Circumplex models for correlation matrices

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.Presented as the 1992 Psychometric Society Presidential Address. I am greatly indebted to Stephen Du Toit for help in the development of the computer program employed here. Part of this research was carried out at the University of South Africa and at the Institute for Statistical Research of the South African Human Sciences Research Council.     

Reexamining the circumplex model of affect

The circumplex model of affect has been among the most widely studied representations of affect. Despite the considerable evidence cited in support of it, methods typically used to evaluate the model have substantial limitations. In this article, the authors attempt to correct past limitations by using a covariance structure model specifically designed to assess circumplex structure. This model was fit to 47 individual correlation matrices from published data sets. Analyses revealed that model fit was typically acceptable and that opposing affective states usually demonstrated strong negative correlations with one another. However, analyses also indicated substantial variability in both model fit and correlations among opposing affective states and suggested several characteristics of studies that partially accounted for this variability. Detailed examination of the locations of affective states for 10 of the correlation matrices with relatively optimal characteristics provided mixed support for the model.     

Evaluation of the circumplex structure of the Activation Deactivation Adjective Check List before and after a short walk

Background and purpose: The Activation Deactivation Adjective Check List (AD ACL; Thayer, 1989, The biopsychology of mood and arousal. New York: Oxford University Press) has been used in several studies to assess affective responses to bouts of physical activity. In recent years, researchers have suggested that the structure of this multidimensional measure should approximate a circumplex. This hypothesis was examined using a circumplex-specific confirmatory method.Methods: Volunteers (n=165) completed the AD ACL before and after a short walk. The data were analyzed using Browne’s Circumplex models for correlation matrices. (Psychometrika, 57, 469–497) circular stochastic process model and CIRCUM software and the analyses were performed at the item level.Results and conclusions: Before the walk, the circumplex provided a close fit to the data, whereas, after the walk, the fit was lower, but still reasonable. At neither time did items theorized to belong to one subscale become interspersed with items theorized to belong to an adjacent subscale. The AD ACL represents a satisfactory, albeit imperfect, option for the assessment of affective responses to physical activity from a circumplex perspective. In the future, closer fit to circumplex structure should be achieved by taking the specific structural postulates of the circumplex model into account from the beginning of the scale development process.     

The measurement of core affect: a Swedish self-report measure derived from the affect circumplex

Three studies were conducted with the aim of developing a new Swedish self-report measure of core affect (the Swedish Core Affect Scale or SCAS). In Study 1,122 participants rated their current mood on 24 unipolar adjective scales. A revised set of 12 bipolar adjective scales was evaluated in Study 2 employing 96 participants who rated their current mood before and after a mood-inducing naturally occurring event. A slightly revised set of adjective scales was used in Study 3, in which another 96 participants rated several induced moods. The results showed that the adjective scale ratings could be aggregated as reliable measures of the independent valence and activation dimensions proposed in the affect circumplex, and that the aggregated measures discriminated mood differences within and between individuals.     

Examining the Invariance of Holland's Vocational Interest Model across Gender

This study examines the structural invariance of Holland's (1973, 1985) vocational interest model across gender. Evidence of gender differences in the fit of Holland's model was sought by submitting 14 (7 male; 7 female) previously published Strong Interest Inventory (SII) General Occupational Themes (GOT) scale correlation matrices to multiple structural analytic techniques. Randomization tests of hypothesized order relations (Hubert Arabie, 1987) and single sample confirmatory factor analyses (CFA) indicated a moderate to strong correspondence between GOT data and Holland's circular order and circumplex models. Randomization tests of differences in model–data fit, and two-sample CFA indicated that these models are a no more or less accurate representation of the observed data for men than for women. Additional analyses aimed at identifying gender differences in the misfit of specific aspects of Holland's model also yielded no evidence of differential fit.     

