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Exact Integral Formulas for False Discovery Rate and the Variance of False Discovery Proportion.

Rovshan G Sadygov1, Justin X Zhu1, Henock M Deberneh1

  • 1Department of Biochemistry and Molecular Biology, The University of Texas Medical Branch, 301 University Blvd, Galveston, Texas 77555, United States.

Journal of Proteome Research
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PubMed
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This study provides exact formulas for positive false discovery proportion (pFDR) and its variance in multiple hypothesis testing. The new integral expressions offer improved accuracy, especially for smaller hypothesis sets in large-scale data analysis.

Keywords:
false discovery rateintegral expression for false discovery rateintegral formula for variance of false discovery proportionmultiple hypothesis testingpositive false discovery ratevariance of false discovery proportion

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Area of Science:

  • Bioinformatics and Computational Biology
  • Statistical Genetics
  • High-Throughput Omics Data Analysis

Background:

  • Multiple hypothesis testing is crucial for large-scale omics data analysis (proteomics, transcriptomics, metabolomics).
  • False Discovery Rate (FDR) and positive FDR (pFDR) are standard error control measures.
  • The pFDR, an expectation of the False Discovery Proportion (FDP), is often approximated by the ratio of expectations, but conditions for this transformation are unclear.

Purpose of the Study:

  • To derive exact integral expressions for the expectation (pFDR) and variance of FDP.
  • To investigate the validity and limitations of the commonly used approximation for pFDR.
  • To provide accurate error estimation methods for large-scale hypothesis testing in omics.

Main Methods:

  • Derivation of exact integral expressions for pFDR and FDP variance.
  • Analysis of the approximation (ratio of expectations) as a limiting case of the integral formula.
  • Development of a recurrence formula for computing pFDR with a specified number of null hypotheses.
  • Approximation of FDP variance for peptide identification in proteomics.

Main Results:

  • Exact integral expressions for pFDR and FDP variance were successfully derived.
  • The widely used ratio of expectations approximation is a specific case of the integral formula for large sample sizes.
  • Simulations show the integral expression is more accurate than the approximation for a small number of hypotheses.
  • For large sample sizes, results from both methods are comparable.

Conclusions:

  • The derived integral expressions provide a more accurate framework for pFDR calculation, particularly when the number of hypotheses is small.
  • This work clarifies the relationship between the exact pFDR and its common approximation, offering guidance for practical application.
  • The findings enhance error control in large-scale omics studies, including proteomics data analysis.