Gramian Matrices in Covariance Structure Models

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 well-known 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. Index terms: confirmatory factor analysis, EQS, Gramian matrices, Heywood cases, improper solutions, LISREL, structural equation models, underidentification.