The rank transform procedure in the two-way layout with interaction
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Abstract
One particular interest of this paper is to develop the properties One particular interest of this paper is to study the asymptotic theory of the rank transformed statistic, computed on ranks or rank scores, for testing for interaction in a two-way layout. Some theorems and sufficient conditions are derived with lemmas and corollaries. Many exact, small sample results are also obtained here for the first time. These results will be useful for the other theoretical studies of the rank transform procedure in experimental designs.
Let Xijn, i = 1... ,1, j = 1 . . . , J, and n = 1 . . . , N, be independent random variables such that Xijn has the continuous distribution function Fij. Further let Fi_ and Fj denote the average distribution function for block i and the treatment j , respectively.
Then the results obtained in this paper can be summarized as follows. When both main effects are present {Fij ^ Fi, and Fij / F_j for at least one i and j) in a general two-way layout, without the restriction of a linear model, then under the null hypothesis of no interaction and the assumptions, which are provided as equations (2.5), (2.6) and (2.8) in this paper, the rank transformed F statistic for interaction converges in distribution to an (I-1)(J-1) degrees of freedom chi-squared random variable divided by (I-1)(J-1).
For a two-by-two factorial design if only if there is one main effect ( Fij = Fi, or Fij = F,j for aU i and j ), imder the null hypothesis of no interaction the rank transformed F statistic for interaction converges in distribution to an (I-1)(J-1) degree of freedom chi-squared random variable divided by (I-1)(J-1).