Generalized Augmentation for Control of the k-Familywise Error Rate
DOI:
https://doi.org/10.6000/1929-6029.2012.01.02.04Keywords:
Augmentation, false negatives rate, GAUGE, generalized familywise error rate, multiple testingAbstract
When performing many hypothesis tests at once a correction for multiplicity is needed to both keep under control the number of false discoveries and be able to detect the true departures from the null hypotheses. A recently introduced method which has been proved to be useful in genomics, neuroimaging and other fields consists in probabilistically controlling that the number of falsely rejected hypotheses does not exceed a pre-specified (low) . We introduce a new multiple testing procedure which is based on the idea of generalized augmentation: at first a number of hypotheses is rejected without any correction, then this number is adjusted by adding or removing rejections. The procedure is shown to keep under control the probability of or more false rejections. We show a small simulation study which suggests that the new procedure is very powerful, especially when the number of tests at stake is large. We conclude with an illustration on a benchmark data set on classification of colon cancer.
References
Hommel G, Hoffman T. Controlled uncertainty. In: Bauer P, Hommel G, Sonnemann E, Eds. Multiple Hypoteses Testing. Heidelberg: Springer 1988; pp. 154-161. http://dx.doi.org/10.1007/978-3-642-52307-6_13 DOI: https://doi.org/10.1007/978-3-642-52307-6_13
Lehmann EL, Romano JP. Generalizations of the Familywise Error Rate. Ann Stat 2005; 33: 1138-54. http://dx.doi.org/10.1214/009053605000000084 DOI: https://doi.org/10.1214/009053605000000084
van der Laan MJ, Dudoit S, Pollard KS. Augmentation procedures for control of the generalized family-wise error rate and tail probabilities for the proportion of false positives. Stat Appl Gen Mol Biol 2004; 3(1). DOI: https://doi.org/10.2202/1544-6115.1042
Pollard KS, van der Laan MJ. Resampling-based multiple testing: Asymptotic control of Type I error and applications to gene expression data. J Stat Plan Inf 2005; 125: 85-100. http://dx.doi.org/10.1016/j.jspi.2003.07.019 DOI: https://doi.org/10.1016/j.jspi.2003.07.019
Finos L, Farcomeni A. k-FWER control without multiplicity correction, with application to detection of genetic determinants of multiple sclerosis in Italian twins. Biomet 2011; 67: 174-81. http://dx.doi.org/10.1111/j.1541-0420.2010.01443.x DOI: https://doi.org/10.1111/j.1541-0420.2010.01443.x
Farcomeni A. Generalized Augmentation to control the False Discovery Exceedance in Multiple Testing. Scand J Stat 2009; 36: 501-17. DOI: https://doi.org/10.1111/j.1467-9469.2008.00633.x
Westfall PH, Young SS. Resampling-based Multiple Testing: Examples and Methods for p-value Adjustment. Wiley 1993.
Farcomeni A. A review of Modern Multiple Hypothesis testing with particular attention to the False Discovery Proportion. Stat Meth Med Res 2008; 17: 347-88. http://dx.doi.org/10.1177/0962280206079046 DOI: https://doi.org/10.1177/0962280206079046
Farcomeni A. Parametric assumptions in single and multiple testing: when should we rely on them? Biomed Stat Clin Epid 2008; 2: 57-69.
Benjamini Y, Hochberg Y. Controlling the false discovery rate: A practical and powerful approach to multiple testing. J Roy Statist Soc Ser B 1995; 57: 289-300. DOI: https://doi.org/10.1111/j.2517-6161.1995.tb02031.x
Sarkar SK. Stepup procedures controlling generalized FWER and generalized FDR. Department of Statistics, Temple University 2005.
Genovese CR, Wasserman L. Operating Characteristics and Extensions of the FDR Procedure. J Roy Statist Soc Ser B 2002; 64: 499-18. http://dx.doi.org/10.1111/1467-9868.00347 DOI: https://doi.org/10.1111/1467-9868.00347
Sarkar SK. FDR-controlling stepwise procedures and their false negatives rates. J Stat Plan Inf 2004; 125: 119-37. http://dx.doi.org/10.1016/j.jspi.2003.06.019 DOI: https://doi.org/10.1016/j.jspi.2003.06.019
Guo W, Romano J. A Generalized Sidak-Holm procedure and control of generalized error rates under independence. Stat Appl Genet Molec Biol 2007; 6(1). http://dx.doi.org/10.2202/1544-6115.1247 DOI: https://doi.org/10.2202/1544-6115.1247
Romano JP, Wolf M. Control of generalized error rates in multiple testing. Ann Stat 2007; 35: 1378-408. http://dx.doi.org/10.1214/009053606000001622 DOI: https://doi.org/10.1214/009053606000001622
Finner H, Roters M. Multiple hypotheses testing and expected number of Type I errors. Ann Stat 2002; 30: 220-38. DOI: https://doi.org/10.1214/aos/1015362191
Esary JD, Proschan F, Walkup DW. Association of Random Variables, with Applications. Ann Math Stat 1967; 38: 1466-74. http://dx.doi.org/10.1214/aoms/1177698701 DOI: https://doi.org/10.1214/aoms/1177698701
Kumar JD, Proschan F. Negative Association of Random Variables with Applications. Ann Statis 1983; 11: 286-95. http://dx.doi.org/10.1214/aos/1176346079 DOI: https://doi.org/10.1214/aos/1176346079
Doukan P. Mixing. Lectures Notes in Statistics, 85; Springer-Verlag 1994. DOI: https://doi.org/10.1007/978-1-4612-2642-0
Farcomeni A. Some Results on the Control of the False Discovery Rate under Dependence. Scand J Stat 2007; 34: 275-97. http://dx.doi.org/10.1111/j.1467-9469.2006.00530.x DOI: https://doi.org/10.1111/j.1467-9469.2006.00530.x
Alon U, Barkai N, Notterman DA, et al. Broad patterns of gene expression revealed by clustering analysis of tumor and normal colon tissue probed by oligonucleotide arrays. Proc Natl Acad Sci USA 1999; 96: 6745-50. http://dx.doi.org/10.1073/pnas.96.12.6745 DOI: https://doi.org/10.1073/pnas.96.12.6745
Cover T, Hart P. Nearest neighbor pattern classification. IEEE trans inf th 1967; IT-13: 21-27. DOI: https://doi.org/10.1109/TIT.1967.1053964
Gordon AY. Explicit formulas for generalized family-wise error rates and unimprovable step-down multiple testing procedures. J Stat Plan Inf 2007; 137: 3497-12. http://dx.doi.org/10.1016/j.jspi.2007.03.027 DOI: https://doi.org/10.1016/j.jspi.2007.03.027
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2012 Alessio Farcomeni
This work is licensed under a Creative Commons Attribution 4.0 International License.
Policy for Journals/Articles with Open Access
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are permitted and encouraged to post links to their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work
Policy for Journals / Manuscript with Paid Access
Authors who publish with this journal agree to the following terms:
- Publisher retain copyright .
- Authors are permitted and encouraged to post links to their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work .
