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.     

Statistical aspects of a three-mode factor analysis model

A special case of Bloxom's version of Tucker's three-mode model is developed statistically. A distinction is made between modes in terms of whether they are fixed or random. Parameter matrices are associated with the fixed modes, while no parameters are associated with the mode representing random observation vectors. The identification problem is discussed, and unknown parameters of the model are estimated by a weighted least squares method based upon a Gauss-Newton algorithm. A goodness-of-fit statistic is presented. An example based upon self-report and peer-report measures of personality shows that the model is applicable to real data. The model represents a generalization of Thurstonian factor analysis; weighted least squares estimators and maximum likelihood estimators of the factor model can be obtained using the proposed theory.This investigation was supported in part by a Research Scientist Development Award (K02-DA00017) and a research grant (DA01070) from the U. S. Public Health Service. The very helpful comments of several anonymous reviewers are gratefully acknowledged.     

Comment on “Correspondence analysis used complementary to loglinear analysis”

In van der Heijden and de Leeuw (1985) it was proposed to use loglinear analysis to detect interactions in a multiway contingency table, and to explore the form of these interactions with correspondence analysis. After performing the exploratory phase of the analysis, we will show here how the results found in this phase can be used for confirmation.This research was conducted while the authors were visiting the Laboratoire de Statistique et Probabilité, Universite Paul Sabatier, Toulouse. This visit was partly made possible by a joint grant of the Netherlands Organisation for the Advancement of Pure Research (Z.W.O.) and the French National Center for Scientific Research (C.N.R.S.). For helpful comments, the authors are indebted to H. Caussinus, J. de Leeuw, and two anonymous reviewers.     

Regularized Finite Mixture Models for Probability Trajectories

Finite mixture models are widely used in the analysis of growth trajectory data to discover subgroups of individuals exhibiting similar patterns of behavior over time. In practice, trajectories are usually modeled as polynomials, which may fail to capture important features of the longitudinal pattern. Focusing on dichotomous response measures, we propose a likelihood penalization approach for parameter estimation that is able to capture a variety of nonlinear class mean trajectory shapes with higher precision than maximum likelihood estimates. We show how parameter estimation and inference for whether trajectories are time-invariant, linear time-varying, or nonlinear time-varying can be carried out for such models. To illustrate the method, we use simulation studies and data from a long-term longitudinal study of children at high risk for substance abuse. This work was supported in part by NIAAA grants R37 AA07065 and R01 AA12217 to RAZ.     

Second-order Non-nonstandard Analysis

Following [3], we build higher-order models of analysis resembling the frameworks of nonstandard analysis. The models are entirely canonical, constructed without Choice. Weak transfer principles are developed and the models are applied to topology, graph theory, and measure theory. A Loeb-like measure is constructed.     

Latent semantic analysis cosines as a cognitive similarity measure: Evidence from priming studies

In distributional semantics models (DSMs) such as latent semantic analysis (LSA), words are represented as vectors in a high-dimensional vector space. This allows for computing word similarities as the cosine of the angle between two such vectors. In two experiments, we investigated whether LSA cosine similarities predict priming effects, in that higher cosine similarities are associated with shorter reaction times (RTs). Critically, we applied a pseudo-random procedure in generating the item material to ensure that we directly manipulated LSA cosines as an independent variable. We employed two lexical priming experiments with lexical decision tasks (LDTs). In Experiment 1 we presented participants with 200 different prime words, each paired with one unique target. We found a significant effect of cosine similarities on RTs. The same was true for Experiment 2, where we reversed the prime-target order (primes of Experiment 1 were targets in Experiment 2, and vice versa). The results of these experiments confirm that LSA cosine similarities can predict priming effects, supporting the view that they are psychologically relevant. The present study thereby provides evidence for qualifying LSA cosine similarities not only as a linguistic measure, but also as a cognitive similarity measure. However, it is also shown that other DSMs can outperform LSA as a predictor of priming effects.     

