Indefinite symmetric matrices that are estimates of positive definite population matrices occur in a variety of contexts such as correlation matrices computed from pairwise present missing data and multinormal based theory for discretized variables. This note describes a methodology for scaling selected off-diagonal rows and columns of such a matrix to achieve positive definiteness. As a contrast to recently developed ridge procedures, the proposed method does not need variables to contain measurement errors. When minimum trace factor analysis is used to implement the theory, only correlations that are associated with Heywood cases are shrunk.

Many test statistics are asymptotically equivalent to quadratic forms of normal variables, which are further equivalent to T = Σ di=1 λiz2i with zi being independent and following N (0, 1). Two approximations to the distribution of T have been implemented in popular software and are widely used in evaluating various models. It is important to know how accurate these approximations are when compared to each other and to the exact distribution of T . The paper systematically studies the quality of the two approximations and examines the effect of λi ’s and the degrees of freedom d by analysis and Monte Carlo. The results imply that one approximation can be as good as the exact distribution when d is large. When the coefficient of variation of the λi ’s is small, another approximation is also adequate for practical model inference. The results are applied to a study of alcoholism and psychological symptoms

The vast majority of structural equation models contain no mean structure, that is, the population means are estimated at the sample means and are then eliminated from modeling consideration. Generalized least squares methods are proposed to estimate potential mean structure parameters and to evaluate whether the given model can be successfully augmented with a mean structure. A simulation evaluates the performance of some alternative tests. A method that takes variability due to estimation of covariance structure parameters into account in the mean structure estimator, as well as in the weight matrix of the generalized least squares function, performs best. In small samples, the F-test and Yuan-Bentler adjusted X2 test perform best.