bluuebunny/crossref_metadata_2025 huggingface dataset:
A Study of Algorithms for Covariance Structure Analysis with Specific Comparisons using Factor Analysis
Authors: Lee S. Y. , Jennrich R. I.
Published: January 2006

Several algorithms for covariance structure analysis are considered in addition to the Fletcher-Powell algorithm. These include the Gauss-Newton, Newton-Raphson, Fisher Scoring, and Fletcher-Reeves algorithms. Two methods of estimation are considered, maximum likelihood and weighted least squares. It is shown that the Gauss-Newton algorithm which in standard form produces weighted least squares estimates can, in iteratively reweighted form, produce maximum likelihood estimates as well. Previously unavailable standard error estimates to be used in conjunction with the Fletcher-Reeves algorithm are derived. Finally all the algorithms are applied to a number of maximum likelihood and weighted least squares factor analysis problems to compare the estimates and the standard errors produced. The algorithms appear to give satisfactory estimates but there are serious discrepancies in the standard errors. Because it is robust to poor starting values, converges rapidly and conveniently produces consistent standard errors for both maximum likelihood and weighted least squares problems, the Gauss-Newton algorithm represents an attractive alternative for at least some covariance structure analyses.
On the Estimation of Parameters in Latent Structure Analysis
Authors: Goodman Leo A.
Published: January 2006

In this note, we describe the iterative procedure introduced earlier by Goodman to calculate the maximum likelihood estimates of the parameters in latent structure analysis, and we provide here a simple and direct proof of the fact that the parameter estimates obtained with the iterative procedure cannot lie outside the allowed interval. Formann recently stated that Goodman's algorithm can yield parameter estimates that lie outside the allowed interval, and we prove in the present note that Formann's contention is incorrect.
Range Restrictions for Product-Moment Correlation Matrices
Authors: Olkin Ingram
Published: January 2006

It is well-known that for a trivariate distribution if two correlations are fixed the remaining one is constrained. Indeed, if one correlation is fixed, then the remaining two are constrained. Both results are extended to the case of a multivariate distribution. The results are applied to some special patterned matrices.
A Maximum Likelihood Method for Fitting the Wandering Vector Model
Authors: De Soete Geert , Carroll J. D.
Published: January 2006

After introducing some extensions of a recently proposed probabilistic vector model for representing paired comparisons choice data, an iterative procedure for obtaining maximum likelihood estimates of the model parameters is developed. The possibility of testing various hypotheses by means of likelihood ratio tests is discussed. Finally, the algorithm is applied to some existing data sets for illustrative purposes.
A Bayesian Approach to Confirmatory Factor Analysis
Authors: Lee Sik-Yum
Published: January 2006

Confirmatory factor analysis is considered from a Bayesian viewpoint, in which prior information on parameter is incorporated in the analysis. An iterative algorithm is developed to obtain the Bayes estimates. A numerical example based on longitudinal data is presented. A simulation study is designed to compare the Bayesian approach with the maximum likelihood method.
Estimation of Covariance Structure Models with Parameters Subject to Functional Restraints
Authors: Lee Sik-Yum
Published: January 2006

This paper demonstrates the feasibility of using the penalty function method to estimate parameters that are subject to a set of functional constraints in covariance structure analysis. Both types of inequality and equality constraints are studied. The approaches of maximum likelihood and generalized least squares estimation are considered. A modified Scoring algorithm and a modified Gauss-Newton algorithm are implemented to produce the appropriate constrained estimates. The methodology is illustrated by its applications to Heywood cases in confirmatory factor analysis, quasi-Weiner simplex model, and multitrait-multimethod matrix analysis.
A Scaling Model with Response Errors and Intrinsically Unscalable Respondents
Authors: Dayton C. Mitchell , Macready George B.
Published: January 2006

Goodman contributed to the theory of scaling by including a category of intrinsically unscalable respondents in addition to the usual scale-type respondents. However, his formulation permits only error-free responses by respondents from the scale types. This paper presents new scaling models which have the properties that: (1) respondents in the scale types are subject to response errors; (2) a test of significance can be constructed to assist in deciding on the necessity for including an intrinsically unscalable class in the model; and (3) when an intrinsically unscalable class is not needed to explain the data, the model reduces to a probabilistic, rather than to a deterministic, form. Three data sets are analyzed with the new models and are used to illustrate stages of hypothesis testing.
Conformity of Two Sociometric Relations
Authors: Wasserman Stanley
Published: January 2006

