Jöreskog-Sörbom fit indices, central and noncentral chi-square variables, structured covariance matrix, biasedness of GFI and AGFI, fit index for GLS estimate,

No abstract

Two new indices each based on two different methods of estimating the population covariance matrix Σ are derived. Certain algebraic results on these indices are also reported. A case study has been presented to show how the proposed indices in comparison to Jöreskog‐Sörbom's GFI are more efficient in detecting the ‘best’ structure out of 13 competing structures hypothesized for the Graduate Record Examination data. Algebraically it has also been shown that the limiting value of GFI fails to reach the desired value of zero under model misspecification. A Monte Carlo study also shows that with increasing sample size the proposed indices as compared to others show more and more sensitivity to model misspecification. Other advantages of the indices as measures of structural closeness of Z to S are discussed.

For all covariance matrices with linear struture, a necessary and sufficient condition for OLSE to be equivalent to GLSE and MLE is shown. For spectrally decomposable class of covariance matrices, Σ, the GLSE with V = Σ-1, MLE and OLSE are shown to be equivalent. In case of intraclass Σ such equivalence is true for any intraclass V. The implications of the results are discussed.