False Discovery Rate Calculator
Solve false discovery rate problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed for accuracy by Daniel Agrici, Founder & Lead Developer
False Discovery Rate Calculator
Calculator
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Formula: BH threshold(i) = (i / m) ร ฮฑ | Adjusted p(i) = min(p(i) ร m / i, 1)
Worked example โ 2 discoveries at FDR 0.05 vs 1 with Bonferroni | BH is more powerful
Formula
BH threshold(i) = (i / m) ร ฮฑ | Adjusted p(i) = min(p(i) ร m / i, 1)
The Benjamini-Hochberg procedure sorts p-values, calculates rank-based thresholds, and finds the largest rank where the p-value falls below its threshold. All hypotheses up to that rank are rejected. Adjusted p-values (q-values) represent the minimum FDR for significance.
Worked Examples
Example 1: Genomics Study with 10 Tests
Problem:A researcher tests 10 genes for differential expression and obtains p-values: 0.001, 0.008, 0.039, 0.041, 0.042, 0.06, 0.10, 0.25, 0.50, 0.80. Apply BH correction at FDR = 0.05.
Solution:Sort p-values and calculate BH thresholds: Rank 1: p=0.001, threshold=0.005 โ Significant Rank 2: p=0.008, threshold=0.010 โ Significant Rank 3: p=0.039, threshold=0.015 โ Not significant Rank 4: p=0.041, threshold=0.020 โ Not significant Largest significant rank: 2 Bonferroni would require p < 0.005 (only 1 discovery)
Result:2 discoveries at FDR 0.05 vs 1 with Bonferroni | BH is more powerful
Example 2: A/B Testing Multiple Variants
Problem:An A/B test platform runs 5 tests with p-values: 0.01, 0.03, 0.04, 0.15, 0.60. Determine which are significant at FDR = 0.10.
Solution:Sort and apply BH procedure at alpha=0.10: Rank 1: p=0.01, threshold=0.02 โ Significant (q=0.05) Rank 2: p=0.03, threshold=0.04 โ Significant (q=0.075) Rank 3: p=0.04, threshold=0.06 โ Significant (q=0.067) Rank 4: p=0.15, threshold=0.08 โ Not significant Rank 5: p=0.60, threshold=0.10 โ Not significant
Result:3 significant variants at FDR 0.10 | Expected false discoveries: 0.3
Frequently Asked Questions
What is the False Discovery Rate (FDR) and why is it important?
The False Discovery Rate is the expected proportion of false positives among all rejected hypotheses in multiple testing scenarios. When you perform many statistical tests simultaneously, like testing thousands of genes for differential expression or running hundreds of A/B test variants, the probability of obtaining false positives by chance alone increases dramatically. At a significance level of 0.05, testing 1,000 hypotheses would yield approximately 50 false positives even if none of the effects are real. FDR control provides a principled way to manage this multiple comparisons problem while retaining more statistical power than ultra-conservative methods like Bonferroni correction, making it essential in genomics, neuroimaging, and large-scale A/B testing.
What is the Benjamini-Hochberg procedure and how does it work?
The Benjamini-Hochberg (BH) procedure, introduced in 1995, is the most widely used method for controlling the False Discovery Rate. It works by first sorting all p-values from smallest to largest and assigning ranks. For each p-value at rank i out of m total tests, it calculates a critical threshold as (i/m) times the desired FDR level alpha. Starting from the largest p-value, the procedure finds the largest rank k where the p-value is less than or equal to its threshold, and rejects all hypotheses with rank 1 through k. The adjusted p-values, also called q-values, represent the minimum FDR at which each test would be declared significant. The BH procedure assumes independence or positive dependence among tests.
What is the difference between FDR and the Bonferroni correction?
The Bonferroni correction controls the Family-Wise Error Rate (FWER), which is the probability of making even one false positive. It divides the significance threshold by the number of tests, becoming extremely conservative as the number of tests grows. With 1,000 tests at alpha equals 0.05, Bonferroni requires p-values below 0.00005. FDR control instead allows a controlled proportion of false positives among discoveries. This difference in philosophy means FDR methods have substantially more statistical power. In practice, if you need absolute certainty that every significant result is real, use Bonferroni. If you can tolerate a small expected fraction of false discoveries in exchange for finding more true effects, use FDR. Most modern high-throughput studies use FDR.
When should I use Benjamini-Yekutieli instead of Benjamini-Hochberg?
The Benjamini-Yekutieli (BY) procedure should be used when the test statistics may have arbitrary or negative dependencies. The standard Benjamini-Hochberg procedure assumes that the test statistics are either independent or positively dependent, which holds in many practical situations like testing for gene expression differences. However, when tests are negatively correlated, which can occur in certain brain imaging analyses or when tests share control samples, the BH procedure may not properly control the FDR. The BY procedure multiplies the BH thresholds by a correction factor based on the harmonic series, making it more conservative but valid under any dependency structure. If you are unsure about the dependency structure of your tests, BY is the safer choice.
What is the difference between p-values and q-values?
A p-value represents the probability of observing results as extreme as the data under the null hypothesis for a single test. A q-value, also called an adjusted p-value or FDR-adjusted p-value, represents the minimum False Discovery Rate at which that particular test would be called significant when performing multiple tests simultaneously. While a p-value of 0.03 means there is a 3 percent chance of a false positive for that individual test, a q-value of 0.03 means that among all tests with q-values at or below 0.03, we expect approximately 3 percent to be false discoveries. Q-values are always at least as large as their corresponding p-values and provide a more honest assessment of statistical significance in the context of multiple comparisons.
References
Reviewed for accuracy by Daniel Agrici, Founder & Lead Developer ยท Editorial policy
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