bluuebunny/crossref_metadata_2025 huggingface dataset:
Orthogonal Rotation Algorithms
Authors: Jennrich Robert I.
Published: January 2006

The quartimax and varimax algorithms for orthogonal rotation attempt to maximize particular simplicity criteria by a sequence of two-factor rotations. Derivations of these algorithms have been fairly complex. A simple general theory for obtaining “two factor at a time” algorithms for any polynomial simplicity criteria satisfying a natural symmetry condition is presented. It is shown that the degree of any symmetric criterion must be a multiple of four. A basic fourth degree algorithm, which is applicable to all symmetric fourth degree criteria, is derived and applied using a variety of criteria. When used with the quartimax and varimax criteria the algorithm is mathematically identical to the standard algorithms for these criteria. A basic eighth degree algorithm is also obtained and applied using a variety of eighth degree criteria. In general the problem of writing a basic algorithm for all symmetric criteria of any specified degree reduces to the problem of maximizing a trigonometric polynomial of degree one-fourth that of the criteria.
Fitting One Matrix to Another Under Choice of a Central Dilation and a Rigid Motion
Authors: Schönemann Peter H. , Carroll Robert M.
Published: January 2006

A least squares method is presented for fitting a given matrix A to another given matrix B under choice of an unknown rotation, an unknown translation, and an unknown central dilation. The procedure may be useful to investigators who wish to compare results obtained with nonmetric scaling techniques across samples or who wish to compare such results with those obtained by conventional factor analytic techniques on the same sample.
Matrix Analysis of Identifiability of Some Finite Markov Models
Authors: Greeno James G. , Millward Richard B. , Merryman Coleman T.
Published: January 2006

Methods developed by Bernbach [1966] and Millward [1969] permit increased generality in analyses of identifiability. Matrix equations are presented that solve part of the identifiability problem for a class of Markov models. Results of several earlier analyses are shown to involve special cases of the equations developed here. And it is shown that a general four-state chain has the same parameter space as an all-or-none model if and only if its representation with an observable absorbing state is lumpable into a Markov chain with three states.
Relations of Factor Score Estimates to their Use
Authors: Tucker Ledyard R
Published: January 2006

Considerations of factor score estimates have concentrated on internal characteristics. This report considers external characteristics of four methods for determining factor score estimates; that is, relations of these estimates to measures on attributes not entered into the factor analysis. These external characteristics are important for many uses of factor score estimates. Findings are that different ones of the methods are appropriate for different uses.
A Dual Scaling Analysis for Paired Compositions
Authors: Bechtel Gordon G.
Published: January 2006

A paired composition is a response (upon a dependent variable) to the ordered pair  of stimuli, treatments, etc. The present paper develops an alternative analysis for the paired compositions layout previously treated by Bechtel's [1967] scaling model. The alternative model relaxes the previous one by including row and column scales that provide an expression of bias for each pair of objects. The parameter estimation and hypothesis testing procedures for this model are illustrated by means of a small group analysis, which represents a new approach to pairwise sociometrics and personality assessment.
Maximum Likelihood Solution to Factor Analysis when Some Factors are Completely Specified
Authors: Gebhardt Friedrich
Published: January 2006

In the factor-analytic model, let some of the factors be known (i.e., the factor loadings are given in advance; they may e.g. be obtained from some previous analyses). However, their covariance matrix may, or may not, be known. The remaining factors (if any) are assumed to be uncorrelated among themselves and to the first set. For this model, the maximum likelihood equations are obtained and an iterative method for the solution is proposed.
Local Dependence Latent Structure Models
Authors: Harper Dean
Published: January 2006

A local independence latent structure model, which assumes m latent classes, requires a minimum of 2m-1 items for the solution of the 2m2 latent parameters. If one adds 3 items to the test and if one assumes local “dependence” between pairs of items, thereby adding\documentclass[12pt]{minimal}
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The Occurrence of an Increase in Correlation by Restriction of Range
Authors: Levin Joseph
Published: January 2006

Given three variables, x, y, z, univariate selection on x, usually reduces the correlation ry z. However, in some cases, when rx y or rx z assumes extreme values, ry z may increase. This reversal is explained by comparison to the effects of suppressor variables and predictors of predictability.
A Note on McDonald’s Generalization of Principal Components Analysis
Authors: Shine Lester C.
Published: January 2006

It is shown that McDonald’s generalization of classical Principal Components Analysis to groups of variables maximally channels the total variance of the original variables through the groups of variables acting as groups. A useful equation is obtained for determining the vectors of correlations of the L2 components with the original variables. A calculation example is given.
Monotone Regression: Continuity and Differentiability Properties
Authors: Kruskal J. B.
Published: January 2006

