The Q-matrix of a cognitively diagnostic test is said to be complete if it allows for the identification of all possible proficiency classes among examinees. Completeness of the Q-matrix is therefore a key requirement for any cognitively diagnostic test. However, completeness of the Q-matrix is often difficult to establish, especially, for tests with a large number of items involving multiple attributes. As an additional complication, completeness is not an intrinsic property of the Q-matrix, but can only be assessed in reference to a specific cognitive diagnosis model (CDM) supposed to underly the data—that is, the Q-matrix of a given test can be complete for one model but incomplete for another. In this article, a method is presented for assessing whether a given Q-matrix is complete for a given CDM. The proposed procedure relies on the theoretical framework of general CDMs and is therefore legitimate for CDMs that can be reparameterized as a general CDM.

The multiple-group categorical factor analysis (FA) model and the graded response model (GRM) are commonly used to examine polytomous items for differential item functioning to detect possible measurement bias in educational testing. In this study, the multiple-group categorical factor analysis model (MC-FA) and multiple-group normal-ogive GRM models are unified under the common framework of discretization of a normal variant. We rigorously justify a set of identified parameters and determine possible identifiability constraints necessary to make the parameters just-identified and estimable in the common framework of MC-FA. By doing so, the difference between categorical FA model and normal-ogive GRM is simply the use of two different sets of identifiability constraints, rather than the seeming distinction between categorical FA and GRM. Thus, we compare the performance on DIF assessment between the categorical FA and GRM approaches through simulation studies on the MC-FA models with their corresponding particular sets of identifiability constraints. Our results show that, under the scenarios with varying degrees of DIF for examinees of different ability levels, models with the GRM type of identifiability constraints generally perform better on DIF detection with a higher testing power. General guidelines regarding the choice of just-identified parameterization are also provided for practical use.

The literature on clustering for continuous data is rich and wide; differently, that one developed for categorical data is still limited. In some cases, the clustering problem is made more difficult by the presence of noise variables/dimensions that do not contain information about the clustering structure and could mask it. The aim of this paper is to propose a model for simultaneous clustering and dimensionality reduction of ordered categorical data able to detect the discriminative dimensions discarding the noise ones. Following the underlying response variable approach, the observed variables are considered as a discretization of underlying first-order latent continuous variables distributed as a Gaussian mixture. To recognize discriminative and noise dimensions, these variables are considered to be linear combinations of two independent sets of second-order latent variables where only one contains the information about the cluster structure while the other one contains noise dimensions. The model specification involves multidimensional integrals that make the maximum likelihood estimation cumbersome and in some cases infeasible. To overcome this issue, the parameter estimation is carried out through an EM-like algorithm maximizing a composite log-likelihood based on low-dimensional margins. Examples of application of the proposal on real and simulated data are performed to show the effectiveness of the proposal.

Under the general common factor model, the reliabilities of factor score estimators might be of more interest than the reliability of the total score (the unweighted sum of item scores). In this paper, lower bounds to the reliabilities of Thurstone’s factor score estimators, Bartlett’s factor score estimators, and McDonald’s factor score estimators are derived and conditions are given under which these lower bounds are equal. The relative performance of the derived lower bounds is studied using classic example data sets. The results show that estimates of the lower bounds to the reliabilities of Thurstone’s factor score estimators are greater than or equal to the estimates of the lower bounds to the reliabilities of Bartlett’s and McDonald’s factor score estimators.

Diagnostic classification models are confirmatory in the sense that the relationship between the latent attributes and responses to items is specified or parameterized. Such models are readily interpretable with each component of the model usually having a practical meaning. However, parameterized diagnostic classification models are sometimes too simple to capture all the data patterns, resulting in significant model lack of fit. In this paper, we attempt to obtain a compromise between interpretability and goodness of fit by regularizing a latent class model. Our approach starts with minimal assumptions on the data structure, followed by suitable regularization to reduce complexity, so that readily interpretable, yet flexible model is obtained. An expectation–maximization-type algorithm is developed for efficient computation. It is shown that the proposed approach enjoys good theoretical properties. Results from simulation studies and a real application are presented.

