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Variance of the number of false discoveries

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  • Art B. Owen

Abstract

Summary. In high throughput genomic work, a very large number d of hypotheses are tested based on n≪d data samples. The large number of tests necessitates an adjustment for false discoveries in which a true null hypothesis was rejected. The expected number of false discoveries is easy to obtain. Dependences between the hypothesis tests greatly affect the variance of the number of false discoveries. Assuming that the tests are independent gives an inadequate variance formula. The paper presents a variance formula that takes account of the correlations between test statistics. That formula involves O(d2) correlations, and so a naïve implementation has cost O(nd2). A method based on sampling pairs of tests allows the variance to be approximated at a cost that is independent of d.

Suggested Citation

  • Art B. Owen, 2005. "Variance of the number of false discoveries," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(3), pages 411-426, June.
    • Handle: RePEc:bla:jorssb:v:67:y:2005:i:3:p:411-426
    DOI: 10.1111/j.1467-9868.2005.00509.x
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    Cited by:

    1. Cheng, Cheng, 2009. "Internal validation inferences of significant genomic features in genome-wide screening," Computational Statistics & Data Analysis, Elsevier, vol. 53(3), pages 788-800, January.
    2. de Uña-Alvarez Jacobo, 2011. "On the Statistical Properties of SGoF Multitesting Method," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 10(1), pages 1-30, April.
    3. Lim Johan & Kim Jayoun & Kim Sang-cheol & Yu Donghyeon & Kim Kyunga & Kim Byung Soo, 2012. "Detection of Differentially Expressed Gene Sets in a Partially Paired Microarray Data Set," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 11(3), pages 1-30, February.
    4. Chang Yu & Daniel Zelterman, 2020. "Distributions associated with simultaneous multiple hypothesis testing," Journal of Statistical Distributions and Applications, Springer, vol. 7(1), pages 1-17, December.
    5. Wang, Jiangzhou & Cui, Tingting & Zhu, Wensheng & Wang, Pengfei, 2023. "Covariate-modulated large-scale multiple testing under dependence," Computational Statistics & Data Analysis, Elsevier, vol. 180(C).
    6. Chen, Xiongzhi & Doerge, R.W., 2020. "A strong law of large numbers related to multiple testing normal means," Statistics & Probability Letters, Elsevier, vol. 159(C).
    7. de Uña-Alvarez Jacobo, 2012. "The Beta-Binomial SGoF method for multiple dependent tests," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 11(3), pages 1-32, May.
    8. Pengfei Wang & Wensheng Zhu, 2022. "Large‐scale covariate‐assisted two‐sample inference under dependence," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(4), pages 1421-1447, December.
    9. Wang, Xia & Shojaie, Ali & Zou, Jian, 2019. "Bayesian hidden Markov models for dependent large-scale multiple testing," Computational Statistics & Data Analysis, Elsevier, vol. 136(C), pages 123-136.
    10. Alessio Farcomeni, 2009. "Generalized Augmentation to Control the False Discovery Exceedance in Multiple Testing," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(3), pages 501-517, September.
    11. Leek Jeffrey T & Storey John D., 2011. "The Joint Null Criterion for Multiple Hypothesis Tests," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 10(1), pages 1-22, June.
    12. Sairam Rayaprolu & Zhiyi Chi, 2021. "False Discovery Variance Reduction in Large Scale Simultaneous Hypothesis Tests," Methodology and Computing in Applied Probability, Springer, vol. 23(3), pages 711-733, September.
    13. Wenguang Sun & T. Tony Cai, 2009. "Large‐scale multiple testing under dependence," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(2), pages 393-424, April.
    14. Gordon, Alexander & Chen, Linlin & Glazko, Galina & Yakovlev, Andrei, 2009. "Balancing type one and two errors in multiple testing for differential expression of genes," Computational Statistics & Data Analysis, Elsevier, vol. 53(5), pages 1622-1629, March.
    15. Chen, Xiongzhi, 2020. "A strong law of large numbers for simultaneously testing parameters of Lancaster bivariate distributions," Statistics & Probability Letters, Elsevier, vol. 167(C).

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