3 Replicability analysis with a primary and a follow- up study For the family of m no replicability null hypotheses H$_{NR,}$$_{1}$, . . . , H$_{NR,m}$ , we consider two relevant error measures: the probability that at least one no replicability null hypothesis is falsely rejected, i.e. the FWER, and the expected fraction of false rejections out of all rejections of no replicability null hypotheses, that is the FDR. Procedure 3.1. The two stage FWER controlling procedure for testing the family of no replicability null hypotheses with parameters ( α$_{1}$, α ) , where 0 < α$_{1}$ < α < 1 : 1. Let R 1 be the set of indices of elementary hypotheses that are selected for testing in a follow-up study based on the data from the primary study. 2. Apply a FWER controlling procedure at level α$_{1}$ , using the data from the pri- mary study only, on the family of null hypotheses H$_{1}$, . . . , H$_{m}$ , and let R p ⊆ { 1 , . . . , m } be the set of indices of rejected hypotheses. Apply a FWER con- trolling procedure at level α − α$_{1}$ , using the data from the follow-up study only, on the family of selected null hypotheses { H$_{j}$ : j ∈ R 1 } , and let R f ⊆ R 1 be the set of indices of rejected hypotheses. Then the set of indices of rejected no replicability null hypotheses is R f ∩ R $_{p}$. Theorem 3.1. For two independent studies, Procedure 3.1 controls the FWER at level α for the family of no replicability null hypotheses H$_{NR,}$$_{1}$, . . . , H$_{NR,m}$ . Proof. Let V$_{p}$ = ∑ j $_{∈R$_{p}$}$(1 − h$_{1}$$_{j}$ ) and V$_{f}$ = ∑ j $_{∈R$_{f}$}$(1 − h$_{2}$$_{j}$ ) be the number of true elementary null hypotheses rejected, respectively, in the primary study and in the follow-up study. Then 6 F W ER ≤ E ( I [ V$_{p}$ + V$_{f}$ > 0]) ≤ E ( I [ V$_{p}$ > 0]) + E ( E ( I [ V$_{f}$ > 0] | p$_{1}$ )) ≤ α$_{1}$ + α − α$_{1}$ = α,