Suppose we made $m$ tests in 2 samples from the population. For simplicity, we compared $m$ traits (i.e. bmi, temperature, height, etc) for equality in means between 2 groups of people (i.e. Germans and Americans) 2 times. In other words, we compared $m$ traits between 2 groups once and then tried to reproduce the research one more time. We fixed level of significance $\alpha$. Our test for difference in means is "quite" reliable, which means that it is parametric and complex (correction for additional factors were required) so we do not expect our model to describe the reality completely, but the 1st type error rate is not different a lot from $\alpha$.

For the first time we performed the study, $n_1$ q-values did not exceed $\alpha$. For the second time (replication study), $n_2$ q-values were less than $\alpha$.

We figured out that after replication 2 studies agreed only in $n$ tests (which means the null hypothesis was rejected for these cases).

How can we say that the agreement in $n$ tests is significantly different from random or not?

More complicated question: suppose that $n$ times the null hypothesis for the means' equality was rejected, but only for $k$ times the direction of difference in means between Germans and Americans was the same. Here it is clear how to test if our $n$ results were just a coincidence (we toss a coin with 0.5 probability and can calculate p-value from it). Having $(n-k)$ -- number of tests that passed FDR control but are for sure wrong since the direction of change is different -- how to estimate the total amount of false discoveries in $n$ pairs of tests?