Multidimensional unfolding methods suffer from the degeneracy problem in almost all circumstances. Most degeneracies are easily recognized: the solutions are perfect but trivial, characterized by approximately equal distances between points from different sets. A definition of an absolutely degenerate solution is proposed, which makes clear that these solutions only occur when an intercept is present in the transformation function. Many solutions for the degeneracy problem have been proposed and tested, but with little success so far. In this paper, we offer a substantial modification of an approach initiated bythat introduced a normalization factor based on thevariance in the usual least squares loss function. Heiser unpublishedthesis, (1981) and showed that the normalization factor proposed by Kruskal and Carroll was not strong enough to avoid degeneracies. The factor proposed in the present paper, based on the coefficient of variation, discourages or penalizes nonmetric transformations of the proximities with small variation, so that the procedure steers away from solutions with small variation in the interpoint distances. An algorithm is described for minimizing the re-adjusted loss function, based on iterative majorization. The results of a simulation study are discussed, in which the optimal range of the penalty parameters is determined. Two empirical data sets are analyzed by our method, clearly showing the benefits of the proposed loss function.

This paper addresses methodological issues that concern the scaling model used in the international comparison of student attainment in the Programme for International Student Attainment (PISA), specifically with reference to whether PISA’s ranking of countries is confounded by model misfit and differential item functioning (DIF). To determine this, we reanalyzed the publicly accessible data on reading skills from the 2006 PISA survey. We also examined whether the ranking of countries is robust in relation to the errors of the scaling model. This was done by studying invariance across subscales, and by comparing ranks based on the scaling model and ranks based on models where some of the flaws of PISA’s scaling model are taken into account. Our analyses provide strong evidence of misfit of the PISA scaling model and very strong evidence of DIF. These findings do not support the claims that the country rankings reported by PISA are robust.

Starting from an explicit scoring rule for time limit tasks incorporating both response time and accuracy, and a definite trade-off between speed and accuracy, a response model is derived. Since the scoring rule is interpreted as a sufficient statistic, the model belongs to the exponential family. The various marginal and conditional distributions for response accuracy and response time are derived, and it is shown how the model parameters can be estimated. The model for response accuracy is found to be the two-parameter logistic model. It is found that the time limit determines the item discrimination, and this effect is illustrated with the Amsterdam Chess Test II.

Loglinear analysis and correspondence analysis provide us with two different methods for the decomposition of contingency tables. In this paper we will show that there are cases in which these two techniques can be used complementary to each other. More specifically, we will show that often correspondence analysis can be viewed as providing a decomposition of the difference between two matrices, each following a specific loglinear model. Therefore, in these cases the correspondence analysis solution can be interpreted in terms of the difference between these loglinear models. A generalization of correspondence analysis, recently proposed by Escofier, will also be discussed. With this decomposition, which includes classical correspondence analysis as a special case, it is possible to use correspondence analysis complementary to loglinear analysis in more instances than those described for classical correspondence analysis. In this context correspondence analysis is used for the decomposition of the residuals of specific restricted loglinear models.

The validity conditions for univariate repeated measures designs are described. Attention is focused on the sphericity requirement. For a v degree of freedom family of comparisons among the repeated measures, sphericity exists when all contrasts contained in the v dimensional space have equal variances. Under nonsphericity, upper and lower bounds on test size and power of a priori, repeated measures, F tests are derived. The effects of nonsphericity are illustrated by means of a set of charts. The charts reveal that small departures from sphericity (.97 ≤ ε < 1.00) can seriously affect test size and power. It is recommended that separate rather than pooled error term procedures be routinely used to test a priori hypotheses.

The case of adaptive testing under a multidimensional response model with large numbers of constraints on the content of the test is addressed. The items in the test are selected using a shadow test approach. The 0–1 linear programming model that assembles the shadow tests maximizes posterior expected Kullback-Leibler information in the test. The procedure is illustrated for five different cases of multidimensionality. These cases differ in (a) the numbers of ability dimensions that are intentional or should be considered as “nuisance dimensions” and (b) whether the test should or should not display a simple structure with respect to the intentional ability dimensions.

