Greenhouse-Geisser Adjustment and the ANOVA-Type Statistic: Cousins or Twins?

Geisser and Greenhouse described a method for repeated measures analysis of variance that has become an important part of statistical tradition. The Greenhouse-Geisser method is based on work of Box, who studied the effects of dependence on the sampling distribution of analysis of variance F ratios and derived adjustments to the degrees of freedom. In separate work on deriving small sample approximations for linear rank statistics in factorial designs, Brunner et al. proposed an F approximation with estimated degrees of freedom based on Box's method of matching moments. In this article we show that these two descendent lines of research, although apparently divergent., actually converge for important special cases. This convergence indicates the close theoretical and practical relationships between the ANOVA-type statistic and the Greenhouse-Geisser F adjustment, which has the useful consequence that software implementations of the latter also can be used to perk form many of the nonparametric tests discussed by Brunner and Puri. Furthermore, the connection indicates that further work and improvements in each area may be useful in the other areas.