Categorical marginal models (CMMs) are flexible tools for modelling dependent or clustered categorical data, when the dependencies themselves are not of interest. A major limitation of maximum likelihood (ML) estimation of CMMs is that the size of the contingency table increases exponentially with the number of variables, so even for a moderate number of variables, say between 10 and 20, ML estimation can become computationally infeasible. An alternative method, which retains the optimal asymptotic efficiency of ML, is maximum empirical likelihood (MEL) estimation. However, we show that MEL tends to break down for large, sparse contingency tables. As a solution, we propose a new method, which we call maximum augmented empirical likelihood (MAEL) estimation and which involves augmentation of the empirical likelihood support with a number of well-chosen cells. Simulation results show good finite sample performance for very large contingency tables.

Multidimensional item response theory (MIRT) is a statistical test theory that precisely estimates multiple latent skills of learners from the responses in a test. Both compensatory and non-compensatory models have been proposed for MIRT: the former assumes that each skill can complement other skills, whereas the latter assumes they cannot. This non-compensatory assumption is convincing in many tests that measure multiple skills; therefore, applying non-compensatory models to such data is crucial for achieving unbiased and accurate estimation. In contrast to tests, latent skills will change over time in daily learning. To monitor the growth of skills, dynamical extensions of MIRT models have been investigated. However, most of them assumed compensatory models, and a model that can reproduce continuous latent states of skills under the non-compensatory assumption has not been proposed thus far. To enable accurate skill tracing under the non-compensatory assumption, we propose a dynamical extension of non-compensatory MIRT models by combining a linear dynamical system and a non-compensatory model. This results in a complicated posterior of skills, which we approximate with a Gaussian distribution by minimizing the Kullback–Leibler divergence between the approximated posterior and the true posterior. The learning algorithm for the model parameters is derived through Monte Carlo expectation maximization. Simulation studies verify that the proposed method is able to reproduce latent skills accurately, whereas the dynamical compensatory model suffers from significant underestimation errors. Furthermore, experiments on an actual data set demonstrate that our dynamical non-compensatory model can infer practical skill tracing and clarify differences in skill tracing between non-compensatory and compensatory models.

Traditional measurement models assume that all item responses correlate with each other only through their underlying latent variables. This conditional independence assumption has been extended in joint models of responses and response times (RTs), implying that an item has the same item characteristics fors all respondents regardless of levels of latent ability/trait and speed. However, previous studies have shown that this assumption is violated in various types of tests and questionnaires and there are substantial interactions between respondents and items that cannot be captured by person- and item-effect parameters in psychometric models with the conditional independence assumption. To study the existence and potential cognitive sources of conditional dependence and utilize it to extract diagnostic information for respondents and items, we propose a diffusion item response theory model integrated with the latent space of variations in information processing rate of within-individual measurement processes. Respondents and items are mapped onto the latent space, and their distances represent conditional dependence and unexplained interactions. We provide three empirical applications to illustrate (1) how to use an estimated latent space to inform conditional dependence and its relation to person and item measures, (2) how to derive diagnostic feedback personalized for respondents, and (3) how to validate estimated results with an external measure. We also provide a simulation study to support that the proposed approach can accurately recover its parameters and detect conditional dependence underlying data.

Cognitive diagnostic models are a powerful family of fine-grained discrete latent variable models in psychometrics. Within this family, the DINA model is a fundamental and parsimonious one that has received significant attention. Similar to other complex latent variable models, identifiability is an important issue for CDMs, including the DINA model. Gu and Xu (Psychometrika 84(2):468–483, 2019) established the necessary and sufficient conditions for strict identifiability of the DINA model. Despite being the strongest possible notion of identifiability, strict identifiability may impose overly stringent requirements on designing the cognitive diagnostic tests. This work studies a slightly weaker yet very useful notion, generic identifiability, which means parameters are identifiable almost everywhere in the parameter space, excluding only a negligible subset of measure zero. We propose transparent generic identifiability conditions for the DINA model, relaxing existing conditions in nontrivial ways. Under generic identifiability, we also explicitly characterize the forms of the measure-zero sets where identifiability breaks down. In addition, we reveal an interesting blessing-of-latent-dependence phenomenon under DINA—that is, dependence between the latent attributes can restore identifiability under some otherwise unidentifiable $\mathbf{Q}$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbf {Q}}$$\end{document}-matrix designs. The blessing of latent dependence provides useful practical implications and reassurance for real-world designs of cognitive diagnostic assessments.

