In this paper, normal/independent distributions, including but not limited to the multivariate t distribution, the multivariate contaminated distribution, and the multivariate slash distribution, are used to develop a robust Bayesian approach for analyzing structural equation models with complete or missing data. In the context of a nonlinear structural equation model with fixed covariates, robust Bayesian methods are developed for estimation and model comparison. Results from simulation studies are reported to reveal the characteristics of estimation. The methods are illustrated by using a real data set obtained from diabetes patients.

The increasing use of diary methods calls for the development of appropriate statistical methods. For the resulting panel data, latent Markov models can be used to model both individual differences and temporal dynamics. The computational burden associated with these models can be overcome by exploiting the conditional independence relations implied by the model. This is done by associating a probabilistic model with a directed acyclic graph, and applying transformations to the graph. The structure of the transformed graph provides a factorization of the joint probability function of the manifest and latent variables, which is the basis of a modified and more efficient E-step of the EM algorithm. The usefulness of the approach is illustrated by estimating a latent Markov model involving a large number of measurement occasions and, subsequently, a hierarchical extension of the latent Markov model that allows for transitions at different levels. Furthermore, logistic regression techniques are used to incorporate restrictions on the conditional probabilities and to account for the effect of covariates. Throughout, models are illustrated with an experience sampling methodology study on the course of emotions among anorectic patients.

A cyclical conditional maximum likelihood estimation procedure is developed for the multidimensional unfolding of two- or three-way dominance data (e.g., preference, choice, consideration) measured on ordered successive category rating scales. The technical description of the proposed model and estimation procedure are discussed, as well as the rather unique joint spaces derived. We then conduct a modest Monte Carlo simulation to demonstrate the parameter recovery of the proposed methodology, as well as investigate the performance of various information heuristics for dimension selection. A consumer psychology application is provided where the spatial results of the proposed model are compared to solutions derived from various traditional multidimensional unfolding procedures. This application deals with consumers intending to buy new luxury sport-utility vehicles (SUVs). Finally, directions for future research are discussed.

It is common practice in IRT to consider items as fixed and persons as random. Both, continuous and categorical person parameters are most often random variables, whereas for items only continuous parameters are used and they are commonly of the fixed type, although exceptions occur. It is shown in the present article that random item parameters make sense theoretically, and that in practice the random item approach is promising to handle several issues, such as the measurement of persons, the explanation of item difficulties, and trouble shooting with respect to DIF. In correspondence with these issues, three parts are included. All three rely on the Rasch model as the simplest model to study, and the same data set is used for all applications. First, it is shown that the Rasch model with fixed persons and random items is an interesting measurement model, both, in theory, and for its goodness of fit. Second, the linear logistic test model with an error term is introduced, so that the explanation of the item difficulties based on the item properties does not need to be perfect. Finally, two more models are presented: the random item profile model (RIP) and the random item mixture model (RIM). In the RIP, DIF is not considered a discrete phenomenon, and when a robust regression approach based on the RIP difficulties is applied, quite good DIF identification results are obtained. In the RIM, no prior anchor sets are defined, but instead a latent DIF class of items is used, so that posterior anchoring is realized (anchoring based on the item mixture). It is shown that both approaches are promising for the identification of DIF.

In order to identify aberrant response-time patterns on educational and psychological tests, it is important to be able to separate the speed at which the test taker operates from the time the items require. A lognormal model for response times with this feature was used to derive a Bayesian procedure for detecting aberrant response times. Besides, a combination of the response-time model with a regular response model in an hierarchical framework was used in an alternative procedure for the detection of aberrant response times, in which collateral information on the test takers’ speed is derived from their response vectors. The procedures are illustrated using a data set for the Graduate Management Admission Test® (GMAT®). In addition, a power study was conducted using simulated cheating behavior on an adaptive test.

Scalability coefficients play an important role in Mokken scale analysis. For a set of items, scalability coefficients have been defined for each pair of items, for each individual item, and for the entire scale. Hypothesis testing with respect to these scalability coefficients has not been fully developed. This study introduces marginal modelling as a framework to derive the standard errors for the scaling coefficients and test hypotheses about these coefficients. Several examples demonstrate the possibilities of marginal modelling in Mokken scale analysis. These possibilities include testing whether Mokken’s criteria for a scale are satisfied, testing whether scalability coefficients of different items are equal, and testing whether scalability coefficients are equal across different groups.