A two-stage procedure is developed for analyzing structural equation models with continuous and polytomous variables. At the first stage, the maximum likelihood estimates of the thresholds, polychoric covariances and variances, and polyserial covariances are simultaneously obtained with the help of an appropriate transformation that significantly simplifies the computation. An asymptotic covariance matrix of the estimates is also computed. At the second stage, the parameters in the structural covariance model are obtained via the generalized least squares approach. Basic statistical properties of the estimates are derived and some illustrative examples and a small simulation study are reported.

Definitions are of course arbitrary. Following Lewis Carroll, one can make words mean what one wishes [cf. Gardner, 1960]. Basically, all that is formally required of a definition is that it be clear: that it enable reliable use of the concept concerned. A more informal, heuristic, desideratum is that it actually influence theorists and researchers to progress in their work.

A strategy for attaining reliability is to make a list of related concepts which might be confused with the one intended, and then to define all the concepts simultaneously in one facet framework. Such a mapping helps to make explicit what the target concept has in common with the others, and how it differs from them. This facet strategy has proved to be useful for distinguishing, for example, among varieties of intelligence [Guttman, 1965; Schlesinger and Guttman, 1969], among varieties of attitude [Guttman, 1959; Elizur, 1970; Jordan, 1971], and even between intelligence and attitude [Guttman, in press]. Evidence is accumulating as to the stimulating effect of the approach for the development of new insights and for the discovery of new forms of lawfulness in data.

Cluster differences scaling is a method for partitioning a set of objects into classes and simultaneously finding a low-dimensional spatial representation of K cluster points, to model a given square table of dissimilarities among n stimuli or objects. The least squares loss function of cluster differences scaling, originally defined only on the residuals of pairs of objects that are allocated to different clusters, is extended with a loss component for pairs that are allocated to the same cluster. It is shown that this extension makes the method equivalent to multidimensional scaling with cluster constraints on the coordinates. A decomposition of the sum of squared dissimilarities into contributions from several sources of variation is described, including the appropriate degrees of freedom for each source. After developing a convergent algorithm for fitting the cluster differences model, it is argued that the individual objects and the cluster locations can be jointly displayed in a configuration obtained as a by-product of the optimization. Finally, the paper introduces a fuzzy version of the loss function, which can be used in a successive approximation strategy for avoiding local minima. A simulation study demonstrates that this strategy significantly outperforms two other well-known initialization strategies, and that it has a success rate of 92 out of 100 in attaining the global minimum.

In this paper, some psychometric models will be presented that belong to the larger class of latent response models (LRMs). First, LRMs are introduced by means of an application in the field of componential item response theory (Embretson, 1980, 1984). Second, a general definition of LRMs (not specific for the psychometric subclass) is given. Third, some more psychometric LRMs, and examples of how they can be applied, are presented. Fourth, a method for obtaining maximum likelihood (ML) and some maximum a posteriori (MAP) estimates of the parameters of LRMs is presented. This method is then applied to the conjunctive Rasch model. Fifth and last, an application of the conjunctive Rasch model is presented. This model was applied to responses to typical verbal ability items (open synonym items).

The purpose of this paper is to introduce a new method for fitting item response theory models with the latent population distribution estimated from the data using splines. A spline-based density estimation system provides a flexible alternative to existing procedures that use a normal distribution, or a different functional form, for the population distribution. A simulation study shows that the new procedure is feasible in practice, and that when the latent distribution is not well approximated as normal, two-parameter logistic (2PL) item parameter estimates and expected a posteriori scores (EAPs) can be improved over what they would be with the normal model. An example with real data compares the new method and the extant empirical histogram approach.

Although several goodness of fit tests have been developed for the Rasch model for dichotomous items, most of them are of a global, asymptotic, and confirmatory type. This paper, based on ideas from a recent thesis by Van den Wollenberg, offers some suggestions for local, small sample, and exploratory techniques: difficulty plots for person groups scoring right and wrong on a specific item, a slope test per item based on a binomial distribution per score group, and a unidimensionality check based on an extended hypergeometric distribution per score group.