Model selection is a popular strategy in structural equation modeling (SEM). To select an “optimal” model, many selection criteria have been proposed. In this study, we derive the asymptotics of several popular selection procedures in SEM, including AIC, BIC, the RMSEA, and a two-stage rule for the RMSEA (RMSEA-2S). All of the results are derived under weak distributional assumptions and can be applied to a wide class of discrepancy functions. The results show that both AIC and BIC asymptotically select a model with the smallest population minimum discrepancy function (MDF) value regardless of nested or non-nested selection, but only BIC could consistently choose the most parsimonious one under nested model selection. When there are many non-nested models attaining the smallest MDF value, the consistency of BIC for the most parsimonious one fails. On the other hand, the RMSEA asymptotically selects a model that attains the smallest population RMSEA value, and the RESEA-2S chooses the most parsimonious model from all models with the population RMSEA smaller than the pre-specified cutoff. The empirical behavior of the considered criteria is also illustrated via four numerical examples.

Behavioral and psychological researchers have shown strong interests in investigating contextual effects (i.e., the influences of combinations of individual- and group-level predictors on individual-level outcomes). The present research provides generalized formulas for determining the sample size needed in investigating contextual effects according to the desired level of statistical power as well as width of confidence interval. These formulas are derived within a three-level random intercept model that includes one predictor/contextual variable at each level to simultaneously cover various kinds of contextual effects that researchers can show interest. The relative influences of indices included in the formulas on the standard errors of contextual effects estimates are investigated with the aim of further simplifying sample size determination procedures. In addition, simulation studies are performed to investigate finite sample behavior of calculated statistical power, showing that estimated sample sizes based on derived formulas can be both positively and negatively biased due to complex effects of unreliability of contextual variables, multicollinearity, and violation of assumption regarding the known variances. Thus, it is advisable to compare estimated sample sizes under various specifications of indices and to evaluate its potential bias, as illustrated in the example.

We propose a new and flexible simulation method for non-normal data with user-specified marginal distributions, covariance matrix and certain bivariate dependencies. The VITA (VIne To Anything) method is based on regular vines and generalizes the NORTA (NORmal To Anything) method. Fundamental theoretical properties of the VITA method are deduced. Two illustrations demonstrate the flexibility and usefulness of VITA in the context of structural equation models. R code for the implementation is provided.

We present a variational maximization–maximization algorithm for approximate maximum likelihood estimation of generalized linear mixed models with crossed random effects (e.g., item response models with random items, random raters, or random occasion-specific effects). The method is based on a factorized variational approximation of the latent variable distribution given observed variables, which creates a lower bound of the log marginal likelihood. The lower bound is maximized with respect to the factorized distributions as well as model parameters. With the proposed algorithm, a high-dimensional intractable integration is translated into a two-dimensional integration problem. We incorporate an adaptive Gauss–Hermite quadrature method in conjunction with the variational method in order to increase computational efficiency. Numerical studies show that under the small sample size conditions that are considered the proposed algorithm outperforms the Laplace approximation.

Person-centered methods are useful for studying individual differences in terms of (dis)similarities between response profiles on multivariate outcomes. Multivariate distance matrix regression (MDMR) tests the significance of associations of response profile (dis)similarities and a set of predictors using permutation tests. This paper extends MDMR by deriving and empirically validating the asymptotic null distribution of its test statistic, and by proposing an effect size for individual outcome variables, which is shown to recover true associations. These extensions alleviate the computational burden of permutation tests currently used in MDMR and render more informative results, thus making MDMR accessible to new research domains.

