False discovery rate:
formula and worked example

The share of discoveries that are false

False discovery rate (FDR) looks at everything a test, model or experiment flagged as positive and asks what fraction of those flags were wrong. If a screening pipeline reports 100 candidate genes and 15 turn out to be noise, the FDR is 15%.

It is the natural measure when you will act on the flagged list: send the samples for follow-up, investigate the alerts, run the next experiment on the hits. The FDR tells you how much of that effort will be wasted.

It is easy to mistake for the false positive rate, but the two have different denominators and answer different questions. The FDR is conditioned on being flagged, whereas the false positive rate is conditioned on being truly negative.

The idea became prominent as laboratories began testing thousands of hypotheses at once, such as expression levels across a genome. Using a 5% significance level on each test would produce hundreds of spurious hits, so researchers needed a measure of the quality of the entire list, not of each item. FDR gave them that, and it now appears in A/B testing platforms, fraud monitoring and many other fields.

Formula and inputs

• False positives: flagged results that are not real.
• True positives: flagged results that are real.

The same counts give precision as true positives divided by the same denominator, so FDR and precision always add to 100%. In the multiple-testing literature the same name is used for the expected value of this proportion, which procedures such as Benjamini-Hochberg aim to hold below a chosen level.

Worked example: 15 false among 100 flagged

A lab's screen flags 100 compounds as active. Follow-up confirms that 85 are genuinely active and 15 are not.

1. 1Add flagged results: 15 + 85 = 100.
2. 2Divide false positives by that total: 15 / 100 = 0.15.
3. 3Multiply by 100: 15%.
4. 4Check: precision is 85%, and 15% + 85% = 100%.

How much false discovery is acceptable

A 15% FDR means roughly one flagged result in seven will not hold up. Whether that is tolerable depends on what follow-up costs and what a wasted lead means. Exploratory research that feeds a second, confirmatory stage can accept a higher FDR; a result that triggers a clinical decision or a public claim needs a much lower one.

Conventions vary. Many fields use targets such as 5% or 10% for exploratory studies, but these are choices, not natural constants, and they should be stated before the data are examined.

The FDR also depends on how many true signals exist. When real effects are rare, the same test procedure produces a much higher FDR than when they are common, because noise supplies most of the flagged results.

In practice it is wise to sense-check the observed proportion against what you expected before the experiment. If a pilot produced 15% false discoveries and the full run is showing 40%, something has changed: a different population, a looser threshold or a data problem. The rate is a diagnostic as well as a score.

When false positives creep up

Raising false positives from 15 to 16.5 lifts the rate from 15% to about 16.26%. The half-count is a modelling convenience, a scaling of the original figure, not a literal tally. The pattern is that the FDR rises by a little more than the proportional change because the denominator grows at the same time, though never by the full 10%.

Frequent confusions

• Confusing it with the false positive rate. A test with a 5% false positive rate can still have a large FDR when true effects are rare.
• Controlling the per-test significance level and assuming the list of hits is mostly correct. With thousands of tests, a 5% level alone yields many false discoveries.
• Estimating FDR from a list that was cleaned using the same data, which understates it.
• Reporting a rate without the number of flagged items. 15% of 10 hits and 15% of 10,000 hits are not equally informative.
• Forgetting that an FDR is an average over the list, not a guarantee about any single result.

Finally, remember that a confirmed list is the only place you can compute the observed rate directly. In most live settings you must estimate it from a validation sample, and the estimate inherits all the sampling noise of that sample.

A planning scenario

A team expects to test 10,000 hypotheses, of which perhaps 100 are real. With 80% power and a 5% per-test false positive rate, the procedure finds about 80 real effects and flags about 495 of the 9,900 nulls, an FDR near 86%. Tightening the threshold or using a correction brings the false alarms down, at the price of fewer true hits. That is the trade a team should see before running the experiment, not after.

The same arithmetic works backwards for budgeting. If a review team can confirm 40 alerts a week and the system's FDR is about 15%, expect around 34 real findings and 6 dead ends. If management then asks for a more sensitive setting that raises the FDR to 40%, the same 40 reviews return only 24 real findings, and the cost of each real finding rises by about 40%.