An Extension of Multiple Correspondence Analysis for Identifying Heterogeneous Subgroups of Respondents

An extension of multiple correspondence analysis is proposed that takes into account cluster-level heterogeneity in respondents’ preferences/choices. The method involves combining multiple correspondence analysis and k-means in a unified framework. The former is used for uncovering a low-dimensional space of multivariate categorical variables while the latter is used for identifying relatively homogeneous clusters of respondents. The proposed method offers an integrated graphical display that provides information on cluster-based structures inherent in multivariate categorical data as well as the interdependencies among the data. An empirical application is presented which demonstrates the usefulness of the proposed method and how it compares to several extant approaches. The work reported in this paper was supported by Grant 290439 and Grant A6394 from the Natural Sciences and Engineering Research Council of Canada to the first and third authors, respectively. We wish to thank Ulf B?ckenholt, Paul Green, and Marc Tomiuk for their insightful comments on an earlier version of this paper. We also wish to thank Byunghwa Yang for generously providing us with his data.     

Functional Generalized Structured Component Analysis

An extension of Generalized Structured Component Analysis (GSCA), called Functional GSCA, is proposed to analyze functional data that are considered to arise from an underlying smooth curve varying over time or other continua. GSCA has been geared for the analysis of multivariate data. Accordingly, it cannot deal with functional data that often involve different measurement occasions across participants and a large number of measurement occasions that exceed the number of participants. Functional GSCA addresses these issues by integrating GSCA with spline basis function expansions that represent infinite-dimensional curves onto a finite-dimensional space. For parameter estimation, functional GSCA minimizes a penalized least squares criterion by using an alternating penalized least squares estimation algorithm. The usefulness of functional GSCA is illustrated with gait data.     

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.     

The asymptotic posterior normality of the latent trait in an IRT model

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.This research was partially supported by Office of Naval Research Cognitive and Neural Sciences Grant N0014-J-90-1940, 442-1548, National Science Foundation Mathematics Grant NSF-DMS-91-01436, and the National Center for Supercomputing Applications. We wish to thank Kumar Joag-dev and Zhiliang Ying for enlightening suggestions concerning the proof of the basic result.The authors wish to thank Kumar Joag-Dev, Brian Junker, Bert Green, Paul Holland, Robert Mislevy, and especially Zhiliang Ying for their useful comments and discussions.     

Bayesian generalized structured component analysis

Generalized structured component analysis (GSCA) is a component-based approach to structural equation modelling, which adopts components of observed variables as proxies for latent variables and examines directional relationships among latent and observed variables. GSCA has been extended to deal with a wider range of data types, including discrete, multilevel or intensive longitudinal data, as well as to accommodate a greater variety of complex analyses such as latent moderation analysis, the capturing of cluster-level heterogeneity, and regularized analysis. To date, however, there has been no attempt to generalize the scope of GSCA into the Bayesian framework. In this paper, a novel extension of GSCA, called BGSCA, is proposed that estimates parameters within the Bayesian framework. BGSCA can be more attractive than the original GSCA for various reasons. For example, it can infer the probability distributions of random parameters, account for error variances in the measurement model, provide additional fit measures for model assessment and comparison from the Bayesian perspectives, and incorporate external information on parameters, which may be obtainable from past research, expert opinions, subjective beliefs or knowledge on the parameters. We utilize a Markov chain Monte Carlo method, the Gibbs sampler, to update the posterior distributions for the parameters of BGSCA. We conduct a simulation study to evaluate the performance of BGSCA. We also apply BGSCA to real data to demonstrate its empirical usefulness.     

