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.     

Patterns of mood change

Two independent sets of data were analyzed, each consisted of self-reports of mood using 10 scales devised by means described in the article. The two data sets were, respectively, based on: (1) unforced mood changes over a 30-day period; and (2) forced mood changes measured following administration of either elation- or depression-inducing statements. Correlation matrices, factor analysis and rearranged correlation matrices were presented to investigate unipolar, bipolar and circumplex hypotheses. The forced data set proved to be the simpler case, with two factors, and the apparent circumplex of the factor plot proved to be the case following rearrangement of the correlation matrices. The unforced data, though confirming the simpler solution in some respects, proved to be more complicated and more difficult to conform to a model, which is by definition two-dimensional. Therefore, natural covariation may be too complex for the circumplex model. However, there is still a possibility that a broad bipolar model can not only fit a circumplex but may have heuristic value     

The nature experiences of a Mississippi watercolorist

Watercolorist Walter Anderson, after a series of hospitalizations for mental illness, began a series of private journeys to a barrier island off the Mississippi coast, where he had a number of ecstatic, mystical experiences vis-à-vis nature and painted the myriad images he found. It is argued that these episodes were not pathological. Anderson's relationships with archetypes and with death are presented as central elements of a spiritual journey.     

On equivariance and invariance of standard errors in three exploratory factor models

Current practice in factor analysis typically involves analysis of correlation rather than covariance matrices. We study whether the standardz-statistic that evaluates whether a factor loading is statistically necessary is correctly applied in such situations and more generally when the variables being analyzed are arbitrarily rescaled. Effects of rescaling on estimated standard errors of factor loading estimates, and the consequent effect onz-statistics, are studied in three variants of the classical exploratory factor model under canonical, raw varimax, and normal varimax solutions. For models with analytical solutions we find that some of the standard errors as well as their estimates are scale equivariant, while others are invariant. For a model in which an analytical solution does not exist, we use an example to illustrate that neither the factor loading estimates nor the standard error estimates possess scale equivariance or invariance, implying that different conclusions could be obtained with different scalings. Together with the prior findings on parameter estimates, these results provide new guidance for a key statistical aspect of factor analysis.We gratefully acknowledge the help of the Associate Editor and three referees whose constructive comments lead to an improved version of the paper. This work was supported by National Institute on Drug Abuse Grants DA01070 and DA00017 and by the University of North Texas Faculty Research Grant Program.     

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.     

Limited information estimation and testing of discretized multivariate normal structural models

Discretized multivariate normal structural models are often estimated using multistage estimation procedures. The asymptotic properties of parameter estimates, standard errors, and tests of structural restrictions on thresholds and polychoric correlations are well known. It was not clear how to assess the overall discrepancy between the contingency table and the model for these estimators. It is shown that the overall discrepancy can be decomposed into a distributional discrepancy and a structural discrepancy. A test of the overall model specification is proposed, as well as a test of the distributional specification (i.e., discretized multivariate normality). Also, the small sample performance of overall, distributional, and structural tests, as well as of parameter estimates and standard errors is investigated under conditions of correct model specification and also under mild structural and/or distributional misspecification. It is found that relatively small samples are needed for parameter estimates, standard errors, and structural tests. Larger samples are needed for the distributional and overall tests. Furthermore, parameter estimates, standard errors, and structural tests are surprisingly robust to distributional misspecification. This research was supported by the Department of Universities, Research and Information Society (DURSI) of the Catalan Government, and by grants BSO2000-0661 and BSO2003-08507 of the Spanish Ministry of Science and Technology.     

Some rao-guttman relationships

An examination of the determinantal equation associated with Rao's canonical factors suggests that Guttman's best lower bound for the number of common factors corresponds to the number of positive canonical correlations when squared multiple correlations are used as the initial estimates of communality. When these initial communality estimates are used, solving Rao's determinantal equation (at the first stage) permits expressing several matrices as functions of factors that differ only in the scale of their columns; these matrices include the correlation matrix with units in the diagonal, the correlation matrix with squared multiple correlations as communality estimates, Guttman's image covariance matrix, and Guttman's anti-image covariance matrix. Further, the factor scores associated with these factors can be shown to be either identical or simply related by a scale change. Implications for practice are discussed, and a computing scheme which would lead to an exhaustive analysis of the data with several optional outputs is outlined.     

