Simultaneous factor analysis of several Gramian matrices

Given several Gramian matrices, a least-square fit to all the matrices by one factor matrix, with a predetermined number of factors, is shown to be the principal axes solution of the average of the matrices.     

Procrustean analysis of matrices with different row order

Procrustean analysis is extended here to the case where matrices have different row order. Using a definition system in the sense of facet theory, fitting transformations are determined for the set of centroids that represent common substantive equivalence classes of all or some matrices. The resulting operators are used to transform the original matrices. An index, δ, which is consistent with this approach expresses the goodness of fit.     

The extension of component analysis to four-mode matrices

A model for four-mode component analysis is developed and presented. The developed model, which is an extension of Tucker's three-mode factor analytic model, allows for the simultaneous analysis of all modes of a four-mode data matrix and the consideration of relationships among the modes. An empirical example based upon viewer perceptions of repetitive advertising shows the four-mode model applicable to real data.This research was supported by the University of Kansas School of Business Research Fund provided by the Fourth National Bank & Trust Company, Wichita. The ideas and opinions expressed herein are solely those of the author.     

Testing independence in two-way contingency tables with data subject to misclassification

The misclassification process is represented by a stochastic matrix containing the probabilities that an individual who belongs in one cell is counted as belonging to another (or perhaps the same) cell. These probabilities are supposed known. If misclassification in the row direction is independent of that along the column variable then the size of the usual chi-square test is unchanged. It is shown how to calculate loss of power in this case and also how to calculate the change in size of the test if the errors are not independent. A modified test criterion is suggested when errors are not independent and a numerical example is included.     

Fitting cross-classification table data to models when observations are subject to classification error

Individuals are classified in a cross-classification table where two behavioral observations on each individual determine the classification. The problem is to test certain structural models assumed to underlie the cross-classified observations. A minimum chi-square test procedure is proposed.     

Constrained latent class analysis: Simultaneous classification and scaling of discrete choice data

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.The authors thank Yosiho Takane, the editor and referees for their valuable suggestions. Authors are listed in reverse alphabetical order.     

Generalization gradients of inhibition after different amounts of training

Five groups of pigeons received seven sessions of variable-interval reinforcement for pecking a blank white key, followed by either 1, 2, 4, 8, or 16 sessions of training on a successive discrimination in which the positive stimulus was the blank white key and the negative stimulus was a black vertical line on the white key. After training, a generalization test was administered along the line-tilt continuum. Relative gradients of inhibition became steeper with increased amounts of training, and reliably nonhorizontal absolute gradients were obtained only from groups of subjects with at least four days of training. Therefore, inhibitory stimulus control improves with added training. Several problems with the concept of “inhibition” are examined and some implications of the results for theoretical analyses of operant discrimination learning are discussed.     

The analysis of multitrait-multimethod matrices via constrained components analysis

Multitrait-Multimethod (MTMM) matrices are often analyzed by means of confirmatory factor analysis (CFA). However, fitting MTMM models often leads to improper solutions, or non-convergence. In an attempt to overcome these problems, various alternative CFA models have been proposed, but with none of these the problem of finding improper solutions was solved completely. In the present paper, an approach is proposed where improper solutions are ruled out altogether and convergence is guaranteed. The approach is based on constrained variants of components analysis (CA). Besides the fact that these methods do not give improper solutions, they have the advantage that they provide component scores which can later on be used to relate the components to external variables. The new methods are illustrated by means of simulated data, as well as empirical data sets.This research has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences to the first author. The authors are obliged to three anonymous reviewers and an associate editor for constructive suggestions on the first version of this paper.     

Orthogonal rotations to maximal agreement for two or more matrices of different column orders

Methods for orthogonal Procrustes rotation and orthogonal rotation to a maximal sum of inner products are examined for the case when the matrices involved have different numbers of columns. An inner product solution offered by Cliff is generalized to the case of more than two matrices. A nonrandom start for a Procrustes solution suggested by Green and Gower is shown to give better results than a random start. The Green-Gower Procrustes solution (with nonrandom start) is generalized to the case of more than two matrices. Simulation studies indicate that both the generalized inner product solution and the generalized Procrustes solution tend to attain their global optima within acceptable computation times. A simple procedure is offered for approximating simple structure for the rotated matrices without affecting either the Procrustes or the inner product criterion.The authors are obliged to Charles Lewis for helpful comments on a previous draft of this paper and to Frank Brokken for preparing a computer program that was used in this study.     

