Mean and Covariance Structure Analysis: Theoretical and Practical Improvements

Abstract The most widely used multivariate statistical models in the social and behavioral sciences involve linear structural relations among observed and latent variables. In practice, these variables are generally nonnormally distributed; hence classical multivariate analysis, based on multinomial error-free variables having no simultaneous interrelations, is not adequate to deal with such data. A promising alternative, based on asymptotically distribution-free (ADF) covariance structure analysis, has been found to be virtually useless in practical model evaluation at finite sample sizes with nonnormal data. We take a new look at the basic statistical theory of structural models under arbitrary distributions, using the methodology of nonlinear regression and generalized least squares estimation. For example, we adopt the use of residual weight matrices from regression theory. We develop a series of estimators and tests based on arbitrary distribution theory. We obtain a type of probabilistic Bartlett correction for various test statistics that can be simply applied in practice. A small simulation study replicates the extremely inadequate performance of one of our own and the original ADF model tests. In contrast, our corrected statistics have approximately correct means at all sample sizes, though their variances tend to be too low at the smallest sample sizes, leading to some “overacceptance” of the true model.