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On the equivalence between the LRT and F-test for testing variance components in a class of linear mixed models

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  • Fares Qeadan

    (University of Utah)

  • Ronald Christensen

    (University of New Mexico)

Abstract

For the special case of balanced one-way random effects ANOVA, it has been established that the generalized likelihood ratio test (LRT) and Wald’s test are largely equivalent in testing the variance component. We extend these results to explore the relationships between Wald’s F test, and the LRT for a much broader class of linear mixed models; the generalized split-plot models. In particular, we explore when the two tests are equivalent and prove that when they are not equivalent, Wald’s F test is more powerful, thus making the LRT test inadmissible. We show that inadmissibility arises in realistic situations with common number of degrees of freedom. Further, we derive the statistical distribution of the LRT under both the null and alternative hypotheses $$H_0$$ H 0 and $$H_1$$ H 1 where $$H_0$$ H 0 is the hypothesis that the between variance component is zero. Providing an exact distribution of the test statistic for the LRT in these models will help in calculating a more accurate p-value than the traditionally used p-value derived from the large sample chi-square mixture approximations.

Suggested Citation

  • Fares Qeadan & Ronald Christensen, 2021. "On the equivalence between the LRT and F-test for testing variance components in a class of linear mixed models," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(3), pages 313-338, April.
    • Handle: RePEc:spr:metrik:v:84:y:2021:i:3:d:10.1007_s00184-020-00777-z
    DOI: 10.1007/s00184-020-00777-z
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    References listed on IDEAS

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    1. Molenberghs, Geert & Verbeke, Geert, 2007. "Likelihood Ratio, Score, and Wald Tests in a Constrained Parameter Space," The American Statistician, American Statistical Association, vol. 61, pages 22-27, February.
    2. Ciprian M. Crainiceanu & David Ruppert, 2004. "Likelihood ratio tests in linear mixed models with one variance component," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(1), pages 165-185, February.
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