Least-squares monotone regression has received considerable discussion and use. Consider the residual sum of squares Q obtained from the least-squares monotone regression of yi on xi. Treating Q as a function of the yi, we prove that the gradient ▽Q exists and is continuous everywhere, and is given by a simple formula. (We also discuss the gradient of d = Q1/2.) These facts, which can be questioned (Louis Guttman, private communication), are important for the iterative numerical solution of models, such as some kinds of multidimensional scaling, in which monotone regression occurs as a subsidiary element, so that the yi and hence indirectly Q are functions of other variables.
A Note on Some Equations of Confirmatory Factor Analysis
Authors: Mulaik Stanley A.
Published: January 2006

The method of deriving the second derivatives of the goodness-of-fit functions of maximum likelihood and least-squares confirmatory factor analysis is discussed. The full set of second derivatives is reported.
Estimation of Latent Ability and Item Parameters when there are Omitted Responses
Authors: Lord Frederic M.
Published: January 2006

Omitted items cannot properly be treated as wrong when estimating ability and item parameters. A convenient method for utilizing the information provided by omissions is presented. Theoretical and empirical justifications are presented for the estimates obtained by the new method.
Non-Parametric Inference for Ordered Alternatives in a Randomized Block Design
Authors: Hettmansperger Thomas P.
Published: January 2006

This paper treats the problem of testing an ordered hypothesis based on the ranks of the data. The statistical procedures for the randomized block design with more than one observation per cell are derived. Multiple comparisons and estimation procedures are also included.
A Note on the Sequential Probability Ratio Test
Authors: Thomas Ewart A. C.
Published: January 2006

Link and Heath [1975] have analysed a random walk model for two-choice reaction time on the assumption that the two probability density functions (p.d.f.s) of the step-size, each p.d.f. corresponding to one stimulus, are mirror reflections of each other; and they have demonstrated the critical role played by the symmetry of the moment generating function (m.g.f.) of the step size in the determination of whether or not error and correct reaction times are equal. It is shown here that, given reflection symmetry, m.g.f. symmetry is necessary and sufficient for the random walk model to be equivalent to a sequential probability ratio test.
On the Construction of Weak Orders from Fragmentary Information
Authors: Fishburn Peter C.
Published: January 2006

An iterative method is proposed for constructing a weak order from a partial order on a set of stimuli that is based on individual pairwise comparison data. The method generalizes Duncan Luce's construction of the weak order induced by a semiorder. Various aspects of the iterative procedure are discussed, including its rationale, the number of iterations required to obtain a weak order, and the extent to which the data support additions to the initial partial order as a function of the number of iterations performed before the additions occur.
Estimating the Weights for Multiple Attributes in a Composite Criterion Using Pairwise Judgments
Authors: Srinivasan V. , Shocker Allan D.
Published: January 2006

This paper presents a new methodology for estimating the weights or saliences of subcriteria (attributes) in a composite criterion measure. The inputs to the estimation procedure consist of (i) a set of stimuli or objects with each stimulus defined by its subcriteria profile (set of attribute values) and (ii) the set of paired comparison dominance (e.g., preference) judgments on the stimuli made by a single judge (expert) in terms of the global criterion. A criterion of fit is developed and its optimization via linear programming is illustrated with an example. The procedure is generalized to estimate a common set of weights when the pairwise judgments on the stimuli are made by more than one judge. The procedure is computationally efficient and has been applied in developing a composite criterion of managerial success yielding high concurrent validity.

This methodology can also be used to perform ordinal multiple regression—i.e., multiple regression with an ordinally scaled dependent variable and a set of intervally scaled predictor variables. The approach is further extended to “internal analysis” (unfolding) using the vector model of preference and to the additive model of “conjoint measurement.”
The Effect of Number of Categories in Rating Scales on Precision of Estimation of Scale Values
Authors: Ramsay J. O.
Published: January 2006

The precision of scale value estimates using maximum likelihood estimation is examined for varying numbers of categories, varying discriminal dispersions and with and without category boundary estimation. It is concluded that this type of estimation is somewhat unsatisfactory if boundaries are estimated unless these are of specific interest. If they are not estimated, it is concluded that using seven or more categories provides very nearly as much precision of estimate as a corresponding continuous judgment task.
A Note on Lawley’s Formulas for Standard Errors in Maximum Likelihood Factor Analysis
Authors: Jennrich Robert I. , Thayer Dorothy T.
Published: January 2006

Evidence is given to indicate that Lawley's formulas for the standard errors of maximum likelihood loading estimates do not produce exact asymptotic results. A small modification is derived which appears to eliminate this difficulty.
Nonlinear Programming Approach to Optimal Scaling of Partially Ordered Categories
Authors: Nishisato Shizuhiko , Arri P. S.
Published: January 2006

