bluuebunny/crossref_metadata_2025 huggingface dataset:
An INAR(1) Negative Multinomial Regression Model for Longitudinal Count Data
Authors: Böckenholt Ulf
Published: January 2006

This paper discusses a regression model for the analysis of longitudinal count data observed in a panel study. An integer-valued first-order autoregressive [INAR(1)] Poisson process is adapted to represent time-dependent correlations among the counts. By combining the INAR(1)-representation with a random effects approach, a new negative multinomial distribution is derived that includes the bivariate negative binomial distribution proposed by Edwards and Gurland (1961) and Subrahmaniam (1966) as a special case. A detailed analysis of the relationship between personality factors and daily emotion experiences illustrates the approach.
Coefficients Alpha and Reliabilities of Unrotated and Rotated Components
Authors: ten Berge Jos M. F. , Hofstee Willem K. B.
Published: January 2006

It has been shown by Kaiser that the sum of coefficients alpha of a set of principal components does not change when the components are transformed by an orthogonal rotation. In this paper, Kaiser's result is generalized. First, the invariance property is shown to hold for any set of orthogonal components. Next, a similar invariance property is derived for the reliability of any set of components. Both generalizations are established by considering simultaneously optimal weights for components with maximum alpha and with maximum reliability, respectively. A short-cut formula is offered to evaluate the coefficients alpha for orthogonally rotated principal components from rotation weights and eigenvalues of the correlation matrix. Finally, the greatest lower bound to reliability and a weighted version are discussed.
Acceleration Model in the Heterogeneous Case of the General Graded Response Model
Authors: Samejima Fumiko
Published: January 2006

A new model, called acceleration model, is proposed in the framework of the heterogenous case of the graded response model, based on processing functions defined for a finite or enumerable number of steps. The model is expected to be useful in cognitive assessment, as well as in more traditional areas of application of latent trait models. Criteria for evaluating models are proposed, and soundness and robustness of the acceleration model are discussed. Graded response models based on individual choice behavior are also discussed, and criticisms on model selection in terms of fitnesses of models to the data are also given.
Round-Robin Analysis of Social Interaction: Exact and Estimated Standard Errors
Authors: Bond Charles F. , Lashley Brian R.
Published: January 2006

Kenny has proposed a variance-components model for dyadic social interaction. His Social Relations model estimates variances and covariances from a round-robin of two-person interactions. The current paper presents a matrix formulation of the Social Relations model. It uses the formulation to derive exact and estimated standard errors for round-robin estimates of Social Relations parameters.
Multidimensional Adaptive Testing
Authors: Segall Daniel O.
Published: January 2006

Maximum likelihood and Bayesian procedures for item selection and scoring of multidimensional adaptive tests are presented. A demonstration using simulated response data illustrates that multidimensional adaptive testing (MAT) can provide equal or higher reliabilities with about one-third fewer items than are required by one-dimensional adaptive testing (OAT). Furthermore, holding test-length constant across the MAT and OAT approaches, substantial improvements in reliability can be obtained from multidimensional assessment. A number of issues relating to the operational use of multidimensional adaptive testing are discussed.
Factor Analysis and AIC
Authors: Akaike Hirotugu
Published: January 2006

The information criterion AIC was introduced to extend the method of maximum likelihood to the multimodel situation. It was obtained by relating the successful experience of the order determination of an autoregressive model to the determination of the number of factors in the maximum likelihood factor analysis. The use of the AIC criterion in the factor analysis is particularly interesting when it is viewed as the choice of a Bayesian model. This observation shows that the area of application of AIC can be much wider than the conventional i.i.d. type models on which the original derivation of the criterion was based. The observation of the Bayesian structure of the factor analysis model leads us to the handling of the problem of improper solution by introducing a natural prior distribution of factor loadings.
Maximum Likelihood Estimation of Multivariate Polyserial and Polychoric Correlation Coefficients
Authors: Poon Wai-Yin , Lee Sik-Yum
Published: January 2006

The method of finding the maximum likelihood estimates of the parameters in a multivariate normal model with some of the component variables observable only in polytomous form is developed. The main stratagem used is a reparameterization which converts the corresponding log likelihood function to an easily handled one. The maximum likelihood estimates are found by a Fletcher-Powell algorithm, and their standard error estimates are obtained from the information matrix. When the dimension of the random vector observable only in polytomous form is large, obtaining the maximum likelihood estimates is computationally rather labor expensive. Therefore, a more efficient method, the partition maximum likelihood method, is proposed. These estimation methods are demonstrated by real and simulated data, and are compared by means of a simulation study.
On Structural Equation Modeling with Data that are not Missing Completely at Random
Authors: Muthén Bengt , Kaplan David , Hollis Michael
Published: January 2006