The problem of comparing two sociometric relations or measurements (A and B) recorded in distinct sociomatrices was originally discussed by Katz and Powell in the early 1950's and Hubert and Baker in the late 1970's. The problem is considered again using a probabilistic model designed specifically for discrete-valued network measurements. The model allows for the presence of various structural tendencies, such as reciprocity and differential popularity. A parameter that isolates the tendency for actors to choose other actors on both relations simultaneously is introduced, and estimated conditional on the presence of other parameters that reflect additional important network properties. The parameter is presented as a symmetric index but is also generalized to the predictive (A on B or B on A) situation. This approach to the problem is illustrated with the same data used by the earlier solutions, and the unique nature of the two relations in the data set (A = received choices, B = perceived choices), as it affects the modeling, is discussed. Significance tests for the parameter and related parameters are described, as well as an extension to more than two relations.
Information and Preference in Partial Orders: A Bimatrix Representation
Authors: Cook Wade D. , Kress Moshe , Seiford Lawrence M.
Published: January 2006

This paper presents a bimatrix structure for examining ordinal partial rankings. A set of axioms is given similar to those of Kemeny and Snell (1962) and Bogart (1973), which uniquely determines the distance between any pair of such rankings. The l1 norm is shown to satisfy this set of axioms, and to be equivalent to the Kemeny and Snell distance on their subspace of weak orderings. Consensus formation is discussed.
Differential Ordering of Objects and Attributes
Authors: Sutcliffe J. P.
Published: January 2006

By reference to nominated attributes, a genus, being a population of objects of one specified kind, may be partitioned into species, being subpopulations of different kinds. A prototype is an object representative of its species within the genus. Using this framework, the paper describes how objects can be relatively differentiated with respect to attributes, and how attributes can be relatively differentiating with respect to objects. Methods and rationale for such differential ordering of objects and attributes are presented by example, formal development, and application.

For a genus Ω comprising n species of object there is a subset P ofn distinct prototypes. With respect to m nominated attributes, each object in Ω has an m-element characterization. Together these determine an n × m objects × attributes matrix, the rows of which are the characterizations of the prototypical objects. Over then species in Ω, an associated relative frequency vector gives the distribution of objects (and of their characterizations). The matrix and vector associate the objects in Ω with points in a metric space (P, δ); and it is with respect to various sums of distances in this attribute space that one can differentially order objects and attributes.

The definition of the distance function δ is generalized across kinds of difference, types of characterization, scale-types of measurement, Minkowski index ≧ 1, and any form of distribution of objects over species. Explanatory and taxonomic applications in psychology and other fields are discussed, with focus on classification, identification, recognition, and search. The Braille code and the identification of its characters provide illustration.
Self-Selection Biases in Correlational Studies Based on Questionnaires
Authors: Bowden Roger J.
Published: January 2006

It is commonly held that even where questionnaire response is poor, correlational studies are affected only by loss of degrees of freedom or precision. We show that this supposition is not true. If the decision to respond is correlated with a substantive variable of interest, then regression or analysis of variance methods based upon the questionnaire results may be adversely affected by self-selection bias. Moreover such bias may arise even where response is 100%. The problem in both cases arises where selection information is passed to the score indirectly via the disturbance or individual effects, rather than entirely via the observable explanatory variables. We suggest tests for the ensuing self-selection bias and possible ways of handling the ensuing problems of inference.
The Occurrence of an Increase in Correlation with Explicit Selection
Authors: McGuire Dennis P.
Published: January 2006

It is generally believed that correlations and standard deviations measured within an explicitly selected group must be smaller than those within an applicant population. A small data set is used to show that this is not always true, and that both validity and reliability estimates within a selected group can exceed those within the applicant population. The increase in correlation within the explicitly selected group is tied to an increase in standard deviation of the predictor in the selected group. This conclusion extends Levin's (1972) result for the case of incidental selection, selection on an unmeasured third variable. If possible, theoretical derivations should not be limited to the case where the predictor standard deviation in the applicant population exceeds that in the selected group. When such a situation occurs in observed data, it cannot be immediately dismissed as an artifact.
A Note on the Computation of the Second-Order Derivatives of the Elementary Symmetric Functions in the Rasch Model
Authors: Formann Anton K.
Published: January 2006

Jansen (1984) gave explicit formulas for the computation of the second-order derivatives of the elementary symmetric functions. But they are only applicable to those pairs of items which have unequal parameters. It is shown here that for equal parameters similar explicit formulas do exist, too, facilitating the application of the Newton-Raphson procedure to estimate the parameters in the Rasch model and related models according to the conditional maximum likelihood principle.
Psychology as a Quantitative Rational Science (Excerpts)
Authors: Thurstone L. L.
Published: January 2006

The purpose of the Psychometric Society is to encourage mathematical psychology. This article justifies the emphasis on quantification, and upon rationalization in science, as well as the Society's conception of the fundamental nature of science.
Scaling and Psychometrika: Spatial and Alternative Representations of Similarity Data
Authors: Torgerson Warren S.
Published: January 2006