Integrating covariates into circumplex structures: an extension procedure for Browne’s circular stochastic process model

Circumplex structures are elements of various psychological domains. Most work focuses on assessing the circular ordering of circumplex indicators and their relationships with covariates. In this article, an extension procedure for Browne’s circumplex model is presented. Our approach models the relationships among circumplex indicators and the relationships of covariates with a latent circumplex simultaneously without affecting the circumplex indicators’ positions on the circumplex. The approach builds upon Browne’s Fourier series parameterization of a correlation function, which is used to model the latent circumplex correlation structure. It extends the shape of the correlation function to the profile of each covariate’s correlations with the circumplex. The model is specified in the framework of structural equation modeling, thereby making it possible to test various hypotheses. Procedures are presented for deriving interval estimates for the parameters that relate the covariates to the circumplex. The model is compared to other approaches for assessing the relationships of a circumplex with covariates. The results of the exemplary applications and a simulation study were in favor of the suggested model. The approach is furthermore illustrated with a real-data example, focusing on the relationships between the interpersonal circumplex and the rivalry and admiration aspects of narcissism.     

Higher-order factors of the big five model of personality: a reanalysis of Digman (1997)

Based on the results from factor analyses conducted on 14 different data sets, Digman proposed a model of two higher-order factors, or metatraits, that subsumed the Big Five personality traits. In the current article, problems in Digman's analyses were explicated, and more appropriate analyses were then conducted using the same 14 correlation matrices from Digman's study. The resultant two-factor model produced improper solutions, poor model fit indices, or both, in almost all of the 14 data sets and thus raised serious doubts about the veracity of Digman's proposed model.     

A Rigorous Test of the Fit of the Circumplex Model to Big Five Personality Data: Theoretical and Methodological Issues and Two Large Sample Empirical Tests

Our objective was to apply the rigorous test developed by Browne (1992) to determine whether the circumplex model fits Big Five personality data. This test has yet to be applied to personality data. Another objective was to determine whether blended items explained correlations among the Big Five traits. We used two working adult samples, the Eugene-Springfield Community Sample and the Professional Worker Career Experience Survey. Fit to the circumplex was tested via Browne's (1992) procedure. Circumplexes were graphed to identify items with loadings on multiple traits (blended items), and to determine whether removing these items changed five-factor model (FFM) trait intercorrelations. In both samples, the circumplex structure fit the FFM traits well. Each sample had items with dual-factor loadings (8 items in the first sample, 21 in the second). Removing blended items had little effect on construct-level intercorrelations among FFM traits. We conclude that rigorous tests show that the fit of personality data to the circumplex model is good. This finding means the circumplex model is competitive with the factor model in understanding the organization of personality traits. The circumplex structure also provides a theoretically and empirically sound rationale for evaluating intercorrelations among FFM traits. Even after eliminating blended items, FFM personality traits remained correlated.     

Power of the Circumplex: Incremental validity of intersection traits in predicting counterproductive work behaviors

This study examines the role of the circumplex model of personality in predicting counterproductive work behaviors (CWBs). Drawing on the fidelity‐bandwidth principle, we investigate the hypotheses that each of the three sets of circumplex traits representing the intersections of conscientiousness–agreeableness, conscientiousness–emotional stability, and agreeableness–emotional stability will account for significant incremental variance over five‐factor model (FFM) traits in predicting CWBs. Results indicated the circumplex traits contribute incremental variance over the FFM traits, are relatively more important than the FFM traits, and account for 1.5–2.0 times more variance in the total R² associated with CWBs. Collectively, these findings show that circumplex traits capture unique information not explained by FFM traits, and this information leads to a better understanding of the dispositional nature of CWBs.     

Further Support for a Hierarchical Model of Affect: Reply to Green and Salovey

Green and Salovey agree with us that happiness and sadness are strongly bipolar, and that Positive and Negative Activation are not. However, Green and Salovey subscribe to the two-dimensional circumplex, a model we find wanting. Actually, the problems they discuss arise from the limitations of the circumplex rather than from issues fundamental to affective structure. And they do not consider the evidence, reported in our target article, that supports an alternative to the circumplex: a three-level hierarchical model. In this reply, we take a closer look at the empirical findings supporting this model, and we emphasize its heuristic promise.     

