In their seminal work on characterizing the manifest probabilities of latent trait models, Cressie and Holland give a theoretically important characterization of the marginal Rasch model. Because their representation of the marginal Rasch model does not involve any latent trait, nor any specific distribution of a latent trait, it opens up the possibility for constructing a Markov chain - Monte Carlo method for Bayesian inference for the marginal Rasch model that does not rely on data augmentation. Such an approach would be highly efficient as its computational cost does not depend on the number of respondents, which makes it suitable for large-scale educational measurement. In this paper, such an approach will be developed and its operating characteristics illustrated with simulated data.

The assumption of latent monotonicity in item response theory models for dichotomous data cannot be evaluated directly, but observable consequences such as manifest monotonicity facilitate the assessment of latent monotonicity in real data. Standard methods for evaluating manifest monotonicity typically produce a test statistic that is geared toward falsification, which can only provide indirect support in favor of manifest monotonicity. We propose the use of Bayes factors to quantify the degree of support available in the data in favor of manifest monotonicity or against manifest monotonicity. Through the use of informative hypotheses, this procedure can also be used to determine the support for manifest monotonicity over substantively or statistically relevant alternatives to manifest monotonicity, rendering the procedure highly flexible. The performance of the procedure is evaluated using a simulation study, and the application of the procedure is illustrated using empirical data.

This paper refers to the exponential family of probability distributions and the conditional maximum likelihood (CML) theory. It is concerned with the determination of the sample size for three groups of tests of linear hypotheses, known as the fundamental trinity of Wald, score, and likelihood ratio tests. The main practical purpose refers to the special case of tests of the class of Rasch models. The theoretical background is discussed and the formal framework for sample size calculations is provided, given a predetermined deviation from the model to be tested and the probabilities of the errors of the first and second kinds.

An approach to generate non-normality in multivariate data based on a structural model with normally distributed latent variables is presented. The key idea is to create non-normality in the manifest variables by applying non-linear linking functions to the latent part, the error part, or both. The algorithm corrects the covariance matrix for the applied function by approximating the deviance using an approximated normal variable. We show that the root mean square error (RMSE) for the covariance matrix converges to zero as sample size increases and closely approximates the RMSE as obtained when generating normally distributed variables. Our algorithm creates non-normality affecting every moment, is computationally undemanding, easy to apply, and particularly useful for simulation studies in structural equation modeling.

This paper develops a theorem that facilitates computing the degrees of freedom of Wald-type chi-square tests for moment restrictions when there is rank deficiency of key matrices involved in the definition of the test. An if and only if (iff) condition is developed for a simple rule of difference of ranks to be used when computing the desired degrees of freedom of the test. The theorem is developed exploiting basics tools of matrix algebra. The theorem is shown to play a key role in proving the asymptotic chi-squaredness of a goodness of fit test in moment structure analysis, and in finding the degrees of freedom of this chi-square statistic.

The monotone homogeneity model (MHM—also known as the unidimensional monotone latent variable model) is a nonparametric IRT formulation that provides the underpinning for partitioning a collection of dichotomous items to form scales. Ellis (Psychometrika 79:303–316, 2014, doi:10.1007/s11336-013-9341-5) has recently derived inequalities that are implied by the MHM, yet require only the bivariate (inter-item) correlations. In this paper, we incorporate these inequalities within a mathematical programming formulation for partitioning a set of dichotomous scale items. The objective criterion of the partitioning model is to produce clusters of maximum cardinality. The formulation is a binary integer linear program that can be solved exactly using commercial mathematical programming software. However, we have also developed a standalone branch-and-bound algorithm that produces globally optimal solutions. Simulation results and a numerical example are provided to demonstrate the proposed method.

Dual scaling (DS) is a multivariate exploratory method equivalent to correspondence analysis when analysing contingency tables. However, for the analysis of rating data, different proposals appear in the DS and correspondence analysis literature. It is shown here that a peculiarity of the DS method can be exploited to detect differences in response styles. Response styles occur when respondents use rating scales differently for reasons not related to the questions, often biasing results. A spline-based constrained version of DS is devised which can detect the presence of four prominent types of response styles, and is extended to allow for multiple response styles. An alternating nonnegative least squares algorithm is devised for estimating the parameters. The new method is appraised both by simulation studies and an empirical application.

The present work explores the connections between cognitive diagnostic models (CDM) and knowledge space theory (KST) and shows that these two quite distinct approaches overlap. It is proved that in fact the Multiple Strategy DINA (Deterministic Input Noisy AND-gate) model and the CBLIM, a competence-based extension of the basic local independence model (BLIM), are equivalent. To demonstrate the benefits that arise from integrating the two theoretical perspectives, it is shown that a fairly complete picture on the identifiability of these models emerges by combining results from both camps. The impact of the results is illustrated by an empirical example, and topics for further research are pointed out.

