Exact person fit indexes for the rasch model for arbitrary alternatives

A new algorithm for obtaining exact person fit indexes for the Rasch model is introduced which realizes most powerful tests for a very general family of alternative hypotheses, including tests concerning DIF as well as model-deviating item correlations. The method is also used as a goodness-of-fit test for whole data sets where the item parameters are assumed to be known. For tests with 30 items at most, exact values are obtained, for longer tests a Monte Carlo-algorithm is proposed. Simulated examples and an empirical investigation demonstrate test power and applicability to item elimination.The author wishes to thank Elisabeth Ponocny-Seliger and the reviewers for many helpful comments. All exact goodness-of-fit tests proposed in this article are implemented in the menu-driven program T-Rasch 1.0 by Ponocny and Ponocny-Seliger (1999) which can be obtained from ProGAMMA (WWW: http://www.gamma.rug.nl) and also performs nonparametric tests.     

An exact and optimal standardized person test for assessing consistency with the rasch model

The Rasch model predicts that an individual's ability level is invariant over subtests of the total test, and thus, all subtests measure the same latent trait. A person test of this invariance hypothesis is discussed that is uniformly most powerful and standardized in the sense that the conditional distribution of the test statistic, given a particular level of ability, does not depend on the absolute value of the examinee's ability parameter. The test can be routinely performed by applying a computer program designed by and obtainable from the author. Finally, a suboptimal test is derived that is extremely easy to use, and an overall group test of the invariance hypothesis discussed. All tests considered do not rely on asymptotic approximations; hence, they may be applied when the test is of only moderate length and the group of examinees is small.     

The derivation of some tests for the rasch model from the multinomial distribution

The present paper is concerned with testing the fit of the Rasch model. It is shown that this can be achieved by constructing functions of the data, on which model tests can be based that have power against specific model violations. It is shown that the asymptotic distribution of these tests can be derived by using the theoretical framework of testing model fit in general multinomial and product-multinomial models. The model tests are presented in two versions: one that can be used in the context of marginal maximum likelihood estimation and one that can be applied in the context of conditional maximum likelihood estimation.I am indebted to Norman Verhelst and Niels Veldhuijzen for their helpful comments. Requests for reprints should be sent to Cees A. W. Glas, Cito, PO Box 1034, 6801 MG Arnhem, THE NETHERLANDS.     

Conditions for rasch-dichotomizability of the unidimensional polytomous rasch model

Jansen and Roskam (1986) discussed the compatibility of the unidimensional polytomous Rasch model with dichotomization of the response continuum. They derived a rather strict condition in which dichotomization of multicategory data that fit the unidimensional polytomous Rasch model, results in dichotomous data which fit the dichotomous Research model with effectively the same subject parameter. In this paper a more general dichotomization condition is derived for the polytomous Rasch model, which appears less restrictive, but upholds that the intrinsic logic of the unidimensional polytomous Rasch model defies dichotomization in general. The robustness of dichotomous analysis investigated in a simulation study. It shows a close relation with the two-parameters (Birnbaum) model. Theoretical and methodological implications are discussed.The authors are indebted to H. Müller (personal communication, August 1986), for giving an example which pointed toward the core equation in this paper. The authors also acknowledge the critical comments of Th. Bezambinder and P. Wakker, and of Psychometrika's reviewers to an earlier version of this paper.     

Higher-order approximations to the distributions of fit indexes under fixed alternatives in structural equation models

Higher-order approximations to the distributions of fit indexes for structural equation models under fixed alternative hypotheses are obtained in nonnormal samples as well as normal ones. The fit indexes include the normal-theory likelihood ratio chi-square statistic for a posited model, the corresponding statistic for the baseline model of uncorrelated observed variables, and various fit indexes as functions of these two statistics. The approximations are given by the Edgeworth expansions for the distributions of the fit indexes under arbitrary distributions. Numerical examples in normal and nonnormal samples with the asymptotic and simulated distributions of the fit indexes show the relative inappropriateness of the normal-theory approximation using noncentral chi-square distributions. A simulation for the confidence intervals of the fit indexes based on the normal-theory Studentized estimators under normality with a small sample size indicates an advantage for the approximation by the Cornish–Fisher expansion over those by the noncentral chi-square distribution and the asymptotic normality. The author is indebted to the reviewers for their comments and suggestions, which have led to the improvement of the previous versions of this paper. This work was partially supported by Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science, and Technology.     

