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.
Similar content being viewed by others
References
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.
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
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
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
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
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
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.
Wilcox RR: Fundamentals of Modern Statistical Methods. Springer, New York, NY, USA; 2001.
Dudoit S, Shaffer JP, Boldrick JC: Multiple hypothesis testing in microarray experiments. Statistical Science 2003,18(1):71-103. 10.1214/ss/1056397487
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
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
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
Xiao Y, Gordon A, Yakovlev A: C++ package for the Cramér-von Mises two-sample test. to appear in Journal of Statistical Software
Hájek J, Šidák Z: Theory of Rank Tests. Academic Press, New York, NY, USA; 1967.
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
Cramér H: On the composition of elementary errors. II: statistical applications. Skandinavisk Aktuarietidskrift 1928, 11: 141-180.
von Mises R: Wahrscheinlichkeitsrechnung und Ihre Anwendung in der Statistik und Theoretischen Physik. Deuticke, Leipzig, Germany; 1931.
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
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.
Shao J, Tu D: The Jackknife and Bootstrap, Springer Series in Statistics. Springer, New York, NY, USA; 1995.
Efron B, Tibshirani R: An Introduction to the Bootstrap. Chapman & Hall/CRC, New York, NY, USA; 1993.
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
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.
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
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
Author information
Authors and Affiliations
Corresponding author
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.
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/BSB/2006/85769