New procedures are presented for measuring invariance and matching factors for fixed variables and for fixed or different subjects. Two of these, the coefficient of invariance for factor loadings and the coefficient of factor similarity, utilize factor scores computed from the different sets of factor loadings and one of the original standard score matrices. Another, the coefficient of subject invariance, is obtained by using one of the sets of factor loadings in conjunction with the different standard score matrices. These coefficients are correlations between factor scores of the appropriate matrices. When the best match of factors is desired, rather than degree of resemblance, the method of assignment is proposed.

A model for longitudinal latent structure analysis is proposed. We assume that test scores for a given mental or attitudinal test are observed for the same individuals at two different points in time. The purpose of the analysis is to fit a model that combines the values of the latent variable at the two time points in a two-dimensional latent density. The correlation coefficient between the two values of the latent variable can then be estimated. The theory and methods are illustrated by a Danish dataset concerning psychic vulnerability.

Three alternative estimation procedures for factor analysis based on the instrumental variables method are presented. These procedures are justified by the method of least squares. Formulas for asymptotic standard errors of factor loadings are derived. The procedures are empirically compared to the method of maximum likelihood. The conclusion, based on the data used in this study, is that two of the procedures seem to work well.

A monotone invariant method of hierarchical clustering based on the Mann-Whitney U-statistic is presented. The effectiveness of the complete-link, single-link, and U-statistic methods in recovering tree structures from error perturbed data are evaluated. The U-statistic method is found to be consistently more effective in recovering the original tree structures than either the single-link or complete-link methods.

Some nonparametric dimensionality assessment procedures, such as DIMTEST and DETECT, use nonparametric estimates of item pair conditional covariances given an appropriately chosen subtest score as their basic building blocks. Such conditional covariances given some subtest score can be regarded as an approximation to the conditional covariances given an appropriately chosen unidimensional latent composite, where the composite is oriented in the multidimensional test space direction in which the subtest score measures best. In this paper, the structure and properties of such item pair conditional covariances given a unidimensional latent composite are thoroughly investigated, assuming a semiparametric IRT modeling framework called a generalized compensatory model. It is shown that such conditional covariances are highly informative about the multidimensionality structure of a test. The theory developed here is very useful in establishing properties of dimensionality assessment procedures, current and yet to be developed, that are based upon estimating such conditional covariances.

In particular, the new theory is used to justify the DIMTEST procedure. Because of the importance of conditional covariance estimation, a new bias reducing approach is presented. A byproduct of likely independent importance beyond the study of conditional covariances is a rigorous score information based definition of an item's and a score's direction of best measurement in the multidimensional test space.

Under consideration is a test battery of binary items. The responses of n individuals are assumed to follow a Rasch model. It is further assumed that the latent individual parameters are distributed within a given population in accordance with a normal distribution. Methods are then considered for estimating the mean and variance of this latent population distribution. Also considered are methods for checking whether a normal population distribution fits the data. The developed methods are applied to data from an achievement test and from an attitude test.

It has long been part of the item response theory (IRT) folklore that under the usual empirical Bayes unidimensional IRT modeling approach, the posterior distribution of examinee ability given test response is approximately normal for a long test. Under very general and nonrestrictive nonparametric assumptions, we make this claim rigorous for a broad class of latent models.

A dynamic factor model is proposed for the analysis of multivariate nonstationary time series in the time domain. The nonstationarity in the series is represented by a linear time dependent mean function. This mild form of nonstationarity is often relevant in analyzing socio-economic time series met in practice. Through the use of an extended version of Molenaar's stationary dynamic factor analysis method, the effect of nonstationarity on the latent factor series is incorporated in the dynamic nonstationary factor model (DNFM). It is shown that the estimation of the unknown parameters in this model can be easily carried out by reformulating the DNFM as a covariance structure model and adopting the ML algorithm proposed by Jöreskog. Furthermore, an empirical example is given to demonstrate the usefulness of the proposed DNFM and the analysis.