The main purpose of this paper is to introduce and study the behavior of minimum $\mathit{\varphi }$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\phi $$\end{document}-divergence estimators as an alternative to the maximum-likelihood estimator in latent class models for binary items. As it will become clear below, minimum $\mathit{\varphi }$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\phi $$\end{document}-divergence estimators are a natural extension of the maximum-likelihood estimator. The asymptotic properties of minimum $\mathit{\varphi }$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\phi $$\end{document}-divergence estimators for latent class models for binary data are developed. Finally, to compare the efficiency and robustness of these new estimators with that obtained through maximum likelihood when the sample size is not big enough to apply the asymptotic results, we have carried out a simulation study.

We propose a two-way Bayesian vector spatial procedure incorporating dimension reparameterization with a variable selection option to determine the dimensionality and simultaneously identify the significant covariates that help interpret the derived dimensions in the joint space map. We discuss how we solve identifiability problems in a Bayesian context that are associated with the two-way vector spatial model, and demonstrate through a simulation study how our proposed model outperforms a popular benchmark model. In addition, an empirical application dealing with consumers’ ratings of large sport utility vehicles is presented to illustrate the proposed methodology. We are able to obtain interpretable and managerially insightful results from our proposed model with variable selection in comparison with the benchmark model.

The Vale–Maurelli (VM) approach to generating non-normal multivariate data involves the use of Fleishman polynomials applied to an underlying Gaussian random vector. This method has been extensively used in Monte Carlo studies during the last three decades to investigate the finite-sample performance of estimators under non-Gaussian conditions. The validity of conclusions drawn from these studies clearly depends on the range of distributions obtainable with the VM method. We deduce the distribution and the copula for a vector generated by a generalized VM transformation, and show that it is fundamentally linked to the underlying Gaussian distribution and copula. In the process we derive the distribution of the Fleishman polynomial in full generality. While data generated with the VM approach appears to be highly non-normal, its truly multivariate properties are close to the Gaussian case. A Monte Carlo study illustrates that generating data with a different copula than that implied by the VM approach severely weakens the performance of normal-theory based ML estimates.

Many screening tests dichotomize a measurement to classify subjects. Typically a cut-off value is chosen in a way that allows identification of an acceptable number of cases relative to a reference procedure, but does not produce too many false positives at the same time. Thus for the same sample many pairs of sensitivities and false positive rates result as the cut-off is varied. The curve of these points is called the receiver operating characteristic (ROC) curve. One goal of diagnostic meta-analysis is to integrate ROC curves and arrive at a summary ROC (SROC) curve. Holling, Böhning, and Böhning (Psychometrika 77:106–126, 2012a) demonstrated that finite semiparametric mixtures can describe the heterogeneity in a sample of Lehmann ROC curves well; this approach leads to clusters of SROC curves of a particular shape. We extend this work with the help of the $t_{\mathit{\alpha }}$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$t_{\alpha }$$\end{document} transformation, a flexible family of transformations for proportions. A collection of SROC curves is constructed that approximately contains the Lehmann family but in addition allows the modeling of shapes beyond the Lehmann ROC curves. We introduce two rationales for determining the shape from the data. Using the fact that each curve corresponds to a natural univariate measure of diagnostic accuracy, we show how covariate adjusted mixtures lead to a meta-regression on SROC curves. Three worked examples illustrate the method.

We investigate the implications of penalizing incorrect answers to multiple-choice tests, from the perspective of both test-takers and test-makers. To do so, we use a model that combines a well-known item response theory model with prospect theory (Kahneman and Tversky, Prospect theory: An analysis of decision under risk, Econometrica 47:263–91, 1979). Our results reveal that when test-takers are fully informed of the scoring rule, the use of any penalty has detrimental effects for both test-takers (they are always penalized in excess, particularly those who are risk averse and loss averse) and test-makers (the bias of the estimated scores, as well as the variance and skewness of their distribution, increase as a function of the severity of the penalty).

Structural equation models (SEM) are widely used for modeling complex multivariate relationships among measured and latent variables. Although several analytical approaches to interval estimation in SEM have been developed, there lacks a comprehensive review of these methods. We review the popular Wald-type and lesser known likelihood-based methods in linear SEM, emphasizing profile likelihood-based confidence intervals (CIs). Existing algorithms for computing profile likelihood-based CIs are described, including two newer algorithms which are extended to construct profile likelihood-based confidence regions (CRs). Finally, we illustrate the use of these CIs and CRs with two empirical examples, and provide practical recommendations on when to use Wald-type CIs and CRs versus profile likelihood-based CIs and CRs. OpenMx example code is provided in an Online Appendix for constructing profile likelihood-based CIs and CRs for SEM.