Other meanings of False discovery rate
STATISTICS
False discovery rate (FDR) is the expected proportion of false positives among the hypotheses rejected by a multiple-testing procedure. Unlike a single-test significance level, FDR addresses the composition of the entire set of reported discoveries and is widely used when many hypotheses are tested simultaneously.
The false discovery rate is the expected fraction of rejected hypotheses that are actually null, usually written as FDR = E[V/R], with V denoting false rejections and R denoting all rejections; the ratio is defined as zero when R = 0.1 A procedure controlling FDR at level q aims to ensure that this expectation is no greater than q.
FDR differs from the family-wise error rate, which is the probability of making at least one false rejection. Family-wise control is more stringent, especially when thousands of tests are performed, whereas FDR control permits some false discoveries in exchange for greater power to detect real effects.2 The target concerns a collection of tests, not the probability that any particular significant result is false.
For example, an FDR of 0.05 means that, over repeated applications of the procedure under comparable conditions, the expected proportion of false discoveries among the reported discoveries is at most 5%. It does not mean that exactly 5% of the current findings are false or that each individual finding has a 5% error probability.
The Benjamini–Hochberg procedure controls FDR by comparing ordered p-values with progressively less stringent thresholds. For m tests, sort the p-values as p(1) ≤ … ≤ p(m), choose the largest index k satisfying p(k) ≤ kq/m, and reject the hypotheses associated with p(1) through p(k).1
This step-up rule is more permissive than procedures that require every p-value to pass the same Bonferroni threshold. Under independent tests, and under several forms of positive dependence, the original procedure controls FDR at the chosen level; more general dependence can require modified methods or stronger assumptions.3 Adjusted p-values, often called q-values in related methods, help present results without requiring readers to repeat the ordering calculation.
FDR procedures are normally applied to a defined family of hypotheses. Changing the family after seeing results, selectively omitting inconvenient tests, or repeatedly testing until a favorable pattern appears can invalidate the stated error guarantee. The procedure also does not repair biased measurements, poorly specified hypotheses, or a study with inadequate power.
FDR control is particularly useful in high-dimensional research, where thousands or millions of hypotheses may arise at once. Common applications include genome-wide association studies, gene-expression analysis, neuroimaging, proteomics, and large-scale screening experiments.4 These fields often seek a workable set of candidates for follow-up rather than a guarantee that no selected candidate is false.
The scientific meaning of an FDR target depends on the tested family and the analysis plan. A discovery selected from a large exploratory screen may require independent replication, biological validation, or a new confirmatory study. Statistical significance after adjustment is not the same as a large effect, practical importance, or causal evidence.
Dependence among tests is a central limitation. Correlated measurements can affect the operating characteristics of procedures, although positive dependence is accommodated by the standard Benjamini–Hochberg result in important settings.3 Procedures based on independent filtering or covariates can improve power when auxiliary information is used transparently and without exploiting the outcomes being tested.
FDR was developed as a compromise between discovery and error avoidance rather than as a simple replacement for the p-value. The distinction became especially influential in fields such as genomics, where conventional family-wise corrections can leave very few findings despite strong aggregate evidence.1
Several related quantities are easy to confuse. The positive false discovery rate conditions on there being at least one rejection, while a local false discovery rate estimates the probability that a particular test is null given its statistic or p-value under a fitted mixture model.5 These are not interchangeable with the frequentist FDR guarantee.
FDR can also be controlled in sequential or online settings, in which hypotheses arrive over time, but those methods must account for the evolving testing process rather than applying a batch correction retrospectively.6 Reporting the tested family, adjustment method, dependence assumptions, and number of rejections makes an FDR analysis substantially easier to evaluate.
FDR guarantees are statements about repeated-use operating characteristics for a specified family of hypotheses; they are not direct posterior probabilities for individual findings.
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