Normal Versus Noncentral Chi-square Asymptotics of Misspecified Models.

The noncentral chi-square approximation of the distribution of the likelihood ratio (LR) test statistic is a critical part of the methodology in structural equation modeling. Recently, it was argued by some authors that in certain situations normal distributions may give a better approximation of the distribution of the LR test statistic. The main goal of this article is to evaluate the validity of employing these distributions in practice. Monte Carlo simulation results indicate that the noncentral chi-square distribution describes behavior of the LR test statistic well under small, moderate, and even severe misspecifications regardless of the sample size (as long as it is sufficiently large), whereas the normal distribution, with a bias correction, gives a slightly better approximation for extremely severe misspecifications. However, neither the noncentral chi-square distribution nor the theoretical normal distributions give a reasonable approximation of the LR test statistics under extremely severe misspecifications. Of course, extremely misspecified models are not of much practical interest. We also use the Thurstone data ( Thurstone & Thurstone, 1941 ) from a classic study of mental ability for our illustration.