bluuebunny/crossref_metadata_2025 huggingface dataset:
Constructing a Covariance Matrix that Yields a Specified Minimizer and a Specified Minimum Discrepancy Function Value
Authors: Cudeck Robert , Browne Michael W.
Published: February 2006

A method is presented for constructing a covariance matrix Σ*0 that is the sum of a matrix Σ(γ0) that satisfies a specified model and a perturbation matrix,E, such that Σ*0=Σ(γ0) +E. The perturbation matrix is chosen in such a manner that a class of discrepancy functions F(Σ*0, Σ(γ0)), which includes normal theory maximum likelihood as a special case, has the prespecified parameter value γ0 as minimizer and a prespecified minimum δ A matrix constructed in this way seems particularly valuable for Monte Carlo experiments as the covariance matrix for a population in which the model does not hold exactly. This may be a more realistic conceptualization in many instances. An example is presented in which this procedure is employed to generate a covariance matrix among nonnormal, ordered categorical variables which is then used to study the performance of a factor analysis estimator.
Extensions of Rasch's Multiplicative Poisson Model
Authors: Jansen Margo G. H. , van Duijn Marijtje A. J.
Published: February 2006

Consideration will be given to a model developed by Rasch that assumes scores observed on some types of attainment tests can be regarded as realizations of a Poisson process. The parameter of the Poisson distribution is assumed to be a product of two other parameters, one pertaining to the ability of the subject and a second pertaining to the difficulty of the test. Rasch's model is expanded by assuming a prior distribution, with fixed but unknown parameters, for the subject parameters. The test parameters are considered fixed. Secondly, it will be shown how additional between- and within-subjects factors can be incorporated. Methods for testing the fit and estimating the parameters of the model will be discussed, and illustrated by empirical examples.
An Efficient Algorithm for TUCKALS3 on Data with Large Numbers of Observation Units
Authors: Kiers Henk A. L. , Kroonenberg Pieter M. , ten Berge Jos M. F.
Published: February 2006

A modification of the TUCKALS3 algorithm is proposed that handles three-way arrays of order I × J × K for any I. When I is much larger than JK, the modified algorithm needs less work space to store the data during the iterative part of the algorithm than does the original algorithm. Because of this and the additional feature that execution speed is higher, the modified algorithm is highly suitable for use on personal computers.
Computing Maximum Likelihood Estimates of Loglinear Models from Marginal Sums with Special Attention to Loglinear Item Response Theory
Authors: Kelderman Henk
Published: February 2006

In this paper algorithms are described for obtaining the maximum likelihood estimates of the parameters in loglinear models. Modified versions of the iterative proportional fitting and Newton-Raphson algorithms are described that work on the minimal sufficient statistics rather than on the usual counts in the full contingency table. This is desirable if the contingency table becomes too large to store. Special attention is given to loglinear IRT models that are used for the analysis of educational and psychological test data. To calculate the necessary expected sufficient statistics and other marginal sums of the table, a method is described that avoids summing large numbers of elementary cell frequencies by writing them out in terms of multiplicative model parameters and applying the distributive law of multiplication over summation. These algorithms are used in the computer program LOGIMO. The modified algorithms are illustrated with simulated data.
Assessing Local Influence for Specific Restricted Likelihood: Application to Factor Analysis
Authors: Kwan C. W. , Fung W. K.
Published: February 2006

In restricted statistical models, since the first derivatives of the likelihood displacement are often nonzero, the commonly adopted formulation for local influence analysis is not appropriate. However, there are two kinds of model restrictions in which the first derivatives of the likelihood displacement are still zero. General formulas for assessing local influence under these restrictions are derived and applied to factor analysis as the usually used restriction in factor analysis satisfies the conditions. Various influence schemes are introduced and a comparison to the influence function approach is discussed. It is also shown that local influence for factor analysis is invariant to the scale of the data and is independent of the rotation of the factor loadings.
Describing the Elephant: Structure and Function in Multivariate Data
Authors: McDonald Roderick P.
Published: February 2006

There is a unity underlying the diversity of models for the analysis of multivariate data. Essentially, they constitute a family models, most generally nonlinear, for structural/functional relations between variables drawn from a behavior domain.
Transformation to Achieve a Longitudinally Stationary Factor Pattern Matrix
Authors: Bieber Stephen L. , Meredith William
Published: February 2006

Meredith's method of extracting a factorially invariant solution is adapted to longitudinal settings. An explorational estimation procedure is presented which attempts to identify the longitudinal factor components of an across occasion variance-covariance matrix. This is effected by transforming an initial factor pattern matrix to stationarity. The estimation is performed in two parts, the first employing a stepwise algorithm to ascertain the dimensionality and existence of the longitudinal components and the second being the direct estimation of the existing factor pattern.
A New Estimator of the Uniqueness in Factor Analysis
Authors: Ihara Masamori , Kano Yutaka
Published: February 2006

