invariance property in K-means clustering for squared dissimilarities, together with the use of multidimensional scaling, is investigated to determine the cluster membership of the observations and to address the problem of selecting the number of clusters in K-means for a dissimilarity matrix. In particular, we analyse the performance of K-means clustering on the full dimensional scaling configuration and on the equivalently partitioned configuration related to a suitable translation of the squared dissimilarities. A Monte Carlo experiment is conducted in which the methodology examined is compared with the results obtained by procedures directly applicable to a dissimilarity matrix.

The PROsetta Stone Project, summarized in this issue by Schalet et al. (Psychometrika 86, 2021), is a major step forward in enabling comparability between different patient-reported outcomes measures. Schalet et al. clearly describe the psychometric methods used in the PROsetta Stone project and other projects from the Patient-Reported Outcomes Measurement Information System (PROMIS): linking based on unidimensional item response theory (IRT), equipercentile linking, and calibrated projection based on multidimensional IRT. Analyses in a validation data set and simulation studies provide strong support that the linking methods are robust when basic assumptions are fulfilled. The links already established will be of great value to the field, and the methodology described by Schalet et al. will hopefully inspire the next series of linking studies. Among potential improvements that should be considered by new studies are: (1) a thorough evaluation of the content of the measures to be linked to better guide the evaluation of measurement assumptions, (2) improvements in the design of linking studies such as selection of the optimal sample to provide data in the score ranges where linking precision is most critical and using counterbalanced designs to control for order effects. Finally, it may be useful to consider how the linking algorithms are used in subsequent data analyses. Analytic strategies based on plausible values or latent regression IRT models may be preferable to the simple transformation of scores from one patient at the time.

With decades of advance research and recent developments in the drug and medical device regulatory approval process, patient-reported outcomes (PROs) are becoming increasingly important in clinical trials. While clinical trial analyses typically treat scores from PROs as observed variables, the potential to use latent variable models when analyzing patient responses in clinical trial data presents novel opportunities for both psychometrics and regulatory science. An accessible overview of analyses commonly used to analyze longitudinal trial data and statistical models familiar in both psychometrics and biometrics, such as growth models, multilevel models, and latent variable models, is provided to call attention to connections and common themes among these models that have found use across many research areas. Additionally, examples using empirical data from a randomized clinical trial provide concrete demonstrations of the implementation of these models. The increasing availability of high-quality, psychometrically rigorous assessment instruments in clinical trials, of which the Patient-Reported Outcomes Measurement Information System (PROMIS®) is a prominent example, provides rare possibilities for psychometrics to help improve the statistical tools used in regulatory science.

An overwhelming majority of articles in psychology compare means, often between multiple groups. However, sometimes we do not know the exact group membership, but only a probability to be in one of the groups. Such information may come from classifiers trained on other datasets, prevalence of group memberships for some parts of the sample, multi-level situations where the group membership is only known as a ratio in an upper level, or expert ratings (e.g., whether a person has a pathological condition or not). We present a simple method that allows to compare group means in the absence of exact knowledge about group membership and investigate the loss of information depending on the probability values theoretically and in a large-scale simulation.

Complex interactive test items are becoming more widely used in assessments. Being computer-administered, assessments using interactive items allow logging time-stamped action sequences. These sequences pose a rich source of information that may facilitate investigating how examinees approach an item and arrive at their given response. There is a rich body of research leveraging action sequence data for investigating examinees’ behavior. However, the associated timing data have been considered mainly on the item-level, if at all. Considering timing data on the action-level in addition to action sequences, however, has vast potential to support a more fine-grained assessment of examinees’ behavior. We provide an approach that jointly considers action sequences and action-level times for identifying common response processes. In doing so, we integrate tools from clickstream analyses and graph-modeled data clustering with psychometrics. In our approach, we (a) provide similarity measures that are based on both actions and the associated action-level timing data and (b) subsequently employ cluster edge deletion for identifying homogeneous, interpretable, well-separated groups of action patterns, each describing a common response process. Guidelines on how to apply the approach are provided. The approach and its utility are illustrated on a complex problem-solving item from PIAAC 2012.

