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Assessing skewness, kurtosis and normality in linear mixed models

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  • Soberón, Alexandra
  • Stute, Winfried

Abstract

Linear mixed models provide a useful tool to fit continuous longitudinal data, with the random effects and error term commonly assumed to have normal distributions. However, this restrictive assumption can result in a lack of robustness and needs to be tested. In this paper, we propose tests for skewness, kurtosis, and normality based on generalized least squares (GLS) residuals. To do it, estimating higher order moments is necessary and an alternative estimation procedure is developed. Compared to other procedures in the literature, our approach provides a closed form expression even for the third and fourth order moments. In addition, no further distributional assumptions on either random effects or error terms are needed to show the consistency of the proposed estimators and tests statistics. Their finite-sample performance is examined in a Monte Carlo study and the methodology is used to examine changes in the life expectancy as well as maternal and infant mortality rate of a sample of OECD countries.

Suggested Citation

  • Soberón, Alexandra & Stute, Winfried, 2017. "Assessing skewness, kurtosis and normality in linear mixed models," Journal of Multivariate Analysis, Elsevier, vol. 161(C), pages 123-140.
    • Handle: RePEc:eee:jmvana:v:161:y:2017:i:c:p:123-140
    DOI: 10.1016/j.jmva.2017.07.010
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    References listed on IDEAS

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    1. Verbeke, Geert & Lesaffre, Emmanuel, 1997. "The effect of misspecifying the random-effects distribution in linear mixed models for longitudinal data," Computational Statistics & Data Analysis, Elsevier, vol. 23(4), pages 541-556, February.
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    3. Jushan Bai & Serena Ng, 2005. "Tests for Skewness, Kurtosis, and Normality for Time Series Data," Journal of Business & Economic Statistics, American Statistical Association, vol. 23, pages 49-60, January.
    4. Ping Wu & Li Xing Zhu, 2010. "An Orthogonality‐Based Estimation of Moments for Linear Mixed Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(2), pages 253-263, June.
    5. Daowen Zhang & Marie Davidian, 2001. "Linear Mixed Models with Flexible Distributions of Random Effects for Longitudinal Data," Biometrics, The International Biometric Society, vol. 57(3), pages 795-802, September.
    6. D. R. Cox, 2002. "Estimation in a simple random effects model with nonnormal distributions," Biometrika, Biometrika Trust, vol. 89(4), pages 831-840, December.
    7. Ye, Rendao & Wang, Tonghui & Gupta, Arjun K., 2014. "Distribution of matrix quadratic forms under skew-normal settings," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 229-239.
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    Cited by:

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