Progress over the past twenty-five years in the development and improvement of models for representation of similarity data is reviewed. The discussion includes comments on class, dimensional, ideal type, and dichotomous attribute representations of underlying similarity structures. Most of the theoretical research in the area appeared, and continues to appear, in Psychometrika.
Assessing the Correspondence of One or More Vectors to a Symmetric Matrix Using Ordinal Regression
Authors: Reynolds Thomas J. , Sutrick Kenneth H.
Published: January 2006

A statistical model for interpreting psychological scaling research, based on the heuristic work of Reynolds (1983), is developed. This new approach has certain advantages over the standard property fitting approach (Chang and Carroll, 1969) currently used to interpret multidimensional scaling spaces (Shepard, 1962; Torgerson, 1965). These advantages are (a) the ability to directly assess the correspondence of a descriptor vector(s) to a symmetric matrix, and (b) to provide a method in which only ordinal properties of such descriptors are required: thus standard rating, ranking, or sorting data collection methods can be used as the basis to interpret the multidimensional space resulting from the distance data.
Probability-of-Coverage Tolerance Intervals for the Beta Binomial
Authors: Jarjoura David
Published: January 2006

True score tolerance intervals, which are designed to cover a chosen proportion of the conditional distribution of true scores given an observed score, are suggested as alternatives to true score confidence intervals. Using large sample theory, a tolerance interval estimator for the beta binomial is derived. An example indicates that with moderate sample sizes, tolerance intervals with high probability of coverage will not be much wider than when the two beta true score parameters are known.
Unbiased Estimators of Ability Parameters, of their Variance, and of their Parallel-Forms Reliability
Authors: Lord Frederic M.
Published: January 2006

Given known item parameters, unbiased estimators are derived i) for an examinee's ability parameter θ and for his proportion-correct true score ς, ii) for the variances of θ and ς across examinees in the group tested, and iii) for the parallel-forms reliability of the maximum likelihood estimator\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}
$$\hat \theta$$
\end{document}.
More on EM for ML Factor Analysis
Authors: Rubin Donald B. , Thayer Dorothy T.
Published: January 2006

We address several issues that are raised by Bentler and Tanaka's [1983] discussion of Rubin and Thayer [1982]. Our conclusions are: standard methods do not completely monitor the possible existence of multiple local maxima; summarizing inferential precision by the standard output based on second derivatives of the log likelihood at a maximum can be inappropriate, even if there exists a unique local maximum; EM and LISREL can be viewed as complementary, albeit not entirely adequate, tools for factor analysis.
Maximum Likelihood Estimation of the Joint Covariance Matrix for Sections of Tests given to Distinct Samples with Application to Test Equating
Authors: Thayer Dorothy T.
Published: January 2006

Consider an old test X consisting of s sections and two new tests Y and Z similar to X consisting of p and q sections respectively. All subjects are given test X plus two variable sections from either test Y or Z. Different pairings of variable sections are given to each subsample of subjects. We present a method of estimating the covariance matrix of the combined test (X1, ..., Xs, Y1, ..., Yp, Z1, ..., Zq) and describe an application of these estimation techniques to linear, observed-score, test equating.
Weakly Parallel Tests in Latent Trait Theory with Some Criticisms of Classical Test Theory
Authors: Samejima Fumiko
Published: January 2006

A new concept of weakly parallel tests, in contrast to strongly parallel tests in latent trait theory, is proposed. Some criticisms of the fundamental concepts in classical test theory, such as the reliability of a test and the standard error of estimation, are given.
Some Properties of Binormamin
Authors: Kaiser Henry F. , Dickman Kern W.
Published: January 2006

Three properties of the binormamin criterion for analytic transformation in factor analysis are discussed. Particular reference is made to Carroll’s oblimin class of criteria.
MACPLUS: A Mathematical Programming Approach to Fitting the ADCLUS Model
Authors: Arabie Phipps , Carroll J. Douglas
Published: January 2006

We present a new algorithm, MAPCLUS (MAthematical Programming CLUStering), for fitting the Shepard-Arabie ADCLUS (for ADditive CLUStering) model. MAPCLUS utilizes an alternating least squares method combined with a mathematical programming optimization procedure based on a penalty function approach, to impose discrete (0,1) constraints on parameters defining cluster membership. This procedure is supplemented by several other numerical techniques (notably a heuristically based combinatorial optimization procedure) to provide an efficient general-purpose computer implemented algorithm for obtaining ADCLUS representations. MAPCLUS is illustrated with an application to one of the examples given by Shepard and Arabie using the older ADCLUS procedure. The MAPCLUS solution uses half as many clusters to achieve nearly the same level of goodness-of-fit. Finally, we consider an extension of the present approach to fitting a three-way generalization of the ADCLUS model, called INDCLUS (INdividual Differences CLUStering).
On the Multivariate Asymptotic Distribution of Sequential Chi-Square Statistics
Authors: Steiger James H. , Shapiro Alexander , Browne Michael W.
Published: January 2006

