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Inner workings of the Kenward–Roger test

Author

Listed:
  • Waseem Alnosaier

    (Institute of Public Administration)

  • David Birkes

    (Oregon State University)

Abstract

For testing a linear hypothesis about fixed effects in a normal mixed linear model, a popular approach is to use a Wald test, in which the test statistic is assumed to have a null distribution that is approximately chi-squared. This approximation is questionable, however, for small samples. In 1997 Kenward and Roger constructed a test that addresses this problem. They altered the Wald test in three ways: (a) adjusting the test statistic, (b) approximating the null distribution by a scaled F distribution, and (c) modifying the formulas to achieve an exact F test in two special cases. Alterations (a) and (b) lead to formulas that are somewhat complicated but can be explained by using Taylor series approximations and a few convenient assumptions. The modified formulas used in alteration (c), however, are more mysterious. Restricting attention to models with linear variance–covariance structure, we provide details of a derivation that justifies these formulas. We show that similar but different derivations lead to different formulas that also produce exact F tests in the two special cases and are equally justifiable. A simulation study was done for testing the equality of treatment effects in block-design models. Tests based on the different derivations performed very similarly. Moreover, the simulations confirm that alteration (c) is worthwhile. The Kenward–Roger test showed greater accuracy in its p values than did the unmodified version of the test.

Suggested Citation

  • Waseem Alnosaier & David Birkes, 2019. "Inner workings of the Kenward–Roger test," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 82(2), pages 195-223, March.
  • Handle: RePEc:spr:metrik:v:82:y:2019:i:2:d:10.1007_s00184-018-0669-9
    DOI: 10.1007/s00184-018-0669-9
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    References listed on IDEAS

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    1. Halekoh, Ulrich & Højsgaard, Søren, 2014. "A Kenward-Roger Approximation and Parametric Bootstrap Methods for Tests in Linear Mixed Models The R Package pbkrtest," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 59(i09).
    2. Green, Paul E, 1974. "On the Design of Choice Experiments Involving Multifactor Alternatives," Journal of Consumer Research, Oxford University Press, vol. 1(2), pages 61-68, Se.
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