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

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  • Yuan, Ke-Hai
  • Hayashi, Kentaro
  • Bentler, Peter M.
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    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.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 98 (2007)
    Issue (Month): 6 (July)
    Pages: 1262-1282

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    Handle: RePEc:eee:jmvana:v:98:y:2007:i:6:p:1262-1282

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    Related research

    Keywords: Kolmogorov-Smirnov distance Noncentral chi-square distribution Normal distribution Quantile-quantile plot Structural model;

    References

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    1. repec:cup:cbooks:9780521496032 is not listed on IDEAS
    2. Wakaki, Hirofumi & Eguchi, Shinto & Fujikoshi, Yasunori, 1990. "A class of tests for a general covariance structure," Journal of Multivariate Analysis, Elsevier, vol. 32(2), pages 313-325, February.
    3. Yanagihara, Hirokazu & Tonda, Tetsuji & Matsumoto, Chieko, 2005. "The effects of nonnormality on asymptotic distributions of some likelihood ratio criteria for testing covariance structures under normal assumption," Journal of Multivariate Analysis, Elsevier, vol. 96(2), pages 237-264, October.
    4. James Steiger & Alexander Shapiro & Michael Browne, 1985. "On the multivariate asymptotic distribution of sequential Chi-square statistics," Psychometrika, Springer, vol. 50(3), pages 253-263, September.
    5. Ke-Hai Yuan & Peter Bentler, 2006. "Mean Comparison: Manifest Variable Versus Latent Variable," Psychometrika, Springer, vol. 71(1), pages 139-159, March.
    6. Albert Satorra & Willem Saris, 1985. "Power of the likelihood ratio test in covariance structure analysis," Psychometrika, Springer, vol. 50(1), pages 83-90, March.
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    Cited by:
    1. Chun, So Yeon & Alexander, Shapiro, 2009. "Normal versus Noncentral Chi-square Asymptotics of Misspecified Models," MPRA Paper 17310, University Library of Munich, Germany.
    2. Shapiro, Alexander, 2009. "Asymptotic normality of test statistics under alternative hypotheses," Journal of Multivariate Analysis, Elsevier, vol. 100(5), pages 936-945, May.
    3. Yuan, Ke-Hai & Chan, Wai, 2008. "Structural equation modeling with near singular covariance matrices," Computational Statistics & Data Analysis, Elsevier, vol. 52(10), pages 4842-4858, June.

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