The multivariate asymptotic distribution of sequential Chi-square test statistics is investigated. It is shown that: (a) when sequential Chi-square statistics are calculated for nested models on the same data, the statistics have an asymptotic intercorrelation which may be expressed in closed form, and which is, in many cases, quite high; and (b) sequential Chi-square difference tests are asymptotically independent. Some Monte Carlo evidence on the applicability of the theory is provided.
A Constrained Spline Estimator of a Hazard Function
Authors: Bloxom Bruce
Published: January 2006

A constrained quadratic spline is proposed as an estimator of the hazard function of a random variable. A maximum penalized likelihood procedure is used to fit the estimator to a sample of psychological response times. The results of a small simulation study suggest that, with a sample size of 500, the procedure may provide a reasonably precise estimate of the shape of a hazard function.
Homogeneity Analysis with k Sets of Variables: An Alternating Least Squares Method with Optimal Scaling Features
Authors: van der Burg Eeke , de Leeuw Jan , Verdegaal Renée
Published: January 2006

Homogeneity analysis, or multiple correspondence analysis, is usually applied to k separate variables. In this paper we apply it to sets of variables by using sums within sets. The resulting technique is called OVERALS. It uses the notion of optimal scaling, with transformations that can be multiple or single. The single transformations consist of three types: nominal, ordinal, and numerical. The corresponding OVERALS computer program minimizes a least squares loss function by using an alternating least squares algorithm. Many existing linear and nonlinear multivariate analysis techniques are shown to be special cases of OVERALS. An application to data from an epidemiological survey is presented.
Multidimensional Rotation and Scaling of Configurations to Optimal Agreement
Authors: Peay Edmund R.
Published: January 2006

An integrated method for rotating and rescaling a set of configurations to optimal agreement in subspaces of varying dimensionalities is developed. The approach relates existing orthogonal rotation techniques as special cases within a general framework based on a partition of variation which provides convenient measures of agreement. In addition to the well-known Procrustes and inner product optimality criteria, a criterion which maximizes the “consensus” among subspaces of the configurations is suggested. Since agreement of subspaces of the configurations can be examined and compared, rotation and rescaling is extended from a data transformation technique to an analytical method.
A Family of Association Coefficients for Metric Scales
Authors: Zegers Frits E. , ten Berge Jos M. F.
Published: January 2006

Four types of metric scales are distinguished: the absolute scale, the ratio scale, the difference scale and the interval scale. A general coefficient of association for two variables of the same metric scale type is developed. Some properties of this general coefficient are discussed. It is shown that the matrix containing these coefficients between any number of variables is Gramian. The general coefficient reduces to specific coefficients of association for each of the four metric scales. Two of these coefficients are well known, the product-moment correlation and Tucker's congruence coefficient. Applications of the new coefficients are discussed.
A Comparison of Latent Trait and Latent Class Analyses of Likert-Type Data
Authors: Masters Geofferey N.
Published: January 2006

This paper brings together and compares two developments in the analysis of Likert attitude scales. The first is the generalization of latent class models to ordered response categories. The second is the introduction of latent trait models with multiplicative parameter structures for the analysis of rating scales. Key similarities and differences between these two methods are described and illustrated by applying a latent trait model and a latent class model to the analysis of a set of “life satisfaction” data. The way in which the latent trait model defines a unit of measurement, takes into account the order of the response categories, and scales the latent classes, is discussed. While the latent class model provides better fit to these data, this is achieved at the cost of a logically inconsistent assignment of individuals to latent classes.
The Determinacy of Common Factors in Large Item Domains
Authors: Rozeboom William W.
Published: January 2006

Bounds are here developed for the multiple correlation of common factors with the items whose factors they are. It is then easy to see, under broad but not completely general conditions, the circumstances under which an infinite item domain does or does not perfectly determine selected subsets of its common factors.
Test Theory without an Answer Key
Authors: Batchelder William H. , Romney A. Kimball
Published: January 2006

A general model is presented for homogeneous, dichotomous items when the answer key is not known a priori. The model is structurally related to the two-class latent structure model with the roles of respondents and items interchanged. For very small sets of respondents, iterative maximum likelihood estimates of the parameters can be obtained by existing methods. For other situations, new estimation methods are developed and assessed with Monte Carlo data. The answer key can be accurately reconstructed with relatively small sets of respondents. The model is useful when a researcher wants to study objectively the knowledge possessed by members of a culturally coherent group that the researcher is not a member of.
The Comparison of Interdependent Correlations between Optimal Linear Composites
Authors: Steiger James H. , Browne Michael W.
Published: January 2006

