Noncentral Chi-Square Versus Normal Distributions in Describing the Likelihood Ratio Statistic: The Univariate Case and Its Multivariate Implication

Multivariate Behavioral Research
Ke-Hai Yuan

Abstract

In the literature of mean and covariance structure analysis, noncentral chi-square distribution is commonly used to describe the behavior of the likelihood ratio (LR) statistic under alternative hypothesis. Due to the inaccessibility of the rather technical literature for the distribution of the LR statistic, it is widely believed that the noncentral chi-square distribution is justified by statistical theory. Actually, when the null hypothesis is not trivially violated, the noncentral chi-square distribution cannot describe the LR statistic well even when data are normally distributed and the sample size is large. Using the one-dimensional case, this article provides the details showing that the LR statistic asymptotically follows a normal distribution, which also leads to an asymptotically correct confidence interval for the discrepancy between the null hypothesis/model and the population. For each one-dimensional result, the corresponding results in the higher dimensional case are pointed out and references are provided. Examples with real data illustrate the difference between the noncentral chi-square distribution and the normal distribution. Monte Carlo results compare the strength of the normal distribution against that o...Continue Reading

References

Mar 1, 1990·Psychological Bulletin·P M Bentler
May 1, 1984·The British Journal of Mathematical and Statistical Psychology·M W Browne
Jan 1, 1996·Pharmacology, Biochemistry, and Behavior·S A Ferguson, M G Paule
Feb 28, 1998·Journal of Affective Disorders·D M SloanM Sajatovic
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Dec 26, 2001·Annual Review of Psychology·Kenneth A Bollen
Jun 14, 2003·The British Journal of Mathematical and Statistical Psychology·Ke-Hai Yuan, Kentaro Hayashi

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Citations

Mar 31, 2015·Psychometrika·Hao Wu, Michael W Browne
Mar 31, 2015·Psychometrika·Hao Wu, Michael W Browne
Feb 1, 2020·Multivariate Behavioral Research·Lauren A Trichtinger, Guangjian Zhang
Nov 30, 2009·Multivariate Behavioral Research·So Yeon Chun, Alexander Shapiro

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Software Mentioned

Amos
EQS
Mplus
SAS Calis
LISREL

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