A modified technique of separable programming was used to maximize the squared correlation ratio of weighted responses to partially ordered categories. The technique employs a polygonal approximation to each single-variable function by choosing mesh points around the initial approximation supplied by Nishisato's method. The major characteristics of this approach are: (i) it does not require any grid refinement; (ii) the entire process of computation quickly converges to the acceptable level of accuracy, and (iii) the method employs specific sets of mesh points for specific variables, whereby it reduces the number of variables for the separable programming technique. Numerical examples were provided to illustrate the procedure.
Descriptive Axioms for Common Factor Theory, Image Theory and Component Theory
Authors: McDonald Roderick P.
Published: January 2006

Through an extension of work by Guttman, common factor theory, image theory, and component theory are derived from distinct minimum subsets of assumptions chosen out of a set of five possible assumptions. It is thence shown that the problem of indeterminacy of factor scores in the common factor model is precisely reflected in the problem of the non-orthogonality of anti-images. Indeed, image scores are determinate for the same reason that the usual estimates of factor scores are determinate, and image scores cannot be used as though they were factor scores for the same reason that factor score estimates cannot be used as though they were factor scores.
Simultaneous Testing of McNemar’s Problem for Several Populations
Authors: Hamdan M. A. , Pirie W. R. , Arnold J. C.
Published: January 2006

McNemar’s problem concerns the hypothesis of equal probabilities for the unlike pairs of correlated binary variables.

We consider four different extensions to this problem, each for testing simultaneous equality of proportions of unlike pairs in c independent populations of correlated binary variables, but each under different assumptions and/or additional hypotheses. For each extension both the likelihood ratio test and the goodness-of-fit chi-square test are given. When c = 1, all cases reduce to McNemar’s problem. For c ≥ 2, however, the tests are quite different, depending on exactly how the hypothesis and alternatives of McNemar are extended.

An example illustrates how widely the results may differ, depending on which extended framework is appropriate.
Functionplane–A New Approach to Simple Structure Rotation
Authors: Katz Jeffrey Owen , Rohlf F. James
Published: January 2006

A new criterion for rotation to an oblique simple structure is proposed. The results obtained are similar to that obtained by Cattell and Muerle's maxplane criterion. Since the proposed criterion is smooth it is possible to locate the local maxima using simple gradient techniques. The results of the application of the Functionplane criterion to three sets of data are given. In each case a better fit to the subjective solution was obtained using the functionplane criterion than was reported for by Hakstian for the oblimax, promax, maxplane, or the Harris-Kaiser methods.
The Effects of Random Error and Subsampling of Dimensions on Recovery of Configurations by Non-Metric Multidimensional Scaling
Authors: Cohen Harvey S. , Jones Lawrence E.
Published: January 2006

Similarity judgments of three-dimensional stimuli were simulated, with the hypothetical subject attending to only some dimensions of stimulus variation (i.e., “subsampling”) on each trial. Recovery of the stimulus configuration by non-metric multidimensional scaling was investigated as a function of subsampling, the amount of random error in the judgments, and the number of stimuli being scaled.

It was found that: (1) dimensions to which the subject often attends were well recovered even when dimensions seldom attended to were not, and (2) measures of recovery based on interpoint distances were inadequate. Several previous Monte Carlo studies were evaluated in light of the results.
A Geometrical Analysis of the Unfolding Model: General Solutions
Authors: Davidson John
Published: January 2006

Given the complete set R of rank orders obtained from any configuration of n stimulus points in r dimensions in accordance with the unfolding model, a configuration from which just these orders may be derived will be described as a solution for R. The space is assumed to be Euclidean. Necessary and sufficient conditions are derived for a configuration to be a solution for R. The geometrical constraints which are necessary and sufficient to determine the subset of pairs of orders and opposites contained in R are also identified and constitute the constraint system for the ordinal vector model. The relationship between the two models is discussed.
On the Relationship Between Ratio of Number of Variables to Number of Factors and Factorial Determinacy
Authors: Meyer Edward P.
Published: January 2006

It is shown that an important distinction can be made between determinacy of common-factors and determinacy of unique-factors and that determinacy can be used as a criterion to establish a lower bound for the ratio of number of variables to number of factors necessary if specified levels of common- and unique-factor determinacy are to be attained.
Improving the Metric Quality of Questionnaire Data
Authors: Gleason Terry C. , Staelin Richard
Published: January 2006

A procedure is proposed whereby questionnaire data, which is usually ordinal in nature and often error-ridden, may be transformed to reduce the error variance in the data and to improve the metric properties of the individual variables. The technique is suggested by a result of Eckart and Young. The properties of the method are investigated by means of a Monte Carlo study. Various matrices were generated representing the usual concept of “true scores”. These matrices were distorted using two levels of random errors and two kinds of categorization. The distorted matrices were in turn transformed by the proposed methods and compared to the “true scores”. In all cases an overall measure of similarity reveals the transformed matrices are better approximations to the “true scores” than the untransformed data. Some properties of the transformation are discussed and some possible applications of the general technique are suggested.
A Further Comparison of Oblique Factor Transformation Methods
Authors: Hakstian A. Ralph , Abell Robert A.
Published: January 2006

