The achievement level is a variable measured with error, that can be estimated by means of the Rasch model. Teacher grades also measure the achievement level but they are expressed on a different scale. This paper proposes a method for combining these two scores to obtain a synthetic measure of the achievement level based on the theory developed for regression with covariate measurement error. In particular, the focus is on ordinal scaled grades, using the SIMEX method for measurement error correction. The result is a measure comparable across subjects with smaller measurement error variance. An empirical application illustrates the method.

Finite mixture models are widely used in the analysis of growth trajectory data to discover subgroups of individuals exhibiting similar patterns of behavior over time. In practice, trajectories are usually modeled as polynomials, which may fail to capture important features of the longitudinal pattern. Focusing on dichotomous response measures, we propose a likelihood penalization approach for parameter estimation that is able to capture a variety of nonlinear class mean trajectory shapes with higher precision than maximum likelihood estimates. We show how parameter estimation and inference for whether trajectories are time-invariant, linear time-varying, or nonlinear time-varying can be carried out for such models. To illustrate the method, we use simulation studies and data from a long-term longitudinal study of children at high risk for substance abuse.

In psychological research, one often aims at explaining individual differences in S-R profiles, that is, individual differences in the responses (R) with which people react to specific stimuli (S). To this end, researchers often postulate an underlying sequential process, which boils down to the specification of a set of mediating variables (M) and the processes that link these mediating variables to the stimuli and responses under study. Obviously, a crucial task is to chart how the individual differences in the S-R profiles are caused by individual differences in the S-M link and/or by individual differences in the M-R link. In this paper we propose a new model, called CLASSI, which was explicitly designed for this task. In particular, the key principle of CLASSI consists of reducing the S, M, and R nodes of a sequential process to a few mutually exclusive types and inducing an S-M and an M-R person typology from the data, with the S-M person types being characterized in terms of if S type then M type rules and the M-R person types in terms of if M type then R type rules. As such, the S-M and M-R person types and their associated if–then rules represent the important individual differences in the S-M and M-R links of the sequential process under study. An algorithm to fit the CLASSI model is described and evaluated in a simulation study. An application of CLASSI to data from the behavioral domain of anger and sadness is discussed. Finally, we relate CLASSI to other methods and discuss possible extensions.

It has been widely reported that in computerized adaptive testing some examinees may get much lower scores than they would normally if an alternative paper-and-pencil version were given. The main purpose of this investigation is to quantitatively reveal the cause for the underestimation phenomenon. The logistic models, including the 1PL, 2PL, and 3PL models, are used to demonstrate our assertions. Our analytical derivation shows that, under the maximum information item selection strategy, if an examinee failed a few items at the beginning of the test, easy but more discriminating items are likely to be administered. Such items are ineffective to move the estimate close to the true θ, unless the test is sufficiently long or a variable-length test is used. Our results also indicate that a certain weighting mechanism is necessary to make the algorithm rely less on the items administered at the beginning of the test.

The use of Candecomp to fit scalar products in the context of INDSCAL is based on the assumption that the symmetry of the data matrices involved causes the component matrices to be equal when Candecomp converges. Ten Berge and Kiers gave examples where this assumption is violated for Gramian data matrices. These examples are believed to be local minima. It is now shown that, in the single-component case, the assumption can only be violated at saddle points. Chances of Candecomp converging to a saddle point are small but still nonzero.

Multidimensional scaling (MDS) models for the analysis of dominance data have been developed in the psychometric and classification literature to simultaneously capture subjects’ preference heterogeneity and the underlying dimensional structure for a set of designated stimuli in a parsimonious manner. There are two major types of latent utility models for such MDS models that have been traditionally used to represent subjects’ underlying utility functions: the scalar product or vector model and the ideal point or unfolding model. Although both models have been widely applied in various social science applications, implicit in the assumption of such MDS methods is that all subjects are homogeneous with respect to their underlying utility function; i.e., they all follow a vector model or an ideal point model. We extend these traditional approaches by presenting a Bayesian MDS model that combines both the vector model and the ideal point model in a generalized framework for modeling metric dominance data. This new Bayesian MDS methodology explicitly allows for mixtures of the vector and the ideal point models thereby accounting for both preference heterogeneity and structural heterogeneity. We use a marketing application regarding physicians’ prescription behavior of antidepressant drugs to estimate and compare a variety of spatial models.

This paper proposes an order-constrained K-means cluster analysis strategy, and implements that strategy through an auxiliary quadratic assignment optimization heuristic that identifies an initial object order. A subsequent dynamic programming recursion is applied to optimally subdivide the object set subject to the order constraint. We show that although the usual K-means sum-of-squared-error criterion is not guaranteed to be minimal, a true underlying cluster structure may be more accurately recovered. Also, substantive interpretability seems generally improved when constrained solutions are considered. We illustrate the procedure with several data sets from the literature.

Eight different variable selection techniques for model-based and non-model-based clustering are evaluated across a wide range of cluster structures. It is shown that several methods have difficulties when non-informative variables (i.e., random noise) are included in the model. Furthermore, the distribution of the random noise greatly impacts the performance of nearly all of the variable selection procedures. Overall, a variable selection technique based on a variance-to-range weighting procedure coupled with the largest decreases in within-cluster sums of squares error performed the best. On the other hand, variable selection methods used in conjunction with finite mixture models performed the worst.

