bluuebunny/crossref_metadata_2025 huggingface dataset:
Parameter Estimation in Latent Trait Models
Authors: Rigdon Steven E. , Tsutakawa Robert K.
Published: January 2006

Latent trait models for binary responses to a set of test items are considered from the point of view of estimating latent trait parameters θ= (θ1, …, θn) and item parameters β=(β1, …, βk), where βj may be vector valued. With θ considered a random sample from a prior distribution with parameter ϕ, the estimation of (θ, β) is studied under the theory of the EM algorithm. An example and computational details are presented for the Rasch model.
A Generalization of Verhelst's Solution for a Constrained Regression Problem in Alscal and Related MDS-Algorithms
Authors: ten Berge Jos M. F.
Published: January 2006

Verhelst derived a solution for a constrained regression problem which occurs in the interval measurement application of ALSCAL and related MDS-algorithms. In the present paper it is shown that Verhelst's solution is based on an implicit nonsingularity assumption. A general solution, which contains Verhelst's solution as a special case, is derived by a simple completing-the-squares type approach instead of partial differentiation with a Lagrange multiplier. In addition, this approach permits the identification of a small interval which uniquely contains the optimal value of a parameter needed to solve the special case where Verhelst's solution is valid.
On the Standard Error of the Modified Biserial Correlation
Authors: Koopman Raymond F.
Published: January 2006

A paradoxical implication of Kraemer's expression for the large-sample standard error of Brogden's form of the biserial correlation is identified, and a new expression is given which does not imply the paradox. However, numerical evidence is presented which calls into question the correctness of the expression.
Relationships between Redundancy Analysis, Canonical Correlation, and Multivariate Regression
Authors: Muller Keith E.
Published: January 2006

This paper attempts to clarify the nature of redundancy analysis and its relationships to canonical correlation and multivariate multiple linear regression. Stewart and Love introduced redundancy analysis to provide non-symmetric measures of the dependence of one set of variables on the other, as channeled through the canonical variates. Van den Wollenberg derived sets of variates which directly maximize the between set redundancy. Multivariate multiple linear regression on component scores (such as principal components) is considered. The problem is extended to include an orthogonal rotation of the components. The solution is shown to be identical to van den Wollenberg's maximum redundancy solution.
A Stochastic Model for Test-Retest Correlations
Authors: Morrison Donald G.
Published: January 2006

A simple stochastic model is formulated in order to determine the optimal time between the first test and the second test when the test-retest method of assessing reliability is used. A forgetting process and a change in true score process are postulated. The optimal time between tests is derived by maximizing the probability that the respondent has not remembered the response on the first test and has not had a change in true score. The resulting test-retest correlation is then found to be a linear function of the true reliability of the test, where the slope of this function is the key probability of not remembering and having no change in true score. Some numerical examples and suggestions for using the results in empirical studies are given. Specific recommendations are presented for improved design and analysis of intentions data.
A Monte Carlo Study of Thirty Internal Criterion Measures for Cluster Analysis
Authors: Milligan Glenn W.
Published: January 2006

A Monte Carlo evaluation of thirty internal criterion measures for cluster analysis was conducted. Artificial data sets were constructed with clusters which exhibited the properties of internal cohesion and external isolation. The data sets were analyzed by four hierarchical clustering methods. The resulting values of the internal criteria were compared with two external criterion indices which determined the degree of recovery of correct cluster structure by the algorithms. The results indicated that a subset of internal criterion measures could be identified which appear to be valid indices of correct cluster recovery. Indices from this subset could form the basis of a permutation test for the existence of cluster structure or a clustering algorithm.
A New Interpretation of Stochastic Test Models
Authors: Colonius Hans
Published: January 2006

A new look at latent trait models is proposed. The event of an item being solved by a person is related to the event that the momentary value of a person-specific random component is at least as large as the corresponding value of an item-specific random component. The Birnbaum logistic test model is shown to be generated by a bivariate extreme value distribution for the components. Some consequences of this interpretation are outlined.
A Note on Identification in the Oblique and Orthogonal Factor Analysis Models
Authors: Algina James
Published: January 2006

Conditions for removing the indeterminancy due to rotation are given for both the oblique and orthogonal factor analysis models. The conditions indicate why published counterexamples to conditions discussed by Jöreskog are not identifiable.
Estimation in the Power Law
Authors: Thomas Hoben
Published: January 2006

Most psychophysicists neglect to consider how error should be characterized in applications of the power law. Failures of the power law to agree with certain theoretical predictions may be due to bias of parameter estimates. A power law with lognormal product structure is proposed and approximately unbiased parameter estimates are given for several common estimation situations.
A Jackknife Test of Homogeneity of Variance with Paired Replicates Data
Authors: Rothstein Stuart M. , Bell William D. , Patrick Jayne A. , Miller Harry
Published: January 2006