A general procedure is provided for comparing correlation coefficients between optimal linear composites. The procedure allows computationally efficient significance tests on independent or dependent multiple correlations, partial correlations, and canonical correlations, with or without the assumption of multivariate normality. Evidence from some Monte Carlo studies on the effectiveness of the methods is also provided.
Synthesized Clustering: A method for Amalgamating Alternative Clustering Bases with Differential Weighting of Variables
Authors: DeSarbo Wayne S. , Carroll J. Douglas , Clark Linda A. , Green Paul E.
Published: January 2006

In the application of clustering methods to real world data sets, two problems frequently arise: (a) how can the various contributory variables in a specific battery be weighted so as to enhance some cluster structure that may be present, and (b) how can various alternative batteries be combined to produce a single clustering that “best” incorporates each contributory set. A new method is proposed (SYNCLUS, SYNthesized CLUStering) for dealing with these two problems.
On Approximate Confidence Intervals for Measures of Concordance
Authors: Palachek Albert D. , Schucany William R.
Published: January 2006

The use of U-statistics based on rank correlation coefficients in estimating the strength of concordance among a group of rankers is examined for cases where the null hypothesis of random rankings is not tenable. The studentized U-statistics is asymptotically distribution-free, and the Student-t approximation is used for small and moderate sized samples. An approximate confidence interval is constructed for the strength of concordance. Monte Carlo results indicate that the Student-t approximation can be improved by estimating the degrees of freedom.
Some Additional Results on Principal Components Analysis of Three-Mode Data by Means of Alternating Least Squares Algorithms
Authors: ten Berge Jos M. F. , de Leeuw Jan , Kroonenberg Pieter M.
Published: January 2006

Kroonenberg and de Leeuw (1980) have developed an alternating least-squares method TUCKALS-3 as a solution for Tucker's three-way principal components model. The present paper offers some additional features of their method. Starting from a reanalysis of Tucker's problem in terms of a rank-constrained regression problem, it is shown that the fitted sum of squares in TUCKALS-3 can be partitioned according to elements of each mode of the three-way data matrix. An upper bound to the total fitted sum of squares is derived. Finally, a special case of TUCKALS-3 is related to the Carroll/Harshman CANDECOMP/PARAFAC model.
A Comparison of the Efficiency and Accuracy of BILOG and LOGIST
Authors: Yen Wendy M.
Published: January 2006

Comparisons are made between BILOG version 2.2 and LOGIST 5.0 Version 2.5 in estimating the item parameters, traits, item characteristic functions (ICFs), and test characteristic functions (TCFs) for the three-parameter logistic model. Data analyzed are simulated item responses for 1000 simulees and one 10-item test, four 20-item tests, and four 40-item tests. LOGIST usually was faster than BILOG in producing maximum likelihood estimates. BILOG almost always produced more accurate estimates of individual item parameters. In estimating ICFs and TCFs BILOG was more accurate for the 10-item test, and the two programs were about equally accurate for the 20- and 40-item tests.
Axiomatic Foundations of a Three-Set Guttman Simplex Model with Applicability to Longitudinal Data
Authors: Collins Linda M. , Cliff Norman
Published: January 2006

In this paper the usual two-set Guttman simplex model is extended to three sets. The axiomatic foundations of this extention are presented. Two cases are discussed. In Case 1 there is a three-set joint order, while in Case 2 there is a two-set joint order consistent across all levels of the third set. Case 2 represents the first clear formulation of a longitudinal developmental scale. The model is discussed in terms of its most straightforward application, longitudinal developmental data, and in terms of other possible applications.
A Note on the Asymptotic Distribution of the Greatest Lower Bound to Reliability
Authors: Shapiro Alexander
Published: January 2006

In a recent article Bentler and Woodward (1983) discussed computational and statistical issues related to the greatest lower bound ρ+ to reliability. Although my work (Shapiro, 1982) was cited frequently some results presented were misunderstood. A sample estimate + of ρ + was considered and it was claimed (Bentler & Woodward) that: “Since + is not a closed form expression ... an exact analytic expression for h has not been found” (p. 247). (h is a vector of partial derivatives of ρ+ as a function of the covariance matrix.) Therefore Bentler and Woodward proposed to use numerical derivatives in order to evaluate the asymptotic variance avar (+) of +.
On the Greatest Lower Bound to Reliability
Authors: Bentler P. M. , Woodward J. Arthur
Published: January 2006

Certain ambiguities in a recent paper on the computation and statistics of the greatest lower bound are clarified.
On Equivalence between a Partial Credit Item and a Set of Independent Rasch Binary Items
Authors: Huynh Huynh
Published: January 2006

Given a Masters partial credit item with n known step difficulties, conditions are stated for the existence of a set of (locally) independent Rasch binary items such that their raw score and the partial credit raw score have identical probability density functions. The conditions are those for the existence of n positive values with predetermined elementary symmetric functions and include the requirement that the n step difficulties form an increasing sequence.
Interval Estimation for the Intraclass Correlation in Dirichlet-Multinomial Data
Authors: Lui Kung-Jong , Cumberland William G. , Mayer Joni A. , Eckhardt Laura
Published: January 2006

