Chebyshev's inequality for nonparametric testing with small N and α in microarray research

Journal of the Royal Statistical Society. Series C: Applied Statistics

Volumn 53, Issue 1, 2004, Pages 95-108

  Beasley, T Mark ^a   Page, Grier P ^a   Brand, Jaap P L ^a   Gadbury, Gary L ^b   Mountz, John D ^a   Allison, David B ^a  

^a UNIVERSITY OF ALABAMA AT BIRMINGHAM   (United States)

^b MISSOURI UNIVERSITY OF SCIENCE AND TECHNOLOGY   (United States)

Author keywords

Chebyshev's inequality; Microarrays; Multiple testing; Nonparametrics; Type I error

Indexed keywords

EID: 1342324984     PISSN: 00359254     EISSN: None     Source Type: Journal    
DOI: 10.1111/j.1467-9876.2004.00428.x     Document Type: Article

References (53)

1. Algina, J., Oshima, T. C. and Lin, W. Y. (1994) Type I error rates for Welch's test and James's second-order test under nonnormality and inequality of variance when there are two groups. J. Educ. Behav. Statist., 19, 275-291.
2. Allison, D. B. and Beasley, T. M. (1998) Method and computer program for controlling the family-wise alpha rate in gene association studies involving multiple phenotypes. Genet. Epidem., 15, 87-101.
3. Allison, D. B. and Coffey, C. S. (2002) Two-stage testing in microarray analysis: what is gained? J. Gerontol. A, 57, B189-B192.
4. Allison, D. B., Gadbury, G. L., Heo, M., Fernández, J. R., Lee, C.-K., Prolla, T. A. and Weindruch, R. (2002) A mixture model approach for the analysis of microarray gene expression data. Comput. Statist. Data Anal., 39, 1-20.
5. Benjamini, Y. and Hochberg, Y. (2000) On the adaptive control of the false discovery rate in multiple testing with independent statistics. J. Educ. Behav. Statist., 25, 60-83.
6. Boneau, C. A. (1960) The effects of violations of assumptions underlying the t test. Psychol. Bull., 57, 49-64.
7. Box, G. E. P. and Watson, G. S. (1962) Robustness to non-normality of regression tests. Biometrika, 49, 93-106.
8. Bradley, J. V. (1968) Distribution-free Statistical Tests. Englewood Cliffs: Prentice Hall.
9. Bradley, J. V. (1978) Robustness? Br. J. Math. Statist. Psychol., 31, 144-152.
10. Brunner, E. and Munzel, U. (2000) The nonparametric Behrens-Fisher problem: asymptotic theory and a small-sample approximation. Biometr. J., 42, 17-25.
11. Campos, A. H., Zhao, Y., Pollman, M. J. and Gibbons, G. H. (2003) DNA microarray profiling to identify angiotensin-responsive genes in vascular smooth muscle cells: potential mediators of vascular disease. Circuln Res., 92, 111-118.
12. Chen, C.-Z., Li, M., de Graaf, D., Monti, S., Göttgens, B., Sanchez, M.-J., Lander, E. S., Golub, T. R., Green, A. R. and Lodish, H. F. (2002) Identification of endoglin as a functional marker that defines long-term repopulating hematopoietic stem cells. Proc. Natn. Acad. Sci. USA, 99, 15468-15473.
13. Cliff, N. (1993) Dominance statistics: ordinal analyses to answer ordinal questions. Psychol. Bull., 114, 494-509.
14. Corominola, H., Conner, L. J., Beavers, L. S., Gadski, R. A., Johnson, D., Caro, J. F. and Rafaeloff-Phail, R. (2001) Identification of novel genes differentially expressed in omental fat of obese subjects and obese type 2 diabetic patients. Diabetes, 50, 2822-2830.
15. Cressie, N. A. and Whitford, H. J. (1986) How to use the two sample Mest. Biometr. J., 28, 131-148.
16. DasGupta, A. (2000) Best constants in Chebychev inequalities with various applications. Metrika, 51, 185-200.
17. Di Bucchianico, A. (1999) Combinatorics, computer algebra and the Wilcoxon-Mann-Whitney test. J. Statist. Planng Inf., 79, 349-364.
18. Dunn, O. J. (1961) Multiple comparison among means. J. Am. Statist. Ass., 56, 52-64.
19. Efron, B. and Tibshirani, R. J. (1993) An Introduction to the Bootstrap. New York: Chapman and Hall.
20. Efron, B. and Tibshirani, R. J. (2003) Empirical Bayes methods and false discovery rates for microarrays. Genet. Epidem., 23, 70-86.
21. Esteban, M. D., Castellanos, M. E., Morales, D. and Vajda, I. (2001) Monte Carlo comparison of four normality tests using different entropy estimates. Communs Statist. Simuln Computn, 30, 761-785.