A closed form estimator of the uniqueness (unique variance) in factor analysis is proposed. It has analytically desirable properties—consistency, asymptotic normality and scale invariance. The estimation procedure is given through the application to the two sets of Emmett's data and Holzinger and Swineford's data. The new estimator is shown to lead to values rather close to the maximum likelihood estimator.
Bayesian Estimation in the Three-Parameter Logistic Model
Authors: Swaminathan Hariharan , Gifford Janice A.
Published: February 2006

A joint Bayesian estimation procedure for the estimation of parameters in the three-parameter logistic model is developed in this paper. Procedures for specifying prior beliefs for the parameters are given. It is shown through simulation studies that the Bayesian procedure (i) ensures that the estimates stay in the parameter space, and (ii) produces better estimates than the joint maximum likelihood procedure as judged by such criteria as mean squared differences between estimates and true values.
Prediction and Classification in Nonlinear Data Analysis: Something Old, Something New, Something Borrowed, Something Blue
Authors: Meulman Jacqueline J.
Published: February 2006

Prediction and classification are two very active areas in modern data analysis. In this paper, prediction with nonlinear optimal scaling transformations of the variables is reviewed, and extended to the use of multiple additive components, much in the spirit of statistical learning techniques that are currently popular, among other areas, in data mining. Also, a classification/clustering method is described that is particularly suitable for analyzing attribute-value data from systems biology (genomics, proteomics, and metabolomics), and which is able to detect groups of objects that have similar values on small subsets of the attributes.
Tucker2 Hierarchical Classes Analysis
Authors: Ceulemans Eva , Van Mechelen Iven
Published: February 2006

This paper presents a new hierarchical classes model, called Tucker2-HICLAS, for binary three-way three-mode data. As any three-way hierarchical classes model, the Tucker2-HICLAS model includes a representation of the association relation among the three modes and a hierarchical classification of the elements of each mode. A distinctive feature of the Tucker2-HICLAS model, being closely related to the Tucker3-HICLAS model (Ceulemans, Van Mechelen & Leenen, 2003), is that one of the three modes is minimally reduced and, hence, that the differences among the association patterns of the elements of this mode are maximally retained in the model. Moreover, as compared to Tucker3-HICLAS, Tucker2-HICLAS implies three rather than four different types of parameters and as such is simpler to interpret. Two types of Tucker2-HICLAS models are distinguished: a disjunctive and a conjunctive type. An algorithm for fitting the Tucker2-HICLAS model is described and evaluated in a simulation study. The model is illustrated with longitudinal data on interpersonal emotions.
The Proper Sequence for Correcting Correlation Coefficients for Range Restriction and Unreliability
Authors: Stauffer Joseph M. , Mendoza Jorge L.
Published: February 2006

Corrections of correlations for range restriction (i.e., selection) and unreliability are common in psychometric work. The current rule of thumb for determining the order in which to apply these corrections looks to the nature of the reliability estimate (i.e., restricted or unrestricted). While intuitive, this rule of thumb is untenable when the correction includes the variable upon which selection is made, as is generally the case. Using classical test theory, we show that it is the nature of the range restriction, not the nature of the available reliability coefficient, that determines the sequence for applying corrections for range restriction and unreliability.
General Ability Measurement: An Application of Multidimensional Item Response Theory
Authors: Segall Daniel O.
Published: February 2006

Two new methods for improving the measurement precision of a general test factor are proposed and evaluated. One new method provides a multidimensional item response theory estimate obtained from conventional administrations of multiple-choice test items that span general and nuisance dimensions. The other method chooses items adaptively to maximize the precision of the general ability score. Both methods display substantial increases in precision over alternative item selection and scoring procedures. Results suggest that the use of these new testing methods may significantly enhance the prediction of learning and performance in instances where standardized tests are currently used.
Rotation in the Dynamic Factor Modeling of Multivariate Stationary Time Series
Authors: Molenaar Peter C. M. , Nesselroade John R.
Published: February 2006

A special rotation procedure is proposed for the exploratory dynamic factor model for stationary multivariate time series. The rotation procedure applies separately to each univariate component series of a q-variate latent factor series and transforms such a component, initially represented as white noise, into a univariate moving-average. This is accomplished by minimizing a so-called state-space criterion that penalizes deviations of the rotated solution from a generalized state-space model with only instantaneous factor loadings. Alternative criteria are discussed in the closing section. The results of an empirical application are presented in some detail.
On Weighted Procrustes and Hyperplane Fitting in Factor Analytic Rotation
Authors: Meredith William
Published: February 2006