A nonparametric test of dispersion with paired replicates data is described which involves jackknifing logarithmic transformations of the ratio of variance estimates for the pre- and post-treatment populations. Results from a Monte Carlo simulation show that the test performs well under Ho and has good power properties. Examples are given of applying the procedure on psychiatric data.
The Extension of Component Analysis to Four-Mode Matrices
Authors: Lastovicka John L.
Published: January 2006

A model for four-mode component analysis is developed and presented. The developed model, which is an extension of Tucker's three-mode factor analytic model, allows for the simultaneous analysis of all modes of a four-mode data matrix and the consideration of relationships among the modes. An empirical example based upon viewer perceptions of repetitive advertising shows the four-mode model applicable to real data.
When are Item Response Models Consistent with Observed Data?
Authors: Holland Paul W.
Published: January 2006

The problem of deciding whether a set of mental test data is consistent with any one of a large class of item response models is considered. The “classical” assumption of locla independence is weakened to a new condition, local nonnegative dependence (LND). Necessary and sufficient conditions are derived for a LND item response model to fit a set of data. This leads to a condition that a set of data must satisfy if it is to be representable by any item response model that assumes both local independence and monotone item characteristic curves. An example is given to show that LND is strictly weaker than local independence. Thus rejection of LND models implies rejection of all item response models that assume local independence for a given set of data.
Structure in Semantic Memory: A Probabilistic Approach using a Continuous Response Task
Authors: Millsap Roger E. , Meredith William
Published: January 2006

In the continuous response (CR) task, the subject is given the name of a familiar category and must respond with category examples within a fixed time limit. The response sequence and the interresponse times are the measured variables. A general model is proposed for performance in the CR task. The time taken to generate a response is identified with the shortest search time from among a set of simultaneous search processes. Each response has a generation probability that is a function of the parameters governing the search process durations. Data involving the category “states in the United States” are used to test the model. The model accounts for the performance of nearly all individuals in these data.
Analytic Smoothing for Equipercentile Equating Under the Common Item Nonequivalent Populations Design
Authors: Kolen Michael J. , Jarjoura David
Published: January 2006

A cubic spline method for smoothing equipercentile equating relationships under the common item nonequivalent populations design is described. Statistical techniques based on bootstrap estimation are presented that are designed to aid in choosing an equating method/degree of smoothing. These include: (a) asymptotic significance tests that compare no equating and linear equating to equipercentile equating; (b) a scheme for estimating total equating error and for dividing total estimated error into systematic and random components. The smoothing technique and statistical procedures are explored and illustrated using data from forms of a professional certification test.
A Model for Repeat Trial Product Tests
Authors: Buchanan Bruce
Published: January 2006

A non-forced choice model is developed that describes subject behavior on repeat trial discrimination tests of the “pick 1 of k” form. The model is developed from the Dirichlet distribution, and it allows for the derivation of individual true scores and of sampling properties for various constructs of interest. These results permit the analysis and comparison of test designs. The model is applied to issues such as forced vs. non-forced choice formats, the best number of alternatives at a choice point, and the selection of expert panels.
Reliability of Test Scores in Nonparametric Item Response Theory
Authors: Sijtsma Klaas , Molenaar Ivo W.
Published: January 2006

Three methods for estimating reliability are studied within the context of nonparametric item response theory. Two were proposed originally by Mokken (1971) and a third is developed in this paper. Using a Monte Carlo strategy, these three estimation methods are compared with four “classical” lower bounds to reliability. Finally, recommendations are given concerning the use of these estimation methods.
Solutions to Some Nonlinear Equations from Nonmetric Data
Authors: Rule Stanley J.
Published: January 2006

A method is presented to provide estimates of parameters of specified nonlinear equations from ordinal data generated from a crossed design. The analytic method, NOPE, is an iterative method in which monotone regression and the Gauss-Newton method of least squares are applied alternatively until a measure of stress is minimized. Examples of solutions from artificial data are presented together with examples of applications of the method to experimental results.
Testing a Unidimensional, Qualitative Unfolding Model for Attitudinal or Developmental Data
Authors: Davison Mark L.
Published: January 2006

Assuming that subject responses rank order stimuli by preference, statistical methods are presented for testing the hypothesis that responses conform to a unidimensional, qualitative unfolding model and to an a priori stimulus ordering. The model postulates that persons and stimulus variables are ordered along a single continuum and that subjects most prefer stimuli nearest their own position. The underlying continuum need not form an interval scale of the stimulus attribute. The general assumptions of the test for the unfolding model make it suitable for the analysis of structure in attitude responses, preference data, and developmental stage data.
On Kristof’s Test for a Linear Relation between True Scores of two Measures
Authors: Healy J. D.
Published: January 2006

