The issue of measurement invariance commonly arises in factor-analytic contexts, with methods for assessment including likelihood ratio tests, Lagrange multiplier tests, and Wald tests. These tests all require advance definition of the number of groups, group membership, and offending model parameters. In this paper, we study tests of measurement invariance based on stochastic processes of casewise derivatives of the likelihood function. These tests can be viewed as generalizations of the Lagrange multiplier test, and they are especially useful for: (i) identifying subgroups of individuals that violate measurement invariance along a continuous auxiliary variable without prespecified thresholds, and (ii) identifying specific parameters impacted by measurement invariance violations. The tests are presented and illustrated in detail, including an application to a study of stereotype threat and simulations examining the tests’ abilities in controlled conditions.

In knowledge space theory, the knowledge state of a student is the set of all problems he is capable of solving in a specific knowledge domain and a knowledge structure is the collection of knowledge states. The basic local independence model (BLIM) is a probabilistic model for knowledge structures. The BLIM assumes a probability distribution on the knowledge states and a lucky guess and a careless error probability for each problem. A key assumption of the BLIM is that the lucky guess and careless error probabilities do not depend on knowledge states (invariance assumption). This article proposes a method for testing the violations of this specific assumption. The proposed method was assessed in a simulation study and in an empirical application. The results show that (1) the invariance assumption might be violated by the empirical data even when the model’s fit is very good, and (2) the proposed method may prove to be a promising tool to detect invariance violations of the BLIM.

The neural correlates of working memory (WM) in schizophrenia (SZ) have been extensively studied using the multisite fMRI data acquired by the Functional Biomedical Informatics Research Network (fBIRN) consortium. Although univariate and multivariate analysis methods have been variously employed to localize brain responses under differing task conditions, important hypotheses regarding the representation of mental processes in the spatio-temporal patterns of neural recruitment and the differential organization of these mental processes in patients versus controls have not been addressed in this context. This paper uses a multivariate state-space model (SSM) to analyze the differential representation and organization of mental processes of controls and patients performing the Sternberg Item Recognition Paradigm (SIRP) WM task. The SSM is able to not only predict the mental state of the subject from the data, but also yield estimates of the spatial distribution and temporal ordering of neural activity, along with estimates of the hemodynamic response. The dynamical Bayesian modeling approach used in this study was able to find significant differences between the predictability and organization of the working memory processes of SZ patients versus healthy subjects. Prediction of some stimulus types from imaging data in the SZ group was significantly lower than controls, reflecting a greater level of disorganization/heterogeneity of their mental processes. Moreover, the changes in accuracy of predicting the mental state of the subject with respect to parametric modulations, such as memory load and task duration, may have important implications on the neurocognitive models for WM processes in both SZ and healthy adults. Additionally, the SSM was used to compare the spatio-temporal patterns of mental activity across subjects, in a holistic fashion and to derive a low-dimensional representation space for the SIRP task, in which subjects were found to cluster according to their diagnosis.

The Deterministic, Gated Item Response Theory Model (DGM, Shu, Unpublished Dissertation. The University of North Carolina at Greensboro, 2010) is proposed to identify cheaters who obtain significant score gain on tests due to item exposure/compromise by conditioning on the item status (exposed or unexposed items). A “gated” function is introduced to decompose the observed examinees’ performance into two distributions (the true ability distribution determined by examinees’ true ability and the cheating distribution determined by examinees’ cheating ability). Test cheaters who have score gain due to item exposure are identified through the comparison of the two distributions. Hierarchical Markov Chain Monte Carlo is used as the model’s estimation framework. Finally, the model is applied in a real data set to illustrate how the model can be used to identify examinees having pre-knowledge on the exposed items.

Most dichotomous item response models share the assumption of latent monotonicity, which states that the probability of a positive response to an item is a nondecreasing function of a latent variable intended to be measured. Latent monotonicity cannot be evaluated directly, but it implies manifest monotonicity across a variety of observed scores, such as the restscore, a single item score, and in some cases the total score. In this study, we show that manifest monotonicity can be tested by means of the order-constrained statistical inference framework. We propose a procedure that uses this framework to determine whether manifest monotonicity should be rejected for specific items. This approach provides a likelihood ratio test for which the p-value can be approximated through simulation. A simulation study is presented that evaluates the Type I error rate and power of the test, and the procedure is applied to empirical data.

In functional magnetic resonance imaging (fMRI), the blood oxygenation level dependent (BOLD) signal is often interpreted as a measure of neural activity. However, because the BOLD signal reflects the complex interplay of neural, vascular, and metabolic processes, such an interpretation is not always valid. There is growing evidence that changes in the baseline neurovascular state can result in significant modulations of the BOLD signal that are independent of changes in neural activity. This paper introduces some of the normalization and calibration methods that have been proposed for making the BOLD signal a more accurate reflection of underlying brain activity for human fMRI studies.

