The existing maximum likelihood theory and its computer software in structural equation modeling are established based on linear relationships among manifest variables and latent variables. However, models with nonlinear relationships are often encountered in social and behavioral sciences. In this article, an EM type algorithm is developed for maximum likelihood estimation of a general nonlinear structural equation model. To avoid computation of the complicated multiple integrals involved, the E-step is completed by a Metropolis-Hastings algorithm. It is shown that the M-step can be completed efficiently by simple conditional maximization. Standard errors of the maximum likelihood estimates are obtained via Louis's formula. The methodology is illustrated with results from a simulation study and two real examples.

The paper surveys 15 years of progress in three psychometric research areas: latent dimensionality structure, test fairness, and skills diagnosis of educational tests. It is proposed that one effective model for selecting and carrying out research is to chose one's research questions from practical challenges facing educational testing, then bring to bear sophisticated probability modeling and statistical analyses to solve these questions, and finally to make effectiveness of the research answers in meeting the educational testing challenges be the ultimate criterion for judging the value of the research. The problem-solving power and the joy of working with a dedicated, focused, and collegial group of colleagues is emphasized. Finally, it is suggested that the summative assessment testing paradigm that has driven test measurement research for over half a century is giving way to a new paradigm that in addition embraces skills level formative assessment, opening up a plethora of challenging, exciting, and societally important research problems for psychometricians.

A comprehensive approach for imposing both row and column constraints on multivariate discrete data is proposed that may be called generalized constrained multiple correspondence analysis (GCMCA). In this method each set of discrete data is first decomposed into several submatrices according to its row and column constraints, and then multiple correspondence analysis (MCA) is applied to the decomposed submatrices to explore relationships among them. This method subsumes existing constrained and unconstrained MCA methods as special cases and also generalizes various kinds of linearly constrained correspondence analysis methods. An example is given to illustrate the proposed method.

This paper introduces a new technique for estimating the parameters of models with continuous latent data. Using the Rasch model as an example, it is shown that existing Bayesian techniques for parameter estimation, such as the Gibbs sampler, are not always easy to implement. Then, a new sampling-based Bayesian technique, called the DA-T-Gibbs sampler, is introduced. The DA-T-Gibbs sampler relies on the particular latent data structure of latent response models to simplify the computations involved in parameter estimation.

A simple and very general algorithm for oblique rotation is identified. While motivated by the rotation problem in factor analysis, it may be used to minimize almost any function of a not necessarily square matrix whose columns are restricted to have unit length. The algorithm has two steps. The first is to compute the gradient of the rotation criterion and the second is to project this onto a manifold of matrices with unit length columns. For this reason it is called a gradient projection algorithm. Because the projection step is very simple, implementation of the algorithm involves little more than computing the gradient of the rotation criterion which for many applications is very simple. It is proven that the algorithm is strictly monotone, that is as long as it is not already at a stationary point, each step will decrease the value of the criterion. Examples from a variety of areas are used to demonstrate the algorithm, including oblimin rotation, target rotation, simplimax rotation, and rotation to similarity and simplicity simultaneously. While it may be, the algorithm is not intended for use as a standard algorithm for well established problems, but rather as a tool for investigating new methods where its generality and simplicity may save an investigator substantial effort.

Procedures are given for determining identified parameters, finding constraints on the covariances, and checking equivalence, in acyclic (recursive) linear path models with correlated error terms (disturbances), by inspection of the path equations, aided by simple recursions. This provides a useful and general alternative to the employment of directed acyclic graph theory for such purposes.

In this paper robustness properties of the maximum likelihood estimator (MLE) and several robust estimators for the logistic regression model when the responses are binary are analysed. It is found that the MLE and the classical Rao's score test can be misleading in the presence of model misspecification which in the context of logistic regression means either misclassification's errors in the responses, or extreme data points in the design space. A general framework for robust estimation and testing is presented and a robust estimator as well as a robust testing procedure are presented. It is shown that they are less influenced by model misspecifications than their classical counterparts. They are finally applied to the analysis of binary data from a study on breastfeeding.