The hypothesis that two variables have a perfect disattenuated correlation and hence measure the same trait, except for errors of measurement, is discussed. Equivalently, the underlying variables, the true scores, are related linearly. We show that several previously proposed ad hoc tests are in fact likelihood ratio tests. The cases when the linear relation is specified and when it is unspecified are both discussed.
A Lower Bound to the Probability of Choosing the Optimal Passing Score for a Mastery Test when there is an External Criterion
Authors: Wilcox Rand R.
Published: January 2006

Recently there has been interest in the problem of determining an optimal passing score for a mastery test when the purpose of the test is to predict success or failure on an external criterion. For the case of constant losses for the two error types, a method of determining an optimal passing score is readily derived using standard techniques. The purpose of this note is to describe a lower bound to the probability of identifying an optimal passing score based on a random sample of N examinees.
An Approach to n-Mode Components Analysis
Authors: Kapteyn Arie , Neudecker Heinz , Wansbeek Tom
Published: January 2006

As an extension of Lastovicka's four-mode components analysis an n-mode components analysis is developed. Using a convenient notation, both a canonical and a least squares solution are derived. The relation between both solutions and their computational aspects are discussed.
Statistics, Data Analysis and Psychometrika: Major Developments
Authors: Appelbaum Mark I.
Published: January 2006

Approximately 15% of all papers appearing in Psychometrika during the period 1960–1984 have been statistical or data analytic in nature. While these papers cover a very wide range of issues and are technically quite similar to those appearing in other major statistical journals, collectively they possess features which make them identifiably Psychometrika.
An Upper Bound for Sstress
Authors: de Leeuw Jan , Bettonvil Bert
Published: January 2006

In this note we derive an upper bound for the minimum for the multidimensional scaling loss function sstress. We conjecture that minimum sstress solution will be biased towards regular positioning of clumps of points over the surface of a sphere.
Empirical Bayes Estimation of Coefficients in the General Linear Model from Data of Deficient Rank
Authors: Braun Henry I. , Jones Douglas H. , Rubin Donald B. , Thayer Dorothy T.
Published: January 2006

Empirical Bayes methods are shown to provide a practical alternative to standard least squares methods in fitting high dimensional models to sparse data. An example concerning prediction bias in educational testing is presented as an illustration.
A More Powerful Method for Testing for Agreement between a Judge and a known Standard
Authors: Wackerly D. D. , Robinson D. H.
Published: January 2006

We assume that a judge's task is to categorize each of N subjects into one of r known classes. The design of primary interest is employed if the judge is presented with s groups, each containing r subjects, such that each group of size r consists of exactly one subject of each of the r types. The probability distribution for the total number of correct choices is developed and used to test the null hypothesis that the judge is “guessing” in favor of the alternative that he or she is operating at a better than chance level. The power of the procedure is shown to be superior to two other procedures which appear in the literature.
Problems with EM Algorithms for ML Factor Analysis
Authors: Bentler P. M. , Tanaka Jeffrey S.
Published: January 2006

Rubin and Thayer recently presented equations to implement maximum likelihood (ML) estimation in factor analysis via the EM algorithm. They present an example to demonstrate the efficacy of the algorithm, and propose that their recovery of multiple local maxima of the ML function “certainly should cast doubt on the general utility of second derivatives of the log likelihood as measures of precision of estimation.” It is shown here, in contrast, that these second derivatives verify that Rubin and Thayer did not find multiple local maxima as claimed. The only known maximum remains the one found by Jöreskog over a decade earlier. The standard errors obtained from the second derivatives and the Fisher information matrix thus remain appropriate where ML assumptions are met. The advantages of the EM algorithm over other algorithms for ML factor analysis remain to be demonstrated.
Expanding the Rasch Model to a General Model having more than One Dimension
Authors: Stegelmann Werner
Published: January 2006

The well-known Rasch model is generalized to a multicomponent model, so that observations of component events are not needed to apply the model. It is shown that the generalized model has retained the property of the specific objectivity of the Rasch model. For a restricted variant of the model, maximum likelihood estimates of its parameters and a statistical test of the model are given. The results of an application to a mathematics test involving six components are described.
The Effect of Affine Transformation on Redundancy Analysis
Authors: Dawson-Saunders Beth K. , Tatsuoka Maurice M.
Published: January 2006

Canonical redundancy analysis provides an estimate of the amount of shared variance between two sets of variables and provides an alternative to canonical correlation. The proof that the total redundancy is equal to the average squared multiple correlation coefficient obtained by regressing each variable in the criterion set on all variables in the predictor set is generalized to the case in which there are a larger number of criterion than predictor variables. It is then shown that the redundancy for the criterion set of variables is invariant under affine transformation of the predictor variables, but not invariant under transformation of the criterion variables.
Gleaning in the Field of Dual Scaling
Authors: Nishisato Shizuhiko
Published: January 2006