A general latent variable model is given which includes the specification of a missing data mechanism. This framework allows for an elucidating discussion of existing general multivariate theory bearing on maximum likelihood estimation with missing data. Here, missing completely at random is not a prerequisite for unbiased estimation in large samples, as when using the traditional listwise or pairwise present data approaches. The theory is connected with old and new results in the area of selection and factorial invariance. It is pointed out that in many applications, maximum likelihood estimation with missing data may be carried out by existing structural equation modeling software, such as LISREL and LISCOMP. Several sets of artifical data are generated within the general model framework. The proposed estimator is compared to the two traditional ones and found superior.
A Multitrait-Multimethod Model with Minimal Assumptions
Authors: Eid Michael
Published: January 2006

A new model of confirmatory factor analysis (CFA) for multitrait-multimethod (MTMM) data sets is presented. It is shown that this model can be defined by only three assumptions in the framework of classical psychometric test theory (CTT). All other properties of the model, particularly the uncorrelatedness of the trait with the method factors are logical consequences of the definition of the model. In the model proposed there are as many trait factors as different traits considered, but the number of method factors is one fewer than the number of methods included in an MTMM study. The covariance structure implied by this model is derived, and it is shown that this model is identified even under conditions under which other CFA-MTMM models are not. The model is illustrated by two empirical applications. Furthermore, its advantages and limitations are discussed with respect to previously developed CFA models for MTMM data.
ANOVA and ANCOVA of Pre- and Post-Test, Ordinal Data
Authors: Davison Mark L. , Sharma Anu R.
Published: January 2006

With random assignment to treatments and standard assumptions, either a one-way ANOVA of post-test scores or a two-way, repeated measures ANOVA of pre- and post-test scores provides a legitimate test of the equal treatment effect null hypothesis for latent variable Θ. In an ANCOVA for pre- and post-test variables X and Y which are ordinal measures of η and Θ, respectively, random assignment and standard assumptions ensure the legitimacy of inferences about the equality of treatment effects on latent variable Θ. Sample estimates of adjusted Y treatment means are ordinal estimators of adjusted post-test means on latent variable Θ.
Generalized Approaches to the Maxbet Problem and the Maxdiff Problem, with Applications to Canonical Correlations
Authors: ten Berge Jos M. F.
Published: January 2006

Van de Geer has reviewed various criteria for transforming two or more matrices to maximal agreement, subject to orthogonality constraints. The criteria have applications in the context of matching factor or configuration matrices and in the context of canonical correlation analysis for two or more matrices. The present paper summarizes and gives a unified treatment of fully general computational solutions for two of these criteria, Maxbet and Maxdiff. These solutions will be shown to encompass various well-known methods as special cases. It will be argued that the Maxdiff solution should be preferred to the Maxbet solution whenever the two criteria coincide. Horst's Maxcor method will be shown to lack the property of monotone convergence. Finally, simultaneous and successive versions of the Maxbet and Maxdiff solutions will be treated as special cases of a fully flexible approach where the columns of the rotation matrices are obtained in successive blocks.
Canonical Analysis of two Convex Polyhedral Cones and Applications
Authors: Tenenhaus Michel
Published: January 2006

Canonical analysis of two convex polyhedral cones consists in looking for two vectors (one in each cone) whose square cosine is a maximum. This paper presents new results about the properties of the optimal solution to this problem, and also discusses in detail the convergence of an alternating least squares algorithm. The set of scalings of an ordinal variable is a convex polyhedral cone, which thus plays an important role in optimal scaling methods for the analysis of ordinal data. Monotone analysis of variance, and correspondence analysis subject to an ordinal constraint on one of the factors are both canonical analyses of a convex polyhedral cone and a subspace. Optimal multiple regression of a dependent ordinal variable on a set of independent ordinal variables is a canonical analysis of two convex polyhedral cones as long as the signs of the regression coefficients are given. We discuss these three situations and illustrate them by examples.
Circumplex Models for Correlation Matrices
Authors: Browne Michael W.
Published: January 2006

Structural models that yield circumplex inequality patterns for the elements of correlation matrices are reviewed. Particular attention is given to a stochastic process defined on the circle proposed by T. W. Anderson. It is shown that the Anderson circumplex contains the Markov Process model for a simplex as a limiting case when a parameter tends to infinity.

Anderson's model is intended for correlation matrices with positive elements. A replacement for Anderson's correlation function that permits negative correlations is suggested. It is shown that the resulting model may be reparametrzed as a factor analysis model with nonlinear constraints on the factor loadings. An unrestricted factor analysis, followed by an appropriate rotation, is employed to obtain parameter estimates. These estimates may be used as initial approximations in an iterative procedure to obtain minimum discrepancy estimates.

