Differential item functioning (DIF) analysis is an important step in establishing the validity of measurements. Most traditional methods for DIF analysis use an item-by-item strategy via anchor items that are assumed DIF-free. If anchor items are flawed, these methods will yield misleading results due to biased scales. In this article, based on the fact that the item’s relative change of difficulty difference (RCD) does not depend on the mean ability of individual groups, a new DIF detection method (RCD-DIF) is proposed by comparing the observed differences against those with simulated data that are known DIF-free. The RCD-DIF method consists of a D-QQ (quantile quantile) plot that permits the identification of internal references points (similar to anchor items), a RCD-QQ plot that facilitates visual examination of DIF, and a RCD graphical test that synchronizes DIF analysis at the test level with that at the item level via confidence intervals on individual items. The RCD procedure visually reveals the overall pattern of DIF in the test and the size of DIF for each item and is expected to work properly even when the majority of the items possess DIF and the DIF pattern is unbalanced. Results of two simulation studies indicate that the RCD graphical test has Type I error rate comparable to those of existing methods but with greater power.

Classic item response models assume that all items with the same difficulty have the same response probability among all respondents with the same ability. These assumptions, however, may very well be violated in practice, and it is not straightforward to assess whether these assumptions are violated, because neither the abilities of respondents nor the difficulties of items are observed. An example is an educational assessment where unobserved heterogeneity is present, arising from unobserved variables such as cultural background and upbringing of students, the quality of mentorship and other forms of emotional and professional support received by students, and other unobserved variables that may affect response probabilities. To address such violations of assumptions, we introduce a novel latent space model which assumes that both items and respondents are embedded in an unobserved metric space, with the probability of a correct response decreasing as a function of the distance between the respondent’s and the item’s position in the latent space. The resulting latent space approach provides an interaction map that represents interactions of respondents and items, and helps derive insightful diagnostic information on items as well as respondents. In practice, such interaction maps enable teachers to detect students from underrepresented groups who need more support than other students. We provide empirical evidence to demonstrate the usefulness of the proposed latent space approach, along with simulation results.

There recently has been growing interest in the study of psychological and neurological processes at an individual level. One goal in such endeavors is to construct person-specific dynamic assessments using time series techniques such as Vector Autoregressive (VAR) models. However, two problems exist with current VAR specifications: (1) VAR models are restricted in that contemporaneous relations are typically modeled either as undirected relations among residuals or directed relations among observed variables, but not both; (2) current estimation frameworks are limited by the reliance on stepwise model building procedures. This study adopts a new modeling approach. We first extended the current unified SEM (uSEM) framework, a widely used structural VAR model, to a hybrid representation (i.e., “huSEM”) to include both undirected and directed contemporaneous effects, and then replaced the stepwise modeling with a LASSO-type regularization for a global search of the optimal sparse model. Our simulation study showed that regularized huSEM performed uniformly the best over alternative VAR representations and/or modeling approaches, with respect to accurately recovering the presence and directionality of hybrid relations and reliably removing false relations when the data are generated to have two types of contemporaneous relations. The present study to our knowledge is the first application of the recently developed regularized SEM technique to the estimation of huSEM, which points to a promising future for statistical learning in psychometric models.

The likelihood ratio test is widely used in exploratory factor analysis to assess the model fit and determine the number of latent factors. Despite its popularity and clear statistical rationale, researchers have found that when the dimension of the response data is large compared to the sample size, the classical Chi-square approximation of the likelihood ratio test statistic often fails. Theoretically, it has been an open problem when such a phenomenon happens as the dimension of data increases; practically, the effect of high dimensionality is less examined in exploratory factor analysis, and there lacks a clear statistical guideline on the validity of the conventional Chi-square approximation. To address this problem, we investigate the failure of the Chi-square approximation of the likelihood ratio test in high-dimensional exploratory factor analysis and derive the necessary and sufficient condition to ensure the validity of the Chi-square approximation. The results yield simple quantitative guidelines to check in practice and would also provide useful statistical insights into the practice of exploratory factor analysis.

Person fit statistics are frequently used to detect aberrant behavior when assuming an item response model generated the data. A common statistic, $l_z$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$l_z$$\end{document}, has been shown in previous studies to perform well under a myriad of conditions. However, it is well-known that $l_z$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$l_z$$\end{document} does not follow a standard normal distribution when using an estimated latent trait. As a result, corrections of $l_z$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$l_z$$\end{document}, called $l_z^{\ast }$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$l_z^*$$\end{document}, have been proposed in the literature for specific item response models. We propose a more general correction that is applicable to many types of data, namely survey or tests with multiple item types and underlying latent constructs, which subsumes previous work done by others. In addition, we provide corrections for multiple estimators of $\theta$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\theta $$\end{document}, the latent trait, including MLE, MAP and WLE. We provide analytical derivations that justifies our proposed correction, as well as simulation studies to examine the performance of the proposed correction with finite test lengths. An applied example is also provided to demonstrate proof of concept. We conclude with recommendations for practitioners when the asymptotic correction works well under different conditions and also future directions.

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.

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.

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.

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.

We consider mediated effects of an exposure, X on an outcome, Y, via a mediator, M, under no unmeasured confounding assumptions in the setting where models for the conditional expectation of the mediator and outcome are partially linear. We propose G-estimators for the direct and indirect effects and demonstrate consistent asymptotic normality for indirect effects when models for the conditional means of M, or X and Y are correctly specified, and for direct effects, when models for the conditional means of Y, or X and M are correct. This marks an improvement, in this particular setting, over previous ‘triple’ robust methods, which do not assume partially linear mean models. Testing of the no-mediation hypothesis is inherently problematic due to the composite nature of the test (either X has no effect on M or M no effect on Y), leading to low power when both effect sizes are small. We use generalized methods of moments (GMM) results to construct a new score testing framework, which includes as special cases the no-mediation and the no-direct-effect hypotheses. The proposed tests rely on an orthogonal estimation strategy for estimating nuisance parameters. Simulations show that the GMM-based tests perform better in terms of power and small sample performance compared with traditional tests in the partially linear setting, with drastic improvement under model misspecification. New methods are illustrated in a mediation analysis of data from the COPERS trial, a randomized trial investigating the effect of a non-pharmacological intervention of patients suffering from chronic pain. An accompanying R package implementing these methods can be found at github.com/ohines/plmed.

The analysis of variance, and mixed models in general, are popular tools for analyzing experimental data in psychology. Bayesian inference for these models is gaining popularity as it allows to easily handle complex experimental designs and data dependence structures. When working on the log of the response variable, the use of standard priors for the variance parameters can create inferential problems and namely the non-existence of posterior moments of parameters and predictive distributions in the original scale of the data. The use of the generalized inverse Gaussian distributions with a careful choice of the hyper-parameters is proposed as a general purpose option for priors on variance parameters. Theoretical and simulations results motivate the proposal. A software package that implements the analysis is also discussed. As the log-transformation of the response variable is often applied when modelling response times, an empirical data analysis in this field is reported.

A maximum likelihood estimation routine is presented for a generalized structural equation model that permits a combination of response variables from various distributions (e.g., normal, Poisson, binomial, etc.). The likelihood function does not have a closed-form solution and so must be numerically approximated, which can be computationally demanding for models with several latent variables. However, the dimension of numerical integration can be reduced if one or more of the latent variables do not directly affect any nonnormal endogenous variables. The method is demonstrated using an empirical example, and the full estimation details, including first-order derivatives of the likelihood function, are provided.