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

• Published in: Journal of multivariate analysis Vol. 98; no. 6; pp. 1262 - 1282
• Main Authors: Yuan, Ke-Hai, Hayashi, Kentaro, Bentler, Peter M.
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 01-07-2007; Elsevier; Taylor & Francis LLC
• Series: Journal of Multivariate Analysis
• Subjects: Asymptotic methods; Distribution theory; Exact sciences and technology; Hypotheses; Kolmogorov-Smirnov distance Noncentral chi-square distribution Normal distribution Quantile-quantile plot Structural model; Kolmogorov–Smirnov distance; Linear inference, regression; Mathematical models; Mathematics; Multivariate analysis; Noncentral chi-square distribution; Normal distribution; Probability and statistics; Probability theory and stochastic processes; Quantile–quantile plot; Sample size; Sampling theory, sample surveys; Sciences and techniques of general use; Statistics; Structural model; Studies; secondary; Normal distribution; 62P25; Noncentral chi-square distribution; Quantile–quantile plot; Structural model; Kolmogorov–Smirnov distance; primary; 62E17; Sample size; Covariance analysis; Noncentral distribution; Kolmogorov-Smirnov distance; Noncentral chi-square distribution; Normal distribution; Quantile-quantile plot; Structural model; Large sample; Gaussian distribution; Likelihood ratio; Misspecified model; Covariance matrix; Multivariate analysis; Mean estimation; Chi square; Empirical model; Statistical method; Hypothesis test; Covariance; primary 62E20;62E17; secondary 62P15;62P25; Distribution function; Quantile
• ISSN: 0047-259X; 1095-7243
• DOI: 10.1016/j.jmva.2006.08.005

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.