Bi-factor and second-order models based on copulas are proposed for item response data, where the items are sampled from identified subdomains of some larger domain such that there is a homogeneous dependence within each domain. Our general models include the Gaussian bi-factor and second-order models as special cases and can lead to more probability in the joint upper or lower tail compared with the Gaussian bi-factor and second-order models. Details on maximum likelihood estimation of parameters for the bi-factor and second-order copula models are given, as well as model selection and goodness-of-fit techniques. Our general methodology is demonstrated with an extensive simulation study and illustrated for the Toronto Alexithymia Scale. Our studies suggest that there can be a substantial improvement over the Gaussian bi-factor and second-order models both conceptually, as the items can have interpretations of discretized maxima/minima or mixtures of discretized means in comparison with discretized means, and in fit to data.

The Wald, likelihood ratio, score, and the recently proposed gradient statistics can be used to assess a broad range of hypotheses in item response theory models, for instance, to check the overall model fit or to detect differential item functioning. We introduce new methods for power analysis and sample size planning that can be applied when marginal maximum likelihood estimation is used. This allows the application to a variety of IRT models, which are commonly used in practice, e.g., in large-scale educational assessments. An analytical method utilizes the asymptotic distributions of the statistics under alternative hypotheses. We also provide a sampling-based approach for applications where the analytical approach is computationally infeasible. This can be the case with 20 or more items, since the computational load increases exponentially with the number of items. We performed extensive simulation studies in three practically relevant settings, i.e., testing a Rasch model against a 2PL model, testing for differential item functioning, and testing a partial credit model against a generalized partial credit model. The observed distributions of the test statistics and the power of the tests agreed well with the predictions by the proposed methods in sufficiently large samples. We provide an openly accessible R package that implements the methods for user-supplied hypotheses.

Applications of structural equation modeling (SEM) may encounter issues like inadmissible parameter estimates, nonconvergence, or unsatisfactory model fit. We propose a new factor rotation method that reparameterizes the factor correlation matrix in exploratory factor analysis (EFA) such that factors can be either exogenous or endogenous. The proposed method is an oblique rotation method for EFA, but it allows directional structural paths among factors. We thus referred it to as FSP (factor structural paths) rotation. In particular, we can use FSP rotation to “translate” an SEM model to incorporate theoretical expectations on both factor loadings and structural parameters. We illustrate FSP rotation with an empirical example and explore its statistical properties with simulated data. The results include that (1) EFA with FSP rotation tends to fit data better and encounters fewer Heywood cases than SEM does when there are cross-loadings and many small nonzero loadings, (2) FSP rotated parameter estimates are satisfactory for small models, and (3) FSP rotated parameter estimates are more satisfactory for large models when the structural parameter matrices are sparse.

Researchers have widely used exploratory factor analysis (EFA) to learn the latent structure underlying multivariate data. Rotation and regularised estimation are two classes of methods in EFA that they often use to find interpretable loading matrices. In this paper, we propose a new family of oblique rotations based on component-wise $L^p$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^p$$\end{document} loss functions $(0<p\le 1)$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(0 < p\le 1)$$\end{document} that is closely related to an $L^p$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^p$$\end{document} regularised estimator. We develop model selection and post-selection inference procedures based on the proposed rotation method. When the true loading matrix is sparse, the proposed method tends to outperform traditional rotation and regularised estimation methods in terms of statistical accuracy and computational cost. Since the proposed loss functions are nonsmooth, we develop an iteratively reweighted gradient projection algorithm for solving the optimisation problem. We also develop theoretical results that establish the statistical consistency of the estimation, model selection, and post-selection inference. We evaluate the proposed method and compare it with regularised estimation and traditional rotation methods via simulation studies. We further illustrate it using an application to the Big Five personality assessment.

