False discovery rate

False discovery rate (FDR) is a statistical method used in multiple hypothesis testing to correct for the problem of Type I errors, which occur when a test incorrectly rejects a true null hypothesis. The FDR is particularly important in fields where large numbers of simultaneous comparisons are made, such as in genomics, proteomics, and other areas of bioinformatics and medical research. It provides a way to control or limit the proportion of false positives among the rejected hypotheses.

Definition

The false discovery rate is defined as the expected proportion of false positives among the rejected hypotheses. Mathematically, if \(V\) represents the number of false positives and \(R\) the total number of rejections (including both true and false positives), the FDR is given by:

\[ \text{FDR} = \mathbb{E}\left[\frac{V}{R}\right] \]

where \(\mathbb{E}\) denotes the expectation. The denominator is set to 1 when no hypotheses are rejected to avoid division by zero.

History

The concept of FDR was introduced by Yoav Benjamini and Yosef Hochberg in 1995. They proposed a method, now known as the Benjamini-Hochberg procedure, for controlling the FDR under independent test assumptions or under certain forms of dependency.

Benjamini-Hochberg Procedure

The Benjamini-Hochberg procedure is a simple, widely used technique for controlling the FDR. It involves ranking the individual p-values obtained from the multiple hypotheses tests in ascending order, and then determining the largest rank \(k\) for which the \(k\)th p-value is less than or equal to \(\frac{k}{m}\cdot q\), where \(m\) is the total number of hypotheses tested and \(q\) is the chosen FDR level.

Applications

FDR control is crucial in many scientific disciplines where researchers perform a large number of parallel tests. In genetics, for example, scientists might test thousands of genes to find those associated with a particular disease. Without FDR control, the number of genes falsely identified as significant could be very high. FDR methods ensure that, on average, a manageable proportion of the findings are false discoveries.

Advantages and Limitations

The main advantage of FDR control is its ability to be more powerful than methods controlling the family-wise error rate (FWER), such as the Bonferroni correction, especially when dealing with large datasets. However, its limitations include assumptions about the independence or specific dependency structures of the tests, which may not always hold in practice.

See Also

• Multiple comparisons problem
• Type I and type II errors
• Bonferroni correction
• Family-wise error rate