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

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 of the noncentral chi-square distribution. The implication to data analysis is discussed whenever relevant. The development is built upon the concepts of basic calculous, linear algebra, and introductory probability and statistics. The aim is to provide the least technical material for quantitative graduate students in social science to understand the condition and limitation of the noncentral chi-square distribution.

ACKNOWLEDGMENTS

The research was supported by Grant DMS04-37167 from the National Science Foundation and Grants DA00017 and DA01070 from the National Institute on Drug Abuse. We are thankful to the editor and two referees for constructive comments that led to a significant improvement of the article over the previous version.

Notes

¹ The β₃ here equals γσ^3/2 in Yuan, Bentler, and Zhang (2005), where the definition of skewness is not the same with that commonly used in textbooks.

^a T _ML = 2.747 corresponds to p value = 0.253 when referred to χ₂ ².

^b T _ML = 2.119 corresponds to p value = 0.347 when referred to χ₂ ².

² According to Table A19 of Snedecor and Cochran (1989, p. 487), the significance level of the sample kurtosis of the 23rd variable is between 1% and 5%; the sample skewness of the 23rd variable is not statistically significant at 5%; both the sample skewness and sample kurtosis of the 17th variable are significant at 5%.