Scale freeness in factor analysis

The notion of scale freeness does not seem to have been well understood in the factor analytic literature. It has been believed that if the loss function that is minimized to obtain estimates of the parameters in the factor model is scale invariant, then the estimates are scale free. It is shown that scale invariance of the loss function is neither a necessary nor a sufficient condition for scale freeness. A theorem that ensures scale freeness in the orthogonal factor model is given in this paper.The authors are grateful for the suggestions of the referees.     

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.     

The Infinitesimal Jackknife with Exploratory Factor Analysis

The infinitesimal jackknife, a nonparametric method for estimating standard errors, has been used to obtain standard error estimates in covariance structure analysis. In this article, we adapt it for obtaining standard errors for rotated factor loadings and factor correlations in exploratory factor analysis with sample correlation matrices. Both maximum likelihood estimation and ordinary least squares estimation are considered.     

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.     

Bayesian estimation in the three-parameter logistic model

A joint Bayesian estimation procedure for the estimation of parameters in the three-parameter logistic model is developed in this paper. Procedures for specifying prior beliefs for the parameters are given. It is shown through simulation studies that the Bayesian procedure (i) ensures that the estimates stay in the parameter space, and (ii) produces better estimates than the joint maximum likelihood procedure as judged by such criteria as mean squared differences between estimates and true values. The research reported here was performed pursuant to Grant No. N0014-79-C-0039 with the Office of Naval Research. A related article by Robert J. Mislevy (1986) appeared when the present paper was in the printing stage.     

Consequentialist Teleology and the Valuation of States of Affairs

Elizabeth Anderson claims that states of affairs are merely extrinsically valuable, since we value them only in virtue of the intrinsically valuable (e.g.) persons in those states of affairs. Since it considers states of affairs to be the sole bearers of intrinsic value, Anderson argues that consequentialism is incoherent because it attempts to globally maximize extrinsic value. I respond to this objection by distinguishing between two forms of consequentialist teleology and arguing that Anderson's claim is either harmless or her argument for the claim is uncompelling. On the first conception of teleology, consequentialists need not hold that states of affairs are the sole bearers of intrinsic value, which allows them to deflect this criticism. On the second account of teleology, even assuming that states of affairs are the sole bearers of intrinsic value, Anderson's argument does not necessarily defeat such views.     

An informal history of the interpersonal circumplex tradition

Circular representations of human nature can be traced from ancient astrology to Leonardo Da Vinci to the modern interpersonal circumplex (IPC) tradition. which began with Harry Stack Sullivan. The Kaiser Foundation Psychology Research Group operationalized key concepts of Sullivan's and Timothy Leary (1957) provided extensions of Sullivanian theory that formed the conceptual basis for contemporary circumplex models The current generation of circumplex investigators is exceptionally well represented by the contributors to this special series. The IPC can serve as an integrative cognitive framework for making sense of diverse realms that are well outside its original focus of convenience. The visual and psychological appeal of this powerful model is briefly illustrated.     

Restricted Maximum Likelihood Estimation for Parameters of the Social Relations Model

In many areas of research, the round-robin design is used to study interpersonal judgments and behaviors. The resulting data are analyzed with the social relations model (SRM), whereby almost all previously published studies have used ANOVA-based methods or multilevel-based methods to obtain SRM parameter estimates. In this article, the SRM is embedded into the linear mixed model framework, and it is shown how restricted maximum likelihood can be employed to estimate the SRM parameters. It is also described how the effect of covariates on the SRM-specific effects can be estimated. An example is presented to illustrate the approach. We also present the results of a simulation study in which the performance of the proposed approach is compared to the ANOVA method.     

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 guttman-harris uniqueness estimates

It is shown that, under very general conditions, uniqueness estimates proposed independently by Guttman [1957] and by Harris [1963] provide tighter upper bounds on the unknown uniqueness values of factor analysis than do existing estimates.The Illinois Department of Mental Health Drug Abuse Program is co-sponsored by the State of Illinois and the Department of Psychiatry, Division of the Biological Sciences and Pritzker School of Medicine, University of Chicago. Send reprint requests to Drug Abuse Programs—Research, East Pavilion, Museum of Science and Industry, 57th Street and South Lake Shore Drive, Chicago, Illinois 60637.     

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.     

Factoring the circumplex

This paper presents a way of factoring the Guttman circumplex to get factor weights whose pattern is close to that postulated by Guttman for the circumplex. An empirical example, including a crude procedure for diagonal estimation, is shown. Other sources of nonuniqueness are discussed, and the limitation to odd-numbered complexity for the tests is indicated. An orthodox factor analysis is applied to the example, and this provides a basis for comparison of the two opposing models.