2.-Statistics/FDR.md

+## Controlling Procedures for False Discovery Rate (FDR)
+
+False Discovery Rate (FDR) controlling procedures are statistical methods used to limit the proportion of false positives (Type I errors) among all significant test results. These methods are less conservative than those controlling the familywise error rate (FWER), such as the Bonferroni correction, and are commonly used when conducting a large number of hypothesis tests in high-dimensional data, such as genomics, ecology, and social sciences.
+
+### What is FDR?
+
+The **False Discovery Rate (FDR)** refers to the expected proportion of false positives (incorrectly rejected null hypotheses) among all the rejected null hypotheses. FDR-controlling procedures aim to balance the detection of true effects while limiting the proportion of false discoveries, making them more flexible than the **familywise error rate (FWER)**, which controls the probability of making at least one Type I error across multiple comparisons.
+
+### Benjamini-Hochberg Procedure
+
+The **Benjamini-Hochberg (BH) procedure** controls the FDR by adjusting the significance threshold in multiple tests, assuming that the tests are independent or positively dependent. It is less conservative than methods like the Bonferroni correction, resulting in greater statistical power.
+
+#### Steps to Apply:
+1. **Rank the p-values**: Sort the p-values in ascending order: $p_1, p_2, \dots, p_m$, where $p_1$ is the smallest p-value and $p_m$ is the largest.
+2. **Calculate the critical value** for each rank $i$:
+
+   $$
+   \text{Critical value} = \frac{i}{m} \cdot \alpha
+   $$
+
+   Where:
+   - $i$: The rank of the p-value,
+   - $m$: The total number of tests,
+   - $\alpha$: The significance level (e.g., 0.05).
+
+3. **Find the largest p-value** ($p_k$) where:
+
+   $$
+   p_k \leq \frac{k}{m} \cdot \alpha
+   $$
+
+   All p-values smaller than or equal to this threshold are considered significant.
+
+#### Common Uses
+- **Genomic studies**: Often used in genome-wide association studies (GWAS) to control the FDR when testing thousands of genetic variants for association with a trait.
+- **Post-hoc analysis**: Applied in multiple comparisons following ANOVA to control for false positives in multiple pairwise tests.
+
+#### Common Pitfalls
+- **Independence Assumption**: This method assumes that the tests are independent or positively dependent. It may not adequately control the FDR for negatively dependent tests.
+
+### Benjamini-Yekutieli Procedure
+
+The **Benjamini-Yekutieli (BY) procedure** is an extension of the BH procedure designed to control the FDR in cases where the tests are dependent. This makes it more conservative but allows for proper FDR control under any form of dependency, such as spatial, temporal, or genetic linkage dependencies.
+
+#### Steps to Apply:
+1. **Rank the p-values**: Sort the p-values in ascending order: $p_1, p_2, \dots, p_m$.
+2. **Calculate the critical value** for each rank $i$ using the adjustment factor $c(m)$:
+
+   $$
+   \text{Critical value} = \frac{i}{m \cdot c(m)} \cdot \alpha
+   $$
+
+   Where $c(m)$ is the sum of the harmonic series:
+
+   $$
+   c(m) = \sum_{i=1}^{m} \frac{1}{i}
+   $$
+
+3. **Find the largest p-value** ($p_k$) where:
+
+   $$
+   p_k \leq \frac{k}{m \cdot c(m)} \cdot \alpha
+   $$
+
+   All p-values smaller than or equal to this threshold are considered significant.
+
+#### Common Uses
+- **Time-series and spatial data**: Used in cases where tests are not independent, such as ecological data with spatial correlations or time-series studies with autocorrelated observations.
+
+#### Common Pitfalls
+- **Loss of Power**: The procedure is more conservative than BH, which can lead to a higher rate of false negatives (Type II errors).
+
+### Storey-Tibshirani Procedure
+
+The **Storey-Tibshirani (ST) procedure** is an adaptive FDR-controlling method that estimates the proportion of true null hypotheses ($\pi_0$), allowing for a more flexible and powerful approach to FDR control, especially in large datasets.
+
+#### Steps to Apply:
+1. **Estimate the proportion of true null hypotheses** ($\pi_0$) using:
+
+   $$
+   \hat{\pi_0} = \frac{\text{Number of large p-values}}{1 - \lambda}
+   $$
+
+   Where $\lambda$ is a threshold (commonly chosen between 0.5 and 1) to define "large" p-values.
+
+2. **Modify the critical value** based on $\hat{\pi_0}$:
+
+   $$
+   \text{Critical value} = \frac{i}{m \cdot \hat{\pi_0}} \cdot \alpha
+   $$
+
+3. **Find the largest p-value** where:
+
+   $$
+   p_k \leq \frac{k}{m \cdot \hat{\pi_0}} \cdot \alpha
+   $$
+
+   All p-values smaller than or equal to this threshold are considered significant.
+
+#### Common Uses
+- **Genomics**: In high-throughput studies where a large proportion of tests are true nulls, the Storey-Tibshirani method is used to improve power by adjusting for $\pi_0$.
+
+#### Common Pitfalls
+- **Estimation of $\pi_0$**: The accuracy of the procedure depends on the correct estimation of $\pi_0$, which can vary based on the choice of $\lambda$.
+
+### Comparison of FDR Controlling Procedures
+
+Here is a comparison of the key FDR controlling procedures in terms of assumptions, conservativeness, and when they are typically used:
+
+$$
+\begin{array}{|c|c|c|c|c|}
+\hline
+\textbf{Procedure} & \textbf{Best For} & \textbf{Conservativeness} & \textbf{Assumption} & \textbf{Key Formula} \\
+\hline
+\text{Benjamini-Hochberg (BH)} & \text{Independent or positively dependent tests} & \text{Less conservative} & \text{Assumes independence or positive dependence} & \frac{i}{m} \cdot \alpha \\
+\hline
+\text{Benjamini-Yekutieli (BY)} & \text{Dependent tests} & \text{More conservative} & \text{Handles complex dependencies} & \frac{i}{m \cdot c(m)} \cdot \alpha \\
+\hline
+\text{Storey-Tibshirani (ST)} & \text{High-dimensional data, adaptive FDR control} & \text{Moderately conservative} & \text{Requires estimation of true null hypotheses ($\pi_0$)} & \frac{i}{m \cdot \hat{\pi_0}} \cdot \alpha \\
+\hline
+\end{array}
+$$
+
+### Applications and Best Practices
+
+1. **Benjamini-Hochberg Procedure (BH)**: Use when tests are independent or positively dependent. Ideal for general multiple testing situations like genomic studies or post-hoc comparisons.
+2. **Benjamini-Yekutieli Procedure (BY)**: Use when tests are dependent, such as in spatially or temporally correlated data. It provides more conservative control of the FDR.
+3. **Storey-Tibshirani Procedure (ST)**: Use in high-dimensional data with many true null hypotheses. Its adaptive nature increases statistical power in large datasets by estimating the proportion of true nulls.
+
+---