Practical applications are reported.
Sequential Sampling Designs for the Two-Parameter Item Response Theory Model
Authors: Berger Martijn P. F.
Published: January 2006

In optimal design research, designs are optimized with respect to some statistical criterion under a certain model for the data. The ideas from optimal design research have spread into various fields of research, and recently have been adopted in test theory and applied to item response theory (IRT) models. In this paper a generalized variance criterion is used for sequential sampling in the two-parameter IRT model. Some general principles are offered to enable a researcher to select the best sampling design for the efficient estimation of item parameters.
The Integration of Multidimensional Scaling and Multivariate Analysis with Optimal Transformations
Authors: Meulman Jacqueline J.
Published: January 2006

The recent history of multidimensional data analysis suggests two distinct traditions that have developed along quite different lines. In multidimensional scaling (MDS), the available data typically describe the relationships among a set of objects in terms of similarity/dissimilarity (or (pseudo-)distances). In multivariate analysis (MVA), data usually result from observation on a collection of variables over a common set of objects. This paper starts from a very general multidimensional scaling task, defined on distances between objects derived from one or more sets of multivariate data. Particular special cases of the general problem, following familiar notions from MVA, will be discussed that encompass a variety of analysis techniques, including the possible use of optimal variable transformation. Throughout, it will be noted how certain data analysis approaches are equivalent to familiar MVA solutions when particular problem specifications are combined with particular distance approximations.
Measuring Misinformation in Repeat Trial Pick 1 of 2 Tests
Authors: Henderson Pamela W. , Buchanan Bruce
Published: January 2006

An extension is described to a product testing model to account for misinformation among subjects. A misinformed subject is one who associates the taste of product A with product B and vice-versa; thus, the subject would tend to perform incorrectly on pick 1 of 2 tests. A likelihood ratio test for the presence of misinformation is described. The model is applied to a data set, and misinformation is found to exist. Biases due to model misspecificationand other implications for product testing are discussed.
Graded Responses and Joining Categories: A Rejoinder to Andrich' “Models For Measurement, Precision, and Nondichotomization of Graded Responses”
Authors: Roskam Edw E.
Published: January 2006

Andrich (1995) claims that the “probability distribution [of graded responses] reflects the precision with which the data are collected” (p. 7), and that an “increase in precision of responses [ . . . ] destroys the joining assumption” (p. 22). He stressed “that Jansen and Roskam simply asserted this equivalence [of the joining assumption and ξ-invariance], and did not derive it.” However, Jansen and Roskam (1986) and Roskam and Jansen (1989)—in the sequel referred to as JR—have neither asserted an equivalence between ξ-invariance and the joining assumption, nor defined ξ-invariance such that it could be considered in terms of estimation.
Nonparametric Goodness-of-Fit Tests for the Rasch Model
Authors: Ponocny Ivo
Published: January 2006

A Monte Carlo algorithm realizing a family of nonparametric tests for the Rasch model is introduced which are conditional on the item and subject marginals. The algorithm is based on random changes of elements of data matrices without changing the marginals; most powerful tests against all alternative hypotheses are given for which a monotone characteristic may be computed from the data matrix; alternatives may also be composed. Computation times are long, but exactp-values are approximated with the quality of approximation only depending on calculation time, but not on the number of persons. The power and the flexibility of the procedure is demonstrated by means of an empirical example where, among others, indicators for increased item similarities, the existence of subscales, violations of sufficiency of the raw score as well as learning processes were found. Many of the features described are implemented in the program T-Rasch 1.0 by Ponocny and Ponocny-Seliger (1999).
A Note on the Equivalence of the Graded Response Model and the Sequential Model
Authors: Bechger Timo M. , Akkermans Wies
Published: January 2006

This paper concerns items that consist of several item steps to be responded to sequentially. The item scoreX is defined as the number of correct responses until the first failure. Samejima's graded response model states that each step h=1,...,m is characterized by a parameter bh, and, for a subject with ability θ, Pr(X ≥ h; θ) = F(θ − bh). Tutz's general sequential model associates with each step a parameterdh, and it states that Pr(X≥h;θ)=Πr=1hG(θ−dr). Tutz's (1991, 1997) conjectures that the models are equivalent if and only ifF(x)=G(x) is an extreme value distribution. This paper presents a proof for this conjecture.
On the Partitioning of Squared Euclidean Distance and its Applications in Cluster Analysis
Authors: Carter Randy L. , Morris Robin , Blashfield Roger K.
Published: January 2006

The partitioning of squared Eucliean distance between two vectors in M-dimensional space into the sum of squared lengths of vectors in mutually orthogonal subspaces is discussed and applications given to specific cluster analysis problems. Examples of how the partitioning idea can be used to help describe and interpret derived clusters, derive similarity measures for use in cluster analysis, and to design Monte Carlo studies with carefully specified types and magnitudes of differences between the underlying population mean vectors are presented. Most of the example applications presented in this paper involve the clustering of longitudinal data, but their use in cluster analysis need not be limited to this arena.
Least-Squares Approximation of an Improper Correlation Matrix by a Proper One
Authors: Knol Dirk L. , ten Berge Jos M. F.
Published: January 2006

