Skip to main content
EURASIP Journal on Bioinformatics and Systems Biology
Download PDF
  • Research Article
  • Open access
  • Published:

The -Version of the Cramér-von Mises Test for Two-Sample Comparisons in Microarray Data Analysis

EURASIP Journal on Bioinformatics and Systems Biology volume 2006, Article number: 85769 (2006) Cite this article

Abstract

Distribution-free statistical tests offer clear advantages in situations where the exact unadjusted -values are required as input for multiple testing procedures. Such situations prevail when testing for differential expression of genes in microarray studies. The Cramér-von Mises two-sample test, based on a certain -distance between two empirical distribution functions, is a distribution-free test that has proven itself as a good choice. A numerical algorithm is available for computing quantiles of the sampling distribution of the Cramér-von Mises test statistic in finite samples. However, the computation is very time- and space-consuming. An counterpart of the Cramér-von Mises test represents an appealing alternative. In this work, we present an efficient algorithm for computing exact quantiles of the -distance test statistic. The performance and power of the -distance test are compared with those of the Cramér-von Mises and two other classical tests, using both simulated data and a large set of microarray data on childhood leukemia. The -distance test appears to be nearly as powerful as its counterpart. The lower computational intensity of the -distance test allows computation of exact quantiles of the null distribution for larger sample sizes than is possible for the Cramér-von Mises test.

[1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25]

References

  1. Grant GR, Manduchi E, Stoeckert CJ: Using nonparametric methods in the context of multiple testing to determine differentially expressed genes. In Methods of Microarray Data Analysis: Papers from CAMDA '00. Edited by: Lin SM, Johnson KF. Kluwer Academic, Norwell, Mass, USA; 2002:37-55.

    Chapter  Google Scholar 

  2. Guan Z, Zhao H: A semiparametric approach for marker gene selection based on gene expression data. Bioinformatics 2005,21(4):529-536. 10.1093/bioinformatics/bti032

    Article  Google Scholar 

  3. Lee M-LT, Gray RJ, Björkbacka H, Freeman MW: Generalized rank tests for replicated microarray data. Statistical Applications in Genetics and Molecular Biology 2005.,4(1): article 3

    Google Scholar 

  4. Qiu X, Xiao Y, Gordon A, Yakovlev A: Assessing stability of gene selection in microarray data analysis. BMC Bioinformatics 2006, 7: 50. 10.1186/1471-2105-7-50

    Article  Google Scholar 

  5. Stamey TA, Warrington JA, Caldwell MC, et al.: Molecular genetic profiling of gleason grade 4/5 prostate cancers compared to benign prostatic hyperplasia. Journal of Urology 2001,166(6):2171-2177. 10.1016/S0022-5347(05)65528-0

    Article  Google Scholar 

  6. Troyanskaya OG, Garber ME, Brown PO, Botstein D, Altman RB: Nonparametric methods for identifying differentially expressed genes in microarray data. Bioinformatics 2002,18(11):1454-1461. 10.1093/bioinformatics/18.11.1454

    Article  Google Scholar 

  7. Srivastava DK, Mudholkar GS: Goodness-of-fit tests for univariate and multivariate normal models. In Handbook of Statistics. Volume 22. Edited by: Khattree R, Rao CR. Elsevier, North-Holland, The Netherlands; 2003:869-906.

    Google Scholar 

  8. Wilcox RR: Fundamentals of Modern Statistical Methods. Springer, New York, NY, USA; 2001.

    Book  MATH  Google Scholar 

  9. Dudoit S, Shaffer JP, Boldrick JC: Multiple hypothesis testing in microarray experiments. Statistical Science 2003,18(1):71-103. 10.1214/ss/1056397487

    Article  MATH  MathSciNet  Google Scholar 

  10. Klebanov L, Gordon A, Xiao Y, Land H, Yakovlev A: A permutation test motivated by microarray data analysis. Computational Statistics and Data Analysis 2006,50(12):3619-3628. 10.1016/j.csda.2005.08.005