Transforming the core array in Tucker three-way component analysis to simplicity is an intriguing way of revealing structures in between standard Tucker three-way PCA, where the core array is unconstrained, and CANDECOMP/PARAFAC, where the core array has a generalized diagonal form. For certain classes of arrays, transformations to simplicity, that is, transformations that produce a large number of zeros, can be obtained explicitly by solving sets of linear equations. The present paper extends these results. First, a method is offered to simplifyJ ×J × 2 arrays. Next, it is shown that the transformation that simplifies anI ×J ×K array can be used to also simplify the (complementary) arrays of order (JK −I) ×J ×K, of orderI × (IK −J) ×K and of orderI ×J × (IJ −K). Finally, the question of what constitutes the maximal simplicity for arrays (the maximal number of zero elements) will be considered. It is shown that cases of extreme simplicity, considered in the past, are, in fact, cases of maximal simplicity.

In a latent class IRT model in which the latent classes are ordered on one dimension, the class specific response probabilities are subject to inequality constraints. The number of these inequality constraints increase dramatically with the number of response categories per item, if assumptions like monotonicity or double monotonicity of the cumulative category response functions are postulated. A Markov chain Monte Carlo method, the Gibbs sampler, can sample from the multivariate posterior distribution of the parameters under the constraints. Bayesian model selection can be done by posterior predictive checks and Bayes factors. A simulation study is done to evaluate results of the application of these methods to ordered latent class models in three realistic situations. Also, an example of the presented methods is given for existing data with polytomous items. It can be concluded that the Bayesian estimation procedure can handle the inequality constraints on the parameters very well. However, the application of Bayesian model selection methods requires more research.

This paper studies the asymptotic distributions of three reliability coefficient estimates: Sample coefficient alpha, the reliability estimate of a composite score following a factor analysis, and the estimate of the maximal reliability of a linear combination of item scores following a factor analysis. Results indicate that the asymptotic distribution for each of the coefficient estimates, obtained based on a normal sampling distribution, is still valid within a large class of nonnormal distributions. Therefore, a formula for calculating the standard error of the sample coefficient alpha, recently obtained by van Zyl, Neudecker and Nel, applies to other reliability coefficients and can still be used even with skewed and kurtotic data such as are typical in the social and behavioral sciences.

Data in social and behavioral sciences are often hierarchically organized though seldom normal, yet normal theory based inference procedures are routinely used for analyzing multilevel models. Based on this observation, simple adjustments to normal theory based results are proposed to minimize the consequences of violating normality assumptions. For characterizing the distribution of parameter estimates, sandwich-type covariance matrices are derived. Standard errors based on these covariance matrices remain consistent under distributional violations. Implications of various covariance estimators are also discussed. For evaluating the quality of a multilevel model, a rescaled statistic is given for both the hierarchical linear model and the hierarchical structural equation model. The rescaled statistic, improving the likelihood ratio statistic by estimating one extra parameter, approaches the same mean as its reference distribution. A simulation study with a 2-level factor model implies that the rescaled statistic is preferable.

The U.S. National Assessment of Educational Progress (NAEP), the Third International Mathematics and Science Study (TIMSS), and the U.S. Adult Literacy Survey collect probability samples of students (or adults) who are administered brief examinations in subject areas such as mathematics and reading (cognitive variables), along with background demographic (primary) and educational environment (secondary) questions. The demographic questions are used in the primary reporting, while the numerous “explanatory” secondary variables, or “covariates”, are only directly utilized in subsequent secondary analyses. The covariates are also used indirectly to create the plausible values (multiple imputations) that are an integral part of analyses because of the use of sparse matrix sampling of cognitive items. The improvement in the precision of the primary reporting due to the inclusion of the covariates is assessed here and contrasted with the precision of reporting using plausible values created using only the primary demographic variables.

The results demonstrate that the improvement in precision depends on the matrix sampling designs for the cognitive assessments. The improvements range from essentially none for the most common designs, to moderate for some less common designs. Consequently, two potential changes in the reporting procedures that could improve the statistical and operational efficiency of primary reporting are (a) eliminate or reduce the collection of covariates and increase the number of cognitive items, (b) to avoid delays, eliminate the covariates from the creation of plausible values used for the primary reports, but include them later when creating public-use files for secondary analyses. The potential improvements in statistical and operational efficiency must be weighed against the intrinsic interest in the covariates, and the potential for small discrepancies in the primary and secondary reporting.

