Normal theory likelihood ratio statistic for mean and covariance structure analysis under alternative hypotheses

Abstract

The normal distribution based likelihood ratio (LR) statistic is widely used in structural equation modeling. Under a sequence of local alternative hypotheses, this statistic has been shown to asymptotically follow a noncentral chi-square distribution. In practice, the population mean vector and covariance matrix as well as the model and sample size are always fixed. It is hard to justify the validity of the noncentral chi-square distribution for the resulting LR statistic even when data are normally distributed and sample size is large. By extending results in the literature, this paper develops normal distributions to describe the behavior of the LR statistic for mean and covariance structure analysis. A sequence of local alternative hypotheses is not necessary for the proposed distributions to be asymptotically valid. When the effect size is medium and above or when the model is not trivially misspecified, empirical results indicate that a refined normal distribution describes the behavior of the LR statistic better than the commonly used noncentral chi-square distribution, as measured by the Kolmogorov-Smirnov distance. Quantile-quantile plots are also provided to better understand the different distributions.

Asymptotic normality of test statistics under alternative hypotheses

“…The analysis is based on somewhat old results which seem to be little known in the statistics literature. These results allow to simplify and generalize the analysis of [9,12] in a significant way. In particular, we show in detail how it can be applied to the analysis of covariance structures.…”

“…Large sample properties of log-likelihood ratio statistics under alternative hypotheses and for nonnested models were studied, e.g., by Vuong [9] (for a more recent discussion of that topic see, e.g., Golden [2] and the references therein). Recently it was argued in Yuan et al [12] that in some cases normal approximations can give better asymptotics, for misspecified models, than the noncentral chi-square for the large sample distribution of the log-likelihood ratio statistic in the analysis of moment (covariance) structures.…”

Normal Versus Noncentral Chi-square Asymptotics of Misspecified Models

“…Recently this issue was discussed in a number of publications with a suggestion that the normal distribution could sometimes be a better alternative for approximating the true distribution of the LR test statistics (e.g., Golden, 2003 ;Olsson, Foss, & Breivik, 2004 ;Yuan, Hayashi, & Bentler, 2007 ;Yuan, 2008).…”

“…These measures were used in Yuan et al (2007). increases implying that for large δ the noncentral chi-square distribution by itself can be approximated by a normal distribution, as it was discussed at the beginning of the section "Theoretical background".…”

Asymptotic expansions in mean and covariance structure analysis

“…The approximations to the distributions of the fit indexes based on residuals and the normal-theory (NT) likelihood ratio statistic are provided by Bentler and Dijkstra [7], Ogasawara [10], Yuan [11], Yuan, Hayashi and Bentler [12].…”