Some historical background and preliminary technical information are first presented, and then a number of hidden, but important, methodological aspects of dual scaling are illustrated and discussed: normed versus projected weights, the amount of information accounted for by each solution, a perfect solution to the problem of multidimensional unfolding, multidimensional quantification space, graphical display, number-of-option problems, option standardization versus item standardization, and asymmetry of symmetric (dual) scaling. Contrary to the common perception that dual scaling and similar quantification methods are now mathematically transparent, the present study demonstrates how much more needs to be clarified for routine use of the method to arrive at valid conclusions. Data analysis must be carried out in such a way that common sense, intuition and sound logic will prevail.
The Analysis of Multitrait-Multimethod Matrices Via Constrained Components Analysis
Authors: Kiers Henk A. L. , Takane Yoshio , ten Berge Jos M. F.
Published: January 2006

Multitrait-Multimethod (MTMM) matrices are often analyzed by means of confirmatory factor analysis (CFA). However, fitting MTMM models often leads to improper solutions, or non-convergence. In an attempt to overcome these problems, various alternative CFA models have been proposed, but with none of these the problem of finding improper solutions was solved completely. In the present paper, an approach is proposed where improper solutions are ruled out altogether and convergence is guaranteed. The approach is based on constrained variants of components analysis (CA). Besides the fact that these methods do not give improper solutions, they have the advantage that they provide component scores which can later on be used to relate the components to external variables. The new methods are illustrated by means of simulated data, as well as empirical data sets.
Loglinear Rasch models for the Analysis of Stability and Change
Authors: Meiser Thorsten
Published: January 2006

Loglinear unidimensional and multidimensional Rasch models are considered for the analysis of repeated observations of polytomous indicators with ordered response categories. Reparameterizations and parameter restrictions are provided which facilitate specification of a variety of hypotheses about latent processes of change. Models of purely quantitative change in latent traits are proposed as well as models including structural change. A conditional likelihood ratio test is presented for the comparison of unidimensional and multiple scales Rasch models. In the context of longitudinal research, this renders possible the statistical test of homogeneity of change against subject-specific change in latent traits. Applications to two empirical data sets illustrate the use of the models.
Regression Estimation and Post-Stratification in Factor Analysis
Authors: Skinner C. J.
Published: January 2006

Regression estimation and poststratification are methods used in survey sampling to estimate a population mean, when additional information is available for some auxiliary variables. The extension of these methods to factor analysis is considered. These methods may be used either to improve the efficiency of estimation or to adjust for the effects of nonrandom selection. The estimation procedure may be formulated in a LISREL framework.
Conjunctive Item Response Theory Kernels
Authors: Jannarone Robert J.
Published: January 2006

Conjunctive item response models are introduced such that (a) sufficient statistics for latent traits are not necessarily additive in item scores; (b) items are not necessarily locally independent; and (c) existing compensatory (additive) item response models including the binomial, Rasch, logistic, and general locally independent model are special cases. Simple estimates and hypothesis tests for conjunctive models are introduced and evaluated as well. Conjunctive models are also identified with cognitive models that assume the existence of several individually necessary component processes for a global ability. It is concluded that conjunctive models and methods may show promise for constructing improved tests and uncovering conjunctive cognitive structure. It is also concluded that conjunctive item response theory may help to clarify the relationships between local dependence, multidimensionality, and item response function form.
Sample Size and Bentler and Bonett's Nonnormed Fit Index
Authors: Bollen Kenneth A.
Published: January 2006

Bentler and Bonett's nonnormed fit index is a widely used measure of goodness of fit for the analysis of covariance structures. This note shows that contrary to what has been claimed the nonnormed fit index is dependent on sample size. Specifically for a constant value of a fitting function, the nonnormed index is inversely related to sample size. A simple alternative fit measure is proposed that removes this dependency. In addition, it is shown that this new measure as well as the old nonnormed fit index can be applied to any fitting function that measures the deviation of the observed covariance matrix from the covariance matrix implied by the parameter estimates for a model.
Tests for Equality of Several Alpha Coefficients when their Sample Estimates are Dependent
Authors: Woodruff David J. , Feldt Leonard S.
Published: January 2006

In a variety of measurement situations, the researcher may wish to compare the reliabilities of several instruments administered to the same sample of subjects. This paper presents eleven statistical procedures which test the equality of m coefficient alphas when the sample alpha coefficients are dependent. Several of the procedures are derived in detail, and numerical examples are given for two. Since all of the procedures depend on approximate asymptotic results, Monte Carlo methods are used to assess the accuracy of the procedures for sample sizes of 50, 100, and 200. Both control of Type I error and power are evaluated by computer simulation. Two of the procedures are unable to control Type I errors satisfactorily. The remaining nine procedures perform properly, but three are somewhat superior in power and Type I error control.
A Nonrandomized Minimax Solution for Passing Scores in the Binomial Error Model
Authors: Huynh Huynh
Published: January 2006

