Higher-order approximations to the distributions of fit indexes under fixed alternatives in structural equation models

Higher-order approximations to the distributions of fit indexes for structural equation models under fixed alternative hypotheses are obtained in nonnormal samples as well as normal ones. The fit indexes include the normal-theory likelihood ratio chi-square statistic for a posited model, the corresponding statistic for the baseline model of uncorrelated observed variables, and various fit indexes as functions of these two statistics. The approximations are given by the Edgeworth expansions for the distributions of the fit indexes under arbitrary distributions. Numerical examples in normal and nonnormal samples with the asymptotic and simulated distributions of the fit indexes show the relative inappropriateness of the normal-theory approximation using noncentral chi-square distributions. A simulation for the confidence intervals of the fit indexes based on the normal-theory Studentized estimators under normality with a small sample size indicates an advantage for the approximation by the Cornish–Fisher expansion over those by the noncentral chi-square distribution and the asymptotic normality.