The equivalence of two methods of parameter estimation for the rasch model

Two methods of estimating parameters in the Rasch model are compared. It is shown that estimates for a certain loglinear model for the score × item × response table are equivalent to the unconditional maximum likelihood estimates for the Rasch model.     

An extension of the rasch model for ratings providing both location and dispersion parameters

An elaboration of a psychometric model for rated data, which belongs to the class of Rasch models, is shown to provide a model with two parameters, one characterising location and one characterising dispersion. The later parameter, derived from the idea of a unit of scale, is also shown to reflect the shape of rating distributions, ranging from unimodal, through uniform, and then to U-shaped distributions. A brief case is made that when a rating distribution is treated as a random error distribution, then the distribution should be unimodal.     

On the robustness of maximum likelihood scaling for violations of the error model

A Monte Carlo study was conducted to investigate the robustness of the assumed error distribution in maximum likelihood estimation models for multidimensional scaling. Data sets generated according to the lognormal, the normal, and the rectangular distribution were analysed with the log-normal error model in Ramsay's MULTISCALE program package. The results show that violations of the assumed error distribution have virtually no effect on the estimated distance parameters. In a comparison among several dimensionality tests, the corrected version of thex ² test, as proposed by Ramsay, yielded the best results, and turned out to be quite robust against violations of the error model.     

Optimal Partitioning of a Data Set Based on the p-Median 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.     

Old Foundations for the Logic of Agency and Action

The paper presents an infinite hierarchy of sound and complete axiomatic systems for Two-Dimensional Modal Tense Logic with Historical Necessity, Agents and Acts. A main novelty of these logics is their capacity to represent formally (i) basic action-sentences asserting that such and such an act is performed/omitted by an agent, as well as (ii) causative action-sentences asserting that by performing/omitting a certain act, an agent causes that such and such a state-of-affairs is realized (e.g. comes about/ceases/remains/remains absent). We illustrate how the formal machinery of our systems can be used to reconstruct a number of interesting ideas in the Logic of Agency and Action that have been proposed by authors like von Wright, von Kutschera, Belnap and Segerberg.     

Tests for linear trend in the smallest eigenvalues of the correlation matrix

A test for linear trend among a set of eigenvalues of a correlation matrix is developed. As a technical implementation of Cattell's scree test, this is a generalization of Anderson's test for the equality of eigenvalues, and extends Bentler and Yuan's work on linear trends in eigenvalues of a covariance matrix. The power of minimumx ² and maximum likelihood ratio tests are compared. Examples show that the linear trend hypothesis is more realistic than the standard hypothesis of equality of eigenvalues, and that the hypothesis is compatible with standard decisions on the number of factors or components to retain in data analysis.This work was supported by National Institute on Drug Abuse Grants DA01070 and DA00017. The assistance of Maia Berkane and several anonymous reviewers is gratefully acknowledged.     

Genetic and environmental influences on the Verbal-Perceptual-Image Rotation (VPR) model of the structure of mental abilities in the Minnesota study of twins reared apart

In previous papers [Johnson, W., & Bouchard Jr., T. J. (2005a). Constructive Replication of the Visual-Perceptual-Image Rotation (VPR) Model in Thurstone's (1941) Battery of 60 Tests of Mental Ability. Intelligence, 33, 417–430.] [Johnson, W., & Bouchard Jr., T. J. (2005b). The Structure of Human Intelligence: It's Verbal, perceptual, and image rotation (VPR), not Fluid and Crystallized. Intelligence, 33, 393–416.] we have proposed the Verbal, perceptual, and image rotation (VPR) model of the structure of mental abilities. The VPR model is hierarchical, with a g factor that contributes strongly to broad verbal, perceptual, and image rotation abilities, which in turn contribute to 8 more specialized abilities. The verbal and perceptual abilities, though separable, are highly correlated, as are the perceptual and mental rotation abilities. The verbal and mental rotation abilities are much less correlated. In this study we used the twin sample in the Minnesota Study of Twins Reared Apart to estimate the genetic and environmental influences and the correlations among them at each order of the VPR model. Genetic influences accounted for 67–79% of the variance throughout the model, with the exception of the second-stratum Content Memory factor, which showed 33% genetic influence. These influences could not be attributed to assessed similarity of rearing environment. Genetic correlations closely mirrored the phenotypic correlations. Together, these findings substantiate the theory that the entire structure of mental abilities is strongly influenced by genes.     

