Extended redundancy analysis (ERA) is a statistical approach to component-based multivariate regression modeling that explores interrelationships among multiple sets of while incorporating regression with a data-reduction technique. The extant models that utilize ERA have assumed the outcome variables with the same data type. Also, ERA models focused on estimating direct pathways only without explicitly addressing mediation effects. In this paper, ERA is extended to handle multiple mediators and mixed types of outcome variables by adopting a Bayesian framework, taking into account correlation structure among all of the outcome variables. The proposed method develops an algorithm that derives the joint posterior distribution of parameters using a Markov chain Monte Carlo algorithm. Simulations and an empirical dataset are provided to illustrate the usefulness of the proposed method.

For classroom teaching and learning, classifying students’ skills into more than two categories (e.g., no, basic, and advanced masteries) is more instructionally relevant. Such classifications require polytomous attributes, for which most existing cognitive diagnosis models (CDMs) are inapplicable. This paper proposes the saturated polytomous cognitive diagnosis model (sp-CDM), a general model that subsumes existing CDMs for polytomous attributes as special cases. The generalization is shown by mathematically illustrating the relationships between the proposed and existing CDMs. Moreover, algorithms to estimate the proposed model is proposed. A simulation study is conducted to evaluate the parameter recovery of the sp-CDM using the proposed estimation algorithms, as well as to illustrate the consequences of improperly fitting constrained or unnecessarily complex polytomous-attribute CDMs. A real-data example involving polytomous attributes is presented to demonstrate the practical utility of the proposed model.

We develop a fully nonparametric, easy-to-use, and powerful test for the missing completely at random (MCAR) assumption on the missingness mechanism of a dataset. The test compares distributions of different missing patterns on random projections in the variable space of the data. The distributional differences are measured with the Kullback-Leibler Divergence, using probability Random Forests (Malley et al., 2011). We thus refer to it as “Projected Kullback–Leibler MCAR” (PKLM) test. The use of random projections makes it applicable even if very few or no fully observed observations are available or if the number of dimensions is large. An efficient permutation approach guarantees the level for any finite sample size, resolving a major shortcoming of most other available tests. Moreover, the test can be used on both discrete and continuous data. We show empirically on a range of simulated data distributions and real datasets that our test has consistently high power and is able to avoid inflated type-I errors. Finally, we provide an R-package PKLMtest with an implementation of our test.

The basic local independence model (BLIM) is a probabilistic model developed in knowledge space theory (KST). Recently, Stefanutti, de Chiusole, et al. (2020, Psychometrika 85, 684–715) proposed the polytomous local independence model (PoLIM), which is an extension of the BLIM to items with more than two response alternatives (polytomous items). In a Commentary to this paper, Chiu et al. (2023, Psychometrika 88, 656–671) claimed that (i) the BLIM is just a deterministic input noisy AND-gate (DINA) model where every item has a single skill and, as a consequence of this, (ii) the “PoLIM is simply a paraphrase of a DINA model in cognitive diagnosis (CD) for polytomous items” (p. 656). This rejoinder shows that such statements are invalid and totally misleading. Its aim is to clarify the nature of the relationship between the BLIM and the DINA, as well as that between the PoLIM and the Polytomous DINA. It builds upon formal results by Heller, et al. (2015, Psychometrika 80(4), 995–1019) on the intimate relation between KST and CD notions, and shows that the BLIM/PoLIM may be conceived as marginal models for whole classes of CD models.

Mediation analysis is one of the most popularly used methods in social sciences and related areas. To estimate the indirect effect, the least-squares regression is routinely applied, which is also the most efficient when the errors are normally distributed. In practice, however, real data sets are often non-normally distributed, either heavy-tailed or skewed, so that the least-squares estimators may behave very badly. To overcome this problem, we propose a robust M-estimation for the indirect effect via a general loss function, with a main focus on the Huber loss which is more slowly varying at large values than the squared loss. We further propose a data-driven procedure to select the optimal tuning constant by minimizing the asymptotic variance of the Huber estimator, which is more robust than the least-squares estimator facing outliers and non-normal data, and more efficient than the least-absolute-deviation estimator. Simulation studies compare the finite sample performance of the Huber loss with the existing competitors in terms of the mean square error, the type I error rate, and the statistical power. Finally, the usefulness of the proposed method is also illustrated using two real data examples.

Multidimensional item response theory (MIRT) offers psychometric models for various data settings, most popularly for dichotomous and polytomous data. Less attention has been devoted to count responses. A recent growth in interest in count item response models (CIRM)—perhaps sparked by increased occurrence of psychometric count data, e.g., in the form of process data, clinical symptom frequency, number of ideas or errors in cognitive ability assessment—has focused on unidimensional models. Some recent unidimensional CIRMs rely on the Conway–Maxwell–Poisson distribution as the conditional response distribution which allows conditionally over-, under-, and equidispersed responses. In this article, we generalize to the multidimensional case, introducing the Multidimensional Two-Parameter Conway–Maxwell–Poisson Model (M2PCMPM). Using the expectation-maximization (EM) algorithm, we develop marginal maximum likelihood estimation methods, primarily for exploratory M2PCMPMs. The resulting discrimination matrices are rotationally indeterminate. Recently, regularization of the discrimination matrix has been used to obtain a simple structure (i.e., a sparse solution) for dichotomous and polytomous data. For count data, we also (1) rotate or (2) regularize the discrimination matrix. We develop an EM algorithm with lasso ($\ell _1$) regularization for the M2PCMPM and compare (1) and (2) in a simulation study. We illustrate the proposed model with an empirical example using intelligence test data.