A nonrandomized minimax solution is presented for passing scores in the binomial error model. The computation does not require prior knowledge regarding an individual examinee or group test data for a population of examinees. The optimum passing score minimizes the maximum risk which would be incurred by misclassifications. A closed-form solution is provided for the case of constant losses, and tables are presented for a variety of situations including linear and quadratic losses. A scheme which allows for correction for guessing is also described.
A Feasible Method for Standard Errors of Estimate in Maximum Likelihood Factor Analysis
Authors: Jennrich R. I. , Clarkson D. B.
Published: January 2006

A jackknife-like procedure is developed for producing standard errors of estimate in maximum likelihood factor analysis. Unlike earlier methods based on information theory, the procedure developed is computationally feasible on larger problems. Unlike earlier methods based on the jackknife, the present procedure is not plagued by the factor alignment problem, the Heywood case problem, or the necessity to jackknife by groups. Standard errors may be produced for rotated and unrotated loading estimates using either orthogonal or oblique rotation as well as for estimates of unique factor variances and common factor correlations. The total cost for larger problems is a small multiple of the square of the number of variables times the number of observations used in the analysis. Examples are given to demonstrate the feasibility of the method.
Inequalities Among Lower Bounds to Reliability: With Applications to Test Construction and Factor Analysis
Authors: Bentler P. M. , Woodward J. Arthur
Published: January 2006

A chain of lower-bound inequalities leading to the greatest lower bound to reliability is established for the internal consistency of a composite of unit-weighted components. The chain includes the maximum split-half coefficient, the lowest coefficient consistent with nonimaginary common factors, and the lowest coefficient consistent with nonimaginary common and unique factors. Optimization theory is utilized to determine the conditions that are requisite for the inequalities. Convergence proofs demonstrate that the coefficients can be attained. Rapid algorithms obtain estimates of the coefficients with sample data. The theory yields methods for splitting items into maximally similar sets and for exploratory factor analysis based on a theoretical solution to the communality problem.
Individual Distributions Under Ordinal Measurement
Authors: Schulman Robert S.
Published: January 2006

For ordinal measurement the concept of an individual propensity distribution is developed. For any given individual the mean of this distribution is his true score, for which estimation procedures are discussed. Two measures of individual dispersion are considered and their distributions derived in the null case. These measures are shown to be counterparts at the individual level of Kendall’s tau and Spearman’s rho. Estimation of the two dispersion measures from sample data is investigated, and the relation of these estimates to the variance of the individual propensity distribution is derived.
Bayesian Estimation in the Two-Parameter Logistic Model
Authors: Swaminathan Hariharan , Gifford Janice A.
Published: January 2006

A Bayesian procedure is developed for the estimation of parameters in the two-parameter logistic item response model. Joint modal estimates of the parameters are obtained and procedures for the specification of prior information are described. Through simulation studies it is shown that Bayesian estimates of the parameters are superior to maximum likelihood estimates in the sense that they are (a) more meaningful since they do not drift out of range, and (b) more accurate in that they result in smaller mean squared differences between estimates and true values.
A Model for Repeat Paired Comparison Preference Tests
Authors: Buchanan Bruce
Published: January 2006

A model is proposed that describes subject behavior on repeat paired comparison preference tests. The model extends prior work in this area in that it explicitly allows for abstentions and permits the derivation of individual true scores for discrimination ability as well as conditional estimates of proportionate preference. With these results, the properties of a paired comparison test can be thoroughly explored. An empirical example is presented, and test design issues are considered. In particular, repeat paired comparison preference tests are shown to be inherently less efficient discrimination tests than are pick 1 of 2 tests.
Quartic Rotation Criteria and Algorithms
Authors: Clarkson Douglas B. , Jennrich Robert I.
Published: January 2006

Most of the currently used analytic rotation criteria for simple structure in factor analysis are summarized and identified as members of a general symmetric family of quartic criteria. A unified development of algorithms for orthogonal and direct oblique rotation using arbitrary criteria from this family is given. These algorithms represent fairly straightforward extensions of present methodology, and appear to be the best methods currently available.
Sequential Social Network Data
Authors: Wasserman Stanley , Iacobucci Dawn
Published: January 2006

