False Discovery Proportion And False Discovery Rate

Why Count the Proportion of False Discoveries?

A genomic analysis may test thousands of hypotheses, such as whether each genetic feature is associated with an outcome. A procedure that makes many rejections can produce some false discoveries even when it performs well by a chosen error criterion. The false discovery proportion describes the share of the rejections that are false in one analysis. The false discovery rate summarizes that proportion over repeated applications of the analysis.

These quantities answer a different question from the familywise error rate (FWER). FWER concerns the probability of making at least one false rejection. FDR instead focuses on the fraction of rejections that are false, while counting an analysis with no rejections as having a false discovery proportion of zero.

Definitions for a Fixed Testing Family

Suppose a study specifies a family of $m$ null hypotheses, $H_1,\ldots,H_m$, and applies a testing procedure to data. The truth or falsity of each hypothesis is treated as fixed; which hypotheses the procedure rejects is random because the data are random. Let $m_0$ be the number of true null hypotheses in the family.

Definition: Let $R$ be the number of hypotheses rejected and $V$ the number of rejected hypotheses that are truly null. The false discovery proportion (FDP) is $$ \operatorname{FDP}=\frac{V}{\max(R,1)}. $$ Thus, if $R>0$, the FDP is $V/R$; if $R=0$, the FDP is defined to be $0$. The false discovery rate (FDR) is the expected FDP under the repeated-sampling distribution of the data: $$ \operatorname{FDR}=E\left[\frac{V}{\max(R,1)}\right]. $$

The FDP is a random quantity before the data are observed and a particular realized proportion after the analysis. FDR is a property of the testing procedure under a specified family, truth configuration, and data-generating distribution. It is not generally the proportion of false rejections in the one observed result, nor does its definition require that the tests be independent.

For example, if an analysis rejects 30 hypotheses and 3 of those are truly null, its realized FDP is $3/30=0.10$. That value alone does not tell us the FDR: the FDR requires averaging the FDP over the possible data outcomes under the procedure. In real applications, the truth of individual hypotheses is usually unknown, so the realized FDP is not directly observable. Simulation with known truth can calculate it; an analysis of real data typically relies on a procedure or model to estimate or control FDR.

Worked Example: Calculating a Realized FDP

Suppose a hypothetical genomic analysis tests 500 hypotheses. It rejects 40, and later, for purposes of evaluating the method, the true state of every hypothesis is known: 6 of the 40 rejections correspond to true null hypotheses. Find the realized FDP.

Here $R=40$ and $V=6$. Since $R>0$, $$ \operatorname{FDP}=\frac{V}{R}=\frac{6}{40}=0.15. $$

The realized false discovery proportion is $0.15$, or 15%. In this hypothetical evaluation, 15% of the rejected hypotheses are false discoveries. This is a statement about this particular analysis, not an estimate of the probability that any specific rejected hypothesis is false.

FDR Is an Expectation, Not a Single-Analysis Ratio

To interpret the expectation, imagine applying the same procedure repeatedly to new data sets generated under the same model, with the same family and the same true and false null hypotheses. Each repetition can produce a different $R$ and $V$, and hence a different FDP. The FDR is the long-run average of those FDP values, with repetitions having no rejections contributing zero.

A useful way to express the definition is to let $I_i$ equal 1 if hypothesis $H_i$ is rejected and 0 otherwise, and let $N_i$ equal 1 if $H_i$ is truly null and 0 otherwise. Then $R=\sum_{i=1}^{m} I_i$, while $V=\sum_{i=1}^{m}N_iI_i$. Because the null truths are fixed, the $N_i$ values do not vary across repetitions.

Justification: Substituting the counts into the FDP and using linearity of expectation gives $$ \operatorname{FDR} =E\left[\frac{\sum_{i=1}^{m}N_iI_i}{\max(R,1)}\right] =\sum_{i=1}^{m}N_iE\left[\frac{I_i}{\max(R,1)}\right]. $$ Only truly null hypotheses contribute to this sum. Each contribution depends on both whether that hypothesis is rejected and the total number of rejections in the same analysis. This is why the FDR is not, in general, obtained by dividing the expected number of false rejections by the expected number of rejections.

Worked Example: Averaging FDP Across Repeated Analyses

Consider a hypothetical family of 100 tests with 50 true null hypotheses. Under a specified procedure and data-generating model, suppose repeated analyses have the following outcomes:

Probability of outcome	Rejections $R$	False rejections $V$	FDP
0.20	0	0	0
0.50	10	1	0.10
0.30	20	4	0.20

The probabilities sum to 1, and each false-rejection count is no greater than the 50 true null hypotheses. The FDR is the probability-weighted mean of the FDP: $$ \operatorname{FDR}=(0.20)(0)+(0.50)(0.10)+(0.30)(0.20)=0.11. $$

Thus the FDR is 0.11, or 11%. This describes the expected false discovery proportion across repeated analyses under the stated setup. For comparison, the expected counts are $E[V]=(0.20)(0)+(0.50)(1)+(0.30)(4)=1.70$ and $E[R]=(0.20)(0)+(0.50)(10)+(0.30)(20)=11$. Their ratio is $1.70/11\approx0.1545$, not 0.11. The ratio of expected counts weights outcomes differently from the average of their FDPs.