An algorithm is presented for the best least-squares fitting correlation matrix approximating a given missing value or improper correlation matrix. The proposed algorithm is based upon a solution for Mosier's oblique Procrustes rotation problem offered by ten Berge and Nevels. A necessary and sufficient condition is given for a solution to yield the unique global minimum of the least-squares function. Empirical verification of the condition indicates that the occurrence of non-optimal solutions with the proposed algorithm is very unlikely. A possible drawback of the optimal solution is that it is a singular matrix of necessity. In cases where singularity is undesirable, one may impose the additional nonsingularity constraint that the smallest eigenvalue of the solution be δ, where δ is an arbitrary small positive constant. Finally, it may be desirable to weight the squared errors of estimation differentially. A generalized solution is derived which satisfies the additional nonsingularity constraint and also allows for weighting. The generalized solution can readily be obtained from the standard “unweighted singular” solution by transforming the observed improper correlation matrix in a suitable way.
Ordinal Consistency and Ordinal True Scores
Authors: Cliff Norman
Published: January 2006

This paper argues that test data are ordinal, that latent trait scores are only determined ordinally, and that test data are used largely for ordinal purposes. Therefore it is desirable to develop a test theory based only on ordinal assumptions. A set of ordinal assumptions is presented, including an ordinal version of local independence. From these assumptions it is first shown that the gamma-correlation between two tests is the product of their gamma-correlations with the true latent order. The theory is generalized to allow for heterogeneous tests by defining a weighted average local independence. The tau-correlations between total score and the latent order can be found in both homogeneous and heterogeneous cases, and a system of differential item weighting to maximize the tau-correlation between weighted items and the latent order is provided. Thus a purely ordinal test theory seems possible.
A Note on Reliability Estimation for a Test with Components of Unknown Functional Lengths
Authors: Liou Michelle
Published: January 2006

This research note proposes two reliability coefficients for tests with components of unknown functional lengths. The derived coefficients are extensions of the techniques devised by Kristof and Feldt and do not require a reduction of test components into parts. Simulation study indicates that the new coefficients yield reasonably stable reliability estimates when the number of test components is small.
Essential Independence and Likelihood-Based Ability Estimation for Polytomous Items
Authors: Junker Brian W.
Published: January 2006

A definition of essential independence is proposed for sequences of polytomous items. For items satisfying the reasonable assumption that the expected amount of credit awarded increases with examinee ability, we develop a theory of essential unidimensionality which closely parallels that of Stout. Essentially unidimensional item sequences can be shown to have a unique (up to change-of-scale) dominant underlying trait, which can be consistently estimated by a monotone transformation of the sum of the item scores. In more general polytomous-response latent trait models (with or without ordered responses), an M-estimator based upon maximum likelihood may be shown to be consistent for θ under essentially unidimensional violations of local independence and a variety of monotonicity/identifiability conditions. A rigorous proof of this fact is given, and the standard error of the estimator is explored. These results suggest that ability estimation methods that rely on the summation form of the log likelihood under local independence should generally be robust under essential independence, but standard errors may vary greatly from what is usually expected, depending on the degree of departure from local independence. An index of departure from local independence is also proposed.
A Stochastic Multidimensional Scaling Procedure for the Empirical Determination of Convex Indifference Curves for Preference/Choice Analysis
Authors: DeSarbo Wayne S. , Jedidi Kamel , Steckel Joel H.
Published: January 2006

The vast majority of existing multidimensional scaling (MDS) procedures devised for the analysis of paired comparison preference/choice judgments are typically based on either scalar product (i.e., vector) or unfolding (i.e., ideal-point) models. Such methods tend to ignore many of the essential components of microeconomic theory including convex indifference curves, constrained utility maximization, demand functions, et cetera. This paper presents a new stochastic MDS procedure called MICROSCALE that attempts to operationalize many of these traditional microeconomic concepts. First, we briefly review several existing MDS models that operate on paired comparisons data, noting the particular nature of the utility functions implied by each class of models. These utility assumptions are then directly contrasted to those of microeconomic theory. The new maximum likelihood based procedure, MICROSCALE, is presented, as well as the technical details of the estimation procedure. The results of a Monte Carlo analysis investigating the performance of the algorithm as a number of model, data, and error factors are experimentally manipulated are provided. Finally, an illustration in consumer psychology concerning a convenience sample of thirty consumers providing paired comparisons judgments for some fourteen brands of over-the-counter analgesics is discussed.
The Asymptotic Posterior Normality of the Latent Trait in an IRT Model
Authors: Chang Hua-Hua , Stout William
Published: January 2006