A new method is proposed for the statistical analysis of dyadic social interaction data measured over time. The data to be studied are assumed to be realizations of a social network of a fixed set of actors interacting on a single relation. The method is based on loglinear models for the probabilities for various dyad (or actor pair) states and generalizes the statistical methods proposed by Holland and Leinhardt (1981), Fienberg, Meyer, & Wasserman (1985), and Wasserman (1987) for social network data. Two statistical models are described: the first is an “associative” approach that allows for the study of how the network has changed over time; the second is a “predictive” approach that permits the researcher to model one time point as a function of previous time points. These approaches are briefly contrasted with earlier methods for the sequential analysis of social networks and are illustrated with an example of longitudinal sociometric data.
A Note to Bollen's Alternative Fit Measure
Authors: Balderjahn Ingo
Published: January 2006

The problem of dependence of the nonnormed fit index on sample size in covariance structure analysis is discussed. Contrary to Bollen (1986) we show that the mean of the nonnormed fit index is independent of sample size for true and almost true models whereas Bollen's alternative index does depend on sample size.
Least-Squares Theory Based on General Distributional Assumptions with an Application to the Incomplete Observations Problem
Authors: Van Praag B. M. S. , Dijkstra T. K. , Van Velzen J.
Published: January 2006

The linear regression model y=β′x+ ε is reanalyzed. Taking the modest position that β′x is an approximation of the “best” predictor of y we derive the asymptotic distribution of b and R2, under mild assumptions.

The method of derivation yields an easy answer to the estimation of β from a data set which contains incomplete observations, where the incompleteness is random.
The Simultaneous Maximization of Congruence for Two or More Matrices under Orthogonal Rotation
Authors: Brokken Frank B.
Published: January 2006

While a rotation procedure currently exists to maximize simultaneously Tucker's coefficient of congruence between corresponding factors of two factor matrices under orthogonal rotation of one factor matrix, only approximate solutions are known for the generalized case where two or more matrices are rotated. A generalization and modification of the existing rotation procedure to simultaneously maximize the congruence is described. An example using four data matrices, comparing the generalized congruence maximization procedure with alternative rotation procedures, is presented. The results show a marked improvement of the obtained congruence using the generalized congruence maximization procedure compared to other procedures, without a significant loss of success with respect to the least squares criterion. A computer program written by the author to perform the rotations is briefly discussed.
Estimating the Imputed Social Cost of Errors of Measurement
Authors: Lord Frederic M.
Published: January 2006

If a loss function is available specifying the social cost of an error of measurement in the score on a unidimensional test, an asymptotic method, based on item response theory, is developed for optimal test design for a specified target population of examinees. Since in the real world such loss functions are not available, it is more useful to reverse this process; thus a method is developed for finding the loss function for which a given test is an optimally designed test for the target population. An illustrative application is presented for one operational test.
Covariance Structure Analysis in Several Populations
Authors: Lee Sik-Yum , Tsui Kwok-Leung
Published: January 2006

This paper is concerned with the study of covariance structural models in several populations. Estimation theory of the parameters that are subject to general functional restraints is developed based on the generalized least squares approach. Asymptotic properties of the constrained estimator are studied; and asymptotic chi-square tests are presented to evaluate appropriate model comparisons. The method of multipliers and the standard reparametrization technique are discussed in obtaining the estimates. The methodology is demonstrated by a set of real data.
The Polyserial Correlation Coefficient
Authors: Olsson Ulf , Drasgow Fritz , Dorans Neil J.
Published: January 2006

The polyserial and point polyserial correlations are discussed as generalizations of the biserial and point biserial correlations. The relationship between the polyserial and point polyserial correlation is derived. The maximum likelihood estimator of the polyserial correlation is compared with a two-step estimator and with a computationally convenient ad hoc estimator. All three estimators perform reasonably well in a Monte Carlo simulation. Some practical applications of the polyserial correlation are described.
Parameter Estimation for the Thurstone Case III Model
Authors: Mackay David B. , Chaiy Seoil
Published: January 2006

A Monte Carlo study was conducted to investigate the ability of three estimation criteria to recover the parameters of Case V and Case III models from comparative judgment data. Significant differences in recovery are shown to exist.
Tests of Sphericity of Normal Distributions and the Analysis of Repeated Measures Designs
Authors: Grieve A. P.
Published: January 2006

The locally best invariant test statistic for testing sphericity of normal distributions is shown to be a simple function of the Box/Geisser-Greenhouse degrees of freedom correction factor in a repeated measures design. Because of this relationship it provides a more intuitively appealing test of the necessary and sufficient conditions for valid F-tests in repeated measures analysis of variance than the likelihood ratio test. The properties of the two tests are compared and tables of the critical values of the Box/Geisser-Greenhouse correction factor are given.
Stepwise Variable Selection in Factor Analysis
Authors: Kano Yutaka , Harada Akira
Published: January 2006

It is very important to choose appropriate variables to be analyzed in multivariate analysis when there are many observed variables such as those in a questionnaire. What is actually done in scale construction with factor analysis is nothing but variable selection.