When a response to a multiple-choice item consists of selecting a single-best answer, it is not possible for examiners to differentiate between a response that is a product of knowledge and one that is largely a product of uncertainty. Certainty-based marking (CBM) is one testing format that requires examinees to express their degree of certainty on the response option they have selected, leading to an item score that depends both on the correctness of an answer and the certainty expressed. The expected score is maximized if examinees truthfully report their level of certainty. However, prospect theory states that people do not always make rational choices of the optimal outcome due to varying risk attitudes. By integrating a psychometric model and a decision-making perspective, the present study looks into the response behaviors of 334 first-year students of physiotherapy on six multiple-choice examinations with CBM in a case study. We used item response theory to model the objective probability of students giving a correct response to an item, and cumulative prospect theory to estimate their risk attitudes when students choose to report their certainty. The results showed that with the given CBM scoring matrix, students’ choices of a certainty level were affected by their risk attitudes. Students were generally risk averse and loss averse when they had a high success probability on an item, leading to an under-reporting of their certainty. Meanwhile, they were risk seeking in case of small success probabilities on the items, resulting in the over-reporting of certainty.

Issues of model selection have dominated the theoretical and applied statistical literature for decades. Model selection methods such as ridge regression, the lasso, and the elastic net have replaced ad hoc methods such as stepwise regression as a means of model selection. In the end, however, these methods lead to a single final model that is often taken to be the model considered ahead of time, thus ignoring the uncertainty inherent in the search for a final model. One method that has enjoyed a long history of theoretical developments and substantive applications, and that accounts directly for uncertainty in model selection, is Bayesian model averaging (BMA). BMA addresses the problem of model selection by not selecting a final model, but rather by averaging over a space of possible models that could have generated the data. The purpose of this paper is to provide a detailed and up-to-date review of BMA with a focus on its foundations in Bayesian decision theory and Bayesian predictive modeling. We consider the selection of parameter and model priors as well as methods for evaluating predictions based on BMA. We also consider important assumptions regarding BMA and extensions of model averaging methods to address these assumptions, particularly the method of Bayesian stacking. Simple empirical examples are provided and directions for future research relevant to psychometrics are discussed.

The emergence of computer-based assessments has made response times, in addition to response accuracies, available as a source of information about test takers’ latent abilities. The development of substantively meaningful accounts of the cognitive process underlying item responses is critical to establishing the validity of psychometric tests. However, existing substantive theories such as the diffusion model have been slow to gain traction due to their unwieldy functional form and regular violations of model assumptions in psychometric contexts. In the present work, we develop an attention-based diffusion model based on process assumptions that are appropriate for psychometric applications. This model is straightforward to analyse using Gibbs sampling and can be readily extended. We demonstrate our model’s good computational and statistical properties in a comparison with two well-established psychometric models.

It is known that a family of fixed-effects item response models with equal discrimination and different guessing parameters has no model identifiability. For this family, some types of information including the Fisher information and a new one are maximized to have model identification. The conditions of monotonicity of these types of information with respect to a tuning parameter are given. In the case of the logistic model with guessing parameters, it is shown that maxima do not exist under some parametrization, where negative lower asymptote can be employed without changing the probabilities of correct responses by examinees.

In this paper, we propose a joint modeling approach to analyze dependency in parallel response data. We define two types of dependency: higher-level dependency and within-item conditional dependency. While higher-level dependency can be estimated with common latent variable modeling approaches, within-item conditional dependency is a unique kind of information that is often not captured with extant methods, despite its potential to shed new insights into the relationship between the two types of response data. We differentiate three ways of modeling within-item conditional dependency by conditioning on raw values, expected values, or residual values of the response data, which have different implications in terms of response processes. The proposed approach is illustrated with the example of analyzing parallel data on response accuracy and brain activations from a Theory of Mind assessment. The consequence of ignoring within-item conditional dependency is investigated with empirical and simulation studies in comparison to conventional dependency analysis that focuses exclusively on relationships between latent variables.

Item response theory scoring based on summed scores is employed frequently in the practice of educational and psychological measurement. Lord and Wingersky (Appl Psychol Meas 8(4):453–461, 1984) proposed a recursive algorithm to compute the summed score likelihood. Cai (Psychometrika 80(2):535–559, 2015) extended the original Lord–Wingersky algorithm to the case of two-tier multidimensional item factor models and called it Lord–Wingersky algorithm Version 2.0. The 2.0 algorithm utilizes dimension reduction to efficiently compute summed score likelihoods associated with the general dimensions in the model. The output of the algorithm is useful for various purposes, for example, scoring, scale alignment, and model fit checking. In the research reported here, a further extension to the Lord–Wingersky algorithm 2.0 is proposed. The new algorithm, which we call Lord–Wingersky algorithm Version 2.5, yields the summed score likelihoods for all latent variables in the model conditional on observed score combinations. The proposed algorithm is illustrated with empirical data for three potential application areas: (a) describing achievement growth using score combinations across adjacent grades, (b) identification of noteworthy subscores for reporting, and (c) detection of aberrant responses.

