Deciding whether follow-up studies have replicated findings in a preliminary large-scale "omics' study"

We propose a formal method to declare that findings from a primary study have been replicated in a follow-up study. Our proposal is appropriate for primary studies that involve large-scale searches for rare true positives (i.e. needles in a haystack). Our proposal assigns an $r$-value to each finding; this is the lowest false discovery rate at which the finding can be called replicated. Examples are given and software is available.