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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. Green, Paul E, 1974. "On the Design of Choice Experiments Involving Multifactor Alternatives," Journal of Consumer Research, Journal of Consumer Research Inc., vol. 1(2), pages 61-68, Se.
    2. 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).
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