We consider the dependence of a broad class of chance-corrected weighted agreement coefficients on the weighting scheme that penalizes rater disagreements. The considered class encompasses many existing coefficients with any number of raters, and one real-valued power parameter defines the weighting scheme that includes linear, quadratic, identity, and radical weights. We obtain the first-order and second-order derivatives of the coefficients with respect to the power parameter and decompose them into components corresponding to all pairs of different category distances. Each component compares its two distances in terms of the ratio of observed to expected-by-chance frequency. A larger ratio for the smaller distance than the larger distance contributes to a positive relationship between the power parameter and the coefficient value; the opposite contributes to a negative relationship. We provide necessary and sufficient conditions for the coefficient value to increase or decrease and the relationship to intensify or weaken as the power parameter increases. We use the first-order and second-order derivatives for corresponding measurement. Furthermore, we show how these two derivatives allow other researchers to obtain quite accurate estimates of the coefficient value for unreported values of the power parameter, even without access to the original data.

Three IRT diagnostic-classification-modeling (DCM)-based multiple choice (MC) item design principles are stated that improve classroom quiz student diagnostic classification. Using proven-optimal maximum likelihood-based student classification, example items demonstrate that adherence to these item design principles increases attribute (skills and especially misconceptions) correct classification rates (CCRs). Simple formulas compute these needed item CCRs. By use of these psychometrically driven item design principles, hopefully enough attributes can be accurately diagnosed by necessarily short MC-item-based quizzes to be widely instructionally useful. These results should then stimulate increased use of well-designed MC item quizzes that target accurately diagnosing skills/misconceptions, thereby enhancing classroom learning.

Restricted latent class models (RLCMs) are an important class of methods that provide researchers and practitioners in the educational, psychological, and behavioral sciences with fine-grained diagnostic information to guide interventions. Recent research established sufficient conditions for identifying RLCM parameters. A current challenge that limits widespread application of RLCMs is that existing identifiability conditions may be too restrictive for some practical settings. In this paper we establish a weaker condition for identifying RLCM parameters for multivariate binary data. Although the new results weaken identifiability conditions for general RLCMs, the new results do not relax existing necessary and sufficient conditions for the simpler DINA/DINO models. Theoretically, we introduce a new form of latent structure completeness, referred to as dyad-completeness, and prove identification by applying Kruskal’s Theorem for the uniqueness of three-way arrays. The new condition is more likely satisfied in applied research, and the results provide researchers and test-developers with guidance for designing diagnostic instruments.

Standard response formats such as rating or visual analogue scales require respondents to condense distributions of latent states or behaviors into a single value. Whereas this is suitable to measure central tendency, it neglects the variance of distributions. As a remedy, variability may be measured using interval-response formats, more specifically the dual-range slider (RS2). Given the lack of an appropriate item response model for the RS2, we develop the Dirichlet dual response model (DDRM), an extension of the beta response model (BRM; Noel & Dauvier in Appl Psychol Meas, 31:47–73, 2007). We evaluate the DDRM’s performance by assessing parameter recovery in a simulation study. Results indicate overall good parameter recovery, although parameters concerning interval width (which reflect variability in behavior or states) perform worse than parameters concerning central tendency. We also test the model empirically by jointly fitting the BRM and the DDRM to single-range slider (RS1) and RS2 responses for two Extraversion scales. While the DDRM has an acceptable fit, it shows some misfit regarding the RS2 interval widths. Nonetheless, the model indicates substantial differences between respondents concerning variability in behavior. High correlations between person parameters of the BRM and DDRM suggest convergent validity between the RS1 and the RS2 interval location. Both the simulation and the empirical study demonstrate that the latent parameter space of the DDRM addresses an important issue of the RS2 response format, namely, the scale-inherent interdependence of interval location and interval width (i.e., intervals at the boundaries are necessarily smaller).

In this article, we present a general theorem and proof for the global identification of composed CFA models. They consist of identified submodels that are related only through covariances between their respective latent factors. Composed CFA models are frequently used in the analysis of multimethod data, longitudinal data, or multidimensional psychometric data. Firstly, our theorem enables researchers to reduce the problem of identifying the composed model to the problem of identifying the submodels and verifying the conditions given by our theorem. Secondly, we show that composed CFA models are globally identified if the primary models are reduced models such as the CT-C$(M-1)$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(M-1)$$\end{document} model or similar types of models. In contrast, composed CFA models that include non-reduced primary models can be globally underidentified for certain types of cross-model covariance assumptions. We discuss necessary and sufficient conditions for the global identification of arbitrary composed CFA models and provide a Python code to check the identification status for an illustrative example. The code we provide can be easily adapted to more complex models.