In educational practice, a test assembly problem is formulated as a system of inequalities induced by test specifications. Each solution to the system is a test, represented by a 0–1 vector, where each element corresponds to an item included (1) or not included (0) into the test. Therefore, the size of a 0–1 vector equals the number of items n in a given item pool. All solutions form a feasible set—a subset of 2n vertices of the unit cube in an n-dimensional vector space. Test assembly is uniform if each test from the feasible set has an equal probability of being assembled. This paper demonstrates several important applications of uniform test assembly for educational practice. Based on Slepian’s inequality, a binary program was analytically studied as a candidate for uniform test assembly. The results of this study establish a connection between combinatorial optimization and probability inequalities. They identify combinatorial properties of the feasible set that control the uniformity of the binary programming test assembly. Computer experiments illustrating the concepts of this paper are presented.

Hierarchical classes models are models for N-way N-mode data that represent the association among the N modes and simultaneously yield, for each mode, a hierarchical classification of its elements. In this paper we present a stochastic extension of the hierarchical classes model for two-way two-mode binary data. In line with the original model, the new probabilistic extension still represents both the association among the two modes and the hierarchical classifications. A fully Bayesian method for fitting the new model is presented and evaluated in a simulation study. Furthermore, we propose tools for model selection and model checking based on Bayes factors and posterior predictive checks. We illustrate the advantages of the new approach with applications in the domain of the psychology of choice and psychiatric diagnosis.

According to the recent Nation’s Report Card, 12th-graders failed to produce gains on the 2005 National Assessment of Educational Progress (NAEP) despite earning better grades on average. One possible explanation is that 12th-graders were not motivated taking the NAEP, which is a low-stakes test. We develop three Bayesian IRT mixture models to describe the results from a group of examinees including both nonguessers and partial guessers. The first assumes that the guesser answers questions based on his or her knowledge up to a certain test item, and guesses thereafter. The second model assumes that the guesser answers relatively easy questions based on his or her knowledge and guesses randomly on the remaining items. The third is constructed to describe more general low-motivation behavior. It assumes that the guesser gives less and less effort as he or she proceeds through the test. The models can provide not only consistent estimates of IRT parameters but also estimates of each examinee’s nonguesser/guesser status and degree of guessing behavior. We show results of a simulation study comparing the performance of the three guessing models to the 2PL-IRT model. Finally, an analysis of real data from a low-stakes test administered to university students is presented.

This paper introduces the generalized logit-linear item response model (GLLIRM), which represents the item-solving process as a series of dichotomous operations or steps. The GLLIRM assumes that the probability function of the item response is a logistic function of a linear composite of basic parameters which describe the operations, and the coefficients depend on three design matrices X, Y and Z. The GLLIRM provides a tool for testing hypotheses on the item-solving process and generalizes existing models. An empirical application is included, in which the model is applied to evaluate sources of difficulty and pairwise item interactions in a logical analysis test.

Samejima (Psychometrika 65:319–335, 2000) proposed the logistic positive exponent family of models (LPEF) for dichotomous responses in the unidimensional latent space. The objective of the present paper is to propose and discuss a graded response model that is expanded from the LPEF, in the context of item response theory (IRT). This specific graded response model belongs to the general framework of graded response model (Samejima, Psychometrika Monograph, No. 17, 1969 and No. 18, 1972; Handbook of modern item response theory, Springer, New York, 1997; Encyclopedia of Social Measurement, Academic Press, San Diego, 2004), and, in particular to the heterogeneous case (Samejima, Psychometrika Monograph, No. 18, 1972). Thus, the model can deal with any number of ordered polytomous responses, such as letter grades (e.g., A, B, C, D, F), etc.

For brevity, hereafter, the model will be called the LPEF graded response model, or LPEFG. This model reflects the opposing two principles contained in the LPEF for dichotomous responses, with the logistic model (Birnbaum, Statistical theories of mental test scores, Addison Wesley, Reading, 1968) as their transition, which provide a reasonable rationale for partial credits in LPEFG, among others.

Most inter-rater reliability studies using nominal scales suggest the existence of two populations of inference: the population of subjects (collection of objects or persons to be rated) and that of raters. Consequently, the sampling variance of the inter-rater reliability coefficient can be seen as a result of the combined effect of the sampling of subjects and raters. However, all inter-rater reliability variance estimators proposed in the literature only account for the subject sampling variability, ignoring the extra sampling variance due to the sampling of raters, even though the latter may be the biggest of the variance components. Such variance estimators make statistical inference possible only to the subject universe. This paper proposes variance estimators that will make it possible to infer to both universes of subjects and raters. The consistency of these variance estimators is proved as well as their validity for confidence interval construction. These results are applicable only to fully crossed designs where each rater must rate each subject. A small Monte Carlo simulation study is presented to demonstrate the accuracy of large-sample approximations on reasonably small samples.

An item response theory model for dealing with test speededness is proposed. The model consists of two random processes, a problem solving process and a random guessing process, with the random guessing gradually taking over from the problem solving process. The involved change point and change rate are considered random parameters in order to model examinee differences in both respects. The proposed model is evaluated on simulated data and in a case study.