It has long been part of the item response theory (IRT) folklore that under the usual empirical Bayes unidimensional IRT modeling approach, the posterior distribution of examinee ability given test response is approximately normal for a long test. Under very general and nonrestrictive nonparametric assumptions, we make this claim rigorous for a broad class of latent models.
Comparing One-Step M-Estimators of Location when there are More than Two Groups
Authors: Wilcox Rand R.
Published: January 2006

Methods for comparing means are known to be highly nonrobust in terms of Type II errors. The problem is that slight shifts from normal distributions toward heavy-tailed distributions inflate the standard error of the sample mean. In contrast, the standard error of various robust measures of location, such as the one-step M-estimator, are relatively unaffected by heavy tails. Wilcox recently examined a method of comparing the one-step M-estimators of location corresponding to two independent groups which provided good control over the probability of a Type I error even for unequal sample sizes, unequal variances, and different shaped distributions. There is a fairly obvious extension of this procedure to pairwise comparisons of more than two independent groups, but simulations reported here indicate that it is unsatisfactory. A slight modification of the procedure is found to give much better results, although some caution must be taken when there are unequal sample sizes and light-tailed distributions. An omnibus test is examined as well.
An Approximation for the Bias Function of the Maximum Likelihood Estimate of a Latent Variable for the General Case where the Item Responses are Discrete
Authors: Samejima Fumiko
Published: January 2006

Lord developed an approximation for the bias function for the maximum likelihood estimate in the context of the three-parameter logistic model. Using Taylor's expansion of the likelihood equation, he obtained an equation that includes the conditional expectation, given true ability, of the discrepancy between the maximum likelihood estimate and true ability. All terms of orders higher than n−1 are ignored where n indicates the number of items. Lord assumed that all item and individual parameters are bounded, all item parameters are known or well-estimated, and the number of items is reasonably large. In the present paper, an approximation for the bias function of the maximum likelihood estimate of the latent trait, or ability, will be developed using the same assumptions for the more general case where item responses are discrete. This will include the dichotomous response level, for which the three-parameter logistic model has been discussed, the graded response level and the nominal response level. Some observations will be made for both dichotomous and graded response levels.
Estimating Flexible Distributions of Ideal-Points with External Analysis of Preferences
Authors: Kamakura Wagner A.
Published: January 2006

Ideal-points are widely used to model choices when preferences are single-peaked. Ideal-point choice models have been typically estimated at the individual-level, or have been based on the assumption that ideal-points are normally distributed over the population of choice makers. We propose two probabilistic ideal-point choice models for the external analysis of preferences that allow for more flexible multimodal distributions of ideal-points, thus acknowledging the existence of subpopulations with distinct preferences. The first model extends the ideal-point probit model for heterogeneous preferences to accommodate a mixture of multivariate normal distributions of ideal-points. The second model assumes that ideal-points are uniformly distributed within finite ranges of the attribute space, leading to a more simplistic formulation and a more flexible distribution. The two models are applied to simulated and actual choice data, and compared to the ideal-point probit model.
Hierarchical Relations Among Three-Way Methods
Authors: Kiers Henk A. L.
Published: January 2006

A number of methods for the analysis of three-way data are described and shown to be variants of principal components analysis (PCA) of the two-way supermatrix in which each two-way slice is “strung out” into a column vector. The methods are shown to form a hierarchy such that each method is a constrained variant of its predecessor. A strategy is suggested to determine which of the methods yields the most useful description of a given three-way data set.
Paired Comparison, Triple Comparison, and Ranking Experiments as Generalized Linear Models, and Their Implementation on GLIM
Authors: Critchlow Douglas E. , Fligner Michael A.
Published: January 2006

A wide variety of paired comparison, triple comparison, and ranking experiments may be viewed as generalized linear models. These include paired comparison models based on both the Bradley-Terry and Thurstone-Mosteller approaches, as well as extensions of these models that allow for ties, order of presentation effects, and the presence of covariates. Moreover, the triple comparison model of Pendergrass and Bradley, as well as models for complete rankings of more than three items, can also be represented as generalized linear models. All such models can be easily fit by maximum likelihood, using the widely available GLIM computer package. Additionally, GLIM enables the computation of likelihood ratio statistics for testing many hypotheses of interest. Examples are presented that cover a variety of cases, along with their implementation on GLIM.
Kernel Smoothing Approaches to Nonparametric Item Characteristic Curve Estimation
Authors: Ramsay J. O.
Published: January 2006