This paper analyzes the sum score based (SSB) formulation of the Rasch model, where items and sum scores of persons are considered as factors in a logit model. After reviewing the evolution leading to the equality between their maximum likelihood estimates, the SSB model is then discussed from the point of view of pseudo-likelihood and of misspecified models. This is then employed to provide new insights into the origin of the known inconsistency of the difficulty parameter estimates in the Rasch model. The main results consist of exact relationships between the estimated standard errors for both models; and, for the ability parameters, an upper bound for the estimated standard errors of the Rasch model in terms of those for the SSB model, which are more easily available.

The aim of this paper is to study the analysis of contingency tables with one heavyweight column or one heavyweight entry by taxicab correspondence analysis (TCA). Given that the mathematics of TCA is simpler than the mathematics of correspondence analysis (CA), the influence of one heavyweight column on the outputs of TCA is studied explicitly without recourse to asymptotics as done by Benzécri (Les Cahiers de L’Analyse des Données 4:413–16, 1979). A reweighting of the heavyweight column is proposed, which can also be applied to CA. A real data set is analyzed.

After a brief review of recent advances in sequential analysis involving sequential generalized likelihood ratio tests, we discuss their use in psychometric testing and extend the asymptotic optimality theory of these sequential tests to the case of sequentially generated experiments, of particular interest in computerized adaptive testing. We then show how these methods can be used to design adaptive mastery tests, which are asymptotically optimal and are also shown to provide substantial improvements over currently used sequential and fixed length tests.

The association structure between manifest variables arising from the single-factor model is investigated using partial correlations. The additional insights to the practitioner provided by partial correlations for detecting a single-factor model are discussed. The parameter space for the partial correlations is presented, as are the patterns of signs in a matrix containing the partial correlations that are not compatible with a single-factor model.

In a recent paper, Hessen (Psychometrika 70(3):497–516, 2005) introduces the class of constant latent odds-ratios models as an extension of the Rasch model for which the sum score is still the sufficient statistic for ability. In this paper the relation between both the general and the general parametric constant latent odds-ratios model and the Rasch model is considered.

Methods of incorporating a ridge type of regularization into partial redundancy analysis (PRA), constrained redundancy analysis (CRA), and partial and constrained redundancy analysis (PCRA) were discussed. The usefulness of ridge estimation in reducing mean square error (MSE) has been recognized in multiple regression analysis for some time, especially when predictor variables are nearly collinear, and the ordinary least squares estimator is poorly determined. The ridge estimation method was extended to PRA, CRA, and PCRA, where the reduced rank ridge estimates of regression coefficients were obtained by minimizing the ridge least squares criterion. It was shown that in all cases they could be obtained in closed form for a fixed value of ridge parameter. An optimal value of the ridge parameter is found by G-fold cross validation. Illustrative examples were given to demonstrate the usefulness of the method in practical data analysis situations.

Recent advances in data recording technology have given researchers new ways of collecting on-line and continuous data for analyzing input-output systems. For example, continuous response digital interfaces are increasingly used in psychophysics. The statistical problem related to these input-output systems reduces to linking time-varying covariates to a continuous response variate. Using real-time data obtained from an experiment in psychoacoustics, we showcase new statistical tools that incorporate dynamical elements of an input-output system. We employ functional data analysis (FDA) methods and a simple differential equation to analyze and model the continuous responses. Furthermore, we outline the issues involved in analyzing input-output systems when the exact form of the underlying mathematical model is not known. Finally, we develop a calibration method to facilitate inter-subject and intra-subject comparisons.

This paper studies correction for chance in coefficients that are linear functions of the observed proportion of agreement. The paper unifies and extends various results on correction for chance in the literature. A specific class of coefficients is used to illustrate the results derived in this paper. Coefficients in this class, e.g. the simple matching coefficient and the Dice/Sørenson coefficient, become equivalent after correction for chance, irrespective of what expectation is used. The coefficients become either Cohen’s kappa, Scott’s pi, Mak’s rho, Goodman and Kruskal’s lambda, or Hamann’s eta, depending on what expectation is considered appropriate. Both a multicategorical generalization and a multivariate generalization are discussed.

Every set of alternate weights (i.e., nonleast squares weights) in a multiple regression analysis with three or more predictors is associated with an infinite class of weights. All members of a given class can be deemed fungible because they yield identical SSE (sum of squared errors) and R2 values. Equations for generating fungible weights are reviewed and an example is given that illustrates how fungible weights can be profitably used to evaluate parameter sensitivity in multiple regression.

Dynamic programming methods for matrix permutation problems in combinatorial data analysis can produce globally-optimal solutions for matrices up to size 30×30, but are computationally infeasible for larger matrices because of enormous computer memory requirements. Branch-and-bound methods also guarantee globally-optimal solutions, but computation time considerations generally limit their applicability to matrix sizes no greater than 35×35. Accordingly, a variety of heuristic methods have been proposed for larger matrices, including iterative quadratic assignment, tabu search, simulated annealing, and variable neighborhood search. Although these heuristics can produce exceptional results, they are prone to converge to local optima where the permutation is difficult to dislodge via traditional neighborhood moves (e.g., pairwise interchanges, object-block relocations, object-block reversals, etc.). We show that a heuristic implementation of dynamic programming yields an efficient procedure for escaping local optima. Specifically, we propose applying dynamic programming to reasonably-sized subsequences of consecutive objects in the locally-optimal permutation, identified by simulated annealing, to further improve the value of the objective function. Experimental results are provided for three classic matrix permutation problems in the combinatorial data analysis literature: (a) maximizing a dominance index for an asymmetric proximity matrix; (b) least-squares unidimensional scaling of a symmetric dissimilarity matrix; and (c) approximating an anti-Robinson structure for a symmetric dissimilarity matrix.