Two generalizations of the Rasch model are compared: the between-item multidimensional model (Adams, Wilson, and Wang, 1997), and the mixture Rasch model (Mislevy & Verhelst, 1990; Rost, 1990). It is shown that the between-item multidimensional model is formally equivalent with a continuous mixture of Rasch models for which, within each class of the mixture, the item parameters are equal to the item parameters of the multidimensional model up to a shift parameter that is specific for the dimension an item belongs to in the multidimensional model. In a simulation study, the relation between both types of models also holds when the number of classes of the mixture is as small as two. The relation is illustrated with a study on verbal aggression.

Recently, the regression extension of latent class analysis (RLCA) model has received much attention in the field of medical research. The basic RLCA model summarizes shared features of measured multiple indicators as an underlying categorical variable and incorporates the covariate information in modeling both latent class membership and multiple indicators themselves. To reduce complexity and enhance interpretability, one usually fixes the number of classes in a given RLCA. Often, goodness of fit methods comparing various estimated models are used as a criterion to select the number of classes. In this paper, we propose a new method that is based on an analogous method used in factor analysis and does not require repeated fitting. Two ideas with application to many settings other than ours are synthesized in deriving the method: a connection between latent class models and factor analysis, and techniques of covariate marginalization and elimination. A Monte Carlo simulation study is presented to evaluate the behavior of the selection procedure and compare to alternative approaches. Data from a study of how measured visual impairments affect older persons’ functioning are used for illustration.

Although RC(M)-association models have become a generally useful tool for the analysis of cross-classified data, the graphical representation resulting from such an analysis can at times be misleading. The relationships present between row category points and column category points cannot be interpreted by inter point distances but only through projection. In order to avoid incorrect interpretation by a distance rule, joint plots should be made that either represent the row categories or the column categories as vectors. In contrast, the present study proposes models in which the distances between row and column points can be interpreted directly, with a large (small) distance corresponding to a small (large) value for the association. The models provide expressions for the odds ratios in terms of distances, which is a feature that makes the proposed models attractive reparametrizations to the usual RC(M)-parametrization. Comparisons to existing data analysis techniques plus an overview of related models and their connections are also provided.

In this paper, we apply sequential one-sided confidence interval estimation procedures with β-protection to adaptive mastery testing. The procedures of fixed-width and fixed proportional accuracy confidence interval estimation can be viewed as extensions of one-sided confidence interval procedures. It can be shown that the adaptive mastery testing procedure based on a one-sided confidence interval with β-protection is more efficient in terms of test length than a testing procedure based on a two-sided/fixed-width confidence interval. Some simulation studies applying the one-sided confidence interval procedure and its extensions mentioned above to adaptive mastery testing are conducted. For the purpose of comparison, we also have a numerical study of adaptive mastery testing based on Wald's sequential probability ratio test. The comparison of their performances is based on the correct classification probability, averages of test length, as well as the width of the “indifference regions.” From these empirical results, we found that applying the one-sided confidence interval procedure to adaptive mastery testing is very promising.

We make theoretical comparisons among five coefficients—Cronbach’s α, Revelle’s β, McDonald’s ωh, and two alternative conceptualizations of reliability. Though many end users and psychometricians alike may not distinguish among these five coefficients, we demonstrate formally their nonequivalence. Specifically, whereas there are conditions under which α, β, and ωh are equivalent to each other and to one of the two conceptualizations of reliability considered here, we show that equality with this conceptualization of reliability and between α and ωh holds only under a highly restrictive set of conditions and that the conditions under which β equals ωh are only somewhat more general. The nonequivalence of α, β, and ωh suggests that important information about the psychometric properties of a scale may be missing when scale developers and users only report α as is almost always the case.

A new methodology is proposed for the simultaneous reduction of units, variables, and occasions of a three-mode data set. Units are partitioned into a reduced number of classes, while, simultaneously, components for variables and occasions accounting for the largest common information for the classification are identified. The model is a constrained three-mode factor analysis and it can be seen as a generalization of the REDKM model proposed by De Soete and Carroll for two-mode data. The least squares fitting problem is mathematically formalized as a constrained problem in continuous and discrete variables. An iterative alternating least squares algorithm is proposed to give an efficient solution to this minimization problem in the crisp and fuzzy classification context. The performances of the proposed methodology are investigated by a simulation study comparing our model with other competing methodologies. Different procedures for starting the proposed algorithm have also been tested. A discussion of some interesting differences in the results follows. Finally, an application to real data illustrates the ability of the proposed model to provide substantive insights into the data complexities.