A weighted collinearity criterion for Procrustean rotation is developed, and it is shown that special cases with respect to the choice of weights and a vector norm are forms of hyperplane fitting, classical oblique Procrustes, etc.; a family of Procrustean transformation procedures is thereby generated. Numerical illustrations utilizing the Holzinger-Swineford data are presented.
Lower Bounds for the Reliability of the Total Score on a Test Composed of Non-Homogeneous Items: I: Algebraic Lower Bounds
Authors: Jackson Paul H. , Agunwamba Christian C.
Published: February 2006

Let ∑x be the (population) dispersion matrix, assumed well-estimated, of a set of non-homogeneous item scores. Finding the greatest lower bound for the reliability of the total of these scores is shown to be equivalent to minimizing the trace of ∑x by reducing the diagonal elements while keeping the matrix non-negative definite. Using this approach, Guttman's bounds are reviewed, a method is established to determine whether his λ4 (maximum split-half coefficient alpha) is the greatest lower bound in any instance, and three new bounds are discussed. A geometric representation, which sheds light on many of the bounds, is described.
Lower Bounds for the Reliability of the Total Score on a Test Composed of Non-Homogeneous Items: II: A Search Procedure to Locate the Greatest Lower Bound
Authors: Woodhouse Brian , Jackson Paul H.
Published: February 2006

Finding the greatest lower bound for the reliability of the total score on a test comprising n non-homogenous items with dispersion matrix Σx is equivalent to maximizing the trace of a diagonal matrix ΣE with elements θI, subject to ΣE and ΣT=Σx − ΣE being non-negative definite. The cases n=2 and n=3 are solved explicity. A computer search in the space of the θi is developed for the general case. When Guttman's λ4 (maximum split-half coefficient alpha) is not the g.l.b., the maximizing set of θi makes the rank of ΣT less than n − 1. Numerical examples of various bounds are given.
A General Solution to Mosier's Oblique Procrustes Problem
Authors: ten Berge Jos M. F. , Nevels Klaas
Published: February 2006

Browne provided a method for finding a solution to the normal equations derived by Mosier for rotating a factor matrix to a best least squares fit with a specified structure. Cramer showed that Browne's solution is not always valid, and proposed a modified algorithm. Both Browne and Cramer assumed the factor matrix to be of full rank. In this paper a general solution is derived, which takes care of rank deficient factor matrices as well. A new algorithm is offered.
Signal/Noise Ratios for Domain-Referenced Tests
Authors: Brennan Robert L. , Kane Michael T.
Published: February 2006

Using the assumption of randomly parallel tests and concepts from generalizability theory, three signal/noise ratios for domain-referenced tests are developed, discussed, and compared. The three ratios have the same noise but different signals depending upon the kind of decision to be made as a result of measurement. It is also shown that these ratios incorporate a definition of noise or error which is different from the classical definition of noise typically used to characterize norm-referenced tests.
Best Univocal Estimates of Orthogonal Common Factors
Authors: Hakstian A. Ralph , Zidek James V. , McDonald Roderick P.
Published: February 2006

A general formulation is presented for obtaining conditionally unbiased, univocal common-factor score estimates that have maximum validity for the true orthogonal factor scores. We note that although this expression is formally different from both Bartlett's formulation and Heermann's approximate expression, all three, while developed from very different rationales, yield identical results given that the common-factor model holds for the data. Although the true factor score validities can be raised by a different non-orthogonal transformation of orthogonalized regression estimates—as described by Mulaik—the resulting estimates lose their univocality.
Goodness of Fit of Trend Curves and Significance of Trend Differences
Authors: Lindquist E. F.
Published: February 2006

Experimental studies of the successive changes (frequently represented by curves describing laws of learning and other similar functional relationships) in a criterion variable accompanying experimental variations in a given “treatment,” and experimental comparisons of such changes for different populations or for different treatments, constitute a large and important class of psychological experiments. In most such experiments, no attempt has been made to analyze or to make allowance for errors of sampling or of observation. In many others, the techniques of error analysis that have been employed have been inefficient, inexact, or inappropriate. This paper suggests tests, using the methods of analysis of variance, of certain hypotheses concerning trends and trend differences in sample means obtained in experiments of this general type. For means of successive independent samples, tests are provided of the hypotheses: (H1) that there is no trend, or that the trend is a horizontal straight line, (H3) that there is a linear trend, (H5) that the trend is as described by a line not derived from the observed means, and (H7) that the trend is as described by a line fitted to the observed means. Tests are also provided of similar hypotheses (H2, H4, H6, and H8, respectively) for means of successive measurements of the same sample. Finally, tests are provided of the null hypotheses that there is no difference in trend in two series of means: (H9) when each mean in each series is based on an independent sample, (H10) when each pair of corresponding means is based on an independent sample, (H11) when each series of means is based on an independent sample, and (H12) when both series are based on a single sample.
Tables for the Determination of the Significance of Skewness and of the Significance of the Difference in the Skewness of Two Independent Distributions
Authors: Taylor Erwin K.
Published: February 2006