When the underlying distribution is discrete with a limited number of categories, methods for interval estimation of the intraclass correlation which assume normality are theoretically inadequate for use. On the basis of large sample theory, this paper develops an asymptotic closed-form interval estimate of the intraclass correlation for the case where there is a natural score associated with each category. This paper employs Monte Carlo simulation to demonstrate that when the underlying intraclass correlation is large, the traditional interval estimator which assumes normality can be misleading. We find that when the number of classes is ≥ 20, the interval estimator proposed here can generally perform reasonably well in a variety of situations. This paper further notes that the proposed interval estimator is invariant with respect to a linear transformation. When the data are on a nominal scale, an extension of the proposed method to account for this case, as well as a discussion on the relationship between the intraclass correlation and a kappa-type measure defined here and on the limitation of the corresponding kappa-type estimator are given.
Logit Models and Logistic Regressions for Social Networks: III. Valued Relations
Authors: Robins Garry , Pattison Philippa , Wasserman Stanley
Published: January 2006

This paper generalizes the p* model for dichotomous social network data (Wasserman & Pattison, 1996) to the polytomous case. The generalization is achieved by transforming valued social networks into three-way binary arrays. This data transformation requires a modification of the Hammersley-Clifford theorem that underpins the p* class of models. We demonstrate that, provided that certain (non-observed) data patterns are excluded from consideration, a suitable version of the theorem can be developed. We also show that the approach amounts to a model for multiple logits derived from a pseudo-likelihood function. Estimation within this model is analogous to the separate fitting of multinomial baseline logits, except that the Hammersley-Clifford theorem requires the equating of certain parameters across logits. The paper describes how to convert a valued network into a data array suitable for fitting the model and provides some illustrative empirical examples.
Bayesian Estimation and Testing of Structural Equation Models
Authors: Scheines Richard , Hoijtink Herbert , Boomsma Anne
Published: January 2006

The Gibbs sampler can be used to obtain samples of arbitrary size from the posterior distribution over the parameters of a structural equation model (SEM) given covariance data and a prior distribution over the parameters. Point estimates, standard deviations and interval estimates for the parameters can be computed from these samples. If the prior distribution over the parameters is uninformative, the posterior is proportional to the likelihood, and asymptotically the inferences based on the Gibbs sample are the same as those based on the maximum likelihood solution, for example, output from LISREL or EQS. In small samples, however, the likelihood surface is not Gaussian and in some cases contains local maxima. Nevertheless, the Gibbs sample comes from the correct posterior distribution over the parameters regardless of the sample size and the shape of the likelihood surface. With an informative prior distribution over the parameters, the posterior can be used to make inferences about the parameters underidentified models, as we illustrate on a simple errors-in-variables model.
Statistical Inference Based on Latent Ability Estimates
Authors: Hoijtink Herbert , Boomsma Anne
Published: January 2006

The quality of approximations to first and second order moments (e.g., statistics like means, variances, regression coefficients) based on latent ability estimates is being discussed. The ability estimates are obtained using either the Rasch, or the two-parameter logistic model. Straightforward use of such statistics to make inferences with respect to true latent ability is not recommended, unless we account for the fact that the basic quantities are estimates. In this paper true score theory is used to account for the latter; the counterpart of observed/true score being estimated/true latent ability. It is shown that statistics based on the true score theory are virtually unbiased if the number of items presented to each examinee is larger than fifteen. Three types of estimators are compared: maximum likelihood, weighted maximum likelihood, and Bayes modal. Furthermore, the (dis)advantages of the true score method and direct modeling of latent ability is discussed.
Decompositions and Biplots in Three-Way Correspondence Analysis
Authors: Carlier André , Kroonenberg Pieter M.
Published: January 2006

In this paper correspondence analysis for three-way contingency tables is presented using three-way generalisations of the singular value decomposition. It is shown that in combination with Lancaster's (1951) additive decomposition of interactions in three-way tables, a detailed analysis is possible of the deviations from independence. Finally, biplots are shown to produce powerful graphical representations of the results from three-way correspondence analyses. An example from child development is used to illustrate the theoretical developments.
A Monotonically Convergent Algorithm for Orthogonal Congruence Rotation
Authors: Kiers Henk A. L. , Groenen Patrick
Published: January 2006

Brokken has proposed a method for orthogonal rotation of one matrix such that its columns have a maximal sum of congruences with the columns of a target matrix. This method employs an algorithm for which convergence from every starting point is not guaranteed. In the present paper, an iterative majorization algorithm is proposed which is guaranteed to converge from every starting point. Specifically, it is proven that the function value converges monotonically, and that the difference between subsequent iterates converges to zero. In addition to the better convergence properties, another advantage of the present algorithm over Brokken's one is that it is easier to program. The algorithms are compared on 80 simulated data sets, and it turned out that the new algorithm performed well in all cases, whereas Brokken's algorithm failed in almost half the cases. The derivation of the algorithm is given in full detail because it involves a series of inequalities that can be of use to derive similar algorithms in different contexts.
Ideal Point Discriminant Analysis
Authors: Takane Yoshio , Bozdogan Hamparsum , Shibayama Tadashi
Published: January 2006