Four prominent oblique transformation techniques—promax, the Harris-Kaiser procedure, biquartimin, and direct oblimin—are examined and compared. Additionally, two newly-developed procedures, falling into the category designated as Case III by Harris and Kaiser [1964], are presented and included in the comparisons. The techniques are compared in light of their freedom from bias in the interfactor correlations, and their ability to yield clear simple structures, over many data sets—some constructed and some “real”—varying widely in terms of number of variables and factors, factorial complexity, and clarity of the hyperplanes. Results are discussed, and implications for practice are noted.
Comment on ‘The Estimation of Proportions in M Groups’
Authors: Leonard Tom
Published: January 2006

Novick, Lewis and Jackson [1973] propose an interesting Bayesian method for the simultaneous estimation of the probabilities of several binomial distributions. They utilise arc-sine transformations and normal approximations and then refer to the normal theory method of Lindley [1972] which utilises the novel concept of ‘exchangeability’. The posterior estimates for the probabilities shrink the usual proportions towards a central value based on collateral information, and should provide substantial improvements.

The purpose of our comment is to discuss the relative merits of other methods which are available whenever exchangeability of the probabilities is appropriate, The main methods are

(1) The Logistic Approach: Leonard [1972a] uses logistic transformations of the parameters and proposes a simple exact method of estimation which is valid even if some of the cell frequencies are zero. This is because the logistic transformations are to the parameters and not to the observed frequencies. The exact posterior modal estimates replace the usual observed logits in such a way that difficulties involving zeros are removed. The are-sine approximation is of course inaccurate in such cases.
Three Steps Towards Robust Regression
Authors: Wainer Howard , Thissen David
Published: January 2006

The three most commonly used statistics, the arithmetic mean, variance, and the product-moment correlation, are most unfortunate choices when data are not strictly Gaussian. A new measure of correlation and a measure of scale are proposed which are substantially more robust than their least squares counterparts. An illustration shows how increased robustness can be obtained through the use of equal regression weights without severe loss in accuracy. The paper also shows how incorporating knowledge about the theoretical structure of the regression coefficients into their estimation can aid substantially in increasing their robustness.
An Interval Estimate for making Statistical Inferences about True Scores
Authors: Lord Frederic M. , Stocking Martha L.
Published: January 2006

A numerical procedure is outlined for obtaining an interval estimate of the regression of true score on observed score. Only the frequency distribution of observed scores is needed for this. The procedure assumes that the conditional distribution of observed scores for fixed true score is binomial. The procedure is applied to several sets of test data.
A Simple Test for Heterogeneity of Variance in Complex Factorial Designs
Authors: Overall John E. , Woodward J. Arthur
Published: January 2006

A simple procedure for testing heterogeneity of variance is developed which generalizes readily to complex, multi-factor experimental designs. Monte Carlo Studies indicate that the Z-variance test statistic presented here yields results equivalent to other familiar tests for heterogeneity of variance in simple one-way designs where comparisons are feasible. The primary advantage of the Z-variance test is in the analysis of factorial effects on sample variances in more complex designs. An example involving a three-way factorial design is presented.
A Class of Factor Analysis Estimation Procedures with Common Asymptotic Sampling Properties
Authors: Swain A. J.
Published: January 2006

A general class of estimation procedures for the factor model is considered. The procedures are shown to yield estimates possessing the same asymptotic sampling properties as those from estimation by maximum likelihood or generalized least squares, both of which are special members of the class. General expressions for the derivatives needed for Newton-Raphson determination of the estimates are derived. Numerical examples are given, and the effect of the choice of estimation procedure is discussed.
Methods for Evaluating Empirical Bayes Point Estimates of Latent Trait Scores
Authors: Kearns Jack , Meredith William
Published: January 2006

Empirical Bayes point estimates of latent trait scores, derived under the assumptions of one of several test theory models, display a certain degree of instability unless the sample size is sufficiently large. A measure of this instability over repeated sampling is the distribution of the overall expected squared error loss which converges, both in probability and in the mean, to the minimum (Bayes) overall expected loss as sample size increases. An asymptotic distribution theory is developed, and the resulting large sample approximation is compared with results obtained from simulated data. Attention is also given to the effects of using a smoothing procedure.
Fitting and Testing Carroll’s Weighted Unfolding Model for Preferences
Authors: Davison Mark L.
Published: January 2006