Evolutionary preference/utility functions: A dynamic perspective

The collection of repeated measures in psychological research is one of the most common data collection formats employed in survey and experimental research. The behavioral decision theory literature documents the existence of the dynamic evolution of preferences that occur over time and experience due to learning, exposure to additional information, fatigue, cognitive storage limitations, etc. We introduce a Bayesian dynamic linear methodology employing an empirical Bayes estimation framework that permits the detection and modeling of such potential changes to the underlying preference utility structure of the respondent. An illustration of revealed stated preference analysis (i.e., conjoint analysis) is given involving students’ preferences for apartments and their underlying attributes and features. We also present the results of several simulations demonstrating the ability of the proposed procedure to recover a variety of different sources of dynamics that may surface with preference elicitation over repeated sequential measurement. Finally, directions for future research are discussed.The authors wish to acknowledge and thank the Editor, the Associate Editor, and two anonymous reviewers for their constructive and insightful comments. Duncan K.H. Fong’s work was sponsored in part by a research grant from the Smeal College.This revised article was published online in August 2005 with the PDF paginated correctly.     

An efficient algorithm for joint correspondence analysis

Joint correspondence analysis is a technique for constructing reduced-dimensional representations of pairwise relationships among categorical variables. The technique was proposed by Greenacre as an alternative to multiple correspondence analysis. Joint correspondence analysis differs from multiple correspondence analysis in that it focuses solely on between-variable relationships. Greenacre described one alternating least-squares algorithm for conducting joint correspondence analysis. Another alternating least-squares algorithm is described in this article. The algorithm is guaranteed to converge, and does so in fewer iterations than does the algorithm proposed by Greenacre. A modification of the algorithm for handling Heywood cases is described. The algorithm is illustrated on two data sets.     

Canonical/redundancy factoring analysis

The interrelationships between two sets of measurements made on the same subjects can be studied by canonical correlation. Originally developed by Hotelling [1936], the canonical correlation is the maximum correlation betweenlinear functions (canonical factors) of the two sets of variables. An alternative statistic to investigate the interrelationships between two sets of variables is the redundancy measure, developed by Stewart and Love [1968]. Van Den Wollenberg [1977] has developed a method of extracting factors which maximize redundancy, as opposed to canonical correlation.A component method is presented which maximizes user specified convex combinations of canonical correlation and the two nonsymmetric redundancy measures presented by Stewart and Love. Monte Carlo work comparing canonical correlation analysis, redundancy analysis, and various canonical/redundancy factoring analyses on the Van Den Wollenberg data is presented. An empirical example is also provided.Wayne S. DeSarbo is a Member of Technical Staff at Bell Laboratories in the Mathematics and Statistics Research Group at Murray Hill, N.J. I wish to express my appreciation to J. Kettenring, J. Kruskal, C. Mallows, and R. Gnanadesikan for their valuable technical assistance and/or for comments on an earlier draft of this paper. I also wish to thank the editor and reviewers of this paper for their insightful remarks.     

A comparison of the classification capabilities of the 1-dimensional kohonen neural network with two pratitioning and three hierarchical cluster analysis algorithms

Neural Network models are commonly used for cluster analysis in engineering, computational neuroscience, and the biological sciences, although they are rarely used in the social sciences. In this study we compare the classification capabilities of the 1-dimensional Kohonen neural network with two partitioning (Hartigan and Späthk-means) and three hierarchical (Ward's, complete linkage, and average linkage) cluster methods in 2,580 data sets with known cluster structure. Overall, the performance of the Kohonen networks was similar to, or better than, the performance of the other methods.     

Structural equations modeling: Fit Indices,sample size,and advanced topics

This article is the second of two parts intended to serve as a primer for structural equations models for the behavioral researcher. The first article introduced the basics: the measurement model, the structural model, and the combined, full structural equations model. In this second article, advanced issues are addressed, including fit indices and sample size, moderators, longitudinal data, mediation, and so forth.     