Cognitive Diagnosis Models (CDMs) are a special family of discrete latent variable models that are widely used in educational and psychological measurement. A key component of CDMs is the Q-matrix characterizing the dependence structure between the items and the latent attributes. Additionally, researchers also assume in many applications certain hierarchical structures among the latent attributes to characterize their dependence. In most CDM applications, the attribute–attribute hierarchical structures, the item-attribute Q-matrix, the item-level diagnostic models, as well as the number of latent attributes, need to be fully or partially pre-specified, which however may be subjective and misspecified as noted by many recent studies. This paper considers the problem of jointly learning these latent and hierarchical structures in CDMs from observed data with minimal model assumptions. Specifically, a penalized likelihood approach is proposed to select the number of attributes and estimate the latent and hierarchical structures simultaneously. An expectation-maximization (EM) algorithm is developed for efficient computation, and statistical consistency theory is also established under mild conditions. The good performance of the proposed method is illustrated by simulation studies and real data applications in educational assessment.

In this paper, the support of the joint probability distribution of categorical variables in the total population is treated as unknown. From a general total population model with unknown support, a general subpopulation model with its support equal to the set of all observed score patterns is derived. In maximum likelihood estimation of the parameters of any such subpopulation model, the evaluation of the log-likelihood function only requires the summation over a number of terms equal to at most the sample size. It is made clear that the parameters of a hypothesized total population model are consistently and asymptotically efficiently estimated by the values that maximize the log-likelihood function of the corresponding subpopulation model. Next, new likelihood ratio goodness-of-fit tests are proposed as alternatives to the Pearson chi-square goodness-of-fit test and the likelihood ratio test against the saturated model. In a simulation study, the asymptotic bias and efficiency of maximum likelihood estimators and the asymptotic performance of the goodness-of-fit tests are investigated.

Latent class models are powerful statistical modeling tools widely used in psychological, behavioral, and social sciences. In the modern era of data science, researchers often have access to response data collected from large-scale surveys or assessments, featuring many items (large J) and many subjects (large N). This is in contrary to the traditional regime with fixed J and large N. To analyze such large-scale data, it is important to develop methods that are both computationally efficient and theoretically valid. In terms of computation, the conventional EM algorithm for latent class models tends to have a slow algorithmic convergence rate for large-scale data and may converge to some local optima instead of the maximum likelihood estimator (MLE). Motivated by this, we introduce the tensor decomposition perspective into latent class analysis with binary responses. Methodologically, we propose to use a moment-based tensor power method in the first step and then use the obtained estimates as initialization for the EM algorithm in the second step. Theoretically, we establish the clustering consistency of the MLE in assigning subjects into latent classes when N and J both go to infinity. Simulation studies suggest that the proposed tensor-EM pipeline enjoys both good accuracy and computational efficiency for large-scale data with binary responses. We also apply the proposed method to an educational assessment dataset as an illustration.

The specification of the $\mathit{Q}$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\varvec{Q}}$$\end{document} matrix in cognitive diagnosis models is important for correct classification of attribute profiles. Researchers have proposed many methods for estimation and validation of the data-driven $\mathit{Q}$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\varvec{Q}}$$\end{document} matrices. However, inference of the number of attributes in the general restricted latent class model remains an open question. We propose a Bayesian framework for general restricted latent class models and use the spike-and-slab prior to avoid the computation issues caused by the varying dimensions of model parameters associated with the number of attributes, K. We develop an efficient Metropolis-within-Gibbs algorithm to estimate K and the corresponding $\mathit{Q}$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\varvec{Q}}$$\end{document} matrix simultaneously. The proposed algorithm uses the stick-breaking construction to mimic an Indian buffet process and employs a novel Metropolis–Hastings transition step to encourage exploring the sample space associated with different values of K. We evaluate the performance of the proposed method through a simulation study under different model specifications and apply the method to a real data set related to a fluid intelligence matrix reasoning test.