In this paper, we take several goodness-of-fit statistics as measures of variable selection and develop backward elimination and forward selection procedures in exploratory factor analysis. Once factor analysis is done for a certain number p of observed variables (the p-variable model is labeled the current model), simple formulas for predicted fit measures such as chi-square, GFI, CFI, IFI and RMSEA, developed in the field of the structural equation modeling, are provided for all models obtained by adding an external variable (so that the number of variables is p + 1) and for those by deleting an internal variable (so that the number is p − 1), provided that the number of factors is held constant.

A program SEFA (Stepwise variable selection in Exploratory Factor Analysis) is developed to actually obtain a list of the fit measures for all such models. The list is very useful in determining which variable should be dropped from the current model to improve the fit of the current model. It is also useful in finding a suitable variable that may be added to the current model. A model with more appropriate variables makes more stable inference in general.

The criteria traditionally often used for variable selection is magnitude of communalities. This criteria gives a different choice of variables and does not improve fit of the model in most cases.
On Equivariance and Invariance of Standard Errors in Three Exploratory Factor Models
Authors: Yuan Ke-Hai , Bentler Peter M.
Published: January 2006

Current practice in factor analysis typically involves analysis of correlation rather than covariance matrices. We study whether the standard z-statistic that evaluates whether a factor loading is statistically necessary is correctly applied in such situations and more generally when the variables being analyzed are arbitrarily rescaled. Effects of rescaling on estimated standard errors of factor loading estimates, and the consequent effect on z-statistics, are studied in three variants of the classical exploratory factor model under canonical, raw varimax, and normal varimax solutions. For models with analytical solutions we find that some of the standard errors as well as their estimates are scale equivariant, while others are invariant. For a model in which an analytical solution does not exist, we use an example to illustrate that neither the factor loading estimates nor the standard error estimates possess scale equivariance or invariance, implying that different conclusions could be obtained with different scalings. Together with the prior findings on parameter estimates, these results provide new guidance for a key statistical aspect of factor analysis.
Internal Consistency of Tests: Analyses Old and New
Authors: Cronbach Lee J.
Published: January 2006

A coefficient derived from communalities of test parts has been proposed as greatest lower bound to Guttman's “immediate retest reliability.” The communalities have at times been calculated from covariances between item sets, which tends to underestimate appreciably. When items are experimentally independent, a consistent estimate of the greatest defensible internal-consistency coefficient is obtained by factoring item covariances. In samples of modest size, this analysis capitalizes on chance; an estimate subject to less upward bias is suggested. For estimating alternate-forms reliability, communality-based coefficients are less appropriate than stratified alpha.
Some Properties of Two Measures of Multivariate Association
Authors: van den Burg Willem , Lewis Charles
Published: January 2006

Two kinds of measures of multivariate association, based on Wilks' Λ and the Bartlett-Nanda-Pillai trace criterion V, respectively, are compared in terms of properties of the univariate R2 which they generalize. A unified set of derivations of the properties is provided which are self-contained and not restricted to decompositions in canonical variates. One conclusion is that a symmetric index based on Λ allows generalization of the multiplicative decomposition of R2 in terms of squared partial correlations, but not the additive decomposition in terms of squared semipartial correlations, while the reverse is true for an asymmetric index based on V.
Using Phantom and Imaginary Latent Variables to Parameterize Constraints in Linear Structural Models
Authors: Rindskopf David
Published: January 2006

The most widely-used computer programs for structural equation models analysis are the LISREL series of Jöreskog and Sörbom. The only types of constraints which may be made directly are fixing parameters at a constant value and constraining parameters to be equal. Rindskopf (1983) showed how these simple properties could be used to represent models with more complicated constraints, namely inequality constraints on unique variances. In this paper, two new concepts are introduced which enable a much wider variety of constraints to be made. The concepts, “phantom” and “imaginary” latent variables, allow fairly general equality and inequality constraints on factor loadings and structural model coefficients.
Orthogonal Rotations to Maximal Agreement for Two or More Matrices of Different Column Orders
Authors: ten Berge Jos M. F. , Knol Dirk L.
Published: January 2006

Methods for orthogonal Procrustes rotation and orthogonal rotation to a maximal sum of inner products are examined for the case when the matrices involved have different numbers of columns. An inner product solution offered by Cliff is generalized to the case of more than two matrices. A nonrandom start for a Procrustes solution suggested by Green and Gower is shown to give better results than a random start. The Green-Gower Procrustes solution (with nonrandom start) is generalized to the case of more than two matrices. Simulation studies indicate that both the generalized inner product solution and the generalized Procrustes solution tend to attain their global optima within acceptable computation times. A simple procedure is offered for approximating simple structure for the rotated matrices without affecting either the Procrustes or the inner product criterion.
Linear Relations among k Sets of Variables
Authors: Van de Geer John P.
Published: January 2006

A family of solutions for linear relations among k sets of variables is proposed. It is shown how these solutions apply for k = 2, and how they can be generalized from there to k ≥ 3.