A multivariate reduced-rank growth curve model with unbalanced data

A multivariate reduced-rank growth curve model is proposed that extends the univariate reducedrank growth curve model to the multivariate case, in which several response variables are measured over multiple time points. The proposed model allows us to investigate the relationships among a number of response variables in a more parsimonious way than the traditional growth curve model. In addition, the method is more flexible than the traditional growth curve model. For example, response variables do not have to be measured at the same time points, nor the same number of time points. It is also possible to apply various kinds of basis function matrices with different ranks across response variables. It is not necessary to specify an entire set of basis functions in advance. Examples are given for illustration.The work reported in this paper was supported by Grant A6394 from the Natural Sciences and Engineering Research Council of Canada to the second author. We thank Jennifer Stephan for her helpful comments on an earlier version of this paper. We also thank Patrick Curran and Terry Duncan for kindly letting us use the NLSY and substance use data, respectively. The substance use data were provided by Grant DA09548 from the National Institute on Drug Abuse.     

Points of view analysis revisited: Fitting multidimensional structures to optimal distance components with cluster restrictions on the variables

Points of view analysis (PVA), proposed by Tucker and Messick in 1963, was one of the first methods to deal explicitly with individual differences in multidimensional scaling, but at some point was apparently superceded by the weighted Euclidean model, well-known as the Carroll and Chang INDSCAL model. This paper argues that the idea behind points of view analysis deserves new attention, especially as a technique to analyze group differences. A procedure is proposed that can be viewed as a streamlined, integrated version of the Tucker and Messick Process, which consisted of a number of separate steps. At the same time, our procedure can be regarded as a particularly constrained weighted Euclidean model. While fitting the model, two types of nonlinear data transformations are feasible, either for given dissimilarities, or for variables from which the dissimilarities are derived. Various applications are discussed, where the two types of transformation can be mixed in the same analysis; a quadratic assignment framework is used to evaluate the results.The research of the first author was supported by the Royal Netherlands Academy of Arts and Sciences (KNAW); the research of the second author by the Netherlands Organization for Scientific Research (NWO Grant 560-267-029). An earlier version of this paper was presented at the European Meeting of the Psychometric Society, Leuven, 1989. We wish to thank Willem J. Heiser for his stimulating comments to earlier versions of this paper, and we are grateful to the Editor and anonymous referees for their helpful suggestions.     

Out-of-bag Prediction Error: A Cross Validation Index for Generalized Structured Component Analysis

Cross validation is a useful way of comparing predictive generalizability of theoretically plausible a priori models in structural equation modeling (SEM). A number of overall or local cross validation indices have been proposed for existing factor-based and component-based approaches to SEM, including covariance structure analysis and partial least squares path modeling. However, there is no such cross validation index available for generalized structured component analysis (GSCA) which is another component-based approach. We thus propose a cross validation index for GSCA, called Out-of-bag Prediction Error (OPE), which estimates the expected prediction error of a model over replications of so-called in-bag and out-of-bag samples constructed through the implementation of the bootstrap method. The calculation of this index is well-suited to the estimation procedure of GSCA, which uses the bootstrap method to obtain the standard errors or confidence intervals of parameter estimates. We empirically evaluate the performance of the proposed index through the analyses of both simulated and real data.     

Restricted multidimensional scaling models for asymmetric proximities

Restricted multidimensional scaling models [Bentler & Weeks, 1978] allowing constraints on parameters, are extended to the case of asymmetric data. Separate functions are used to model the symmetric and antisymmetric parts of the data. The approach is also extended to the case in which data are presumed to be linearly related to squared distances. Examples of several models are provided, using journal citation data. Possible extensions of the models are considered. This research was supported in part by USPHS Grant 0A01070, P. M. Bentler, principal investigator, and NIMH Grant MH-24819, E. J. Anthony and J. Worland, principal investigators. The authors wish to thank E. W. Holman and several anonymous reviewers for their valuable suggestions concerning this research.     

Regularized Generalized Structured Component Analysis

Generalized structured component analysis (GSCA) has been proposed as a component-based approach to structural equation modeling. In practice, GSCA may suffer from multi-collinearity, i.e., high correlations among exogenous variables. GSCA has yet no remedy for this problem. Thus, a regularized extension of GSCA is proposed that integrates a ridge type of regularization into GSCA in a unified framework, thereby enabling to handle multi-collinearity problems effectively. An alternating regularized least squares algorithm is developed for parameter estimation. A Monte Carlo simulation study is conducted to investigate the performance of the proposed method as compared to its non-regularized counterpart. An application is also presented to demonstrate the empirical usefulness of the proposed method.     