The estimation of variance-covariance and correlation matrices from incomplete data

Employing simulated data, several methods for estimating correlation and variance-covariance matrices are studied for observations missing at random from data matrices. The effect of sample size, number of variables, percent of missing data and average intercorrelations of variables are examined for several proposed methods.The author is indebted to Professors Leonard A. Marascuilo, Gus W. Haggstrom, especially Henry F. Kaiser for their invaluable suggestions throughout this work. Appreciation is also extended to the Computer Center facility of the University of California at Berkeley for the use of computer time to complete the necessary computations.     

Estimation and visualization of confusability matrices from adaptive measurement data

We present a simple but effective method based on Luce’s choice axiom [Luce, R.D. (1959). Individual choice behavior: A theoretical analysis. New York: John Wiley & Sons] for consistent estimation of the pairwise confusabilities of items in a multiple-choice recognition task with arbitrarily chosen choice-sets. The method combines the exact (non-asymptotic) Bayesian way of assessing uncertainty with the unbiasedness emphasized in the classical frequentist approach.We apply the method to data collected using an adaptive computer game designed for prevention of reading disability. A player’s estimated confusability of phonemes (or more accurately, phoneme-grapheme connections) and larger units of language is visualized in an easily understood way with color cues and explicit indication of the accuracy of the estimates. Visualization of learning-related changes in the player’s performance is considered.The empirical validity of the choice axiom is evaluated using the game data itself. The axiom appears to hold reasonably well although a small systematic violation is observable for the smallest choice-set sizes.     

Structural analysis of covariance and correlation matrices

A general approach to the analysis of covariance structures is considered, in which the variances and covariances or correlations of the observed variables are directly expressed in terms of the parameters of interest. The statistical problems of identification, estimation and testing of such covariance or correlation structures are discussed.Several different types of covariance structures are considered as special cases of the general model. These include models for sets of congeneric tests, models for confirmatory and exploratory factor analysis, models for estimation of variance and covariance components, regression models with measurement errors, path analysis models, simplex and circumplex models. Many of the different types of covariance structures are illustrated by means of real data.1978 Psychometric Society Presidential Address.This research has been supported by the Bank of Sweden Tercentenary Foundation under the project entitledStructural Equation Models in the Social Sciences, Karl G. Jöreskog, project director.     

Road-sign utility as judged by drivers with different amounts of driving experience

This study starts from the hypothesis that sign perception depends on the motivational valences of signs. Two groups of drivers with different amounts of driving experience rated Swedish road signs on dimensions assumed to indicate perceived sign utility. The ratings were performed against a scenario that included traffic officials' intentions to simplify the sign system. One dimension was amount of reduction in frequency of occurrence; subjects indicated percentages of reduction they believed to be reasonable. A second dimension was appropriateness of complete elimination of a sign from the traffic environment; subjects rated the degree to which elimination was reasonable. Subjective frequency of sign occurrence was rated last. Construct validation procedures showed strong correlations between reduction and elimination ratings only for the higher-experience group. Contrary to expectation, both groups showed the same high level of positive attitudes toward all signs. As expected, frequency was a judgmental base for utility, though weakly, except for permissive signs. It was more extensively related to utility for the higher-experience group, particularly for percentage ratings. Curvilinear relationships obtained between frequency and utility for permissive and road guidance signs.     

ConPar: a method for identifying groups of concordant subject proximity matrices for subsequent multidimensional scaling analyses

A common representation of data within the context of multidimensional scaling (MDS) is a collection of symmetric proximity (similarity or dissimilarity) matrices for each of M subjects. There are a number of possible alternatives for analyzing these data, which include: (a) conducting an MDS analysis on a single matrix obtained by pooling (averaging) the M subject matrices, (b) fitting a separate MDS structure for each of the M matrices, or (c) employing an individual differences MDS model. We discuss each of these approaches, and subsequently propose a straightforward new method (CONcordance PARtitioning—ConPar), which can be used to identify groups of individual-subject matrices with concordant proximity structures. This method collapses the three-way data into a subject×subject dissimilarity matrix, which is subsequently clustered using a branch-and-bound algorithm that minimizes partition diameter. Extensive Monte Carlo testing revealed that, when compared to K-means clustering of the proximity data, ConPar generally provided better recovery of the true subject cluster memberships. A demonstration using empirical three-way data is also provided to illustrate the efficacy of the proposed method.