Mood and belief-value correlation

This is an investigation of belief-value distortion as measured by the correlation between sets of beliefs and values for a given attitude object. Correlations were computed for each subject separately. Initial mood levels and mood changes subsequent to ratings of attitudes to nuclear power were investigated in the first two studies. In Study 1 mood changes were positively correlated with attitude. In Study 2, mood levels were positively correlated both with attitude, optimism, and tendency towards extreme belief-value correlations (belief-value distortion). In a third study, attitudes were studied both towards nuclear power and alcohol. There were no clear relations between the two fields of contents. Mood changes were small and inconsistent. However, tendency to act correlated with both optimism and belief-value distortion, also with the level of attitude kept constant. A model relating mood, optimism, belief-value distortion, and action is suggested.     

Evaluation of factor analytic research procedures by means of simulated correlation matrices

In order to study the effectiveness of factor analytic methods, a procedure was developed for computing simulated correlation matrices which are more similar to real data correlation matrices than are those matrices computed from the factor analysis structural model. In the present investigation, three methods of factor extraction were studied as applied to 54 simulated correlation matrices which varied in proportion of variance derived from a major factor domain, number of factors in the major domain, and closeness of the simulation procedure to the factor analysis structural model. While the factor extraction methods differed little from one another in quality of results for matrices more dissimilar to the factor analytic model, major differences in quality of results were associated with fewer factors in the major domain, higher proportion of variance from the major domain, and closeness of the simulation procedure to the factor analysis structural model.The research was jointly supported by the University of Illinois and the office of Naval Research under contracts Nonr 1834(39) and N00014-67-A-0305-0003.     

A note on structural models for the circumplex

Jöreskog (1974) developed a latent variable model for the covariance structure of the circumplex which, under certain conditions, includes a model for a patterned correlation matrix (Browne, 1977). This model is of limited usefulness, however, in that it employs a known matrix that is rank deficient for many problems. Furthermore, the model is inappropriate for the circumplex which contains negative covariances. This paper presents alternative models for the perfect circumplex and quasi-circumplex that avoids these difficulties, and that includes the important model for a patterned correlation circumplex matrix. Two numerical examples are provided.This research was supported in part by a grant from the Graduate School of the University of Minnesota. I wish to thank M. W. Browne for suggesting the final model presented in this paper. James Steiger and the Editor also made several valuable suggestions.     

A Simple, Exact Test for the Holland Hexagon

A method is presented for testing the full circumplex model implied by Holland's (1992) theory of vocational interests. The procedure is based on the Spearman rank-order correlation coefficient applied to the hypothesized ranks among correlations defined by the hexagon. In keeping with Holland's stated preference for simple measures and techniques (Holland, 1992, pp. 6–7), the method is easy to compute and apply and, unlike existing techniques, is readily amenable to comparing the fit found in several matrices and for use in meta-analysis.     

Evidence for extending the circumplex model of personality trait language to self-reported moods

We show that self-ratings by 141 undergraduates on 17 items form a circumplex that closely agrees with Conte and Plutchik's (1981) circumplex of personality traits. The order of items on a circumplex, generated by self-ratings following a here-and-now instruction set, scarcely differs from that of a circumplex representing the same ratings averaged over a 2-week period. The configuration becomes more elliptical as one moves from the state-like measures (here-and-now ratings) to pure trait measures (averages), but measurement bias is shown to be the most probable cause of the distortion. Allowing for the limitations of small samples of items and raters, the findings show that the circumplex model represents a valid theoretical construct and does not arise from measurement error or from an illusory implicit personality theory.     

Mood-Dependent Cognitive Change in a Man With Bipolar Disorder Who Cycles Every 24 Hours

A case study of a bipolar patient whose mood changes every 24 hours is described to illustrate the changes in cognitive processing and content during different phases of bipolar disorder. The participant completed a battery of questionnaires and tasks on 4 separate occasions: twice when depressed and twice when manic. Depression tended to be associated with the recall of more general and unpleasant autobiographical memories and greater dependence on others as assessed by self-report and behavioral methods. In contrast, mania tended to be associated with more specific and pleasant autobiographical memories, and antidependence. These preliminary findings set the groundwork for further systematic studies of intraindividual variation in cognitive processing in bipolar disorder. The relevance to future research, theory, and clinical practice are discussed.     

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.