It is essential to control self-reported trait measurements for response style effects to ensure a valid interpretation of estimates. Traditional psychometric models facilitating such control consider item responses as the result of two kinds of response processes—based on the substantive trait, or based on response styles—and they assume that both of these processes have a constant influence across the items of a questionnaire. However, this homogeneity over items is not always given, for instance, if the respondents’ motivation declines throughout the questionnaire so that heuristic responding driven by response styles may gradually take over from cognitively effortful trait-based responding. The present study proposes two dynamic IRTree models, which account for systematic continuous changes and additional random fluctuations of response strategies, by defining item position-dependent trait and response style effects. Simulation analyses demonstrate that the proposed models accurately capture dynamic trajectories of response processes, as well as reliably detect the absence of dynamics, that is, identify constant response strategies. The continuous version of the dynamic model formalizes the underlying response strategies in a parsimonious way and is highly suitable as a cognitive model for investigating response strategy changes over items. The extended model with random fluctuations of strategies can adapt more closely to the item-specific effects of different response processes and thus is a well-fitting model with high flexibility. By using an empirical data set, the benefits of the proposed dynamic approaches over traditional IRTree models are illustrated under realistic conditions.

This article presents a joint modeling framework of ordinal responses and response times (RTs) for the measurement of latent traits. We integrate cognitive theories of decision-making and confidence judgments with psychometric theories to model individual-level measurement processes. The model development starts with the sequential sampling framework which assumes that when an item is presented, a respondent accumulates noisy evidence over time to respond to the item. Several cognitive and psychometric theories are reviewed and integrated, leading us to three psychometric process models with different representations of the cognitive processes underlying the measurement. We provide simulation studies that examine parameter recovery and show the relationships between latent variables and data distributions. We further test the proposed models with empirical data measuring three traits related to motivation. The results show that all three models provide reasonably good descriptions of observed response proportions and RT distributions. Also, different traits favor different process models, which implies that psychological measurement processes may have heterogeneous structures across traits. Our process of model building and examination illustrates how cognitive theories can be incorporated into psychometric model development to shed light on the measurement process, which has had little attention in traditional psychometric models.

The Pearson and likelihood ratio statistics are commonly used to test goodness of fit for models applied to data from a multinomial distribution. The goodness-of-fit test based on Pearson’s Chi-squared statistic is sometimes considered to be a global test that gives little guidance to the source of poor fit when the null hypothesis is rejected, and it has also been recognized that the global test can often be outperformed in terms of power by focused or directional tests. For the cross-classification of a large number of manifest variables, the GFfit statistic focused on second-order marginals for variable pairs i, j has been proposed as a diagnostic to aid in finding the source of lack of fit after the model has been rejected based on a more global test. When data are from a table formed by the cross-classification of a large number of variables, the common global statistics may also have low power and inaccurate Type I error level due to sparseness in the cells of the table. The sparseness problem is rarely encountered with the GFfit statistic because it is focused on the lower-order marginals. In this paper, a new and extended version of the GFfit statistic is proposed by decomposing the Pearson statistic from the full table into orthogonal components defined on marginal distributions and then defining the new version, $GFfi{t}_{\perp }^{(ij)}$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$GFfit_{\perp }^{(ij)}$$\end{document}, as a partial sum of these orthogonal components. While the emphasis is on lower-order marginals, the new version of $GFfi{t}_{\perp }^{(ij)}$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$GFfit_{\perp }^{(ij)}$$\end{document} is also extended to higher-order tables so that the $GFfi{t}_{\perp }$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$GFfit_{\perp }$$\end{document} statistics sum to the Pearson statistic. As orthogonal components of the Pearson $X^2$\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} statistic, $GFfi{t}_{\perp }^{(ij)}$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$GFfit_{\perp }^{(ij)}$$\end{document} statistics have advantages over other lack-of-fit diagnostics that are currently available for cross-classified tables: the $GFfi{t}_{\perp }^{(ij)}$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$GFfit_{\perp }^{(ij)}$$\end{document} generally have higher power to detect lack of fit while maintaining good Type I error control even if the joint frequencies are very sparse, as will be shown in simulation results; theoretical results will establish that $GFfi{t}_{\perp }^{(ij)}$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$GFfit_{\perp }^{(ij)}$$\end{document} statistics have known degrees of freedom and are asymptotically independent with known joint distribution, a property which facilitates less conservative control of false discovery rate (FDR) or familywise error rate (FWER) in a high-dimensional table which would produce a large number of bivariate lack-of-fit diagnostics. Computation of $GFfi{t}_{\perp }^{(ij)}$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$GFfit_{\perp }^{(ij)}$$\end{document} statistics is also computationally stable. The extended $GFfi{t}_{\perp }^{(ij)}$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$GFfit_{\perp }^{(ij)}$$\end{document} statistic can be applied to a variety of models for cross-classified tables. An application of the new GFfit statistic as a diagnostic for a latent variable model is presented.