A fast numerical algorithm for the estimation of diffusion model parameters

In this paper, we describe a new algorithmic approach for parameter estimation in Ratcliff's [(1978). A theory of memory retrieval. Psychological Review, 85 (2), 59-108] diffusion model. This problem, especially if inter-trial variabilities of parameters are included in the model, is computationally very expensive; the parameter estimation procedure often takes a long time even with today's high-speed computers. The algorithm described here makes the calculation of the cumulative distribution functions for predicted process durations computationally much less expensive. This improvement is achieved by solving the Kolmogorov backward equation numerically instead of employing the previously used closed form solution. Additionally, the algorithm can determine the optimum fit for one of the model parameters (the starting point z) directly, thereby reducing the dimension of the parameter search space by one. The resulting method is shown to be notably faster than the standard (closed-form solution) method for parameter estimation.     

From the Universe of Knowledge to the Universe of Concepts: The Structural Revolution in Classification for Information Retrieval

During the twentieth century, bibliographic classification theory underwent a structural revolution. The first modern bibliographic classifications were top-down systems that started at the universe of knowledge and subdivided that universe downward to minute subclasses. After the invention of faceted classification by S.R. Ranganathan, the ideal was to build bottom-up classifications that started with the universe of concepts and built upward to larger and larger faceted classes. This ideal has not been achieved, and the two kinds of classification systems are not mutually exclusive. This paper examines the process by which this structural revolution was accomplished by looking at the spread of facet theory after 1924 when Ranganathan attended the School of Librarianship, London, through selected classification textbooks that were published after that date. To this end, the paper examines the role of W.C.B. Sayers as a teacher and author of three editions of The Manual of Classification for Librarians and Bibliographers. Sayers influenced both Ranganathan and the various members of the Classification Research Group (CRG) who were his students. Further, the paper contrasts the methods of evaluating classification systems that arose between Sayers’s Canons of Classification in 1915–1916 and J. Mills’s A Modern Outline of Library Classification in 1960 in order to demonstrate the speed with which one kind of classificatory structure was overtaken by another.

After Kohlberg: Some implications of an Ethics of Virtue for the theory of moral education and development

It is beyond serious dispute that post-war reflection upon and research into moral education and development has been well nigh dominated by an extensive and ambitious research programme influenced and initiated by the modem cognitive developmental theorist Lawrence Kohlberg — a programme which can also be seen, moreover, as standing in a tradition of philosophical reflection about the nature of moral life going back to such significant enlightenment thinkers as Kant and Rousseau. It will also be familiar, however, that a powerful critique of this essentially liberal conception of the nature of moral life and values has lately gathered momentum under the influence of contemporary post-analytical and communitarian social and moral theorists variously under the spell of Aristotle. In the first place, then, this paper argues that a basically Kohlbergian approach to thinking about moral education is difficult — if not impossible — to sustain in the face of this neo-Aristotelian critique; secondly, however, it attempts to explore the possibilities of an alternative virtue-theoretical basis for understanding the nature of moral life and education.     

Searching for the Roots of the Circumstantial Ad Hominem

This paper looks into the known evidence on the origins of the type of argument called the circumstantial ad hominemargument in modern logic textbooks, and introduces some new evidence. This new evidence comes primarily from recent historical work by Jaap Mansfeld and Jonathan Barnes citing many cases where philosophers in the ancient world were attacked on the grounds that their personal actions failed to be consistent with their philosophical teachings. On the total body of evidence, two hypotheses about the roots of the circumstantial ad hominem are considered. One is that it came from Aristotle through Locke. The other is that it may have had separate roots in these ancient philosophical writings that criticized philosophers for not practicing what they preached.     

A study of the spin-Hamiltonian parameters for Cr5+ at the rhombically-distorted tetrahedral P5+ site of Ca2PO4Cl crystal using a two-mechanism model

The complete high-order perturbation formulae of spin-Hamiltonian (SH) parameters (g factors g_i and hyperfine structure constants A_i , where i = x, y, z) containing contributions from both the crystal-field (CF) and charge-transfer (CT) mechanisms (the latter mechanism is neglected in the widely-used CF theory) are established for d¹ ions in rhombic tetrahedra. From these formulae, the SH parameters of Cr⁵⁺ ion at the rhombically-distorted tetrahedral P⁵⁺ site of Ca₂PO₄Cl crystal are calculated. The CF and CT energy levels used in the calculation are obtained from the optical spectra of the studied Ca₂PO₄Cl : Cr⁵⁺ crystal. The calculated results showed reasonable agreement with the experimental values. The signs of the hyperfine structure constants A_i and the relative importance of the CT mechanism to SH parameters are acquired from the calculations.