A new method of multiple discriminant analysis was developed that allows a mixture of continuous and discrete predictors. The method can be justified under a wide class of distributional assumptions on the predictor variables. The method can also handle three different sampling situations, conditional, joint and separate. In this method both subjects (cases or any other sampling units) and criterion groups are represented as points in a multidimensional euclidean space. The probability of a particular subject belonging to a particular criterion group is stated as a decreasing function of the distance between the corresponding points. A maximum likelihood estimation procedure was developed and implemented in the form of a FORTRAN program. Detailed analyses of two real data sets were reported to demonstrate various advantages of the proposed method. These advantages mostly derive from model evaluation capabilities based on the Akaike Information Criterion (AIC).
Heuristic Method for Label Placement in Scatterplots
Authors: Noma Elliot
Published: January 2006

Kuhfeld (1986) has proposed several alternative scatterplot routines with labeled points. To this list is added a heuristic routine that repositions labels to eliminate overlap while attempting to maximize the number of labels presented in a metric plot.
The Mixed Model for Multivariate Repeated Measures: Validity Conditions and an Approximate Test
Authors: Boik Robert J.
Published: January 2006

Repeated measures on multivariate responses can be analyzed according to either of two models: a doubly multivariate model (DMM) or a multivariate mixed model (MMM). This paper reviews both models and gives three new results concerning the MMM. The first result is, primarily, of theoretical interest; the second and third have implications for practice. First, it is shown that, given multivariate normality, a condition called multivariate sphericity of the covariance matrix is both necessary and sufficient for the validity of the MMM analysis. To test for departure from multivariate sphericity, the likelihood ratio test can be employed. The second result is an approximation to the null distribution of the likelihood ratio test statistic, useful for moderate sample sizes. Third, for situations satisfying multivariate normality, but not multivariate sphericity, a multivariate ε correction factor is derived. The ε correction factor generalizes Box's ε and can be used to construct an adjusted MMM test.
The Derivation of Some Tests for the Rasch Model from the Multinomial Distribution
Authors: Glas Cees A. W.
Published: January 2006

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.
On the Sensitivity of a Composite to its Weights
Authors: Koopman Raymond F.
Published: January 2006

A method is presented for investigating two aspects of the sensitivity of a linear composite to the values of its weights.
Models for Ordinal Hierarchical Classes Analysis
Authors: Leenen Iwin , Van Mechelen Iven , De Boeck Paul
Published: January 2006

This paper proposes an ordinal generalization of the hierarchical classes model originally proposed by De Boeck and Rosenberg (1998). Any hierarchical classes model implies a decomposition of a two-way two-mode binary array M into two component matrices, called bundle matrices, which represent the association relation and the set-theoretical relations among the elements of both modes in M. Whereas the original model restricts the bundle matrices to be binary, the ordinal hierarchical classes model assumes that the bundles are ordinal variables with a prespecified number of values. This generalization results in a classification model with classes ordered along ordinal dimensions. The ordinal hierarchical classes model is shown to subsume Coombs and Kao's (1955) model for nonmetric factor analysis. An algorithm is described to fit the model to a given data set and is subsequently evaluated in an extensive simulation study. An application of the model to student housing data is discussed.
Simultaneous Analysis of Multivariate Polytomous Variates in Several Groups
Authors: Lee Sik-Yum , Poon Wai-Yin , Bentler P. M.
Published: January 2006

The frequencies of m independent p-way contingency tables are analyzed by a model that assumes that the ordinal categorical data in each of m groups are generated from a latent continuous multivariate normal distribution. The parameters of these multivariate distributions and of the relations between the ordinal and latent variables are estimated by maximum likelihood. Goodness-of-fit statistics based on the likelihood ratio criterion and the Pearsonian chisquare are provided to test the hypothesis that the proposed model is correct, that is, it fits the observed sample data. Hypotheses on the invariance of means, variances, and polychoric correlations of the latent variables across populations are tested by Wald statistics. The method is illustrated on an example involving data on three five-point ordinal scales obtained from male and female samples.
Simple Structure in Component Analysis Techniques for Mixtures of Qualitative and Quantitative Variables
Authors: Kiers Henk A. L.
Published: January 2006

