The LMPCA program: A graphical user interface for fitting the linked-mode PARAFAC-PCA model to coupled real-valued data

In behavioral research, PARAFAC analysis, a three-mode generalization of standard principal component analysis (PCA), is often used to disclose the structure of three-way three-mode data. To get insight into the underlying mechanisms, one often wants to relate the component matrices resulting from such a PARAFAC analysis to external (two-way two-mode) information, regarding one of the modes of the three-way data. To this end, linked-mode PARAFAC-PCA analysis can be used, in which the three-way and the two-way data set, which have one mode in common, are simultaneously analyzed. More specifically, a PARAFAC and a PCA model are fitted to the three-way and the two-way data, respectively, restricting the component matrix for the common mode to be equal in both models. Until now, however, no software program has been publicly available to perform such an analysis. Therefore, in this article, the LMPCA program, a free and easy-to-use MATLAB graphical user interface, is presented to perform a linked-mode PARAFAC-PCA analysis. The LMPCA software can be obtained from the authors at http://ppw.kuleuven.be/okp/software/LMPCA. For users who do not have access to MATLAB, a stand-alone version is provided.     

Parallel Factor Analysis of gait waveform data: A multimode extension of Principal Component Analysis

Gait data are typically collected in multivariate form, so some multivariate analysis is often used to understand interrelationships between observed data. Principal Component Analysis (PCA), a data reduction technique for correlated multivariate data, has been widely applied by gait analysts to investigate patterns of association in gait waveform data (e.g., interrelationships between joint angle waveforms from different subjects and/or joints). Despite its widespread use in gait analysis, PCA is for two-mode data, whereas gait data are often collected in higher-mode form. In this paper, we present the benefits of analyzing gait data via Parallel Factor Analysis (Parafac), which is a component analysis model designed for three- or higher-mode data. Using three-mode joint angle waveform data (subjects×time×joints), we demonstrate Parafac's ability to (a) determine interpretable components revealing the primary interrelationships between lower-limb joints in healthy gait and (b) identify interpretable components revealing the fundamental differences between normal and perturbed subjects' gait patterns across multiple joints. Our results offer evidence of the complex interconnections that exist between lower-limb joints and limb segments in both normal and abnormal gaits, confirming the need for the simultaneous analysis of multi-joint gait waveform data (especially when studying perturbed gait patterns).     

Implicit Theories of Personality: Further Evidence of Extreme Response Style

Five groups of 19 subjects made ratings on 11 personality trait scales of overlapping subsets of 59 artificial stimulus persons who were described by one to five personality trait adjectives. The data were analyzed per group of subjects (blockwise) and per type of stimulus person (questionnaire-wise) by means of three-mode principal component analyses. This yielded highly similar structures for the scales, and in the blockwise analyses, for the stimulus persons. This similarity was substantiated by external three-mode analyses, which showed that all stimulus persons can be mapped into one overall configuration. In all analyses it was found that differences between subjects consisted of differences in extremity of judgment, which suggests the operation of response style.     

Three-way metric unfolding via alternating weighted least squares

Three-way unfolding was developed by DeSarbo (1978) and reported in DeSarbo and Carroll (1980, 1981) as a new model to accommodate the analysis of two-mode three-way data (e.g., nonsymmetric proximities for stimulus objects collected over time) and three-mode, three-way data (e.g., subjects rendering preference judgments for various stimuli in different usage occasions or situations). This paper presents a revised objective function and new algorithm which attempt to prevent the common type of degenerate solutions encountered in typical unfolding analysis. We begin with an introduction of the problem and a review of three-way unfolding. The three-way unfolding model, weighted objective function, and new algorithm are presented. Monte Carlo work via a fractional factorial experimental design is described investigating the effect of several data and model factors on overall algorithm performance. Finally, three applications of the methodology are reported illustrating the flexibility and robustness of the procedure.We wish to thank the editor and reviewers for their insightful comments.     

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.     

Differentiation of affective and denotative meaning systems and their influence in personality ratings.

The present study presents an empirical example of the dichotomy of affective and denotative meaning systems and their influence on individual differences in personality ratings. The three-mode factor analytic technique with a newly developed transformation methodology for the scale mode was applied to data collected by Hogenraad from 50 French-speaking Belgians, rating 40 personality concepts against 40 semantic differential scales. Results indicated that three affective dimensions (evaluation, potency, and activity) proved to be dominant in the indigenous factor structure of personality impressions and that three dimensions in the "other" space, orthogonal to affect, are clearly interpretable denotative semantic features of personalities. Three idealized individual differences on interactions of these two meaning systems with four concept factors were highlighted by the final rotated inner core matrix. The present methodology along with the semantic differential technique and three-mode factor analysis can be applied to various types of subjects and/or concept domains for better understanding of intra- and intercultural differences.     