Several of the more popular statistics texts used in psychology are reviewed with respect to their treatment of the measurement and interpretation of skewness. Some areas in psychology where measures of skewness of distributions may yield significant information are indicated. Tables of 1%, 2% and 5% points of moment measures of skewness and tables of the 1% and 5% points of the difference in skewness of two uncorrelated distributions are presented. These tables are approximations based on the approximate normality of skewness in large samples from normal populations. The limited confidence with which these tables can be used in the absence of exact knowledge of the distribution function of the underlying statistics is indicated.
A Modified Aitken Pivotal Condensation Method for Partial Regression and Multiple Correlation
Authors: Van Boven Alice
Published: February 2006

In this modification of the Aitken Pivotal Condensation Method for obtaining partial regression weights and multiple correlation coefficients, the unit matrices used by Aitken are eliminated, with a resultant decrease in the computational work. Through a simplified arrangement of successive matrices, each one is reduced by the same simple rule. Other modifications also contribute to ease of computation and facilitate checking.
Dynamics of the Cerebral Cortex Automatic Development of Equilibrium in Self-Organizing Systems
Authors: Ashby W. R.
Published: February 2006

The nervous system, and particularly the cerebral cortex, is examined in certain of its activities, it being treated as a purely physical dynamic system. It is shown mathematically that if, as seems likely, the cerebral cortex can undergo manifold changes of neuronic organization, then it follows that these changes must automatically lead to the development of more and more states of equilibrium, the process being unavoidable and largely irreversible. The theorem may be established on basic physical principles without appeal to special physiological details. The possible significance of this observation is indicated.
A One-Way Random Effects Model for Trimmed Means
Authors: Wilcox Rand R.
Published: February 2006

The random effects ANOVA model plays an important role in many psychological studies, but the usual model suffers from at least two serious problems. The first is that even under normality, violating the assumption of equal variances can have serious consequences in terms of Type I errors or significance levels, and it can affect power as well. The second and perhaps more serious concern is that even slight departures from normality can result in a substantial loss of power when testing hypotheses. Jeyaratnam and Othman (1985) proposed a method for handling unequal variances, under the assumption of normality, but no results were given on how their procedure performs when distributions are nonnormal. A secondary goal in this paper is to address this issue via simulations. As will be seen, problems arise with both Type I errors and power. Another secondary goal is to provide new simulation results on the Rust-Fligner modification of the Kruskal-Wallis test. The primary goal is to propose a generalization of the usual random effects model based on trimmed means. The resulting test of no differences among J randomly sampled groups has certain advantages in terms of Type I errors, and it can yield substantial gains in power when distributions have heavy tails and outliers. This last feature is very important in applied work because recent investigations indicate that heavy-tailed distributions are common. Included is a suggestion for a heteroscedastic Winsorized analog of the usual intraclass correlation coefficient.
Some Critical Observations of the Test Information Function as a Measure of Local Accuracy in Ability Estimation
Authors: Samejima Fumiko
Published: February 2006

The test information function serves important roles in latent trait models and in their applications. Among others, it has been used as the measure of accuracy in ability estimation. A question arises, however, if the test information function is accurate enough for all meaningful levels of ability relative to the test, especially when the number of test items is relatively small (e.g., less than 50). In the present paper, using the constant information model and constant amounts of test information for a finite interval of ability, simulated data were produced for eight different levels of ability and for twenty different numbers of test items ranging between 10 and 200. Analyses of these data suggest that it is desirable to consider some modification of the test information function when it is used as the measure of accuracy in ability estimation.
A Simplification of a Result by Zellini on the Maximal Rank of Symmetric Three-Way Arrays
Authors: Rocci Roberto , ten Berge Jos M. F.
Published: February 2006

Zellini (1979, Theorem 3.1) has shown how to decompose an arbitrary symmetric matrix of order n × n as a linear combination of 1/2n(n + 1) fixed rank one matrices, thus constructing an explicit tensor basis for the set of symmetric n × n matrices. Zellini's decomposition is based on properties of persymmetric matrices. In the present paper, a simplified tensor basis is given, by showing that a symmetric matrix can also be decomposed in terms of 1/2n(n + 1) fixed binary matrices of rank one. The decomposition implies that an n × n × p array consisting of p symmetric n × n slabs has maximal rank 1/2n(n + 1). Likewise, an unconstrained INDSCAL (symmetric CANDECOMP/PARAFAC) decomposition of such an array will yield a perfect fit in 1/2n(n + 1) dimensions. When the fitting only pertains to the off-diagonal elements of the symmetric matrices, as is the case in a version of PARAFAC where communalities are involved, the maximal number of dimensions can be further reduced to 1/2n(n − 1). However, when the saliences in INDSCAL are constrained to be nonnegative, the tensor basis result does not apply. In fact, it is shown that in this case the number of dimensions needed can be as large as p, the number of matrices analyzed.
The Unique Correspondence of the Item Response Function and Item Category Response Functions in Polytomously Scored Item Response Models
Authors: Chang Hua-Hua , Mazzeo John
Published: February 2006