This article considers the problem of power and sample size calculations for normal outcomes within the framework of multivariate linear models. The emphasis is placed on the practical situation that not only the values of response variables for each subject are just available after the observations are made, but also the levels of explanatory variables cannot be predetermined before data collection. Using analytic justification, it is shown that the proposed methods extend the existing approaches to accommodate the extra variability and arbitrary configurations of the explanatory variables. The major modification involves the noncentrality parameters associated with the F approximations to the transformations of Wilks likelihood ratio, Pillai trace and Hotelling-Lawley trace statistics. A treatment of multivariate analysis of covariance models is employed to demonstrate the distinct features of the proposed extension. Monte Carlo simulation studies are conducted to assess the accuracy using a child’s intellectual development model. The results update and expand upon current work in the literature.

In the present paper, a new family of item response theory (IRT) models for dichotomous item scores is proposed. Two basic assumptions define the most general model of this family. The first assumption is local independence of the item scores given a unidimensional latent trait. The second assumption is that the odds-ratios for all item-pairs are constant functions of the latent trait. Since the latter assumption is characteristic of the whole family, the models are called constant latent odds-ratios (CLORs) models. One nonparametric special case and three parametric special cases of the general CLORs model are shown to be generalizations of the one-parameter logistic Rasch model. For all CLORs models, the total score (the unweighted sum of the item scores) is shown to be a sufficient statistic for the latent trait. In addition, conditions under the general CLORs model are studied for the investigation of differential item functioning (DIF) by means of the Mantel-Haenszel procedure.

A necessary and sufficient condition is given in this paper for the existence and uniqueness of the maximum likelihood (the so-called joint maximum likelihood) estimate of the parameters of the Partial Credit Model. This condition is stated in terms of a structural property of the pattern of the data matrix that can be easily verified on the basis of a simple iterative procedure. The result is proved by using an argument of Haberman (1977).

The rater agreement literature is complicated by the fact that it must accommodate at east two different properties of rating data: the number of raters (two versus more than two) and the rating scale level (nominal versus metric). While kappa statistics are most widely used for nominal scales, intraclass correlation coefficients have been preferred for metric scales. In this paper, we suggest a dispersion-weighted kappa framework for multiple raters that integrates some important agreement statistics by using familiar dispersion indices as weights for expressing disagreement. These weights are applied to ratings identifying cells in the traditional inter-judge contingency table. Novel agreement statistics can be obtained by applying less familiar indices of dispersion in the same way.

The authors introduce subset conjunction as a classification rule by which an acceptable alternative must satisfy some minimum number of criteria. The rule subsumes conjunctive and disjunctive decision strategies as special cases.

Subset conjunction can be represented in a binary-response model, for example, in a logistic regression, using only main effects or only interaction effects. This results in a confounding of the main and interaction effects when there is little or no response error. With greater response error, a logistic regression, even if it gives a good fit to data, can produce parameter estimates that do not reflect the underlying decision process. The authors propose a model in which the binary classification of alternatives into acceptable/unacceptable categories is based on a probabilistic implementation of a subset-conjunctive process. The satisfaction of decision criteria biases the odds toward one outcome or the other. The authors then describe a two-stage choice model in which a (possibly large) set of alternatives is first reduced using a subset-conjunctive rule, after which an alternative is selected from this reduced set of items. They describe methods for estimating the unobserved consideration probabilities from classification and choice data, and illustrate the use of the models for cancer diagnosis and consumer choice. They report the results of simulations investigating estimation accuracy, incidence of local optima, and model fit.

Human performance in cognitive testing and experimental psychology is expressed in terms of response speed and accuracy. Data analysis is often limited to either speed or accuracy, and/or to crude summary measures like mean response time (RT) or the percentage correct responses. This paper proposes the use of mixed regression for the psychometric modeling of response speed and accuracy in testing and experiments. Mixed logistic regression of response accuracy extends logistic item response theory modeling to multidimensional models with covariates and interactions. Mixed linear regression of response time extends mixed ANOVA to unbalanced designs with covariates and heterogeneity of variance. Related to mixed regression is conditional regression, which requires no normality assumption, but is limited to unidimensional models. Mixed and conditional methods are both applied to an experimental study of mental rotation. Univariate and bivariate analyzes show how within-subject correlation between response and RT can be distinguished from between-subject correlation, and how latent traits can be detected, given careful item design or content analysis. It is concluded that both response and RT must be recorded in cognitive testing, and that mixed regression is a versatile method for analyzing test data.

Data in social and behavioral sciences are often hierarchically organized. Special statistical procedures that take into account the dependence of such observations have been developed. Among procedures for 2-level covariance structure analysis, Muthén’s maximum likelihood (MUML) has the advantage of easier computation and faster convergence. When data are balanced, MUML is equivalent to the maximum likelihood procedure. Simulation results in the literature endorse the MUML procedure also for unbalanced data. This paper studies the analytical properties of the MUML procedure in general. The results indicate that the MUML procedure leads to correct model inference asymptotically when level-2 sample size goes to infinity and the coefficient of variation of the level-1 sample sizes goes to zero. The study clearly identifies the impact of level-1 and level-2 sample sizes on the standard errors and test statistic of the MUML procedure. Analytical results explain previous simulation results and will guide the design or data collection for the future applications of MUML.