A minimum-cost network-flow solution to the case V thurstone scaling problem

This paper presents an approach for determining unidimensional scale estimates that are relatively insensitive to limited inconsistencies in paired comparisons data. The solution procedure, shown to be a minimum-cost network-flow problem, is presented in conjunction with a sensitivity diagnostic that assesses the influence of a single pairwise comparison on traditional Thurstone (ordinary least squares) scale estimates. When the diagnostic indicates some source of distortion in the data, the network technique appears to be more successful than Thurstone scaling in preserving the interval scale properties of the estimates.My special thanks go to Alvin Silk, Thomas Magnanti, and Roy Welsch for their support and advice throughout the formative stages of this paper, and to V. Srinivasan for his helpful comments on a later draft of this paper. I also wish to thank the Editor, Associate Editor, and two reviewers for their constructive suggestions.James M. Lattin is Associate Professor of Marketing and Management Science and the James and Doris McNamara Faculty Fellow for 1988-1989.     

Interval estimation for the intraclass correlation in dirichlet-multinomial data

When the underlying distribution is discrete with a limited number of categories, methods for interval estimation of the intraclass correlation which assume normality are theoretically inadequate for use. On the basis of large sample theory, this paper develops an asymptotic closed-form interval estimate of the intraclass correlation for the case where there is a natural score associated with each category. This paper employs Monte Carlo simulation to demonstrate that when the underlying intraclass correlation is large, the traditional interval estimator which assumes normality can be misleading. We find that when the number of classes is 20, the interval estimator proposed here can generally perform reasonably well in a variety of situations. This paper further notes that the proposed interval estimator is invariant with respect to a linear transformation. When the data are on a nominal scale, an extension of the proposed method to account for this case, as well as a discussion on the relationship between the intraclass correlation and a kappa-type measure defined here and on the limitation of the corresponding kappa-type estimator are given.The authors wish to thank the Editor, the Associate Editor, and the three referees for many valuable comments and suggestions to improve the clarity of this paper. The works for the first, the third, and the fourth authors were partially supported by grant #R01AR43025-01 from the National Institute of Arthritis and Musculoskeletal and Skin Diseases.     

Generalized Structured Component Analysis with Latent Interactions

Generalized structured component analysis (GSCA) is a component-based approach to structural equation modeling. In practice, researchers may often be interested in examining the interaction effects of latent variables. However, GSCA has been geared only for the specification and testing of the main effects of variables. Thus, an extension of GSCA is proposed to effectively deal with various types of interactions among latent variables. In the proposed method, a latent interaction is defined as a product of interacting latent variables. As a result, this method does not require the construction of additional indicators for latent interactions. Moreover, it can easily accommodate both exogenous and endogenous latent interactions. An alternating least-squares algorithm is developed to minimize a single optimization criterion for parameter estimation. A Monte Carlo simulation study is conducted to investigate the parameter recovery capability of the proposed method. An application is also presented to demonstrate the empirical usefulness of the proposed method.     

Latent variable modeling in heterogeneous populations

Common applications of latent variable analysis fail to recognize that data may be obtained from several populations with different sets of parameter values. This article describes the problem and gives an overview of methodology that can address heterogeneity. Artificial examples of mixtures are given, where if the mixture is not recognized, strongly distorted results occur. MIMIC structural modeling is shown to be a useful method for detecting and describing heterogeneity that cannot be handled in regular multiple-group analysis. Other useful methods instead take a random effects approach, describing heterogeneity in terms of random parameter variation across groups. These random effects models connect with emerging methodology for multilevel structural equation modeling of hierarchical data. Examples are drawn from educational achievement testing, psychopathology, and sociology of education. Estimation is carried out by the LISCOMP program.Presidential address delivered at the Psychometric Society meetings in Los Angeles, USA and Leuven, Belgium, July 1989. The research was supported by Grant No. SES-8821668 from the National Science Foundation and by Grant No. OERI-G-86-003 from the Office for Educational Research and Improvement, Department of Education. I thank Leigh Burstein, Mike Hollis, Linda Muthén, and Albert Satorra for helpful discussions and Tammy Tam, Jin-Wen Yang, Suk-Woo Kim, and Lynn Short for computational assistance. Designs were created by Arlette Collier, Rita Ling and Jennifer Edic-Bryant.     

