An Extended GFfit Statistic Defined on Orthogonal Components of Pearson's Chi-Square

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, [Formula: see text], as a partial sum of these orthogonal components. While the emphasis is on lower-order marginals, the new version of [Formula: see text] is also extended to higher-order tables so that the [Formula: see text] statistics sum to the Pearson statistic. As orthogonal components of the Pearson [Formula: see text] statistic, [Formula: see text] statistics have advantages over other lack-of-fit diagnostics that are currently available for cross-classified tables: the [Formula: see text] 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 [Formula: see text] 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 [Formula: see text] statistics is also computationally stable. The extended [Formula: see text] 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.

Use of the Lagrange Multiplier Test for Assessing Measurement Invariance Under Model Misspecification

This article studies the Type I error, false positive rates, and power of four versions of the Lagrange multiplier test to detect measurement noninvariance in item response theory (IRT) models for binary data under model misspecification. The tests considered are the Lagrange multiplier test computed with the Hessian and cross-product approach, the generalized Lagrange multiplier test and the generalized jackknife score test. The two model misspecifications are those of local dependence among items and nonnormal distribution of the latent variable. The power of the tests is computed in two ways, empirically through Monte Carlo simulation methods and asymptotically, using the asymptotic distribution of each test under the alternative hypothesis. The performance of these tests is evaluated by means of a simulation study. The results highlight that, under mild model misspecification, all tests have good performance while, under strong model misspecification, the tests performance deteriorates, especially for false positive rates under local dependence and power for small sample size under misspecification of the latent variable distribution. In general, the Lagrange multiplier test computed with the Hessian approach and the generalized Lagrange multiplier test have better performance in terms of false positive rates while the Lagrange multiplier test computed with the cross-product approach has the highest power for small sample sizes. The asymptotic power turns out to be a good alternative to the classic empirical power because it is less time consuming. The Lagrange tests studied here have been also applied to a real data set.

Dynamic latent variable models for the analysis of cognitive abilities in the elderly population

Cognitive functioning is a key indicator of overall individual health. Identifying factors related to cognitive status, especially in later life, is of major importance. We concentrate on the analysis of the temporal evolution of cognitive abilities in the elderly population. We propose to model the individual cognitive functioning as a multidimensional latent process that accounts also for the effects of individual-specific characteristics (gender, age, and years of education). The proposed model is specified within the generalized linear latent variable framework, and its efficient estimation is obtained using a recent approximation technique, called dimensionwise quadrature. It provides a fast and streamlined approximate inference for complex models, with better or no degradation in accuracy compared with standard techniques. The methodology is applied to the cognitive assessment data from the Health and Retirement Study combined with the Asset and Health Dynamic study in the years between 2006 and 2010. We evaluate the temporal relationship between two dimensions of cognitive functioning, that is, episodic memory and general mental status. We find a substantial influence of the former on the evolution of the latter, as well as evidence of severe consequences on both cognitive abilities among less-educated and older individuals.

A Composite Likelihood Inference in Latent Variable Models for Ordinal Longitudinal Responses

The paper proposes a composite likelihood estimation approach that uses bivariate instead of multivariate marginal probabilities for ordinal longitudinal responses using a latent variable model. The model considers time-dependent latent variables and item-specific random effects to be accountable for the interdependencies of the multivariate ordinal items. Time-dependent latent variables are linked with an autoregressive model. Simulation results have shown composite likelihood estimators to have a small amount of bias and mean square error and as such they are feasible alternatives to full maximum likelihood. Model selection criteria developed for composite likelihood estimation are used in the applications. Furthermore, lower-order residuals are used as measures-of-fit for the selected models.

A factor mixture analysis model for multivariate binary data

The paper proposes a latent variable model for binary data coming from an unobserved heterogeneous population. The heterogeneity is taken into account by replacing the traditional assumption of Gaussian distributed factors by a finite mixture of multivariate Gaussians. The aim of the proposed model is twofold: it allows to achieve dimension reduction when the data are dichotomous and, simultaneously, it performs model based clustering in the latent space. Model estimation is obtained by means of a maximum likelihood method via a generalized version of the EM algorithm. In order to evaluate the performance of the model a simulation study and two real applications are illustrated.

Latent variable models for multivariate longitudinal ordinal responses

The paper proposes a full information maximum likelihood estimation method for modelling multivariate longitudinal ordinal variables. Two latent variable models are proposed that account for dependencies among items within time and between time. One model fits item-specific random effects which account for the between time points correlations and the second model uses a common factor. The relationships between the time-dependent latent variables are modelled with a non-stationary autoregressive model. The proposed models are fitted to a real data set.