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Testing for replicability in a follow-up study when the primary study hypotheses are two-sided

When testing for replication of results from a primary study with two-sided hypotheses in a follow-up study, we are usually interested in discovering the features with discoveries in the same direction in the two studies. The direction of testing in the follow-up study for each feature can therefore be decided by the primary study. We prove that in this case the methods suggested in Heller, Bogomolov, and Benjamini (2014) for control over false replicability claims are valid. Specifically, we prove that if we input into the procedures in Heller, Bogomolov, and Benjamini (2014) the one-sided p-values in the directions favoured by the primary study, then we achieve directional control over the desired error measure (family-wise error rate or false discovery rate).