In this paper, we propose a class of locally dependent latent trait models for responses to psychological and educational tests. Typically, item response models treat an individual's multiple response to stimuli as conditional independent given the individual's latent trait. In this paper, instead the focus is on models based on a family of conditional distributions, or kernel, that describes joint multiple item responses as a function of student latent trait, not assuming conditional independence. Specifically, we examine a hybrid kernel which comprises a component for one-way item response functions and a component for conditional associations between items given latent traits. The class of models allows the extension of item response theory to cover some new and innovative applications in psychological and educational research. An EM algorithm for marginal maximum likelihood of the hybrid kernel model is proposed. Furthermore, we delineate the relationship of the class of locally dependent models and the log-linear model by revisiting the Dutch identity (Holland, 1990).

A computerized adaptive test (CAT) is usually administered to small groups of examinees at frequent time intervals. It is often the case that examinees who take the test earlier share information with examinees who will take the test later, thus increasing the risk that many items may become known. Item overlap rate for a group of examinees refers to the number of overlapping items encountered by these examinees divided by the test length. For a specific item pool, different item selection algorithms may yield different item overlap rates. An important issue in designing a good CAT item selection algorithm is to keep item overlap rate below a preset level. In doing so, it is important to investigate what the lowest rate could be for all possible item selection algorithms. In this paper we rigorously prove that if every item has an equal possibility to be selected from the pool in a fixed-length CAT, the number of overlapping items among any α randomly sampled examinees follows the hypergeometric distribution family for α ≥ 1. Thus, the expected values of the number of overlapping items among any randomly sampled α examinees can be calculated precisely. These values may serve as benchmarks in controlling item overlap rates for fixed-length adaptive tests.

The main purpose of this article is to develop a Bayesian approach for structural equation models with ignorable missing continuous and polytomous data. Joint Bayesian estimates of thresholds, structural parameters and latent factor scores are obtained simultaneously. The idea of data augmentation is used to solve the computational difficulties involved. In the posterior analysis, in addition to the real missing data, latent variables and latent continuous measurements underlying the polytomous data are treated as hypothetical missing data. An algorithm that embeds the Metropolis-Hastings algorithm within the Gibbs sampler is implemented to produce the Bayesian estimates. A goodness-of-fit statistic for testing the posited model is presented. It is shown that the proposed approach is not sensitive to prior distributions and can handle situations with a large number of missing patterns whose underlying sample sizes may be small. Computational efficiency of the proposed procedure is illustrated by simulation studies and a real example.

A fundamental assumption of most IRT models is that items measure the same unidimensional latent construct. For the polytomous Rasch model two ways of testing this assumption against specific multidimensional alternatives are discussed. One, a marginal approach assuming a multidimensional parametric latent variable distribution, and, two, a conditional approach with no distributional assumptions about the latent variable. The second approach generalizes the Martin-Löf test for the dichotomous Rasch model in two ways: to polytomous items and to a test against an alternative that may have more than two dimensions. A study on occupational health is used to motivate and illustrate the methods.

Multilevel factor analysis models are widely used in the social sciences to account for heterogeneity in mean structures. In this paper we extend previous work on multilevel models to account for general forms of heterogeneity in confirmatory factor analysis models. We specify various models of mean and covariance heterogeneity in confirmatory factor analysis and develop Markov Chain Monte Carlo (MCMC) procedures to perform Bayesian inference, model checking, and model comparison.

We test our methodology using synthetic data and data from a consumption emotion study. The results from synthetic data show that our Bayesian model perform well in recovering the true parameters and selecting the appropriate model. More importantly, the results clearly illustrate the consequences of ignoring heterogeneity. Specifically, we find that ignoring heterogeneity can lead to sign reversals of the factor covariances, inflation of factor variances and underappreciation of uncertainty in parameter estimates. The results from the emotion study show that subjects vary both in means and covariances. Thus traditional psychometric methods cannot fully capture the heterogeneity in our data.