The option characteristic curve, the relation between ability and probability of choosing a particular option for a test item, can be estimated by nonparametric smoothing techniques. What is smoothed is the relation between some function of estimated examinee ability rankings and the binary variable indicating whether or not the option was chosen. This paper explores the use of kernel smoothing, which is particularly well suited to this application. Examples show that, with some help from the fast Fourier transform, estimates can be computed about 500 times as rapidly as when using commonly used parametric approaches such as maximum marginal likelihood estimation using the three-parameter logistic distribution. Simulations suggest that there is no loss of efficiency even when the population curves are three-parameter logistic. The approach lends itself to several interesting extensions.
Kruskal's Polynomial for 2 × 2 × 2 Arrays and a Generalization to 2 × n × n Arrays
Authors: ten Berge Jos M. F.
Published: January 2006

A remarkable difference between the concept of rank for matrices and that for three-way arrays has to do with the occurrence of non-maximal rank. The set of n × n matrices that have a rank less than n has zero volume. Kruskal pointed out that a 2 × 2 × 2 array has rank three or less, and that the subsets of those 2 × 2 × 2 arrays for which the rank is two or three both have positive volume. These subsets can be distinguished by the roots of a certain polynomial. The present paper generalizes Kruskal's results to 2 × n × n arrays. Incidentally, it is shown that two n × n matrices can be diagonalized simultaneously with positive probability.
Equivalent Models in Covariance Structure Analysis
Authors: Luijben Thom C. W.
Published: January 2006

Defining equivalent models as those that reproduce the same set of covariance matrices, necessary and sufficient conditions are stated for the local equivalence of two expanded identified models M1 and M2 when fitting the more restricted model M0. Assuming several regularity conditions, the rank deficiency of the Jacobian matrix, composed of derivatives of the covariance elements with respect to the union of the free parameters of M1 and M2 (which characterizes model M12), is a necessary and sufficient condition for the local equivalence of M1 and M2. This condition is satisfied, in practice, when the analysis dealing with the fitting of M0, predicts that the decreases in the chi-square goodness-of-fit statistic for the fitting of M1 or M2, or M12 are all equal for any set of sample data, except on differences due to rounding errors.
Constrained Latent Class Analysis: Simultaneous Classification and Scaling of Discrete Choice Data
Authors: Böckenholt Ulf , Böckenholt Ingo
Published: January 2006

A reparameterization of a latent class model is presented to simultaneously classify and scale nominal and ordered categorical choice data. Latent class-specific probabilities are constrained to be equal to the preference probabilities from a probabilistic ideal-point or vector model that yields a graphical, multidimensional representation of the classification results. In addition, background variables can be incorporated as an aid to interpreting the latent class-specific response probabilities. The analyses of synthetic and real data sets illustrate the proposed method.
SAS Macro Programs for Statistical Graphics
Authors: Friendly Michael
Published: January 2006

The purpose of this announcement is to describe a collection of general macro programs for statistical graphics for use with the SAS System that have been made available in conjunction with the book, SAS system for statistical graphics, first edition (Friendly, 1991). The primary goals of the book are to survey the kinds of graphic displays that are useful for different questions and data, and to show how can these displays be done with the SAS System. It emphasizes displays that reveal aspects of data not easily captured in numerical summaries or tabular formats and diagnostic displays that help determine if assumptions of an analysis are met.

All of the programs use keywords for the required and optional parameters and supply default values where possible to parameters not specified when the macro is invoked.
Convergence Properties of an Iterative Procedure of Ipsatizing and Standardizing a Data Matrix, with Applications to Parafac/Candecomp Preprocessing
Authors: ten Berge Jos M. F. , Kiers Henk A. L.
Published: January 2006

Centering a matrix row-wise and rescaling it column-wise to a unit sum of squares requires an iterative procedure. It is shown that this procedure converges to a stable solution. This solution need not be centered row-wise if the limiting point of the interations is a matrix of rank one. The results of the present paper bear directly on several types of preprocessing methods in Parafac/Candecomp.
Incorporating Prior Theory in Covariance Structure Analysis: A Bayesian Approach
Authors: Fornell Claes , Rust Roland T.
Published: January 2006

A Bayesian approach to the testing of competing covariance structures is developed. The method provides approximate posterior probablities for each model under consideration without prior specification of individual parameter distributions. The method is based on ayesian updating using cross-validated pseudo-likelihoods. Given that the observed variables are the samefor all competing models, the approximate posterior probabilities may be obtained easily from the chi square values and other known constants, using only a hand calculator. The approach is illustrated using and example which illustrates how the prior probabilities can alter the results concerning which model specification is preferred.
Exploratory Longitudinal Factor Analysis in Multiple Populations
Authors: Tisak John , Meredith William
Published: January 2006

For multiple populatios, a longtidinal factor analytic model which is entirely exploratory, that is, no explicit identification constraints, is proposed. Factorial collapse and period/practice effects are allowed. An invariant and/or stationary factor pattern is permitted. This model is formulated stochastically. To implement this model a stagewise EM algorithm is developed. Finally a numerical illustration utilizing Nesselroade and Baltes' data is presented.
A Latent Trait Theory Via a Stochastic Learning Theory for a Knowledge Space
Authors: Falmagne Jean-Claude
Published: January 2006