Several methods have been developed for the analysis of a mixture of qualitative and quantitative variables, and one, called PCAMIX, includes ordinary principal component analysis (PCA) and multiple correspondence analysis (MCA) as special cases. The present paper proposes several techniques for simple structure rotation of a PCAMIX solution based on the rotation of component scores and indicates how these can be viewed as generalizations of the simple structure methods for PCA. In addition, a recently developed technique for the analysis of mixtures of qualitative and quantitative variables, called INDOMIX, is shown to construct component scores (without rotational freedom) maximizing the quartimax criterion over all possible sets of component scores. A numerical example is used to illustrate the implication that when used for qualitative variables, INDOMIX provides axes that discriminate between the observation units better than do those generated from MCA.
Statistical Inference for Multiple Choice Tests
Authors: Hsu John S. J. , Leonard Tom , Tsui Kam-Wah
Published: January 2006

Finite sample inference procedures are considered for analyzing the observed scores on a multiple choice test with several items, where, for example, the items are dissimilar, or the item responses are correlated. A discrete p-parameter exponential family model leads to a generalized linear model framework and, in a special case, a convenient regression of true score upon observed score. Techniques based upon the likelihood function, Akaike's information criteria (AIC), an approximate Bayesian marginalization procedure based on conditional maximization (BCM), and simulations for exact posterior densities (importance sampling) are used to facilitate finite sample investigations of the average true score, individual true scores, and various probabilities of interest. A simulation study suggests that, when the examinees come from two different populations, the exponential family can adequately generalize Duncan's beta-binomial model. Extensions to regression models, the classical test theory model, and empirical Bayes estimation problems are mentioned. The Duncan, Keats, and Matsumura data sets are used to illustrate potential advantages and flexibility of the exponential family model, and the BCM technique.
A Stochastic Multidimensional Scaling Procedure for the Spatial Representation of Three-Mode, Three-Way Pick Any/J Data
Authors: Jedidi Kamel , DeSarbo Wayne S.
Published: January 2006

This paper presents a new stochastic multidimensional scaling procedure for the analysis of three-mode, three-way pick any/J data. The method provides either a vector or ideal-point model to represent the structure in such data, as well as “floating” model specifications (e.g., different vectors or ideal points for different choice settings), and various reparameterization options that allow the coordinates of ideal points, vectors, or stimuli to be functions of specified background variables. A maximum likelihood procedure is utilized to estimate a joint space of row and column objects, as well as a set of weights depicting the third mode of the data. An algorithm using a conjugate gradient method with automatic restarts is developed to estimate the parameters of the models. A series of Monte Carlo analyses are carried out to investigate the performance of the algorithm under diverse data and model specification conditions, examine the statistical properties of the associated test statistic, and test the robustness of the procedure to departures from the independence assumptions. Finally, a consumer psychology application assessing the impact of situational influences on consumers' choice behavior is discussed.
A Multidimensional Latent Trait Model for Measuring Learning and Change
Authors: Embretson Susan E.
Published: January 2006

A latent trait model is presented for the repeated measurement of ability based on a multidimensional conceptualization of the change process. A simplex structure is postulated to link item performance under a given measurement condition or occasion to initial ability and to one or more modifiabilities that represent individual differences in change. Since item discriminations are constrained to be equal within a measurement condition, the model belongs to the family of multidimensional Rasch models. Maximum likelihood estimators of the item parameters and abilities are derived, and an example provided that shows good recovery of both item and ability parameters. Properties of the model are explored, particularly for several classical issues in measuring change.
An Extension of the Rating Scale Model with an Application to the Measurement of Change
Authors: Fischer G. H. , Parzer P.
Published: January 2006

The polytomous unidimensional Rasch model with equidistant scoring, also known as the rating scale model, is extended in such a way that the item parameters are linearly decomposed into certain basic parameters. The extended model is denoted as the linear rating scale model (LRSM). A conditional maximum likelihood estimation procedure and a likelihood-ratio test of hypotheses within the framework of the LRSM are presented. Since the LRSM is a generalization of both the dichotomous Rasch model and the rating scale model, the present algorithm is suited for conditional maximum likelihood estimation in these submodels as well. The practicality of the conditional method is demonstrated by means of a dichotomous Rasch example with 100 items, of a rating scale example with 30 items and 5 categories, and in the light of an empirical application to the measurement of treatment effects in a clinical study.
A Maximin Model for Test Design with Practical Constraints
Authors: van ver Linden Wim J. , Boekkooi-Timminga Ellen
Published: January 2006

A maximin model for IRT-based test design is proposed. In the model only the relative shape of the target test information function is specified. It serves as a constraint subject to which a linear programming algorithm maximizes the information in the test. In the practice of test construction, several demands as linear constraints in the model. A worked example of a text construction problem with practical constraints is presented. The paper concludes with a discussion of some alternative models of test construction.
An Algorithm for Comparing Medians
Authors: Charlin Ventura L. , Wilcox Rand R.
Published: January 2006

An algorithm is presented for computing a test statistic for comparing medians based on a small-sample size estimate of the variance of the usual sample median.