A quadratic programming algorithm is presented for fitting Carroll’s weighted unfolding model for preferences to known multidimensional scale values. The algorithm can be applied directly to pairwise preferences; it permits nonnegativity constraints on subject weights; and it provides a means of testing various preference model hypotheses. While basically metric, it can be combined with Kruskal’s monotone regression to fit ordinal data. Monte Carlo results show that (a) adequacy of “true” preference recovery depends on the number of data points and the amount of error, and (b) the proportion of data variance accounted for by the model sometimes only approximately reflects “true” recovery.
Item Synonymization: A Method for Determining the Total Meaning of Pencil-Paper Reactions
Authors: Lentz Theodore F. , Whitmer Edith F.
Published: January 2006

Items have been studied heretofore for their value as elements of particular tests to the neglect of more fundamental research into the multiple potentiality of items. This article proposes a method of grouping items into “synonymies” comprising all of the items which correlate with a given key item. These synonymies can be used for interpretation of the total meaning of the key item: (1) by inspection of the constituent items and (2) by correlational study of obtained single scores of individual persons. The method is illustrated by four items with inter- and intra-correlations, and characteristics of an ideal background reservoir of items are pointed out.
Transformation of a Three-Mode Multidimensional Scaling Solution to Indscal Form
Authors: MacCallum Robert C.
Published: January 2006

Relations between Tucker's three-mode multidimensional scaling and Carroll and Chang's INDSCAL are discussed. The possibility is raised that it may be profitable to attempt to transform a three-mode solution to the general form of an INDSCAL solution. Operationally, this involves transforming the three-mode core matrix so that each section is, as nearly as possible, a diagonal matrix. A technique is developed for accomplishing such a transformation, and is applied to two sets of data from the literature. Results indicate that the process is both feasible and valuable, providing useful information on the relative appropriateness of the two models.
Component Models for Three-Way Data: An Alternating Least Squares Algorithm with Optimal Scaling Features
Authors: Sands Richard , Young Forrest W.
Published: January 2006

A review of the existing techniques for the analysis of three-way data revealed that none were appropriate to the wide variety of data usually encountered in psychological research, and few were capable of both isolating common information and systematically describing individual differences. An alternating least squares algorithm was proposed to fit both an individual difference model and a replications component model to three-way data which may be defined at the nominal, ordinal, interval, ratio, or mixed measurement level; which may be discrete or continuous; and which may be unconditional, matrix conditional, or row conditional. This algorithm was evaluated by a Monte Carlo study. Recovery of the original information was excellent when the correct measurement characteristics were assumed. Furthermore, the algorithm was robust to the presence of random error. In addition, the algorithm was used to fit the individual difference model to a real, binary, subject conditional data set. The findings from this application were consistent with previous research in the area of implicit personality theory and uncovered interesting systematic individual differences in the perception of political figures and roles.
Principal Component Analysis of Three-Mode Data by Means of Alternating Least Squares Algorithms
Authors: Kroonenberg Pieter M. , de Leeuw Jan
Published: January 2006

A new method to estimate the parameters of Tucker’s three-mode principal component model is discussed, and the convergence properties of the alternating least squares algorithm to solve the estimation problem are considered. A special case of the general Tucker model, in which the principal component analysis is only performed over two of the three modes is briefly outlined as well. The Miller & Nicely data on the confusion of English consonants are used to illustrate the programs TUCKALS3 and TUCKALS2 which incorporate the algorithms for the two models described.
Statistical Inference for False Positive and False Negative Error Rates in Mastery Testing
Authors: Huynh Huynh
Published: January 2006

This paper describes an asymptotic inferential procedure for the estimates of the false positive and false negative error rates. Formulas and tables are described for the computations of the standard errors. A simulation study indicates that the asymptotic standard errors may be used even with samples of 25 cases as long as the Kuder-Richardson Formula 21 reliability is reasonably large. Otherwise, a large sample would be required.
Comparing Latent Distributions
Authors: Andersen Erling B.
Published: January 2006

We consider the problem of comparing m latent population distributions when the observed values are scores on a test battery with binary items. The latent densities are assumed to be normal densities, and we consider a test for equality of the means as well as a test equality of the variances. In addition, we consider a longitudinal model, where the test battery has been applied to the same individuals at different points in time. This model allows for correlations between the latent variable at different time points, and methods are discussed for estimating the correlation coefficient.
Some Small Sample Results for Maximum Likelihood Estimation in Multidimensional Scaling
Authors: Ramsay J. O.
Published: January 2006