The item response function (IRF) for a polytomously scored item is defined as a weighted sum of the item category response functions (ICRF, the probability of getting a particular score for a randomly sampled examinee of ability θ). This paper establishes the correspondence between an IRF and a unique set of ICRFs for two of the most commonly used polytomous IRT models (the partial credit models and the graded response model). Specifically, a proof of the following assertion is provided for these models: If two items have the same IRF, then they must have the same number of categories; moreover, they must consist of the same ICRFs. As a corollary, for the Rasch dichotomous model, if two tests have the same test characteristic function (TCF), then they must have the same number of items. Moreover, for each item in one of the tests, an item in the other test with an identical IRF must exist. Theoretical as well as practical implications of these results are discussed.
An Investigation of Hierarchical Bayes Procedures in Item Response Theory
Authors: Kim Seock-Ho , Cohen Allan S. , Baker Frank B. , Subkoviak Michael J. , Leonard Tom
Published: February 2006

Hierarchical Bayes procedures for the two-parameter logistic item response model were compared for estimating item and ability parameters. Simulated data sets were analyzed via two joint and two marginal Bayesian estimation procedures. The marginal Bayesian estimation procedures yielded consistently smaller root mean square differences than the joint Bayesian estimation procedures for item and ability estimates. As the sample size and test length increased, the four Bayes procedures yielded essentially the same result.
On the Distribution of the Maximum Likelihood Estimator of Cronbach's Alpha
Authors: van Zyl J. M. , Neudecker H. , Nel D. G.
Published: February 2006

The asymptotic normal distribution of the maximum likelihood estimator of Cronbach's alpha (under normality) is derived for the case when no assumptions are made about the covariances among items. The asymptotic distribution is also considered for the special case of compound symmetry and compared to the exact distribution.
Logistic Positive Exponent Family of Models: Virtue of Asymmetric Item Characteristic Curves
Authors: Samejima Fumiko
Published: February 2006

The paper addresses and discusses whether the tradition of accepting point-symmetric item characteristic curves is justified by uncovering the inconsistent relationship between the difficulties of items and the order of maximum likelihood estimates of ability. This inconsistency is intrinsic in models that provide point-symmetric item characteristic curves, and in this paper focus is put on the normal ogive model for observation. It is also questioned if in the logistic model the sufficient statistic has forfeited the rationale that is appropriate to the psychological reality. It is observed that the logistic model can be interpreted as the case in which the inconsistency in ordering the maximum likelihood estimates is degenerated.

The paper proposes a family of models, called the logistic positive exponent family, which provides asymmetric item chacteristic curves. A model in this family has a consistent principle in ordering the maximum likelihood estimates of ability. The family is divided into two subsets each of which has its own principle, and includes the logistic model as a transition from one principle to the other. Rationale and some illustrative examples are given.
Orthogonal Principal Planes
Authors: Filzmoser Peter
Published: February 2006

Factor analysis and principal component analysis result in computing a new coordinate system, which is usually rotated to obtain a better interpretation of the results. In the present paper, the idea of rotation to simple structure is extended to two dimensions. While the classical definition of simple structure is aimed at rotating (one-dimensional) factors, the extension to a simple structure for two dimensions is based on the rotation of planes. The resulting planes (principal planes) reveal a better view of the data than planes spanned by factors from classical rotation and hence allow a more reliable interpretation. The usefulness of the method as well as the effectiveness of a proposed algorithm are demonstrated by simulation experiments and an example.
Generalized Latent Trait Models
Authors: Moustaki Irini , Knott Martin
Published: February 2006

In this paper we discuss a general model framework within which manifest variables with different distributions in the exponential family can be analyzed with a latent trait model. A unified maximum likelihood method for estimating the parameters of the generalized latent trait model will be presented. We discuss in addition the scoring of individuals on the latent dimensions. The general framework presented allows, not only the analysis of manifest variables all of one type but also the simultaneous analysis of a collection of variables with different distributions. The approach used analyzes the data as they are by making assumptions about the distribution of the manifest variables directly.
Psychometric Engineering as Art
Authors: Thissen David
Published: February 2006

The Psychometric Society is “devoted to the development of Psychology as a quantitative rational science”. Engineering is often set in contradistinction with science; art is sometimes considered different from science. Why, then, juxtapose the words in the title:psychometric, engineering, and art? Because an important aspect of quantitative psychology is problem-solving, and engineering solves problems. And an essential aspect of a good solution is beauty—hence, art. In overview and with examples, this presentation describes activities that are quantitative psychology as engineering and art—that is, as design. Extended illustrations involve systems for scoring tests in realistic contexts. Allusions are made to other examples that extend the conception of quantitative psychology as engineering and art across a wider range of psychometric activities.
A Scaled Difference Chi-Square Test Statistic for Moment Structure Analysis
Authors: Satorra Albert , Bentler Peter M.
Published: February 2006