Functional principal component analysis (FPCA) and functional multiple-set canonical correlation analysis (FMCCA) are data reduction techniques for functional data that are collected in the form of smooth curves or functions over a continuum such as time or space. In FPCA, low-dimensional components are extracted from a single functional dataset such that they explain the most variance of the dataset, whereas in FMCCA, low-dimensional components are obtained from each of multiple functional datasets in such a way that the associations among the components are maximized across the different sets. In this paper, we propose a unified approach to FPCA and FMCCA. The proposed approach subsumes both techniques as special cases. Furthermore, it permits a compromise between the techniques, such that components are obtained from each set of functional data to maximize their associations across different datasets, while accounting for the variance of the data well. We propose a single optimization criterion for the proposed approach, and develop an alternating regularized least squares algorithm to minimize the criterion in combination with basis function approximations to functions. We conduct a simulation study to investigate the performance of the proposed approach based on synthetic data. We also apply the approach for the analysis of multiple-subject functional magnetic resonance imaging data to obtain low-dimensional components of blood-oxygen level-dependent signal changes of the brain over time, which are highly correlated across the subjects as well as representative of the data. The extracted components are used to identify networks of neural activity that are commonly activated across the subjects while carrying out a working memory task.

A method is proposed that combines dimension reduction and cluster analysis for categorical data by simultaneously assigning individuals to clusters and optimal scaling values to categories in such a way that a single between variance maximization objective is achieved. In a unified framework, a brief review of alternative methods is provided and we show that the proposed method is equivalent to GROUPALS applied to categorical data. Performance of the methods is appraised by means of a simulation study. The results of the joint dimension reduction and clustering methods are compared with the so-called tandem approach, a sequential analysis of dimension reduction followed by cluster analysis. The tandem approach is conjectured to perform worse when variables are added that are unrelated to the cluster structure. Our simulation study confirms this conjecture. Moreover, the results of the simulation study indicate that the proposed method also consistently outperforms alternative joint dimension reduction and clustering methods.

Propensity score methods are an important tool to help reduce confounding in non-experimental studies and produce more accurate causal effect estimates. Most propensity score methods assume that covariates are measured without error. However, covariates are often measured with error. Recent work has shown that ignoring such error could lead to bias in treatment effect estimates. In this paper, we consider an additional complication: that of differential measurement error across treatment groups, such as can occur if a covariate is measured differently in the treatment and control groups. We propose two flexible Bayesian approaches for handling differential measurement error when estimating average causal effects using propensity score methods. We consider three scenarios: systematic (i.e., a location shift), heteroscedastic (i.e., different variances), and mixed (both systematic and heteroscedastic) measurement errors. We also explore various prior choices (i.e., weakly informative or point mass) on the sensitivity parameters related to the differential measurement error. We present results from simulation studies evaluating the performance of the proposed methods and apply these approaches to an example estimating the effect of neighborhood disadvantage on adolescent drug use disorders.

This article considers the application of the simulation-extrapolation (SIMEX) method for measurement error correction when the error variance is a function of the latent variable being measured. Heteroskedasticity of this form arises in educational and psychological applications with ability estimates from item response theory models. We conclude that there is no simple solution for applying SIMEX that generally will yield consistent estimators in this setting. However, we demonstrate that several approximate SIMEX methods can provide useful estimators, leading to recommendations for analysts dealing with this form of error in settings where SIMEX may be the most practical option.

The elicitation of an ordinal judgment on multiple alternatives is often required in many psychological and behavioral experiments to investigate preference/choice orientation of a specific population. The Plackett–Luce model is one of the most popular and frequently applied parametric distributions to analyze rankings of a finite set of items. The present work introduces a Bayesian finite mixture of Plackett–Luce models to account for unobserved sample heterogeneity of partially ranked data. We describe an efficient way to incorporate the latent group structure in the data augmentation approach and the derivation of existing maximum likelihood procedures as special instances of the proposed Bayesian method. Inference can be conducted with the combination of the Expectation-Maximization algorithm for maximum a posteriori estimation and the Gibbs sampling iterative procedure. We additionally investigate several Bayesian criteria for selecting the optimal mixture configuration and describe diagnostic tools for assessing the fitness of ranking distributions conditionally and unconditionally on the number of ranked items. The utility of the novel Bayesian parametric Plackett–Luce mixture for characterizing sample heterogeneity is illustrated with several applications to simulated and real preference ranked data. We compare our method with the frequentist approach and a Bayesian nonparametric mixture model both assuming the Plackett–Luce model as a mixture component. Our analysis on real datasets reveals the importance of an accurate diagnostic check for an appropriate in-depth understanding of the heterogenous nature of the partial ranking data.