Tucker2 hierarchical classes analysis

This paper presents a new hierarchical classes model, called Tucker2-HICLAS, for binary three-way three-mode data. As any three-way hierarchical classes model, the Tucker2-HICLAS model includes a representation of the association relation among the three modes and a hierarchical classification of the elements of each mode. A distinctive feature of the Tucker2-HICLAS model, being closely related to the Tucker3-HICLAS model (Ceulemans, Van Mechelen & Leenen, 2003), is that one of the three modes is minimally reduced and, hence, that the differences among the association patterns of the elements of this mode are maximally retained in the model. Moreover, as compared to Tucker3-HICLAS, Tucker2-HICLAS implies three rather than four different types of parameters and as such is simpler to interpret. Two types of Tucker2-HICLAS models are distinguished: a disjunctive and a conjunctive type. An algorithm for fitting the Tucker2-HICLAS model is described and evaluated in a simulation study. The model is illustrated with longitudinal data on interpersonal emotions. The first author is a Researcher of the Fund for Scientific Research—Flanders (Belgium). The research reported in this paper was partially supported by the Research Council of K.U. Leuven (GOA/2000/02). The authors are grateful to Iwin Leenen for the fruitful discussions.     

Relations between multidimensional scaling and three-mode factor analysis

A combination is achieved of two lines of psychometric interest: a) multidimensional scaling and b) factor analysis. This is accomplished with the use of three-mode factor analysis of scalar product matrices, one for each subject. Two of the modes are the groups of objects scaled and the third mode is the sample of subjects. Results are an object space, a person space, and a system for changing weights given to dimensions and of angles between dimensions in the object space for individuals located at different places in the person space. The development is illustrated with data from an adjective similarity study.Supported in part by the Personnel and Training Branch of the Office of Naval Research under Contract Number 00014-67-A-0305-0003. A number of very helpful comments were made by an anonymous editorial reviewer forPsychometrika.During 1970–71 at the L. L. Thurstone Psychometric Laboratory, University of North Carolina.     

The Use of Exploratory Factor Analysis and Principal Components Analysis in Communication Research

Exploratory factor analysis is a popular statistical technique used in communication research. Although exploratory factor analysis (EFA) and principal components analysis (PCA) are different techniques, PCA is often employed incorrectly to reveal latent constructs (i.e., factors) of observed variables, which is the purpose of EFA. PCA is more appropriate for reducing measured variables into a smaller set of variables (i.e., components) by keeping as much variance as possible out of the total variance in the measured variables. Furthermore, the popular use of varimax rotation raises some concerns about the relationships among the factors that researchers claim to discover. This paper discusses the distinct purposes of PCA and EFA, using two data sets as examples to highlight the differences in results between these procedures, and also reviews the use of each technique in three major communication journals: Communication Monographs, Human Communication Research, and Communication Research.     

Tucker3 hierarchical classes analysis

This paper presents a new model for binary three-way three-mode data, called Tucker3 hierarchical classes model (Tucker3-HICLAS). This new model generalizes Leenen, Van Mechelen, De Boeck, and Rosenberg's (1999) individual differences hierarchical classes model (INDCLAS). Like the INDCLAS model, the Tucker3-HICLAS model includes a hierarchical classification of the elements of each mode, and a linking structure among the three hierarchies. Unlike INDCLAS, Tucker3-HICLAS (a) does not restrict the hierarchical classifications of the three modes to have the same rank, and (b) allows for more complex linking structures among the three hierarchies. An algorithm to fit the Tucker3-HICLAS model is described and evaluated in an extensive simulation study. An application of the model to hostility data is discussed.The first author is a Research Assistant of the Fund for Scientific Research-Flanders (Belgium). The research reported in this paper was partially supported by the Research Council of K.U. Leuven (GOA/2000/02). We are grateful to Kristof Vansteelandt for providing us with an interesting data set.     

EP Component Identification and Measurement by Principal Components-Analysis

Between the acquisition of Evoked Potential (EP) data and their interpretation lies a major problem: What to measure? An approach to this kind of problem is outlined here in terms of Principal Components Analysis (PCA). An important second theme is that experimental manipulation is important to functional interpretation. It would be desirable to have a system of EP measurement with the following characteristics: (1) represent the data in a concise, parsimonous way; (2) determine EP components from the data without assuming in advance any particular waveforms for the components; (3) extract components which are independent of each other; (4) measure the amounts (contributions) of various components in observed EPs; (5) use measures that have greater reliability than measures at any single time point or peak; and (6) identify and measure conponents that overlap in time. PCA has these desirable characteristics. Simulations are illustrated. PCA′s beauty also has some warts that are discussed. In addition to discussing the usual two-mode model of PCA, an extension of PCA to a three-mode model is described that provides separate parameters for (1) waveforms over time, (2) coefficients for spatial distribution, and (3) scores telling the amount of each component in each EP. PCA is compared with more traditional approaches. Some biophysical considerations are briefly discussed. Choices to be made in applying PCA are considered. Other issues include misallocation of variance, overlapping components, validation, and latency changes.     