A family of scaling corrections aimed to improve the chi-square approximation of goodness-of-fit test statistics in small samples, large models, and nonnormal data was proposed in Satorra and Bentler (1994). For structural equations models, Satorra-Bentler's (SB) scaling corrections are available in standard computer software. Often, however, the interest is not on the overall fit of a model, but on a test of the restrictions that a null model say M0 implies on a less restricted one M1. If T0 and T1 denote the goodness-of-fit test statistics associated to M0 andM1, respectively, then typically the difference Td=T0−T1 is used as a chi-square test statistic with degrees of freedom equal to the difference on the number of independent parameters estimated under the models M0 andM1. As in the case of the goodness-of-fit test, it is of interest to scale the statistic Td in order to improve its chi-square approximation in realistic, that is, nonasymptotic and nonormal, applications. In a recent paper, Satorra (2000) shows that the difference between two SB scaled test statistics for overall model fit does not yield the correct SB scaled difference test statistic. Satorra developed an expression that permits scaling the difference test statistic, but his formula has some practical limitations, since it requires heavy computations that are not available in standard computer software. The purpose of the present paper is to provide an easy way to compute the scaled difference chi-square statistic from the scaled goodness-of-fit test statistics of models M0 and M1. A Monte Carlo study is provided to illustrate the performance of the competing statistics.
Factor Analysis with (mixed) Observed and Latent Variables in the Exponential Family
Authors: Wedel Michel , Kamakura Wagner A.
Published: February 2006

We develop a general approach to factor analysis that involves observed and latent variables that are assumed to be distributed in the exponential family. This gives rise to a number of factor models not considered previously and enables the study of latent variables in an integrated methodological framework, rather than as a collection of seemingly unrelated special cases. The framework accommodates a great variety of different measurement scales and accommodates cases where different latent variables have different distributions. The models are estimated with the method of simulated likelihood, which allows for higher dimensional factor solutions to be estimated than heretofore. The models are illustrated on synthetic data. We investigate their performance when the distribution of the latent variables is mis-specified and when part of the observations are missing. We study the properties of the simulation estimators relative to maximum likelihood estimation with numerical integration. We provide an empirical application to the analysis of attitudes.
Errata for “The Proper Sequence for Correcting Correlation Coefficients for Range Restriction and Unreliability”
Authors: Stauffer Joseph M. , Mendoza Jorge L.
Published: February 2006

Corrections of correlations for range restriction (i.e., selection) and unreliability are common in psychometric work. The current rule of thumb for determining the order in which to apply these corrections looks to the nature of the reliability estimate (i.e., restricted or unrestricted). While intuitive, this rule of thumb is untenable when the correction includes the variable upon which selection is made, as is generally the case. Using classical test theory, we show that it is the nature of the range restriction, not the nature of the available reliability coefficient, that determines the sequence for applying corrections for range restriction and unreliability.
Test Theory without True Scores?
Authors: Cliff Norman
Published: February 2006

This paper traces the course of the consequences of viewing test responses as simply providing dichotomous data concerning ordinal relations. It begins by proposing that the score matrix is best considered to be items-plus-persons by items-plus-persons, and recording the wrongs as well as the rights. This shows how an underlying order is defined, and was used to provide the basis for a tailored testing procedure. It also was used to define a number of measures of test consistency. Test items provide person dominance relations, and the relations provided by one item can be in one of three relations with a second one: redundant, contradictory, or unique. Summary statistics concerning the number of relations of each kind are easy to get and provide useful information about the test, information which is related to but different from the usual statistics. These concepts can be extended to form the basis of a test theory which is based on ordinal statistics and frequency counts and which invokes the concept of true scores only in a limited sense.
Estimation for the Multiple Factor Model when Data are Missing
Authors: Finkbeiner Carl
Published: February 2006

A maximum likelihood method of estimating the parameters of the multiple factor model when data are missing from the sample is presented. A Monte Carlo study compares the method with 5 heuristic methods of dealing with the problem. The present method shows some advantage in accuracy of estimation over the heuristic methods but is considerably more costly computationally.
Estimating an Unobserved Component of a Serial Response Time Model
Authors: Bloxom Bruce
Published: February 2006

A method is developed for estimating the response time distribution of an unobserved component in a two-component serial model, assuming the components are stochastically independent. The estimate of the component’s density function is constrained only to be unimodal and non-negative. Numerical examples suggest that the method can yield reasonably accurate estimates with sample sizes of 300 and, in some cases, with sample sizes as small as 100.
Considerations in Psychometric Modeling of Response Time
Authors: Bloxom Bruce
Published: February 2006

