Abstract: In this paper, we consider the problem of simultaneously testing many two-sided hypotheses when rejections of null hypotheses are accompanied by claims of the direction of the alternative. The fundamental goal is to construct methods that control the mixed directional familywise error rate (mdFWER), which is the probability of making any type 1 or type 3 (directional) error. In particular, attention is focused on cases where the hypotheses are ordered as so that is tested only if have all been previously rejected. In this situation, one can control the usual familywise error rate under arbitrary dependence by the basic procedure which tests each hypothesis at level a, and no other multiplicity adjustment is needed. However, we show that this is far too liberal if one also accounts for directional errors. But, by imposing certain dependence assumptions on the test statistics, one can retain the basic procedure. Through a simulation study and a clinical trial example, we numerically illustrate good performance of the proposed procedures compared to the existing mdFWER c6ontrolling procedures. The proposed procedures are also implemented in the R-package FixSeqMTP.

Key words and phrases: Directional error, fixed sequence multiple testing, mixed directional familywise error rate, monotone likelihood ratio, positive dependence, type 1 error.