The family of polychoric models (PM) categories ordinal data with latent multivariate normal variables. This modeling framework is commonly used to study the association between ordinal variables, often leading to a polychoric correlation model (PCM). Moreover, PM subsumes several well-known psychometric models, such as the structural equation modeling (SEM) with ordinal data. That said, the identifiability of PM has not been addressed in the literature. Meanwhile, in recent years researchers have suggested that the latent variables underlying PM could be generalized to the family of elliptical distributions, such as the multivariate logistic and t distributions. This article concerns the identifiability of PM and PCM with latent elliptical distributions, for which we show that PM is not identifiable and PCM is identifiable. In particular, we prove the identifiability of the polychoric t correlation model based on the copula representation. We then move on to find the set of identifiability constraints of PM through an “equivalence-classes approach of identifiability,” and demonstrate its use in two applications: one concerns the identifiability of PM on Likert scales and on comparative judgment, and the other concerns the identifiability of ordinal SEM and item factor analysis. Possible implications induced by these identifiability constraints are discussed.

We consider a test which allows students to attempt a multiple-choice question multiple times (i.e., multiple attempts). The most extreme form of multiple attempts is the answer-until-correct (AUC) procedure. Previous research has demonstrated that multiple-attempt procedures such as AUC could potentially enhance learning and increase reliability. However, for multiple-choice items, guessing is still non-ignorable. Traditional models of sequential item response theory (SIRT) could model multiple-attempt procedures but fail to take guessing into account. The purpose of this study is to propose SIRT models for multiple-choice, multiple-attempt items (SIRT-MM). First, we defined a family of SIRT-MM models to account for the idiosyncrasies of items, answer options, and examinee behavior. We also explained how these models could improve person parameter estimates by taking into account partial (mis)information of examinees. Second, we conducted model comparisons between the SIRT-MM models, the graded response model, and the nominal response model. Third, we discussed how the item and person parameters can be estimated, and evaluated item and person parameter recovery of SIRT-MM models under different conditions through a simulation study. Finally, we applied the SIRT-MM models to a real dataset and demonstrated their utility through model selection, person parameter recovery, and information functions.

This article introduces a novel methodology to model the hierarchical dependence structure of manifest variables (MVs). This is done by reconstructing their correlation matrix considering a hierarchy of latent factors which forms an ultrametric correlation matrix. The proposed ultrametric factor analysis model will be shown to obtain reliable, unidimensional, and unique hierarchical factors. This approach addresses the limitations of traditional factor analysis methods that often oversimplify the multidimensional and complex relationships among MVs. The article provides an in-depth mathematical description of the proposed model, as well as an algorithm for parameter estimation. An extensive simulation study with $3,000$ generated samples validates the proposal across twelve different scenarios. Finally, we demonstrate the potential of the proposed model using a real data set that is a benchmark in psychological research.

In principle, structural equation modeling (SEM) is capable of emulating all approaches based on the general linear model. Yet, modeling sum scores in a structural equation model is not straightforward. Existing approaches to studying sum scores in a structural equation model are limited in terms either of model specification or of model assessment. This paper introduces a specification to SEM that allows for directly modeling sum scores and that overcomes existing approaches’ limitations in dealing with sum scores in the SEM context. The sum score model we present builds on the recently proposed refined Henseler–Ogasawara (H–O) specification of composites. It allows us to estimate models with sum scores in an integrative way. It can mimic the results of existing approaches and provides a means of assessing whether a sum score fully transmits the effects of or on the variables that make up the sum score. In addition, it allows for taking into account random measurement error in the variables that form the sum score. Consequently, this model specification offers researchers an improved way of judging and defending the use of sum scores empirically and conceptually.

We study how the Conditioning on Added Regression Predictions (CARP) statistics from different item pairs can be aggregated into a single overall test of monotone homogeneity. As a pairwise statistic, we use the mean conditional covariance (MCC) or its standardized value ($Z$). We use three different estimates of the covariance matrix of the pairwise test statistics: (1) the covariance matrix of the MCCs, based on the sample moments; (2) the covariance matrix of the MCCs or $Z$s, based on bootstrapping; and (3) the covariance matrix of the $Z$s, equated to the identity matrix. We consider various aggregation methods, including (a) the chi-bar-square statistic; (b) the preselected standardized partial sum of pairwise statistics; (c) the product of preselected $p$-values; (d) the minimum of preselected $p$-values; and (e–h) the same statistics, but now conditioned on post-selecting only the negative values in the test sample. We study the Type 1 error rate and power of the ensuing 20 tests based on simulations. The tests with the highest power among the tests that control the Type I error rate are based on $Z$-statistics with the identity matrix: the conditional likelihood ratio test, the conditionalized product of $p$-values, the conditionalized sum of Z-values, and the preselected product of $p$-values.

Conditional dependence (CD) reflects potential interactions between persons and items in measurement, offering valuable information for deriving personalized diagnoses, evaluations, and feedback. The recent integration of psychometric models with latent space provides an effective way to visualize and quantify person–item interactions unexplained by latent variables and item parameters. In such applications, it is important to recognize the relative nature of CD, as models with different structures and complexities (e.g., due to factor dimensionality and item parameters) produce varying systematic explanations of person and item effects, leading to differing residual variations in both quantitative and qualitative sense. To demonstrate this relativity, we extend the previously developed unidimensional Rasch-based latent space item response model by incorporating between-item multidimensionality and item discrimination parameters. The resulting model can be reduced to simpler models with appropriate constraints, allowing us to explore the relativity in CD by comparing them. Simulation studies demonstrate that (1) the most complex proposed model properly recovers its parameters, (2) it outperforms the traditional IRT models by accounting for CD, and (3) the models in comparison exhibit distinctive extents of CD. The study continues with empirical examples that further illustrate relative changes in the extent and configurations of CD with practical implications.