4.5 - Q-Values | STAT 555

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Storey's method also leads to a direct estimate of FDP. If we reject at level \(\alpha\) we expect the number of false discoveries to be \(\alpha m_0\). So the estimate of FDP is \(\alpha\hat{m}_0 / R\).

This leads directly to the Storey q-value [1] which is often interpreted as either an FDR-adjusted p-value or FDP(p) where p is any observed p-value in the experiment.

We start by sorting the p-values as we do for the BH or Holmes procedures.

Note that if we reject for \(p\leq p_{(i)}\) then the total rejections will be at least i (with equality unless two or more of the p-values are equal to \(p_{(i)}\)). Let R(\(\alpha\)) be the number of rejections when we reject for all \(p\leq\alpha\). Then define the q-values by:

\[q(1)= p_{(1)}\hat{m}_0/R(p_{(1)})\]

\[q(i+1)=max(q(i),p_{(i+1)}\hat{m}_0/R(p_{(i+1)})\]

This associates a q-value with each feature, which estimates the FDP if you reject the null hypothesis for this feature and all features which are this significant or more. Often we pick a cut-off for the q-value and reject the null hypothesis for all features with q-value less than or equal to our cut-off.

[1] Storey, John D. "The positive false discovery rate: a Bayesian interpretation and the q-value." Annals of statistics(2003): 2013-2035. https://projecteuclid.org/download/pdf_1/euclid.aos/1074290335

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Lessons

• Lesson 1: Introduction to Cell Biology
• Lesson 2: Basic Statistical Inference for Bioinformatics Studies
• Lesson 3: Designing Bioinformatics Experiments
• Lesson 4: Multiple Testing
  • 4.1 - Mistakes in Statistical Testing
  • 4.2 - Controlling Family-wise Error Rate
  • 4.3 -1995 - Two Huge Steps for Biological Inference
  • 4.4 - Estimating \(m_0\) (or \(\pi_0\))
  • 4.5 - q-Values
  • 4.6 - Using the Histogram of p-values
• Lesson 5: Microarray Preprocessing
• Lesson 6: Statistics for Differential Expression in Microarray Studies
• Lesson 7: Linear Models for Differential Expression in Microarray Studies
• Lesson 8: Tables and Count Data
• Lesson 9: RNA Seq Data
• Lesson 10: Clustering
• Lesson 11: Gene Set Analysis
• Lesson 12: Single Nucleotide Polymorphisms
• Lesson 13: ChIP-Seq Data
• Lesson 14: Classification
• Lesson 15: Cross-validation, Bootstraps and Consensus
• Lesson 16 - Multivariate Statistics and Dimension Reduction

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