Semiparametric models express a set of distributions of event times in terms of (a) a single parameter which varies across distributions and (b) a single function which does not vary across distributions and which has an unspecified form. These models appear to be attractive alternatives to parametric models of response times in psychometrics. However, our use of such models may require incorporating additional functions which do not vary across distributions and may require expressing the models in terms of the joint distribution of response class and response time.
Increasing Item Complexity: A Possible Cause of Scale Shrinkage for Unidimensional Item Response Theory
Authors: Yen Wendy M.
Published: February 2006

When the three-parameter logistic model is applied to tests covering a broad range of difficulty, there frequently is an increase in mean item discrimination and a decrease in variance of item difficulties and traits as the tests become more difficult. To examine the hypothesis that this unexpected scale shrinkage effect occurs because the items increase in complexity as they increase in difficulty, an approximate relationship is derived between the unidimensional model used in data analysis and a multidimensional model hypothesized to be generating the item responses. Scale shrinkage is successfully predicted for several sets of simulated data.
On Component Analyses
Authors: Meredith William , Millsap Roger E.
Published: February 2006

Principal components analysis can be redefined in terms of the regression of observed variables upon component variables. Two criteria for the adequacy of a component representation in this context are developed and are shown to lead to different component solutions. Both criteria are generalized to allow weighting, the choice of weights determining the scale invariance properties of the resulting solution. A theorem is presented giving necessary and sufficient conditions for equivalent component solutions under different choices of weighting. Applications of the theorem are discussed that involve the components analysis of linearly derived variables and of external variables.
Two New Test Statistics for the Rasch Model
Authors: van den Wollenberg Arnold L.
Published: February 2006

It is shown that the presently available statistical tests for the Rasch model are insensitive to violation of the unidimensionality axiom. Two new test statistics are presented. The first one, Q1, is sensitive to the same effects as the presently available statistics, but has some desirable properties of a nonstatistical nature. The second statistic, Q2, is sensitive to violation of local stochastic independence and unidimensionality and thus fills an existing gap.
Marginal Maximum Likelihood Estimation for the One-Parameter Logistic Model
Authors: Thissen David
Published: February 2006

Two algorithms are described for marginal maximum likelihood estimation for the one-parameter logistic model. The more efficient of the two algorithms is extended to estimation for the linear logistic model. Numerical examples of both procedures are presented.
The Scale Grid: Some Interrelations of Data Models
Authors: Coombs Clyde H.
Published: February 2006

Perhaps an appropriate subtitle for this paper would be “Some Speculations on the Interrelations of Psychological Methodologies.” The Scale Grid is a name I have given to a model which presumes to define the underlying continuities between such diverse areas as psychophysics, objective testing, attitude studies including questionnaire and interview techniques, learning experiments, rating scale methods, essay examinations, and projective instruments. The intent of the Scale Grid is to make explicit the fundamental similarities and differences of the methodologies among these various areas of psychological research.
Bayesian Factor Analysis for Multilevel Binary Observations
Authors: Ansari Asim , Jedidi Kamel
Published: February 2006

Multilevel covariance structure models have become increasingly popular in the psychometric literature in the past few years to account for population heterogeneity and complex study designs. We develop practical simulation based procedures for Bayesian inference of multilevel binary factor analysis models. We illustrate how Markov Chain Monte Carlo procedures such as Gibbs sampling and Metropolis-Hastings methods can be used to perform Bayesian inference, model checking and model comparison without the need for multidimensional numerical integration. We illustrate the proposed estimation methods using three simulation studies and an application involving student's achievement results in different areas of mathematics.
Graphical Regression Models for Polytomous Variables
Authors: Anderson Carolyn J. , Böckenholt Ulf
Published: February 2006

When modeling the relationship between two nominal categorical variables, it is often desirable to include covariates to understand how individuals differ in their response behavior. Typically, however, not all the relevant covariates are available, with the result that the measured variables cannot fully account for the associations between the nominal variables. Under the assumption that the observed and unobserved variables follow a homogeneous conditional Gaussian distribution, this paper proposes RC(M) regression models to decompose the residual associations between the polytomous variables. Based on Goodman's (1979, 1985)RC(M) association model, a distinctive feature of RC(M) regression models is that they facilitate the joint estimation of effects due to manifest and omitted (continuous) variables without requiring numerical integration. The RC(M) regression models are illustrated using data from the High School and Beyond study (Tatsuoka & Lohnes, 1988).
An Irt-Based Model for Dichotomous Longitudinal Data
Authors: Fischer Gerhard H.
Published: February 2006

