On stepwise control of directional errors under independence and some dependence

• We reformulate the problem of type 1 and directional errors control of stepwise procedures (for controlling the mdFWER or mdFDR) as the usual problem of type 1 errors control (for controlling FWER or FDR).
• We introduce a modified Holm procedure with the proven control of the FWER under independence.
• We introduce different stepwise procedures for controlling the FWER under various dependencies including positive dependence and block dependence.
• We introduce different stepwise procedures for controlling the FDR under various dependencies.

In this paper, the problem of error control of stepwise multiple testing procedures is considered. For two-sided hypotheses, control of both type 1 and type 3 (or directional) errors is required, and thus mixed directional familywise error rate control and mixed directional false discovery rate control are each considered by incorporating both types of errors in the error rate. Mixed directional familywise error rate control of stepwise methods in multiple testing has proven to be a challenging problem, as demonstrated in Shaffer (1980). By an appropriate formulation of the problem, some new stepwise procedures are developed that control type 1 and directional errors under independence and various dependencies.