The reliable change index has been used to evaluate the significance of individual change in health-related quality of life. We estimate reliable change for two measures (physical function and emotional distress) in the Patient-Reported Outcomes Measurement Information System (PROMIS®) 29-item health-related quality of life measure (PROMIS-29 v2.1). Using two waves of data collected 3 months apart in a longitudinal observational study of chronic low back pain and chronic neck pain patients receiving chiropractic care, and simulations, we compare estimates of reliable change from classical test theory fixed standard errors with item response theory standard errors from the graded response model. We find that unless true change in the PROMIS physical function and emotional distress scales is substantial, classical test theory estimates of significant individual change are much more optimistic than estimates of change based on item response theory.

Item response theory (IRT) models are non-linear latent variable models for discrete measures, whereas factor analysis (FA) is a latent variable model for continuous measures. In FA, the standard error (SE) of individuals’ scores is common for all individuals. In IRT, the SE depends on the individual’s score, and the SE function is to be provided. The empirical standard deviation of the scores across discrete ranges should also be computed to inform the extent to which IRT SEs overestimate or underestimate the variability of the scores. Within the target range of scores the test was designed to measure, one should expect IRT SEs to be smaller and more precise than FA SEs, and therefore preferable to assess clinical change. Outside the target range, IRT SEs may be too large and more imprecise than FA SEs, and FA more precise to assess change. As a result, whether FA or IRT characterize reliable change more accurately in a sample will depend on the proportion of individuals within or outside the IRT target score range. An application is provided to illustrate these concepts.

The model-implied instrumental variable (MIIV) estimator is an equation-by-equation estimator of structural equation models that is more robust to structural misspecifications than full information estimators. Previous studies have concentrated on endogenous variables that are all continuous (MIIV-2SLS) or all ordinal. We develop a unified MIIV approach that applies to a mixture of binary, ordinal, censored, or continuous endogenous observed variables. We include estimates of factor loadings, regression coefficients, variances, and covariances along with their asymptotic standard errors. In addition, we create new goodness of fit tests of the model and overidentification tests of single equations. Our simulation study shows that the proposed MIIV approach is more robust to structural misspecifications than diagonally weighted least squares (DWLS) and that both the goodness of fit model tests and the overidentification equations tests can detect structural misspecifications. We also find that the bias in asymptotic standard errors for the MIIV estimators of factor loadings and regression coefficients are often lower than the DWLS ones, though the differences are small in large samples. Our analysis shows that scaling indicators with low reliability can adversely affect the MIIV estimators. Also, using a small subset of MIIVs reduces small sample bias of coefficient estimates, but can lower the power of overidentification tests of equations.

A social network comprises both actors and the social connections among them. Such connections reflect the dependence among social actors, which is essential for individuals’ mental health and social development. In this article, we propose a mediation model with a social network as a mediator to investigate the potential mediation role of a social network. In the model, the dependence among actors is accounted for by a few mutually orthogonal latent dimensions which form a social space. The individuals’ positions in such a latent social space are directly involved in the mediation process between an independent and dependent variable. After showing that all the latent dimensions are equivalent in terms of their relationship to the social network and the meaning of each dimension is arbitrary, we propose to measure the whole mediation effect of a network. Although individuals’ positions in the latent space are not unique, we rigorously articulate that the proposed network mediation effect is still well defined. We use a Bayesian estimation method to estimate the model and evaluate its performance through an extensive simulation study under representative conditions. The usefulness of the network mediation model is demonstrated through an application to a college friendship network.

The present paper introduces a new explanatory item response model to account for the learning that takes place during a psychometric test due to the repeated use of the operations involved in the items. The proposed model is an extension of the operation-specific learning model (Fischer and Formann in Appl Psychol Meas 6:397–416, 1982; Scheiblechner in Z für Exp Angew Psychol 19:476–506, 1972; Spada in Spada and Kempf (eds.) Structural models of thinking and learning, Huber, Bern, Germany, pp 227–262, 1977). The paper discusses special cases of the model, which, together with the general formulation, differ in the type of response in which the model states that learning occurs: (1) correct and incorrect responses equally (non-contingent learning); (2) correct responses only (contingent learning); and (3) correct and incorrect responses to a different extent (differential contingent learning). A Bayesian framework is adopted for model estimation and evaluation. A simulation study is conducted to examine the performance of the estimation and evaluation methods in recovering the true parameters and selecting the true model. Finally, an empirical study is presented to illustrate the applicability of the model to detect learning effects using real data.