To capture the cognitive organization of a set of questions or problems pertaining to a body of information, Doignon and Falmagne have proposed, and analyzed in a number of papers, the concept of a knowledge space, that is, a distinguished collection of subsets of questions, representing the possible knowledge states. This collection of sets is assumed to satisfy a number of conditions. Since this concept is a deterministic one, the problem of empirical testing arises. A stochastic version of a knowledge space is developed in this paper, in which the knowledge states are considered as possible epochs in a subject's learning history. The knowledge space is decomposed as a union of a number of possible learning paths, called gradations. The model specifies how a subject is channelled through and progresses along a gradation. A probabilistic axiom of the “local indepencence” type relates the knowledge states to the observable responses. The predictions of this model are worked out in details in the case of parametric assumptions involving gamma distributions. An application of the model to artificial data is described, based on maximum likelihood methods. The statistical analysis is shown to be capable of revealing the combinatoric core of the model.
Fitting the Off-Diagonal Dedicom Model in the Least-Squares Sense by a Generalization of the Harman and Jones Minres Procedure of Factor Analysis
Authors: ten Berge Jos M. F. , Kiers Henk A. L.
Published: January 2006

Harshman's DEDICOM model providesa framework for analyzing square but asymmetric materices of directional relationships among n objects or persons in terms of a small number of components. One version of DEDICOM ignores the diagonal entries of the matrices. A straight-forward computational solution for this model is offered in the present paper. The solution can be interpreted as a generalized Minres procedure suitable for handing asymmetric matrices.
Estimating Multiple Classification Latent Class Models
Authors: Maris E.
Published: January 2006

This paper presents a new class of models for persons-by-items data. The essential new feature of this class is the representation of the persons: every person is represented by its membership to multiple latent classes, each of which belongs to one latent classification. The models can be considered as a formalization of the hypothesis that the responses come about in a process that involves the application of a number of mental operations. Two algorithms for maximum likelihood (ML) and maximum a posteriori (MAP) estimation are described. They both make use of the tractability of the complete data likelihood to maximize the observed data likelihood. Properties of the MAP estimators (i.e., uniqueness and goodness-of-recovery) and the existence of asymptotic standard errors were examined in a simulation study. Then, one of these models is applied to the responses to a set of fraction addition problems. Finally, the models are compared to some related models in the literature.
The Theoretical Detect Index of Dimensionality and its Application to Approximate Simple Structure
Authors: Zhang Jinming , Stout William
Published: January 2006

In this paper, a theoretical index of dimensionality, called the theoretical DETECT index, is proposed to provide a theoretical foundation for the DETECT procedure. The purpose of DETECT is to assess certain aspects of the latent dimensional structure of a test, important to practitioner and research alike. Under reasonable modeling restrictions referred to as “approximate simple structure”, the theoretical DETECT index is proven to be maximized at the correct dimensionality-based partition of a test, where the number of item clusters in this partition corresponds to the number of substantively separate dimensions present in the test and by “correct” is meant that each cluster in this partition contains only items that correspond to the same separate dimension. It is argued that the separation into item clusters achieved by DETECT is appropriate from the applied perspective of desiring a partition into clusters that are interpretable as substantively distinct between clusters and substantively homogeneous within cluster. Moreover, the maximum DETECT index value is a measure of the amount of multidimensionality present. The estimation of the theoretical DETECT index is discussed and a genetic algorithm is developed to effectively execute DETECT. The study of DETECT is facilitated by the recasting of two factor analytic concepts in a multidimensional item response theory setting: a dimensionally homogeneous item cluster and an approximate simple structure test.
Estimating the Polyserial Correlation Coefficient
Authors: Bedrick Edward J. , Breslin Frederick C.
Published: January 2006

We develop simple noniterative estimators of the polyserial correlation coefficient. A general relationship between the polyserial correlation and the point polyserial correlation is exploited to give extensions of Pearson's, Brogden's, and Lord's biserial estimators to the multicategory setting. The small sample and asymptotic properties of these estimators are studied in some detail. A comparison with maximum likelihood estimates shows that Lord's polyserial estimator is fairly efficient across three probability models.
Analysis of Residuals for the Multinomial Item Response Model
Authors: Reiser Mark
Published: January 2006

Using the item response model as developed on the multinomial distribution, asymptotic variances are obtained for residuals associated with response patterns and first-, and second-order marginal frequencies of manifest variables. When the model does not fit well, an examination of these residuals may reveal the source of the poor fit. Finally, a limited-information test of fit for the model is developed by using residuals defined for the first-, and second-order marginals. Model evaluation based on residuals for these marginals is particularly useful when the response pattern frequencies are sparse.
The Relative Sensitivities of Same-Different and Identification Judgment Models to Perceptual Dependence
Authors: Ennis Daniel M. , Ashby F. Gregory
Published: January 2006

