On the modified s-method and directional errors

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Abstract
Scheffe (1970) introduced a modified S-method for testing all linear contrasts, which is a stagewise rejective two stage test procedure. If a contrast hypothesis α'μ = 0 is rejected using this method, the problem arises whether or not one can decide for α'μ < 0 or aα'μ > 0 without violating the multiple level α. We give a general proof that the modified S-method still controls the experimentwise level of significance α. Furthermore, we show that this procedure can also be used as a very powerful peocedure for the special case of comparing three means with unequal sample size, which is of great practical importance. Finally we note that a similar result holds true for various closed directional range procedures for the comparison of three means.