Handling Missing Covariates in Conditional Mixture Models Under Missing at Random Assumptions

Mixture modeling is a popular method that accounts for unobserved population heterogeneity using multiple latent classes that differ in response patterns. Psychologists use conditional mixture models to incorporate covariates into between-class and/or within-class regressions. Although psychologists often have missing covariate data, conditional mixtures are currently fit with a conditional likelihood, treating covariates as fixed and fully observed. Under this exogenous-x approach, missing covariates are handled primarily via listwise deletion. This sacrifices efficiency and does not allow missingness to depend on observed outcomes. Here we describe a modified joint likelihood approach that (a) allows inference about parameters of the exogenous-x conditional mixture even with nonnormal covariates, unlike a conventional multivariate mixture; (b) retains all cases under missing at random assumptions; (c) yields lower bias and higher efficiency than the exogenous-x approach under a variety of conditions with missing covariates; and (d) is straightforward to implement in available commercial software. The proposed approach is illustrated with an empirical analysis predicting membership in latent classes of conduct problems. Recommendations for practice are discussed.     

Equivalent Dynamic Models

Equivalences of two classes of dynamic models for weakly stationary multivariate time series are discussed: dynamic factor models and autoregressive models. It is shown that exploratory dynamic factor models can be rotated, yielding an infinite set of equivalent solutions for any observed series. It also is shown that dynamic factor models with lagged factor loadings are not equivalent to the currently popular state-space models, and that restriction of attention to the latter type of models may yield invalid results. The known equivalent vector autoregressive model types, standard and structural, are given a new interpretation in which they are conceived of as the extremes of an innovating type of hybrid vector autoregressive models. It is shown that consideration of hybrid models solves many problems, in particular with Granger causality testing.     

Effects on indscal of non-orthogonal perceptions of object space dimensions

A general question is raised concerning the possible consequences of employing the very popular INDSCAL multidimensional scaling model in cases where the assumptions of that model may be violated. Simulated data are generated which violate the INDSCAL assumption that all individuals perceive the dimensions of the common object space to be orthogonal. INDSCAL solutions for these various sets of data are found to exhibit extremely high goodness of fit, but systematically distorted object spaces and negative subject weights. The author advises use of Tucker's three-mode model for multidimensional scaling, which can account for non-orthogonal perceptions of the object space dimensions. It is shown that the INDSCAL model is a special case of the three-mode model.     

Analyzing Job Analysis Data Using Mixture Rasch Models

An important piece of validity evidence to support the use of credentialing exams comes from performing a job analysis of the profession. One common job analysis method is the task inventory method, where people working in the field are surveyed using rating scales about the tasks thought necessary to safely and competently perform the job. This article describes how mixture Rasch models can be used to analyze these data, and how results from these analyses can help to identify whether different groups of people may be responding to job tasks differently. Three examples from different credentialing programs illustrate scenarios that can be found when applying mixture Rasch models to job analysis data. Discussion of what these results may imply for the development of credentialing exams and other analyses of job analysis data is provided.     

Causal models and the assessment-treatment relationship in behavior therapy

The relationship between causal models in behavior therapy and strategies for designing intervention programs on the basis of assessment data is considered. The evolution away from simple univariate causal models is noted and complex causal models for behavior disorders are stressed. However, complex causal models currently preclude empirically based functional analytic and keystone target behavior assessment strategies for intervention design. Future directions for intervention design involving a combination of diagnostic, functional, analytic, and keystone strategies and markers for causal paths are discussed.     

Evaluating evidence for a global mindset factor across multiple ability domains

The current studies help to clarify the nature of growth mindsets by evaluating how strongly people hold a global belief that generalizes across multiple ability domains (e.g., math, writing). Study 1 (N = 651) showed that a bifactor model, consisting of a common global belief and beliefs specific to each domain, fit the data reasonably well. Global mindset beliefs and domain-specific mindset beliefs predicted domain-specific outcomes, whereas global mindset more strongly predicted global outcomes than domain-specific factors. Study 2 (N = 1,422) used an augmented bifactor model with newly developed global mindset items that only served as indicators of the global factor. Results showed high convergence between the new global mindset items and the global factor from a bifactor model.