The family of solutions depends on three independent choices: (i) to what extent a solution may be influenced by differences in variances of components within each set; (ii) to what extent the sets may be differentially weighted with respect to their contribution to the solution—including orthogonality constraints; (iii) whether or not individual sets of variables may be replaced by an orthogonal and unit normalized basis.

Solutions are compared with respect to their optimality properties. For each solution the appropriate stationary equations are given. For one example it is shown how the determinantal equation of the stationary equations can be interpreted.
Rating Scale Analysis with Latent Class Models
Authors: Rost Jürgen
Published: January 2006

A general approach for analyzing rating data with latent class models is described, which parallels rating models in the framework of latent trait theory. A general rating model as well as a two-parameter model with location and dispersion parameters, analogous to Andrich's Dislocmodel are derived, including parameter estimation via the EM-algorithm. Two examples illustrate the application of the models and their statisticalcontrol. Model restrictions through equality constrains are discussed and multiparameter generalizations are outlined.
Eta, Disco, Odisco, and F
Authors: Guttman Louis
Published: January 2006

Two coefficients are proposed for measuring the extent of overlap in distributions as a direct function of the variance between the arithmetic means (“disco” and “odisco”). They are designed to answer such questions as: “Given the value of a numerical variable x, to which population should an individual be assigned so that minimum error would be incurred?” This is just the reverse of the question addressed by ANOVA. These coefficients are shown to be analytic in x and they are related to Pearson's eta and Fisher's F. Extensions of these coefficients (designed for univariate, one-way discrimination) to k-way and multivariate discriminant analysis and measurement of “interaction” are suggested.
Comparing Item Characteristic Curves
Authors: Rosenbaum Paul R.
Published: January 2006

Test items are often evaluated and compared by contrasting the shapes of their item characteristics curves (ICC's) or surfaces. The current paper develops and applies three general (i.e., nonparametric) comparisons of the shapes of two item characteristic surfaces: (i) proportional latent odds, (ii) uniform relative difficulty, and (iii) item sensitivity. Two items may be compared in these ways while making no assumption about the shapes of item characteristic surfaces for other items, and no assumption about the dimensionality of the latent variable. Also studied is a method for comparing the relative shapes of two item characteristic curves in two examinee populations.
Retrospective Questions and Correlations
Authors: Sikkel Dirk , Jelierse Gerard
Published: January 2006

In survey research it is not uncommon to ask questions of the following type: “How many times did you undertake action a in reference period T of length τ?.” The relationship is established between τ and the correlation of the number of reported actions with some background variable. To this end it is assumed that the process of actions satisfies a renewal model with individual heterogeneity. Also a model has to be formulated for possible recall effects. Applications are given in the field of medical consumption.
On the Inhomogeneity of a Test Compounded of Two Rasch Homogeneous Subscales
Authors: Formann Anton K. , Rop Ilse
Published: January 2006

Van den Wollenberg stated a theorem specifying the conditions for a test, which is composed of two Rasch homogeneous subscales, as also behaving Rasch homogeneously. In our note, simple numerical counterexamples are given for which this theorem fails.
A Dynamic Factor Model for the Analysis of Multivariate Time Series
Authors: Molenaar Peter C. M.
Published: January 2006

As a method to ascertain the structure of intra-individual variation, P-technique has met difficulties in the handling of a lagged covariance structure. A new statistical technique, coined dynamic factor analysis, is proposed, which accounts for the entire lagged covariance function of an arbitrary second order stationary time series. Moreover, dynamic factor analysis is shown to be applicable to a relatively short stretch of observations and therefore is considered worthwhile for psychological research. At several places the argumentation is clarified through the use of examples.
A New Proof for Monotone Likelihood Ratio for the Sum of Independent Bernoulli Random Variables
Authors: Huynh Huynh
Published: January 2006

By use of an inequality of Marcus and Lopes for elementary symmetric functions, a new proof is presented for the following result by Ghurye and Wallace: Given that the independent random variables Xj are Bernoulli with success probability pj (θ) strictly between 0 and 1 and nondecreasing in θ, the sum ΣXj has monotone likelihood ratio.
A General Model for Preferential and Triadic Choice in Terms of Central F Distribution Functions
Authors: Ennis Daniel M. , Johnson Norman L.
Published: January 2006

A model for preferential and triadic choice is derived in terms of weighted sums of central F distribution functions. This model is a probabilistic generalization of Coombs' (1964) unfolding model and special cases, such as the model of Zinnes and Griggs (1974), can be derived easily from it. This new form extends previous work by Mullen and Ennis (1991) and provides more insight into the same problem that they discussed.