Gramian Matrices in Covariance Structure Models

Abstract

Covariance structure models frequently contain out-of-range estimates that make no sense from either substantive or statistical points of view. Negative variance estimates are the most wellknown of these improper solutions, but correlations that are out of range also occur. Methods to minimize improper estimates have been accomplished by reparameterization and estimation under simple inequality constraints; but these solutions, discussed previously in this journal (Marsh, 1989), do not guarantee that the covariance matrices involved represent variances and covariances of real numbers, as required. A general approach to avoiding improper solutions in structural equation models is proposed. Although this approach does not resolve inadequacies in the data or theoretical model that may generate an improper solution, it solves the long-standing problem of obtaining proper estimates.One of the oldest, and most widely recognized, problems in the application of covariance structure analysis or structural equation models is the frequent occurrence of negative variance estimates, especially for unique (error) factors. For example, Bollen (1987) estimated a one-factor factor analysis model for three variables. Two of the unique variances were estimated as positive, but one was negative. Such estimates, often called Heywood cases (after Heywood, 1931), present serious problems, because the variances being estimated are supposed to represent the variances of persons' scores on real and not imaginary variables. Because real random variables must have non-negative variances, a negative variance estimate has no meaning, that is, it is inadmissible or improper. Yet, such variance estimates occur frequently in covariance structure analysis, especially when the sample size is small and a model has an insufficient number of indicators having high loadings on each factor (e.g., Anderson & Gerbing, 1984;Boomsma, 1985). ~~lhn9s (1987) improper estimate was eliminated when outlier cases were deleted.A recent discussion of this problem, and an evaluation of some suggested solutions, can be found in Dillon, I~ur~ar, ~ Mulani (1987). Their discussion, however, was incomplete. It did not deal with various other ways that &dquo;offending estimates'&dquo; can occur in covariance structure analysis. Rindskopf (1984) discussed several sources of this problem; Marsh (1989) also addressed the issue. Since this paper was completed, an excellent review of the topic was provided by Wothke (1993). The purpose of the present paper is to summarize the issues, and then to provide a solution to the fundamental problem that Wothke believes still remains: &dquo;There is a definite need for the development and teaching of a more general statistical solution to the estimation problems discussed her~99 (p. 288).Although negative variance estimates are the most widely known inadmissible estimates, correlations that are out of range occur frequently as well. Marsh (1989) reported a number of confirmatory multitrait-multimethod models that had factor correlatio...

A psychometric evaluation of the Mayer–Salovey–Caruso Emotional Intelligence Test Version 2.0

“…It is well known that models with dtwo-indicator factorsT tend to create problems with respect to the production of non-positive definite matrices (Bentler & Jamshidian, 1994;Worthke, 1993), which are technically inadmissible in SEM. Mayer et al (2003) stated, bIn the four factor solution only, the two within-area latent variable covariances (i.e., between Perceiving and Facilitating, and between Understanding and Managing) were additionally constrained to be equal so as to reduce a high covariance between Perceiving and Facilitating branch scoresQ (p. 103).…”

Is More Ever Too Much? The Number of Indicators per Factor in Confirmatory Factor Analysis

“…Even when there is convergence, the obtained solution may not be interpretable. Of particular relevance to the present investigation, the solution may be improper such that one of more of the parameter estimation matrices is not positive definite (Bentler & Jamshidian, 1994;Wothke, 1993). Thus, for example, a variance or residual variance estimate may be negative (typically called a Heywood case) or a factor correlation (standardized factor covariance) may have an absolute value greater than 1.0. van Driel (1978; see also Bollen, 1989;Dillon, Kumar, & Mulani, 1987) recommended that the formal requirement of positive definiteness be dropped, thus allowing researchers to distinguish between three classifications of improper solution: (1) boundary cases in which the confidence interval around the offending parameter contains proper values (e.g., the confidence interval around a negative uniqueness includes positive values) so that the problem may merely reflect sampling fluctuations;…”

Constrained versus unconstrained estimation in structural equation modeling.

“…Using a mixture distribution as the reference distribution for T C is then impossible, because we simply do not know when it becomes the right reference distribution and how many components it has. This point has also been made by Bentler and Jamshidian (1994), who wrote “because the true population model is unknown, it would not be possible to decide whether the usual chi-square test is appropriate in the given situation or whether [the mixture distributions] method must be used instead” (pp. 91–92).…”

“…3An even more advanced approach would be to keep each covariance matrix in the model positive definite, but we are not aware of any program that is able to do this for general SEMs. A method for doing so was proposed by Bentler and Jamshidian (1994) but has not to our knowledge been implemented.…”