In this paper implicit function-based parameterizations for orthogonal and oblique rotation matrices are proposed. The parameterizations are used to construct Newton algorithms for minimizing differentiable rotation criteria applied to m factors and p variables. The speed of the new algorithms is compared to that of existing algorithms and to that of Newton algorithms based on alternative parameterizations. Several rotation criteria were examined and the algorithms were evaluated over a range of values for m. Initial guesses for Newton algorithms were improved by subconvergence iterations of the gradient projection algorithm. Simulation results suggest that no one algorithm is fastest for minimizing all criteria for all values of m. Among competing algorithms, the gradient projection algorithm alone was faster than the implicit function algorithm for minimizing a quartic criterion over oblique rotation matrices when m is large. In all other conditions, however, the implicit function algorithms were competitive with or faster than the fastest existing algorithms. The new algorithms showed the greatest advantage over other algorithms when minimizing a nonquartic component loss criterion.

The infinitesimal jackknife provides a simple general method for estimating standard errors in covariance structure analysis. Beyond its simplicity and generality what makes the infinitesimal jackknife method attractive is that essentially no assumptions are required to produce consistent standard error estimates, not even the requirement that the population sampled has the covariance structure assumed. Commonly used covariance structure analysis software uses parametric methods for estimating parameters and standard errors. When the population sampled has the covariance structure assumed, but fails to have the distributional form assumed, the parameter estimates usually remain consistent, but the standard error estimates do not. This has motivated the introduction of a variety of nonparametric standard error estimates that are consistent when the population sampled fails to have the distributional form assumed. The only distributional assumption these require is that the covariance structure be correctly specified. As noted, even this assumption is not required for the infinitesimal jackknife. The relation between the infinitesimal jackknife and other nonparametric standard error estimators is discussed. An advantage of the infinitesimal jackknife over the jackknife and the bootstrap is that it requires only one analysis to produce standard error estimates rather than one for every jackknife or bootstrap sample.

This paper studies three models for cognitive diagnosis, each illustrated with an application to fraction subtraction data. The objective of each of these models is to classify examinees according to their mastery of skills assumed to be required for fraction subtraction. We consider the DINA model, the NIDA model, and a new model that extends the DINA model to allow for multiple strategies of problem solving. For each of these models the joint distribution of the indicators of skill mastery is modeled using a single continuous higher-order latent trait, to explain the dependence in the mastery of distinct skills. This approach stems from viewing the skills as the specific states of knowledge required for exam performance, and viewing these skills as arising from a broadly defined latent trait resembling the θ of item response models. We discuss several techniques for comparing models and assessing goodness of fit. We then implement these methods using the fraction subtraction data with the aim of selecting the best of the three models for this application. We employ Markov chain Monte Carlo algorithms to fit the models, and we present simulation results to examine the performance of these algorithms.

Conjoint choice experiments are used widely in marketing to study consumer preferences amongst alternative products. We develop a class of choice models, belonging to the class of Poisson race models, that describe a ‘random utility’ which lends itself to a process-based description of choice. The models incorporate a dependence structure which captures the relationship between the attributes of the choice alternatives and which appropriately moderates the randomness inherent in the race. The new models are applied to conjoint choice data and are shown to have performance markedly superior to that of independent Poisson race models and of the multinomial logit model.

Although the K-means algorithm for minimizing the within-cluster sums of squared deviations from cluster centroids is perhaps the most common method for applied cluster analyses, a variety of other criteria are available. The p-median model is an especially well-studied clustering problem that requires the selection of p objects to serve as cluster centers. The objective is to choose the cluster centers such that the sum of the Euclidean distances (or some other dissimilarity measure) of objects assigned to each center is minimized. Using 12 data sets from the literature, we demonstrate that a three-stage procedure consisting of a greedy heuristic, Lagrangian relaxation, and a branch-and-bound algorithm can produce globally optimal solutions for p-median problems of nontrivial size (several hundred objects, five or more variables, and up to 10 clusters). We also report the results of an application of the p-median model to an empirical data set from the telecommunications industry.

The CANDECOMP/PARAFAC (CP) model decomposes a three-way array into a prespecified number of R factors and a residual array by minimizing the sum of squares of the latter. It is well known that an optimal solution for CP need not exist. We show that if an optimal CP solution does not exist, then any sequence of CP factors monotonically decreasing the CP criterion value to its infimum will exhibit the features of a so-called “degeneracy”. That is, the parameter matrices become nearly rank deficient and the Euclidean norm of some factors tends to infinity. We also show that the CP criterion function does attain its infimum if one of the parameter matrices is constrained to be column-wise orthonormal.