Parafac2 is the most flexible Simultaneous Component Analysis (SCA) model that produces an essentially unique solution. In this paper, we discuss how Parafac2’s special sign indeterminacy affects applications of SCA, and we reveal how an external criterion variable can be used to ensure that estimated Parafac2 weights are meaningfully signed across the levels of the nesting mode. We present an example with real data from clinical psychology that illustrates the importance of Parafac2’s special sign indeterminacy, as well as the effectiveness of our proposed solution. We also discuss the implications of our results for general applications of SCA.

To date, most methods for direct blockmodeling of social network data have focused on the optimization of a single objective function. However, there are a variety of social network applications where it is advantageous to consider two or more objectives simultaneously. These applications can broadly be placed into two categories: (1) simultaneous optimization of multiple criteria for fitting a blockmodel based on a single network matrix and (2) simultaneous optimization of multiple criteria for fitting a blockmodel based on two or more network matrices, where the matrices being fit can take the form of multiple indicators for an underlying relationship, or multiple matrices for a set of objects measured at two or more different points in time. A multiobjective tabu search procedure is proposed for estimating the set of Pareto efficient blockmodels. This procedure is used in three examples that demonstrate possible applications of the multiobjective blockmodeling paradigm.

We introduce a new statistical procedure for the identification of unobserved categories that vary between individuals and in which objects may span multiple categories. This procedure can be used to analyze data from a proposed sorting task in which individuals may simultaneously assign objects to multiple piles. The results of a synthetic example and a consumer psychology study involving categories of restaurant brands illustrate how the application of the proposed methodology to the new sorting task can account for a variety of categorization phenomena including multiple category memberships and for heterogeneity through individual differences in the saliency of latent category structures.

The common way to calculate confidence intervals for item response theory models is to assume that the standardized maximum likelihood estimator for the person parameter θ is normally distributed. However, this approximation is often inadequate for short and medium test lengths. As a result, the coverage probabilities fall below the given level of significance in many cases; and, therefore, the corresponding intervals are no longer confidence intervals in terms of the actual definition. In the present work, confidence intervals are defined more precisely by utilizing the relationship between confidence intervals and hypothesis testing. Two approaches to confidence interval construction are explored that are optimal with respect to criteria of smallness and consistency with the standard approach.

Nonlinear dynamic factor analysis models extend standard linear dynamic factor analysis models by allowing time series processes to be nonlinear at the latent level (e.g., involving interaction between two latent processes). In practice, it is often of interest to identify the phases—namely, latent “regimes” or classes—during which a system is characterized by distinctly different dynamics. We propose a new class of models, termed nonlinear regime-switching state-space (RSSS) models, which subsumes regime-switching nonlinear dynamic factor analysis models as a special case. In nonlinear RSSS models, the change processes within regimes, represented using a state-space model, are allowed to be nonlinear. An estimation procedure obtained by combining the extended Kalman filter and the Kim filter is proposed as a way to estimate nonlinear RSSS models. We illustrate the utility of nonlinear RSSS models by fitting a nonlinear dynamic factor analysis model with regime-specific cross-regression parameters to a set of experience sampling affect data. The parallels between nonlinear RSSS models and other well-known discrete change models in the literature are discussed briefly.

To facilitate the interpretation of canonical correlation analysis (CCA) solutions, procedures have been proposed in which CCA solutions are orthogonally rotated to a simple structure. In this paper, we consider oblique rotation for CCA to provide solutions that are much easier to interpret, though only orthogonal rotation is allowed in the existing formulations of CCA. Our task is thus to reformulate CCA so that its solutions have the freedom of oblique rotation. Such a task can be achieved using Yanai’s (Jpn. J. Behaviormetrics 1:46–54, 1974; J. Jpn. Stat. Soc. 11:43–53, 1981) generalized coefficient of determination for the objective function to be maximized in CCA. The resulting solutions are proved to include the existing orthogonal ones as special cases and to be rotated obliquely without affecting the objective function value, where ten Berge’s (Psychometrika 48:519–523, 1983) theorems on suborthonormal matrices are used. A real data example demonstrates that the proposed oblique rotation can provide simple, easily interpreted CCA solutions.

Findings suggest that in psychological tests not only the responses but also the times needed to give the responses are related to characteristics of the test taker. This observation has stimulated the development of latent trait models for the joint distribution of the responses and the response times. Such models are motivated by the hope to improve the estimation of the latent traits by additionally considering response time. In this article, the potential relevance of the response times for psychological assessment is explored for the model of van der Linden (Psychometrika 72:287–308, 2007) that seems to have become the standard approach to response time modeling in educational testing. It can be shown that the consideration of response times increases the information of the test. However, one also can prove that the contribution of the response times to the test information is bounded and has a simple limit.

We investigate the performance of three statistics, R1, R2 (Glas in Psychometrika 53:525–546, 1988), and M2 (Maydeu-Olivares & Joe in J. Am. Stat. Assoc. 100:1009–1020, 2005, Psychometrika 71:713–732, 2006) to assess the overall fit of a one-parameter logistic model (1PL) estimated by (marginal) maximum likelihood (ML). R1 and R2 were specifically designed to target specific assumptions of Rasch models, whereas M2 is a general purpose test statistic. We report asymptotic power rates under some interesting violations of model assumptions (different item discrimination, presence of guessing, and multidimensionality) as well as empirical rejection rates for correctly specified models and some misspecified models. All three statistics were found to be more powerful than Pearson’s X2 against two- and three-parameter logistic alternatives (2PL and 3PL), and against multidimensional 1PL models. The results suggest that there is no clear advantage in using goodness-of-fit statistics specifically designed for Rasch-type models to test these models when marginal ML estimation is used.