Probabilistic models of same-different and identification judgments are compared (within each paradigm) with regard to their sensitivity to perceptual dependence or the degree to which the underlying psychological dimensions are correlated. Three same-different judgment models are compared. One is a step function or decision bound model and the other two are probabilistic variants of a similarity model proposed by Shepard. Three types of identification models are compared: decision bound models, a probabilistic multidimensional scaling model, and probabilistic models based on the Shepard-Luce choice rule. The decision bound models were found to be most sensitive to perceptual dependence, especially when there is considerable distributional overlap. The same-different model based on the city-block metric and an exponential decay similarity function, and the corresponding identification model were found to be particularly insensitive to perceptual dependence. These results suggest that if a Shepard-type similarity function accurately describes behavior, then under typical experimental conditions it should be difficult to see the effects of perceptual dependence. This result provides strong support for a perceptual independence assumption when using these models. These theoretical results may also play an important role in studying different decision rules employed at different stages of identification training.
Probability of Directional Errors with Disordinal (Qualitative) Interaction
Authors: Shaffer Juliet Popper
Published: January 2006

In a factorial design with two or more factors, there is nonzero interaction when the differences among the levels of one factor vary with levels of other factors. The interaction is disordinal or qualitative with respect to a specific factor, say A, if the difference between at least two levels of A is positive for some and negative for some levels of the other factors. Using standard methods of analysis, there is a potentially large probability of drawing incorrect conclusions about the signs of differences in the presence of disordinal interaction. The maximum probability of such incorrect conclusions, or directional errors, is derived for two-factor designs in which the factor of interest has two levels and the number of levels of the other factor varies.
An Efficient Algorithm for PARAFAC of Three-Way Data with Large Numbers of Observation Units
Authors: Kiers Henk A. L. , Krijnen Wim P.
Published: January 2006

The CANDECOMP algorithm for the PARAFAC analysis of n × m × p three-way arrays is adapted to handle arrays in which n > mp more efficiently. For such arrays, the adapted algorithm needs less memory space to store the data during the iterations, and uses less computation time than the original CANDECOMP algorithm. The size of the arrays that can be handled by the new algorithm is in no way limited by the number of observation units (n) in the data.
Canonical Analysis of Contingency Tables with Linear Constraints
Authors: Böckenholt Ulf , Böcknholt Ingo
Published: January 2006

A generalized least squares approach is presented for incorporating linear constraints on the standardized row and column scores obtained from a canonical analysis of a contingency table. The method is easy to implement and may simplify considerably the interpretation of a data matrix. The approach is compared to a restricted maximum likelihood procedure.
A Latent Trait Model for Dichotomous Choice Data
Authors: Hoijtink Herbert
Published: January 2006

The PARELLA model is a probabilistic parallelogram model that can be used for the measurement of latent attitudes or latent preferences. The data analyzed are the dichotomous responses of persons to stimuli, with a one (zero) indicating agreement (disagreement) with the content of the stimulus. The model provides a unidimensional representation of persons and items. The response probabilities are a function of the distance between person and stimulus: the smaller the distance, the larger the probability that a person will agree with the content of the stimulus. An estimation procedure based on expectation maximization and marginal maximum likelihood is developed and the quality of the resulting parameter estimates evaluated.
Projection of a Binary Criterion into a Model of Hierarchical Classes
Authors: Van Mechelen Iven , De Boeck Paul
Published: January 2006

A formal analysis is made of how to project an attribute criterion into the hierarchical classes model for object by attribute data proposed by De Boeck and Rosenberg. The projection is conceptualized as the prediction of the attribute criterion by means of a logical rule defined on the basis of attribute combinations from the model. Eliminative and constructive strategies are proposed to find logical rules with maximal predictive power and minimal formula complexity. Logical analyses of a real data set are reported and compared with a logistic regression to demonstrate the usefulness of the logical strategies, and to show the complementarity of logical and probabilistic approaches.
The Expected Likelihood of Transitivity of Preference
Authors: Gehrlein WIlliam V.
Published: January 2006

May's model of pairwise preference determination is used to assess the expected likelihood that a subject's pairwise preference comparisons on three alternatives will be transitive. A closed form representation for this expected likelihood is obtained for each situation considered. When the subject is assumed to act precisely according to rankings on attributes with May's model, the computed expected likelihoods the relatively large. When the subject becomes a probabilistic chooser, as defined in a specific manner, expected likelihoods of transitivity decrease significantly from corresponding values with May's model. For a probabilistic Chooser, there is a significant likelihood that the subject might yield transitive pairwise preferences substantially different than the results suggested by May's model.