Some aspects of the small sample behavior of maximum likelihood estimates in multidimensional scaling are investigated by Monte Carlo. An investigation of Model M2 in the MULTISCALE program package shows that the chi-square test of dimensionality requires a correction of tabled chi-square values to be unbiased. A formula for this correction in the case of two dimensions is estimated. The power of the test of dimensionality is acceptable with as few as two replications for 15 stimuli and as few as five replications for 10 stimuli. The biases in the exponent and standard error estimates in this model are also investigated.
A Simple Model for the Utility of Gambling
Authors: Fishburn Peter C.
Published: January 2006

A model of the utility of gambling is presented in a modified von Neumann-Morgenstern format. Axioms imply a utility function that preserves preferences between sure things and between gambles. The addition of a utility of gambling term to the expected utility of a gamble preserves preference comparisons between gambles and sure things. Aspects of the utility of gambling are noted, and comparisons are made to standard concepts of risk attitudes.
Reliability of Multiple Classifications
Authors: Huynh Huynh
Published: January 2006

Cohen's kappa index is reformulated for multiple classifications based on exchangeable random variables. It is found that kappa is between 0 and 1 inclusive. Two characterizations for kappa are stated in terms of the relationship between such random variables. Within the normal test score model, kappa increases with test reliability and test length. Furthermore, when based on binary classifications, kappa is an inverse U-shaped function of the cutoff score. These trends also hold for the beta-binomial test score model.
Statistical Aspects of a Three-Mode Factor Analysis Model
Authors: Bentler P. M. , Lee Sik-Yum
Published: January 2006

A special case of Bloxom's version of Tucker's three-mode model is developed statistically. A distinction is made between modes in terms of whether they are fixed or random. Parameter matrices are associated with the fixed modes, while no parameters are associated with the mode representing random observation vectors. The identification problem is discussed, and unknown parameters of the model are estimated by a weighted least squares method based upon a Gauss-Newton algorithm. A goodness-of-fit statistic is presented. An example based upon self-report and peer-report measures of personality shows that the model is applicable to real data. The model represents a generalization of Thurstonian factor analysis; weighted least squares estimators and maximum likelihood estimators of the factor model can be obtained using the proposed theory.
Rotational Equivalence of Factor Loading Matrices with Specified Values
Authors: Jennrich Robert I.
Published: January 2006

Under mild assumptions, when appropriate elements of a factor loading matrix are specified to be zero, all orthogonally equivalent matrices differ at most by column sign changes. Here a variety of results are given for the more complex case when the specified values are not necessarily zero. A method is given for constructing reflections to preserve specified rows and columns. When the appropriate k(k − 1)/2 elements have been specified, sufficient conditions are stated for the existence of 2k orthogonally equivalent matrices.
Additive Similarity Trees
Authors: Sattath Shmuel , Tversky Amos
Published: January 2006

Similarity data can be represented by additive trees. In this model, objects are represented by the external nodes of a tree, and the dissimilarity between objects is the length of the path joining them. The additive tree is less restrictive than the ultrametric tree, commonly known as the hierarchical clustering scheme. The two representations are characterized and compared. A computer program, ADDTREE, for the construction of additive trees is described and applied to several sets of data. A comparison of these results to the results of multidimensional scaling illustrates some empirical and theoretical advantages of tree representations over spatial representations of proximity data.
Maximally Correlated Orthogonal Composites and Oblique Factor Analytic Solutions
Authors: Price James M. , Nicewander W. Alan
Published: January 2006

Kaiser presented a method for finding a set of derived orthogonal variables which correlate maximally with a set of original variables. A simpler, more complete derivation of Kaiser's result is given and compared to related types of transformations. The transformation derived here suggests a direct method for finding the orthogonal factor solution which is maximally similar to a given oblique solution.
On Green's Best Linear Composites with a Specified Structure, and Oblique Estimates of Factor Scores
Authors: Ten Berge Jos M. F.
Published: January 2006

Green solved the problem of least-squares estimation of several criteria subject to the constraint that the estimates have an arbitrary fixed covariance or correlation matrix. In the present paper an omission in Green's proof is discussed and resolved. Furthermore, it is shown that some recently published solutions for estimating oblique factor scores are special cases of Green's solution for the case of fixed covariance matrices.
Multidimensional Scaling Models for Reaction Times and Same-Different Judgments
Authors: Takane Yoshio , Sergent Justine
Published: January 2006

A method for joint analysis of reaction times and same-different judgments is discussed. A set of stimuli is assumed to have some parametric representation which uniquely defines dissimilarities between the stimuli. Those dissimilarities are then related to the observed reaction times and same-different judgments through a model of psychological processes. Three representation models of dissimilarities are considered, the Minkowski power distance model, the linear model, and Tversky's feature matching model. Maximum likelihood estimation procedures are developed and implemented in the form of a FORTRAN program. An example is given to illustrate the kind of analyses that can be performed by the proposed method.
Univariate Repeated Measures Techniques Applied to Multivariate Data
Authors: Thomas D. Roland
Published: January 2006