The present research evaluates the stability of self-esteem as assessed by a daily version of the Rosenberg (Society and the adolescent self-image, Princeton University Press, Princeton, 1965) general self-esteem scale (RGSE). The scale was administered to 391 undergraduates for five consecutive days. The longitudinal data were analyzed using the integrated LC-LSTM framework that allowed us to evaluate: (1) the measurement invariance of the RGSE, (2) its stability and change across the 5-day assessment period, (3) the amount of variance attributable to stable and transitory latent factors, and (4) the criterion-related validity of these factors. Results provided evidence for measurement invariance, mean-level stability, and rank-order stability of daily self-esteem. Latent state-trait analyses revealed that variances in scores of the RGSE can be decomposed into six components: stable self-esteem (40 %), ephemeral (or temporal-state) variance (36 %), stable negative method variance (9 %), stable positive method variance (4 %), specific variance (1 %) and random error variance (10 %). Moreover, latent factors associated with daily self-esteem were associated with measures of depression, implicit self-esteem, and grade point average.

In this paper, we study the identification of a particular case of the 3PL model, namely when the discrimination parameters are all constant and equal to 1. We term this model, 1PL-G model. The identification analysis is performed under three different specifications. The first specification considers the abilities as unknown parameters. It is proved that the item parameters and the abilities are identified if a difficulty parameter and a guessing parameter are fixed at zero. The second specification assumes that the abilities are mutually independent and identically distributed according to a distribution known up to the scale parameter. It is shown that the item parameters and the scale parameter are identified if a guessing parameter is fixed at zero. The third specification corresponds to a semi-parametric 1PL-G model, where the distribution G generating the abilities is a parameter of interest. It is not only shown that, after fixing a difficulty parameter and a guessing parameter at zero, the item parameters are identified, but also that under those restrictions the distribution G is not identified. It is finally shown that, after introducing two identification restrictions, either on the distribution G or on the item parameters, the distribution G and the item parameters are identified provided an infinite quantity of items is available.

Convergence of the expectation-maximization (EM) algorithm to a global optimum of the marginal log likelihood function for unconstrained latent variable models with categorical indicators is presented. The sufficient conditions under which global convergence of the EM algorithm is attainable are provided in an information-theoretic context by interpreting the EM algorithm as alternating minimization of the Kullback–Leibler divergence between two convex sets. It is shown that these conditions are satisfied by an unconstrained latent class model, yielding an optimal bound against which more highly constrained models may be compared.

We apply the Hawkes process to the analysis of dyadic interaction. The Hawkes process is applicable to excitatory interactions, wherein the actions of each individual increase the probability of further actions in the near future. We consider the representation of the Hawkes process both as a conditional intensity function and as a cluster Poisson process. The former treats the probability of an action in continuous time via non-stationary distributions with arbitrarily long historical dependency, while the latter is conducive to maximum likelihood estimation using the EM algorithm. We first outline the interpretation of the Hawkes process in the dyadic context, and then illustrate its application with an example concerning email transactions in the work place.

Rubin and Thayer (Psychometrika, 47:69–76, 1982) proposed the EM algorithm for exploratory and confirmatory maximum likelihood factor analysis. In this paper, we prove the following fact: the EM algorithm always gives a proper solution with positive unique variances and factor correlations with absolute values that do not exceed one, when the covariance matrix to be analyzed and the initial matrices including unique variances and inter-factor correlations are positive definite. We further numerically demonstrate that the EM algorithm yields proper solutions for the data which lead the prevailing gradient algorithms for factor analysis to produce improper solutions. The numerical studies also show that, in real computations with limited numerical precision, Rubin and Thayer’s (Psychometrika, 47:69–76, 1982) original formulas for confirmatory factor analysis can make factor correlation matrices asymmetric, so that the EM algorithm fails to converge. However, this problem can be overcome by using an EM algorithm in which the original formulas are replaced by those guaranteeing the symmetry of factor correlation matrices, or by formulas used to prove the above fact.

It is shown that for any full column rank matrix X0 with more rows than columns there is a neighborhood \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\mathcal{N}$\end{document} of X0 and a continuous function f on \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\mathcal{N}$\end{document} such that f(X) is an orthogonal complement of X for all X in \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\mathcal{N}$\end{document}. This is used to derive a distribution free goodness of fit test for covariance structure analysis. This test was proposed some time ago and is extensively used. Unfortunately, there is an error in the proof that the proposed test statistic has an asymptotic χ2 distribution. This is a potentially serious problem, without a proof the test statistic may not, in fact, be asymptoticly χ2. The proof, however, is easily fixed using a continuous orthogonal complement function. Similar problems arise in other applications where orthogonal complements are used. These can also be resolved by using continuous orthogonal complement functions.