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One-sided inferences are often of interest when comparing treatments with a control. 12) and suppose that the kth treatment is a control with which the first k - 1 treatments are to be compared. The parametric functions of interest are yi = 4 - 6, [l Ii-5 k - 1). The correlation coefficients of the LS estimators jj = Yj - Yk(1 I i 5 k - 1) are given by The resulting (1 - a)-level simultaneous lower one-sided confidence SINGLE-STEP PROCEDURES FOR NONHIEIURCHICAL FAMILIES 33 intervals for the treatment-control differences are given by This is the well-known Dunnett's (1955) procedure for one-sided comparisons with a control.
15) where I = L'a is in 2,the row space of L with dimension m. 10). A more general proof of this result when L a ' is singular is given by Altschul and Marcuson (1979). 1 (Roy and Bose 1953). If LVL' is a nonsingular matrix, then 2 2 sup T , = sup T , = #€It" IEY - %A (+ - Yo"L')--Y? 17)) is chosen equal to then that test has level a. fei, Proof. Since L and V are full rank, it follows tuat LVL' is full rank and there exists a nonsingular m x m matrix M such that MM' = LVL'. Also there is a one-to-one correspondence between every I E 2' and every a E W" such that I = L'a.
3 it follows that we can test any hypotheses on the y,’s using this confidence region and the Type I W E Ia for all such tests. In particular. we can test one-sided hypotheses Hbr’ : y, S 0 and H6,’ : ‘y, 2 0. $,SF,) and this rejection implies the decision ’y, > 0 (respectively, <0), i E I. The Type I FWE for all such tests is the same as the Type I11 FWE for all directional decisions and hence the conclusion of the theorem follows. El ’ An alternative proof of the theorem can be given by noting that the directional decision procedure given above can be looked upon as a UI procedure derived by representing Ho = n,,, H,, = n l E (HLl+) l fl H6;’).