Repeated measures designs in psychology have traditionally been analyzed by the univariate mixed model approach, in which the repeated measures effect is tested against an error term based on the subject by treatment interaction. This paper considers the extension of this analysis to designs in which the individual repeated measures are multivariate. Sufficient conditions for a valid multivariate mixed model analysis are given, and a test is described to determine whether or not given data satisfy these conditions.
Clustering with Relational Constraint
Authors: Ferligoj Anuška , Batagelj Vladimir
Published: January 2006

The paper deals with clustering problems where grouping is constrained by a symmetric and reflexive relation. For solving clustering problems with relational constraints two methods are adapted: the “standard” hierarchical clustering procedure based on the Lance and Williams formula, and local optimization procedure, CLUDIA. To illustrate these procedures, clusterings of the European countries are given based on the developmental indicators where the relation is determined by the geographical neighbourhoods of countries.
Constrained Canonical Correlation
Authors: DeSarbo Wayne S. , Hausman Robert E. , Lin Shen , Thompson Wesley
Published: January 2006

This paper explores some of the problems associated with traditional canonical correlation. A response surface methodology is developed to examine the stability of the derived linear functions, where one wishes to investigate how much the coefficients can change and still be in an ɛ-neighborhood of the globally optimum canonical correlation value. In addition, a discrete (or constrained) canonical correlation method is formulated where the derived coefficients of these linear functions are constrained to be in some small set, e.g., {1, 0, −1}, to aid in the interpretation of the results. An example concerning the psychographic responses of Wharton MBA students of the University of Pennsylvania regarding driving preferences and life-style considerations is provided.
Modified Biserial Correlation Coefficients
Authors: Kraemer Helena Chmura
Published: January 2006

Asymptotic distribution theory of Brogden's form of biserial correlation coefficient is derived and large sample estimates of its standard error obtained. Its efficiency relative to the biserial correlation coefficient is examined. Other modifications of the statistic are evaluated, and on the basis of these results, recommendations for choice of estimator of biserial correlation are presented.
The Asymptotic Distribution of Commonality Components
Authors: Hedges Larry V. , Olkin Ingram
Published: January 2006

Commonality components have been defined as a method of partitioning squared multiple correlations. In this paper, the asymptotic joint distribution of all 2k − 1 squared multiple correlations is derived. The asymptotic joint distribution of linear combinations of squared multiple correlations is obtained as a corollary. In particular, the asymptotic joint distribution of commonality components are derived as a special case. Simultaneous and nonsimultaneous asymptotic confidence intervals for commonality components can be obtained from this distribution.
On the Matrix Formulation of Kaiser's Varimax Criterion
Authors: Neudecker H.
Published: January 2006

The author provides a full-fledged matrix derivation of Sherin's matrix formulation of Kaiser's varimax criterion. He uses matrix differential calculus in conjunction with the Hadamard (or Schur) matrix product. Two results on Hadamard products are presented.
An Improvement of the Harman-Fukuda-Method for the Minres Solution in Factor Analysis
Authors: Hafner Robert
Published: January 2006

The method proposed by Harman and Fukuda to treat the so-called Heywood case in the minres method in factor analysis i.e., the case where the resulting communalities are greater than one, involves the frequent solution of eigenvalue problems. A simple method to treat this problem requiring less computing time and enjoying higher numerical stability is described in this paper.
A Note on the Estimation of the Level of Predictive Precision of a Fitted Linear Equation
Authors: Mendoza Jorge L.
Published: January 2006

A procedure that utilizes the sample multiple correlation to form a lower bound for the level of predictive precision of a fitted regression equation is suggested. The procedure is shown to yield probability statements which are true at least 100(1−α)% of the time.
Matching Models in the Analysis of Cross-Classifications
Authors: Hubert Lawrence J.
Published: January 2006

Inference models motivated by the combinatorial chance literature and the concept of object matching may be used in the analysis of a contingency table if the conditional assumption of fixed row and column totals is imposed. More specifically, by developing a matching reinterpretation for several problems of interest in the prediction analysis of cross-classifications—as defined by Hildebrand, Laing and Rosenthal, appropriate significance tests can be given that may differ from those justified by the more common multinomial models. In the course of the paper the distinction between a degree-1 statistic (based on the relationship between single objects) and a degree-2 statistic (based on the relationship between object pairs) is reviewed in some detail. Also, several specializations are presented to topics of current methodological importance in psychology; for instance, a number of references are made to the measurement of nominal scale response agreement between two raters.
On Meredith’s Solution for Weighted Procrustes Rotation
Authors: Nevels Klaas
Published: January 2006