Horn’s parallel analysis is a widely used method for assessing the number of principal components and common factors. We discuss the theoretical foundations of parallel analysis for principal components based on a covariance matrix by making use of arguments from random matrix theory. In particular, we show that (i) for the first component, parallel analysis is an inferential method equivalent to the Tracy–Widom test, (ii) its use to test high-order eigenvalues is equivalent to the use of the joint distribution of the eigenvalues, and thus should be discouraged, and (iii) a formal test for higher-order components can be obtained based on a Tracy–Widom approximation. We illustrate the performance of the two testing procedures using simulated data generated under both a principal component model and a common factors model. For the principal component model, the Tracy–Widom test performs consistently in all conditions, while parallel analysis shows unpredictable behavior for higher-order components. For the common factor model, including major and minor factors, both procedures are heuristic approaches, with variable performance. We conclude that the Tracy–Widom procedure is preferred over parallel analysis for statistically testing the number of principal components based on a covariance matrix.

We construct a framework for meta-analysis and other multi-level data structures that codifies the sources of heterogeneity between studies or settings in treatment effects and examines their implications for analyses. The key idea is to consider, for each of the treatments under investigation, the subject’s potential outcome in each study or setting were he to receive that treatment. We consider four sources of heterogeneity: (1) response inconsistency, whereby a subject’s response to a given treatment would vary across different studies or settings, (2) the grouping of nonequivalent treatments, where two or more treatments are grouped and treated as a single treatment under the incorrect assumption that a subject’s responses to the different treatments would be identical, (3) nonignorable treatment assignment, and (4) response-related variability in the composition of subjects in different studies or settings. We then examine how these sources affect heterogeneity/homogeneity of conditional and unconditional treatment effects. To illustrate the utility of our approach, we re-analyze individual participant data from 29 randomized placebo-controlled studies on the cardiovascular risk of Vioxx, a Cox-2 selective nonsteroidal anti-inflammatory drug approved by the FDA in 1999 for the management of pain and withdrawn from the market in 2004.

This paper discusses the issue of differential item functioning (DIF) in international surveys. DIF is likely to occur in international surveys. What is needed is a statistical approach that takes DIF into account, while at the same time allowing for meaningful comparisons between countries. Some existing approaches are discussed and an alternative is provided. The core of this alternative approach is to define the construct as a large set of items, and to report in terms of summary statistics. Since the data are incomplete, measurement models are used to complete the incomplete data. For that purpose, different models can be used across countries. The method is illustrated with PISA’s reading literacy data. The results indicate that this approach fits the data better than the current PISA methodology; however, the league tables are nearly identical. The implications for monitoring changes over time are discussed.

A new framework for sequential multiblock component methods is presented. This framework relies on a new version of regularized generalized canonical correlation analysis (RGCCA) where various scheme functions and shrinkage constants are considered. Two types of between block connections are considered: blocks are either fully connected or connected to the superblock (concatenation of all blocks). The proposed iterative algorithm is monotone convergent and guarantees obtaining at convergence a stationary point of RGCCA. In some cases, the solution of RGCCA is the first eigenvalue/eigenvector of a certain matrix. For the scheme functions x, $|x|$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\vert }x{\vert }$$\end{document}, $x^2$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x^{2}$$\end{document} or $x^4$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x^{4}$$\end{document} and shrinkage constants 0 or 1, many multiblock component methods are recovered.

Samejima’s graded response model (GRM) has gained popularity in the analyses of ordinal response data in psychological, educational, and health-related assessment. Obtaining high-quality point and interval estimates for GRM parameters attracts a great deal of attention in the literature. In the current work, we derive generalized fiducial inference (GFI) for a family of multidimensional graded response model, implement a Gibbs sampler to perform fiducial estimation, and compare its finite-sample performance with several commonly used likelihood-based and Bayesian approaches via three simulation studies. It is found that the proposed method is able to yield reliable inference even in the presence of small sample size and extreme generating parameter values, outperforming the other candidate methods under investigation. The use of GFI as a convenient tool to quantify sampling variability in various inferential procedures is illustrated by an empirical data analysis using the patient-reported emotional distress data.