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Expected Power for the False Discovery Rate with Independence.

D H Glueck¹, K E Muller, A Karimpour-Fard

¹Department of Preventive Medicine and Biometrics, University of Colorado Denver and Health Sciences Center, Denver, Colorado, USA.

Communications in Statistics: Theory and Methods

|September 28, 2011

Summary

This study provides theoretical power expressions for the Benjamini-Hochberg procedure, a key tool in multiple comparisons. The new analytic results offer a simulation-free approach for understanding statistical power in hypothesis testing.

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Area of Science:

Statistics
Statistical Inference
Multiple Comparisons

Background:

The Benjamini-Hochberg (BH) procedure is a cornerstone for controlling the false discovery rate in multiple hypothesis testing.
Existing power analyses for the BH procedure primarily rely on computationally intensive simulations.
A need exists for theoretical frameworks to provide exact power calculations and deeper insights into the BH procedure's performance.

Purpose of the Study:

To derive theoretical expressions for the expected power of the Benjamini-Hochberg procedure.
To establish a simulation-free method for evaluating the statistical power of multiple comparison procedures.
To provide a foundation for more precise power calculations in complex statistical scenarios.

Main Methods:

Development of theoretical expressions for expected power by making specific assumptions about the number and truthfulness of hypotheses and the distributions of test statistics.
Derivation of bounds for multidimensional rejection regions.
Utilizing the law of total probability and a representation of the joint density function of largest p-values to determine the distribution of rejections.

Main Results:

The study presents analytic expressions for expected power under the assumption of independent hypotheses.
Theoretical distributions for the total number of rejections and the number of true rejections are derived.
Provides a mathematical framework for understanding the power of the Benjamini-Hochberg procedure.

Conclusions:

The derived theoretical expressions offer a significant advancement over simulation-based power analyses for the Benjamini-Hochberg procedure.
This work provides valuable tools for researchers needing to design studies with adequate statistical power.
The theoretical framework can be extended to other false discovery rate controlling procedures and non-independent settings.