Meredith developed a criterion for weighted procrustes rotation. The solution was given using Lagrange multipliers. We show that the solution can be obtained more simply, and that our approach enables us to transform the orthogonalization problem, also considered by Meredith, to that of minimizing a quadratic form on the unit sphere.
Simultaneous Factor Analysis of Dichotomous Variables in Several Groups
Authors: Muthén Bengt , Christoffersson Anders
Published: January 2006

A new method is proposed for a simultaneous factor analysis of dichotomous responses from several groups of individuals. The method makes it possible to compare factor loading pattern, factor variances and covariances, and factor means over groups. The method uses information from first and second order proportions and estimates the model by generalized least-squares. Hypotheses regarding different degrees of invariance over groups may be evaluated by a large-sample chi-square test.
Matrix Reorganization and Dynamic Programming: Applications to Paired Comparisons and Unidimensional Seriation
Authors: Hubert L. J. , Golledge R. G.
Published: January 2006

A recursive dynamic programming strategy is discussed for optimally reorganizing the rows and simultaneously the columns of an n × n proximity matrix when the objective function measuring the adequacy of a reorganization has a fairly simple additive structure. A number of possible objective functions are mentioned along with several numerical examples using Thurstone's paired comparison data on the relative seriousness of crime. Finally, the optimization tasks we propose to attack with dynamic programming are placed in a broader theoretical context of what is typically referred to as the quadratic assignment problem and its extension to cubic assignment.
Marginal Maximum Likelihood Estimation of Item Parameters: Application of an EM Algorithm
Authors: Bock R. Darrell , Aitkin Murray
Published: January 2006

Maximum likelihood estimation of item parameters in the marginal distribution, integrating over the distribution of ability, becomes practical when computing procedures based on an EM algorithm are used. By characterizing the ability distribution empirically, arbitrary assumptions about its form are avoided. The Em procedure is shown to apply to general item-response models lacking simple sufficient statistics for ability. This includes models with more than one latent dimension.
Optimal Item Difficulty for the Three-Parameter Normal Ogive Response Model
Authors: Wolfe John H.
Published: January 2006

In tailored testing, it is important to determine the optimal difficulty of the next item to present to the examinee. This paper shows that the difference that maximizes information for the three-parameter normal ogive response model is approximately 1.7 times the optimal difference θ − b for the three-parameter logistic model. Under the normal model, calculation of the optimal difficulty for minimizing the Bayes risk is equivalent to maximizing an associated information function.
A New Computational Method to Fit the Weighted Euclidean Distance Model
Authors: de Leeuw Jan , Pruzansky Sandra
Published: January 2006

This paper describes a computational method for weighted euclidean distance scaling which combines aspects of an “analytic” solution with an approach using loss functions. We justify this new method by giving a simplified treatment of the algebraic properties of a transformed version of the weighted distance model. The new algorithm is much faster than INDSCAL yet less arbitrary than other “analytic” procedures. The procedure, which we call SUMSCAL (subjective metric scaling), gives essentially the same solutions as INDSCAL for two moderate-size data sets tested.
A Multidimensional Scaling Model for the Size-Weight Illusion
Authors: Dunn Terrence R. , Harshman Richard A.
Published: January 2006

The kinds of individual differences in perceptions permitted by the weighted euclidean model for multidimensional scaling (e.g., INDSCAL) are much more restricted than those allowed by Tucker’s Three-mode Multidimensional Scaling (TMMDS) model or Carroll’s Idiosyncratic Scaling (IDIOSCAL) model. Although, in some situations the more general models would seem desirable, investigators have been reluctant to use them because they are subject to transformational indeterminacies which complicate interpretation. In this article, we show how these indeterminacies can be removed by constructing specific models of the phenomenon under investigation. As an example of this approach, a model of the size-weight illusion is developed and applied to data from two experiments, with highly meaningful results. The same data are also analyzed using INDSCAL. Of the two solutions, only the one obtained by using the size-weight model allows examination of individual differences in the strength of the illusion; INDSCAL can not represent such differences. In this sample, however, individual differences in illusion strength turn out to be minor. Hence the INDSCAL solution, while less informative than the size-weight solution, is nonetheless easily interpretable.
An Extension of the Rasch Model for Ratings Providing Both Location and Dispersion Parameters
Authors: Andrich David
Published: January 2006

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 Various Causes of Improper Solutions in Maximum Likelihood Factor Analysis
Authors: van Driel Otto P.
Published: January 2006

In the applications of maximum likelihood factor analysis the occurrence of boundary minima instead of proper minima is no exception at all. In the past the causes of such improper solutions could not be detected. This was impossible because the matrices containing the parameters of the factor analysis model were kept positive definite. By dropping these constraints, it becomes possible to distinguish between the different causes of improper solutions. In this paper some of the most important causes are discussed and illustrated by means of artificial and empirical data.