False Discovery Rates (FDR): What They Are and How They Work

If you’ve ever dabbled in high-throughput sequencing, genomics, or large-scale hypothesis testing, then chances are you’ve heard of False Discovery Rates (FDR). Maybe you’ve even used them. But have you ever paused to ask:

• Where do False Discovery Rates come from?

• What do they actually mean?

• How do they really work?

Let’s dive in.

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🚨 The Core Idea

Before we get into technicalities, here’s the essence of FDR:

False Discovery Rates are a statistical tool to weed out bad data that looks good.

In other words, they help us control the number of false positives when testing multiple hypotheses.

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The Problem with Multiple Comparisons

When you perform thousands of statistical tests—common in areas like genomics or drug screening—the chance of getting some significant results just by random chance becomes quite high.

This is the multiple comparisons problem.

• Even if nothing is truly significant, you might still find several low p-values just by chance.

• A raw p-value threshold like 0.05 is not enough to control for this.

This is where the False Discovery Rate steps in.

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📘 What is FDR?

In statistics:

• A false discovery is a test result that appears significant, but is actually a false positive.

• The False Discovery Rate (FDR) is the expected proportion of these false positives among all the results you call "significant".

Unlike stricter corrections (like the Bonferroni correction, which controls the family-wise error rate), FDR offers a balance: it controls for error while maintaining reasonable power (i.e., it doesn't hide too many true positives).

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🎯 Benjamini-Hochberg (BH) Procedure

The most commonly used method to control FDR is the Benjamini-Hochberg method. Although people often refer to FDR and BH interchangeably, it's important to note:

Technically, FDR is a concept, and Benjamini-Hochberg is a method to estimate and control it.

Key Concept

Before diving into the method, remember this:

• If your samples come from the same distribution, p-values will be uniformly distributed.

• If some samples come from different distributions, those p-values will be skewed toward 0 — indicating potential discoveries.

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🧪 How the Benjamini-Hochberg Method Works

At a high level, the BH method:

1. Sorts all p-values from smallest to largest.

2. Assigns a rank to each p-value.

3. Adjusts each p-value based on its rank and the total number of tests.

4. Flags p-values as significant if they are below a certain threshold (e.g., FDR < 0.05).

Mathematically, for each p-value p(i) ranked i out of n total tests:

padj(i)=p(i)⋅nip_{adj}(i) = \frac{p(i) \cdot n}{i}padj​(i)=ip(i)⋅n​

Then, compare the adjusted p-values to your FDR cutoff.

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📉 An Example (Intuitive)

Let’s say you run 100 tests and sort the p-values. The BH method will make the lowest p-values stay close to their original values but inflate the rest. This adjustment reduces the chance of reporting too many false positives.

For example:

If your FDR threshold is 0.05, none of these would be considered significant after correction—even if they seemed significant before.

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✅ Why FDR Matters

In high-dimensional data analysis, FDR correction:

• Keeps your false positive rate low.

• Lets you discover meaningful signals without being overly conservative.

• Helps interpret results more honestly and reproducibly.

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🔚 The End (But Also the Beginning)

So there you have it:

• FDR is not a statistical luxury—it’s a necessity when dealing with large-scale hypothesis testing.

• The Benjamini-Hochberg method gives us a practical and powerful way to control it.

• Adjusted p-values help you separate the truly significant from the seemingly significant.

Next time you're working on gene expression, drug screening, or any project with thousands of comparisons—don't forget your FDR correction.

Your conclusions depend on it.