A note on step-up test in orthogonal saturated designs

Let Xi be the ith order statistic of β^j2 for 1⩽j⩽k, where β^j's follow independent normal distributions with respective means βj and common variance σ2. In this paper, we provide a stochastic ordering of the random vector T=(X2/X1,2X3/(X1+X2),…,(m-1)Xm/∑j=1m-1Xj) as the βj's change for any given 2⩽m⩽k. With this result and the assumption of effect sparsity, we construct a step-up simultaneous testing procedure that strongly controls experimentwise error rate for a sequence of null hypotheses regarding the number of negligible effects (zero βj's) in orthogonal saturated designs.