Relation to Familywise Error Rate

The FWER is the probability of at least one false rejection, $P(V\geq 1)$. It treats an analysis with one false rejection and an analysis with many false rejections alike: each counts as an error event. FDR accounts for the fraction of rejections that are false, so the number of total rejections also matters.

If all hypotheses in the family are true nulls, then $V=R$ in every analysis. When $R>0$, the FDP is 1; when $R=0$, it is 0. Therefore, in the all-null case, $$ \operatorname{FDR}=P(R>0)=P(V\geq1)=\operatorname{FWER}. $$ With a mixture of true and false nulls, the two criteria need not agree.

Worked Example: FDR and FWER Can Differ

Suppose a hypothetical procedure is applied repeatedly to a family of 100 tests containing one true null and 99 false nulls. In 10% of analyses, it rejects 100 hypotheses, including the one true null; in the remaining 90%, it rejects 100 hypotheses and none is truly null. Assume these are the only possible outcomes.

In the first outcome, $R=100$, $V=1$, and $\operatorname{FDP}=1/100=0.01$. In the second, $R=100$, $V=0$, and $\operatorname{FDP}=0$. Thus $$ \operatorname{FDR}=(0.10)(0.01)+(0.90)(0)=0.001. $$

The FWER is $P(V\geq1)=0.10$, because a false rejection occurs in 10% of analyses. Here FDR is 0.001, or 0.1%, while FWER is 0.10, or 10%. The FDP is small in the analyses with a false rejection because that one false rejection is among 100 total rejections. Neither measure is inherently the right target for every scientific purpose; they summarize different error concerns.

Assumptions, Interpretation, and Failure Modes

The definitions require a specified family of hypotheses, a rule for deciding which hypotheses are rejected, a truth configuration for those hypotheses, and a probability distribution for the data under which the expectation is taken. They do not require independence, a particular test statistic, or a specific multiple-testing procedure. Those features can matter for whether a procedure achieves a desired FDR bound, but they are not part of the definition of FDP or FDR.

In genomic applications, the family and truth configuration may be known in a simulation but not in an observed study. Consequently, one observed list of discoveries can provide a realized FDP only if the true status of its rejected hypotheses is known. An estimated FDR is not the same thing as directly counting false discoveries. It depends on the estimation or testing procedure and the assumptions that justify it.

A useful diagnostic is to check the scale and denominator before interpreting any reported number. For $R>0$, the FDP must be between 0 and 1 because $0\leq V\leq R$. If $R=0$, the FDP is 0 by definition, not undefined. The FDR also lies between 0 and 1 because it is the expectation of a quantity in that range. A reported value outside these bounds signals a calculation or reporting error.

A common mistake is to say, “The FDR is 10%, so 10% of the discoveries in this particular data set are false.” That treats a repeated-sampling expectation as though it were the observed FDP. The correction is to distinguish the quantities: the FDP is the false fraction in one analysis, while the FDR is the expected FDP across repeated analyses under the specified setup. Another mistake is to calculate $E[V]/E[R]$ and call it FDR. The worked repeated-analysis example shows why that ratio need not equal $E[V/\max(R,1)]$.

Choosing between FDR and FWER depends on the consequences of errors. FWER is often a natural target when even one false claim is especially costly. FDR can be more informative when an analysis reports many discoveries and the scientific goal concerns the expected proportion of false claims among them. FDR does not guarantee a particular false proportion in every realized analysis, and its definition alone does not establish that any procedure controls it.

Check Your Understanding

Answer each question, then compare with the solutions below.

An analysis rejects 25 hypotheses, 5 of which are truly null. What is the realized FDP?
A procedure has $R=0$ in 40% of repeated analyses and FDP 0.05 in the other 60%. What is its FDR?
Explain why $E[V]/E[R]$ is not generally the definition of FDR.
If every hypothesis in a family is a true null, how are FDR and FWER related?
Can the FDR definition be used when tests are dependent? What additional issue might dependence raise?

Solutions. 1. The FDP is $V/R=5/25=0.20$, or 20%. 2. The FDR is $(0.40)(0)+(0.60)(0.05)=0.03$, or 3%. 3. FDR averages the FDP within each analysis and then averages across analyses; dividing expected counts instead uses $E[V]/E[R]$, which can weight outcomes differently. 4. When all nulls are true, FDP is 1 if any hypothesis is rejected and 0 otherwise, so $\operatorname{FDR}=P(V\geq1)=\operatorname{FWER}$. 5. Yes. The definition applies under dependence because it uses the joint distribution of outcomes; dependence can, however, affect whether a particular procedure controls FDR at its intended level.