Building an identifiable latent class model with covariate effects on underlying and measured variables

In recent years, latent class models have proven useful for analyzing relationships between measured multiple indicators and covariates of interest. Such models summarize shared features of the multiple indicators as an underlying categorical variable, and the indicators' substantive associations with predictors are built directly and indirectly in unique model parameters. In this paper, we provide a detailed study on the theory and application of building models that allow mediated relationships between primary predictors and latent class membership, but that also allow direct effects of secondary covariates on the indicators themselves. Theory for model identification is developed. We detail an Expectation-Maximization algorithm for parameter estimation, standard error calculation, and convergent properties. Comparison of the proposed model with models underlying existing latent class modeling software is provided. A detailed analysis of how visual impairments affect older persons' functioning requiring distance vision is used for illustration.This work was supported by National Institute on Aging (NIA) Program Project P01-AG-10184-03 and National Institutes of Mental Health grant R01-MH-56639-01A1. Dr. Bandeen-Roche is a Brookdale National Fellow. The authors wish to thank Drs. Gary Rubin and Sheila West for kindly making the Salisbury Eye Evaluation data available. We also thank the Editor, the Associate Editor, and three referees for their valuable comments.     

The validity of personality,service orientation,and reading comprehension measures as predictors of Flight Attendant training performance

The present study examined the predictive power of Big Five personality dimensions, a situational measure of service orientation and a reading comprehension measure as predictors of Flight Attendant training criteria. The criteria consisted of multiple ratings made by trainers of trainees across a number of training dimensions. At least partial data was collected on 424 Flight Attendant trainees. Results indicated significant correlations between all three predictors and a number of the training criteria.The authors would like to thank the Editor and two anonymous reviewers for their help revising this article.     

Testing a unidimensional,qualitative unfolding model for attitudinal or developmental data

Assuming that subject responses rank order stimuli by preference, statistical methods are presented for testing the hypothesis that responses conform to a unidimensional, qualitative unfolding model and to an a priori stimulus ordering. The model postulates that persons and stimulus variables are ordered along a single continuum and that subjects most prefer stimuli nearest their own position. The underlying continuum need not form an interval scale of the stimulus attribute. The general assumptions of the test for the unfolding model make it suitable for the analysis of structure in attitude responses, preference data, and developmental stage data.This research was supported by a grant from the U.S. Public Health Service (Grant No. 1-R01-MH27861-01) to the University of Minnesota. I wish to thank Sanford Weisberg for his helpful suggestions. I also wish to thank Karen Kitchener and Patricia King for letting me use their data.     

Multilevel Dynamic Generalized Structured Component Analysis for Brain Connectivity Analysis in Functional Neuroimaging Data

We extend dynamic generalized structured component analysis (GSCA) to enhance its data-analytic capability in structural equation modeling of multi-subject time series data. Time series data of multiple subjects are typically hierarchically structured, where time points are nested within subjects who are in turn nested within a group. The proposed approach, named multilevel dynamic GSCA, accommodates the nested structure in time series data. Explicitly taking the nested structure into account, the proposed method allows investigating subject-wise variability of the loadings and path coefficients by looking at the variance estimates of the corresponding random effects, as well as fixed loadings between observed and latent variables and fixed path coefficients between latent variables. We demonstrate the effectiveness of the proposed approach by applying the method to the multi-subject functional neuroimaging data for brain connectivity analysis, where time series data-level measurements are nested within subjects.