The LLRA (linear logistic model with relaxed assumptions; Fischer, 1974, 1977a, 1977b, 1983a) was developed, within the framework of generalized Rasch models, for assessing change in dichotomous item score matrices between two points in time; it allows to quantify change on latent trait dimensions and to explain change in terms of treatment effects, treatment interactions, and a trend effect. A remarkable feature of the model is that unidimensionality of the item set is not required. The present paper extends this model to designs with any number of time points and even with different sets of items presented on different occasions, provided that one unidimensional subscale is available per latent trait. Thus unidimensionality assumptions within subscales are combined with multidimensionality of the item set. Conditional maximum likelihood methods for parameter estimation and hypothesis testing are developed, and a necessary and sufficient condition for unique identification of the model, given the data, is derived. Finally, a sample application is presented.
Clustering N Objects into K Groups under Optimal Scaling of Variables
Authors: van Buuren Stef , Heiser Willem J.
Published: February 2006

We propose a method to reduce many categorical variables to one variable with k categories, or stated otherwise, to classify n objects into k groups. Objects are measured on a set of nominal, ordinal or numerical variables or any mix of these, and they are represented as n points in p-dimensional Euclidean space. Starting from homogeneity analysis, also called multiple correspondence analysis, the essential feature of our approach is that these object points are restricted to lie at only one of k locations. It follows that these k locations must be equal to the centroids of all objects belonging to the same group, which corresponds to a sum of squared distances clustering criterion. The problem is not only to estimate the group allocation, but also to obtain an optimal transformation of the data matrix. An alternating least squares algorithm and an example are given.
A Factor Simplicity Index
Authors: Lorenzo-Seva Urbano
Published: February 2006

We propose an index for assessing the degree of factor simplicity in the context of principal components and exploratory factor analysis. The new index, which is called Loading Simplicity, is based on the idea that the communality of each variable should be related to few components, or factors, so that the loadings in each variable are either zero or as far from zero as possible. This index does not depend on the scale of the factors, and its maximum and minimum are only related to the degree of simplicity in the loading matrix. The aim of the index is to enable the degree of simplicity in loading matrices to be compared.
Nonparametric IRT: Testing the Bi-isotonicity of Isotonic Probabilistic Models (ISOP)
Authors: Scheiblechner Hartmann
Published: February 2006

Nonparametric tests for testing the validity of polytomous ISOP-models (unidimensional ordinal probabilistic polytomous IRT-models) are presented. Since the ISOP-model is a very general nonparametric unidimensional rating scale model the test statistics apply to a great multitude of latent trait models. A test for the comonotonicity of item sets of two or more items is suggested. Procedures for testing the comonotonicity of two item sets and for item selection are developed. The tests are based on Goodman-Kruskal's gamma index of ordinal association and are generalizations thereof. It is an essential advantage of polytomous ISOP-models within probabilistic IRT-models that the tests of validity of the model can be performed before and without the model being fitted to the data. The new test statistics have the further advantage that no prior order of items or subjects needs to be known.
Four Simultaneous Component Models for the Analysis of Multivariate Time Series from More than One Subject to Model Intraindividual and Interindividual Differences
Authors: Timmerman Marieke E. , Kiers Henk A. L.
Published: February 2006

A class of four simultaneous component models for the exploratory analysis of multivariate time series collected from more than one subject simultaneously is discussed. In each of the models, the multivariate time series of each subject is decomposed into a few series of component scores and a loading matrix. The component scores series reveal the latent data structure in the course of time. The interpretation of the components is based on the loading matrix. The simultaneous component models model not only intraindividual variability, but interindividual variability as well. The four models can be ordered hierarchically from weakly to severely constrained, thus allowing for big to small interindividual differences in the model. The use of the models is illustrated by an empirical example.
Probability Matrix Decomposition Models
Authors: Maris Eric , De Boeck Paul , Van Mechelen Iven
Published: February 2006

In this paper, we consider a class of models for two-way matrices with binary entries of 0 and 1. First, we consider Boolean matrix decomposition, conceptualize it as a latent response model (LRM) and, by making use of this conceptualization, generalize it to a larger class of matrix decomposition models. Second, probability matrix decomposition (PMD) models are introduced as a probabilistic version of this larger class of deterministic matrix decomposition models. Third, an algorithm for the computation of the maximum likelihood (ML) and the maximum a posteriori (MAP) estimates of the parameters of PMD models is presented. This algorithm is an EM-algorithm, and is a special case of a more general algorithm that can be used for the whole class of LRMs. And fourth, as an example, a PMD model is applied to data on decision making in psychiatric diagnosis.
Decomposition of a Rasch Partial Credit Item into Independent Binary and Indecomposable Trinary Items
Authors: Huynh Huynh
Published: February 2006

For each Rasch (Masters) partial credit item, there exists a set of independent Rasch binary and indecomposable trinary items for which the sum of the scores and the partial credit score have identical probability density functions. If each indecomposable trinary item is further expressed as the sum of two binary items, then the binary items are positively dependent and cannot be both of the Rasch type.