Clusterwise simultaneous component analysis for analyzing structural differences in multivariate multiblock data

Many studies yield multivariate multiblock data, that is, multiple data blocks that all involve the same set of variables (e.g., the scores of different groups of subjects on the same set of variables). The question then rises whether the same processes underlie the different data blocks. To explore the structure of such multivariate multiblock data, component analysis can be very useful. Specifically, 2 approaches are often applied: principal component analysis (PCA) on each data block separately and different variants of simultaneous component analysis (SCA) on all data blocks simultaneously. The PCA approach yields a different loading matrix for each data block and is thus not useful for discovering structural similarities. The SCA approach may fail to yield insight into structural differences, since the obtained loading matrix is identical for all data blocks. We introduce a new generic modeling strategy, called clusterwise SCA, that comprises the separate PCA approach and SCA as special cases. The key idea behind clusterwise SCA is that the data blocks form a few clusters, where data blocks that belong to the same cluster are modeled with SCA and thus have the same structure, and different clusters have different underlying structures. In this article, we use the SCA variant that imposes equal average cross-products constraints (ECP). An algorithm for fitting clusterwise SCA-ECP solutions is proposed and evaluated in a simulation study. Finally, the usefulness of clusterwise SCA is illustrated by empirical examples from eating disorder research and social psychology.     

A reformulation of the general Euclidean model for the external analysis of preference data

This paper discusses least squares methods for fitting a reformulation of the general Euclidean model for the external analysis of preference data. The reformulated subject weights refer to a common set of reference vectors for all subjects and hence are comparable across subjects. If the rotation of the stimulus space is fixed, the subject weight estimates in the model are uniquely determined. Weight estimates can be guaranteed nonnegative. While the reformulation is a metric model for single stimulus data, the paper briefly discusses extensions to nonmetric, pairwise, and logistic models. The reformulated model is less general than Carroll's earlier formulation.The author is grateful to Christopher J. Nachtsheim for his helpful suggestions.     

Model Selection with the Linear Mixed Model for Longitudinal Data

Model building or model selection with linear mixed models (LMMs) is complicated by the presence of both fixed effects and random effects. The fixed effects structure and random effects structure are codependent, so selection of one influences the other. Most presentations of LMM in psychology and education are based on a multilevel or hierarchical approach in which the variance-covariance matrix of the random effects is assumed to be positive definite with nonzero values for the variances. When the number of fixed effects and random effects is unknown, the predominant approach to model building is a step-up method in which one starts with a limited model (e.g., few fixed and random intercepts) and then additional fixed effects and random effects are added based on statistical tests. A model building approach that has received less attention in psychology and education is a top-down method. In the top-down method, the initial model has a single random intercept but is loaded with fixed effects (also known as an “overelaborate” model). Based on the overelaborate fixed effects model, the need for additional random effects is determined. There has been little if any examination of the ability of these methods to identify a true population model (i.e., identifying the model that generated the data). The purpose of this article is to examine the performance of the step-up and top-down model building approaches for exploratory longitudinal data analysis. Student achievement data sets from the Chicago longitudinal study serve as the populations in the simulations.     

Internal Employment Opportunity and External Employment Opportunity: Independent or Interactive Retention Effects?

Contemporary attrition/retention frameworks suggest that internal mobility opportunities (e.g., promotions) and external mobility opportunities (i.e., alternative sources of employment) impact turnover decisions. The present study used data collected' from 413 U.S. Air Force enlistees to estimate the joint effects of internal mobility opportunities (i.e., training/career broadening opportunities) and external mobility opportunities (e.g., existing job offers) on reenlistment. Logistic regression analysis revealed that internal and external mobility opportunities had additive effects on the study's attrition criterion. Moderated regressions failed to detect interactive effects among the study's mobility variables. Implications of the findings for turnover frameworks and retention of military personnel are discussed.     

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.     

The CHIC Model: A Global Model for Coupled Binary Data

Often problems result in the collection of coupled data, which consist of different N-way N-mode data blocks that have one or more modes in common. To reveal the structure underlying such data, an integrated modeling strategy, with a single set of parameters for the common mode(s), that is estimated based on the information in all data blocks, may be most appropriate. Such a strategy implies a global model, consisting of different N-way N-mode submodels, and a global loss function that is a (weighted) sum of the partial loss functions associated with the different submodels. In this paper, such a global model for an integrated analysis of a three-way three-mode binary data array and a two-way two-mode binary data matrix that have one mode in common is presented. A simulated annealing algorithm to estimate the model parameters is described and evaluated in a simulation study. An application of the model to real psychological data is discussed. T. Wilderjans is a Research Assistant of the Fund for Scientific Research—Flanders (Belgium). The research reported in this paper was partially supported by the Research Council of K.U. Leuven (GOA/2005/04). We are grateful to Kristof Vansteelandt for providing us with an interesting data set. We also thank three anonymous reviewers for their useful comments.     

Indclas: A three-way hierarchical classes model

A three-way three-mode extension of De Boeck and Rosenberg's (1988) two-way two-mode hierarchical classes model is presented for the analysis of individual differences in binary object × attribute arrays. In line with the two-way hierarchical classes model, the three-way extension represents both the association relation among the three modes and the set-theoretical relations among the elements of each model. An algorithm for fitting the model is presented and evaluated in a simulation study. The model is illustrated with data on psychiatric diagnosis. Finally, the relation between the model and extant models for three-way data is discussed.The research reported in this paper was partially supported by NATO (Grant CRG.921321 to Iven Van Mechelen and Seymour Rosenberg).