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Define counts before probabilities
Let V be the number of false rejections and R the total number of rejections. Family-wise error rate is P(V ≥ 1). False discovery proportion is V/R when R is positive and zero when R is zero. False discovery rate is the expectation of that proportion.
Compare two fictional outcomes
| Outcome | V | R | Any false rejection? | False discovery proportion |
|---|---|---|---|---|
| One false result out of one rejection | 1 | 1 | Yes | 1 |
| One false result out of twenty rejections | 1 | 20 | Yes | 0.05 |
Keep a guarantee separate from one outcome
A procedure-level bound on an expectation is not a certificate for the fraction in one realized report. Likewise, a bound on the probability of any error does not reveal which individual result is false. Both targets need a specified testing procedure and its assumptions.
Inspect the all-true-null special case
If every tested null is true, any rejection is false. Then V = R and the false discovery proportion is one whenever R > 0, otherwise zero. In that special case, its expectation equals the probability of at least one rejection. The two targets differ more generally when genuine effects are possible.