Testing Pattern Hypotheses for Correlation Matrices

McDonald 1974 obtained Maximum Likelihood (ML) estimates of the free parameters, and an asymptotic likelihood-ratio test, for the hypothesis that one or more elements of a covariance matrix are zero, and/or that groups of two or more of its elements are equal. Estimation was by Newton's method, starting from a closed-form Least Squares (LS) solution that is typically close to the ML solution point. The hypothesis can also be tested using the general model for the analysis of covariance structures given by Jöreskog 1970, but the combination of a closed-form LS starting point and the classical Newton method given by McDonald yields estimates in about a quarter of the computer time needed by JOreskog's program, ACØVS.