    Article  MATH  MathSciNet  Google Scholar 

  11. Burr EJ: Small-sample distribution of the two-sample Cramér-von Mises criterion for small equal samples. The Annals of Mathematical Statistics 1963, 34: 95-101. 10.1214/aoms/1177704245

    Article  MATH  MathSciNet  Google Scholar 

  12. Schmid F, Trede M: A distribution free test for the two sample problem for general alternatives. Computational Statistics and Data Analysis 1995,20(4):409-419. 10.1016/0167-9473(95)92844-N

    Article  MATH  Google Scholar 

  13. Xiao Y, Gordon A, Yakovlev A: C++ package for the Cramér-von Mises two-sample test. to appear in Journal of Statistical Software

  14. Hájek J, Šidák Z: Theory of Rank Tests. Academic Press, New York, NY, USA; 1967.

    MATH  Google Scholar 

  15. Anderson TW: On the distribution of the two-sample Cramér-von Mises criterion. The Annals of Mathematical Statistics 1962, 33: 1148-1159. 10.1214/aoms/1177704477

    Article  MATH  Google Scholar 

  16. Cramér H: On the composition of elementary errors. II: statistical applications. Skandinavisk Aktuarietidskrift 1928, 11: 141-180.

    Google Scholar 

  17. von Mises R: Wahrscheinlichkeitsrechnung und Ihre Anwendung in der Statistik und Theoretischen Physik. Deuticke, Leipzig, Germany; 1931.

    Google Scholar 

  18. Burr EJ:Distribution of the two-sample Cramér-von Mises and Watson's . The Annals of Mathematical Statistics 1964, 35: 1091-1098. 10.1214/aoms/1177703267

    Article  MATH  MathSciNet  Google Scholar 

  19. Zajta AJ, Pandikow W: A table of selected percentiles for the Cramér-von Mises Lehmann test: equal sample sizes. Biometrika 1977,64(1):165-167.

    MATH  MathSciNet  Google Scholar 

  20. Shao J, Tu D: The Jackknife and Bootstrap, Springer Series in Statistics. Springer, New York, NY, USA; 1995.

    Book  Google Scholar 

  21. Efron B, Tibshirani R: An Introduction to the Bootstrap. Chapman & Hall/CRC, New York, NY, USA; 1993.

    Book  MATH  Google Scholar 

  22. Anderson TW, Darling DA: Asymptotic theory of certain "goodness of fit" criterion based on stochastic processes. The Annals of Mathematical Statistics 1952, 23: 193-212. 10.1214/aoms/1177729437

    Article  MATH  MathSciNet  Google Scholar 

  23. Csorgo S, Faraway JJ: The exact and asymptotic distributions of Cramér-von Mises statistics. Journal of the Royal Statistical Society. Series B 1996, 58: 221-234.

    MathSciNet  Google Scholar 

  24. Büning H: Robustness and power of modified Lepage, Kolmogorov-Smirnov and Cramér-von Mises two-sample tests. Journal of Applied Statistics 2002,29(6):907-924. 10.1080/02664760220136212

    Article  MATH  MathSciNet  Google Scholar 

  25. Schmid F, Trede M:An -variant of the Cramér-von Mises test. Statistics and Probability Letters 1996,26(1):91-96. 10.1016/0167-7152(95)00256-1

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Biostatistics and Computational Biology, University of Rochester, 601 Elmwood Avenue, P.O. Box 630, Rochester, NY, 14642, USA

    Yuanhui Xiao, Alexander Gordon & Andrei Yakovlev

  2. Department of Mathematics and Statistics, Georgia State University, Atlanta, GA, 30303, USA

    Yuanhui Xiao

  3. Department of Mathematics and Statistics, University of North Carolina at Charlotte, 9201 University City Boulevard, Charlotte, NC, 28223, USA

    Alexander Gordon

Authors
  1. Yuanhui Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander Gordon
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andrei Yakovlev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuanhui Xiao.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Xiao, Y., Gordon, A. & Yakovlev, A. The -Version of the Cramér-von Mises Test for Two-Sample Comparisons in Microarray Data Analysis. J Bioinform Sys Biology 2006, 85769 (2006). https://doi.org/10.1155/BSB/2006/85769

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/BSB/2006/85769

Keywords