Although the Bock–Aitkin likelihood-based estimation method for factor analysis of dichotomous item response data has important advantages over classical analysis of item tetrachoric correlations, a serious limitation of the method is its reliance on fixed-point Gauss-Hermite (G-H) quadrature in the solution of the likelihood equations and likelihood-ratio tests. When the number of latent dimensions is large, computational considerations require that the number of quadrature points per dimension be few. But with large numbers of items, the dispersion of the likelihood, given the response pattern, becomes so small that the likelihood cannot be accurately evaluated with the sparse fixed points in the latent space. In this paper, we demonstrate that substantial improvement in accuracy can be obtained by adapting the quadrature points to the location and dispersion of the likelihood surfaces corresponding to each distinct pattern in the data. In particular, we show that adaptive G-H quadrature, combined with mean and covariance adjustments at each iteration of an EM algorithm, produces an accurate fast-converging solution with as few as two points per dimension. Evaluations of this method with simulated data are shown to yield accurate recovery of the generating factor loadings for models of up to eight dimensions. Unlike an earlier application of adaptive Gibbs sampling to this problem by Meng and Schilling, the simulations also confirm the validity of the present method in calculating likelihood-ratio chi-square statistics for determining the number of factors required in the model. Finally, we apply the method to a sample of real data from a test of teacher qualifications.

The maximum and minimum of a sample from a probability distribution are extremely important random variables in many areas of psychological theory, methodology, and statistics. For instance, the behavior of the mean of the maximum or minimum processing time, as a function of the number of component random processing times (n), has been studied extensively in an effort to identify the underlying processing architecture (e.g., Townsend & Ashby, 1983; Colonius & Vorberg, 1994). Little is known concerning how measures of variability of the maximum or minimum change with n. Here, a new measure of random variability, the quantile spread, is introduced, which possesses sufficient strength to define distributional orderings and derive a number of results concerning variability of the maximum and the minimum statistics. The quantile spread ordering may be useful in many venues. Several interesting open problems are pointed out.

Weekly cycles in emotion were examined by combining item response modeling and spectral analysis approaches in an analysis of 179 college students' reports of daily emotions experienced over 7 weeks. We addressed the measurement of emotion using an item response model. Spectral analysis and multilevel sinusoidal models were used to identify interindividual differences in intraindividual cyclic change. Simulations and incomplete data designs were used to examine how well this combination of analysis techniques might work when applied to other practical data problems. Empirically, we found systematic individual differences in the extent to which individuals' emotions follow a weekly cycle, and in how such cycles are exhibited. Weekly cycles accounted for very little variance in day to day emotions at the individual level. Analytically, we illustrate how measurement, change, and interindividual difference models from different traditions may be combined in a practical manner to describe some of the complexities of human behavior.

Fully and partially ranked data arise in a variety of contexts. From a Bayesian perspective, attention has focused on distance-based models; in particular, the Mallows model and extensions thereof. In this paper, a class of prior distributions, the Binary Tree, is developed on the symmetric group. The attractive features of the class are: it provides a closed-form solution to the posterior distribution; and a simple way to interpret the parameters of the prior distribution. The advantages of the proposed method are illustrated by comparing it to metric-based models using data analyzed by other researchers in this context.

Van Breukelen offers a promising method for modeling both response speed and response accuracy. However, the underlying conception of both dependent measures is somewhat flawed, leading the author to conclude that the approach possesses limitations that, under revised assumptions, may not hold. The central misconception, and a set of related misconceptions, is addressed, and it is suggested that this approach holds a good deal of promise for application in the perceptual and cognitive sciences.

A central theme of research on human development and psychopathology is whether a therapeutic intervention or a turning-point event, such as a family break-up, alters the trajectory of the behavior under study. This paper lays out and applies a method for using observational longitudinal data to make more confident causal inferences about the impact of such events on developmental trajectories. The method draws upon two distinct lines of research: work on the use of finite mixture modeling to analyze developmental trajectories and work on propensity scores. The essence of the method is to use the posterior probabilities of trajectory group membership from a finite mixture modeling framework, to create balance on lagged outcomes and other covariates established prior to t for the purpose of inferring the impact of first-time treatment at t on the outcome of interest. The approach is demonstrated with an analysis of the impact of gang membership on violent delinquency based on data from a large longitudinal study conducted in Montreal.