False Discovery Rate (FDR) Calculator
False Discovery Rate (FDR): —
Understanding the False Discovery Rate (FDR)
In statistical hypothesis testing, when multiple tests are performed simultaneously, there's an increased chance of encountering false positives (Type I errors). The False Discovery Rate (FDR) is a crucial metric used to control these errors. It represents the expected proportion of rejected null hypotheses that are actually false discoveries (i.e., Type I errors).
Why is FDR Important?
When you conduct a single hypothesis test at a significance level of $\alpha$ (e.g., 0.05), the probability of a Type I error is $\alpha$. However, if you perform many tests, the probability of getting at least one Type I error can become very high. FDR provides a way to manage this by controlling the proportion of false rejections among all rejections.
Key Terms:
- Total Number of Hypothesis Tests (m): The total number of independent statistical tests performed.
- Number of Rejected Null Hypotheses (R): The total count of hypotheses for which the null hypothesis was rejected at a given significance level.
- Number of True Positives (S): The count of true discoveries, meaning hypotheses that were correctly rejected because the null hypothesis was indeed false.
- Number of False Positives (V): These are Type I errors – instances where the null hypothesis was rejected, but it was actually true.
Calculating FDR:
The False Discovery Rate is calculated using the following formula:
$$ \text{FDR} = \frac{V}{R} $$
Where:
- $V$ is the number of False Positives.
- $R$ is the total number of Rejected Null Hypotheses.
Since $R = S + V$ (total rejections = true positives + false positives), we can also express $V$ as $V = R – S$. Substituting this into the FDR formula, we get:
$$ \text{FDR} = \frac{R – S}{R} $$
If $R=0$, the FDR is considered 0, as no discoveries were made, thus no false discoveries could have occurred.
Example Calculation:
Suppose you conduct 1000 hypothesis tests ($m=1000$). Based on your chosen significance level, you reject 50 null hypotheses ($R=50$). Through further analysis or prior knowledge, you determine that 40 of these rejections are true discoveries ($S=40$).
Using the formula:
$$ \text{FDR} = \frac{R – S}{R} = \frac{50 – 40}{50} = \frac{10}{50} = 0.2 $$
This means that, on average, 20% of the rejected null hypotheses are expected to be false discoveries.
When to Use FDR:
FDR is particularly useful in fields like genomics, neuroimaging, and high-throughput screening where researchers perform thousands or even millions of hypothesis tests. Controlling the family-wise error rate (FWER) in such scenarios can be too conservative, leading to a low power to detect true effects. FDR offers a more balanced approach, allowing for more discoveries while still providing a reasonable control over the proportion of errors.