null hypotheses H$_{NR,}$$_{1}$, . . . , H$_{NR,m}$ in either one of the following situations: 

1. The set of p -values within each study has property PRDS, and the selection rule in step 1 of Procedure 3.2 is Bonferroni at level w$_{1}$q$_{1}$ when the primary study is study one, and at level (1 − w$_{1}$ ) q$_{1}$ when the primary study is study two. 

2. Arbitrary dependence among the p -values within each study, and the expressions in step 2 of Procedure 3.2 are modified as follows: in the terms rq$_{1}$/m and R$_{2}$q$_{1}$/R$_{1}$, q$_{1}$ is replaced by q$_{1}$/ ( ∑ m i $_{=1}$1 /i ) , and in the terms r ( q − q$_{1}$ ) /R$_{1}$ and R$_{2}$ ( q − q$_{1}$ ) /R$_{1}$, q − q$_{1}$ is replaced by ( q − q$_{1}$ ) / ( ∑ R$_{1}$ i $_{=1}$1 /i ) . 

Proof. The proof of Theorem 4.1 in Appendix C relies only on the facts that Procedure 3.2 used in step 1 and in step 2 of Procedure 4.1 is valid. Therefore, the same proof shows that item 1 follows from item 2 of Theorem 3.3, and item 2 follows from item 1 of Theorem G.2. 

