A long-standing problem concerning the iterative analysis of orthogonal saturated designs has been resolved. Consider an unreplicated factorial design yielding independent, normally distributed estimators of k parameters but no independent variance estimator. Many authors have proposed iterative step-down tests for the analysis of such designs. In essence, the first such methods were proposed by Birnbaum [3] and Daniel [4]. It is well known that iterative methods are more powerful than corresponding closed step-down tests (see Voss [12]), and so the iterative tests are more popular. This popularity has grown despite lack of a proof that the iterative methods strongly control the family-wise error rate, while some corresponding closed step-down tests with modestly less power have been known to provide such control. Venter and Steele [11] claimed that certain iterative step-down tests strongly control error rates, but they failed to provide a proof. Recently Holm, Mark and Adolfsson [5] provided the first iterative step-down test for analysis of orthogonal saturated designs shown to strongly control the familywise error rate. Using the same technical approach, but without the need for explicit consideration of coverage bounds, we establish strong control of the familywise error rate for a large class of iterative step-down tests, including the iterative tests of Zahn [19, 20] and Ventor and Steel [11], iterative variations on the tests of Daniel [4], Birnbaum [3], Voss [12], Voss and Wang [14], Lenth [8] and Ye, Hamada and Wu [18], and a generalization of the Holm, Mark and Adolfsson [5] test for orthogonal saturated designs. Also included are the iterative step-down tests of Langsrud and Naes [6] for nearly saturated designs. Since our approach does not explicitly involve coverage bounds, implementation is relatively simple. Also, the normality assumption can be relaxed.
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