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<Material>A critical review on and around sphericity tests

  • CHINO Naohito
    Association for Research in Humanities Aichigakuin University

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  • 反復測度デザイン概説-2-球形検定とその周辺についての批判的レビュ-
  • ハンプク ソクド デザイン ガイセツ 2 キュウケイ ケンテイ ト ソノ シュ

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A critical review on and around sphericity tests was made in the context of univariate and multivariate mixed models in order to clarify the present state of researches and problems which remained to be solved. We overviewed (1) mathematical statistical studies of Mauchly's LR criterion, (2) studies on the Mauchly's criterion in the context of repeated measures ANOVA as well as the Thomas' criterion in MMM (multivariate mixed model), (3) studies on the power of the Mauchly's criterion in the neighborhood of the null hypothesis, and the LBI test (locally best in variance test), and (4) robustness studies on the sphericity tests under non-normality conditions. It was concluded, from the overview, that there existed various problems yet to be solved concerning the sphericity tests. These are (1) the exact distribution of the LBI criterion when the number of variates is greater than three, and (2) exact distribution of the LR multivariate sphericity criterion proposed by Thomas (1983). Moreover, the necessity for robustness studies was suggested under the non-normal distributions with light tail such as multivariate beta distribution and so on.


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