22. Gauss, C. F. (1823) Theoria combinationis observationum erroribus minimus obnoxiae, pars prior. Comm. Soc. Reg. Sci. Gotting. Rec., 5.
23. Gibson, G. (2002) Microarrays in ecology and evolution: a preview. Molec. Ecol., 11, 17-24.
24. Goddard, M. E. (2001) The validity of genetic models underlying quantitative traits. Livstck Prod. Sci., 72, 117-127.
25. Good, P. I. (2000) Permutation Tests: a Practical Guide to Resampling Methods for Testing Hypotheses, 2nd edn. London: Springer.
26. Han, H., Bearss, D. J., Browne, L. W., Calaluce, R., Nagle, R. B. and Von Hoff, D. D. (2002) Identification of differentially expressed genes in pancreatic cancer cells using cDNA microarray. Cancer Res., 62, 2890-2896.
27. Hernan, R., Fasheh, R., Calabrese, C., Frank, A. J., Maclean, K. H., Allard, D., Barraclough, R. and Gilbertson, R. J. (2003) ERBB2 up-regulates S100A4 and several other prometastatic genes in medulloblastoma. Cancer Res., 63, 140-148.
28. Hotelling, H. (1961) The behavior of some standard statistical tests under non-standard conditions. In Proc. 4th Berkeley Symp. Mathematical Statistics and Probability, vol. 1 (ed. J. Neyman), pp. 319-359. Berkeley: University of California Press.
29. Karian, Z. A. and Dudewicz, E. J. (2000) Fitting Statistical Distributions: the Generalized Lambda Distribution and Generalized Bootstrap Methods. New York: CRC.
30. Kayo, T., Allison, D. B., Weindruch, R. and Prolla, T. A. (2001) Influences of aging and caloric restriction on the transcriptional profile of skeletal muscle from rhesus monkeys. Proc. Natn. Acad. Sci. USA, 98, 5093-5098.
31. Kerr, M. K. and Churchill, G. A. (2001) Statistical design and the analysis of gene expression microarray data. Genet. Res., 77, 123-128.
32. Lee, C. K., Weindruch, R. and Prolla, T. A. (2000) Gene-expression profile of the ageing brain in mice. Nat. Genet., 25, 294-297.
33. Lee, J.-M., Williams, M. E., Tingey, S. V. and Rafalski, J. A. (2002) DNA array profiling of gene expression changes during maize embryo development. Funct. Integr. Genom., 2, 13-27.
34. Li, W., Liang, X., Leu, J. I., Kovalovich, K., Ciliberto, G. and Traub, R. (2001) Global changes in interleukin-6-dependent gene expression patterns in mouse livers after partial hepatectomy. Hepatology, 6, 1377-1386.
35. Li, Y., Li, Y., Tang, R., Xu, H., Qiu, M., Chen, Q., Chen, J., Fu, Z., Ying, K., Xie, Y. and Mao, Y. (2002) Discovery and analysis of hepatocellular carcinoma genes using cDNA microarrays. J. Cancer Res. Clin. Oncol., 128, 369-379.
36. López, I. P., Marti, A., Milagro, F. I., Zulet, M., Moreno-Aliaga, M. J., Martinez, J. A. and De Miguel, C. (2003) DNA microarray analysis of genes differentially expressed in diet-induced (cafeteria) obese rats. Obes. Res., 11, 188-194.
37. Mallows, C. L. (1956) Generalizations of Tchebycheff's inequalities (with discussion). J. R. Statist. Soc. B, 18, 139-176.
38. Miyazawa, K., Mori, A., Yamamoto, K. and Okudaira, H. (1998) Transcriptional roles of CCAAT/enhancer binding protein-beta, nuclear factor-kappaB, and C-promoter binding factor 1 in interleukin (IL) -1beta-induced IL-6 synthesis by human rheumatoid fibroblast-like synoviocytes. J. Biol. Chem., 273, 7620-7627.
39. Mountz, J. D., Hsu, H. C., Matsuki, Y. and Zhang, H. G. (2001) Apoptosis and rheumatoid arthritis: past, present, and future directions. Curr. Rheum. Rep., 3, 70-78.
40. Mountz, J. D. and Zhang, H. G. (2001) Regulation of apoptosis of synovial fibroblasts. Curr. Dir. Autoimmun., 3, 216-239.
41. Myers, C., Charboneau, A., Cheung, I., Hanks, D. and Boudreau, N. (2002) Sustained expression of homeobox D10 inhibits angiogenesis. Am. J. Pathol., 161, 2009-2016.
42. Phadtare, S., Kato, I. and Inouye, M. (2002) DNA microarray analysis of the expression profile of escherichia coli in response to treatment with 4,5-Dihydroxy-2-Cyclopenten-l-One. J. Bacteriol., 184, 6725-6729.
43. Richmond, T. and Somerville, S. (2000) Chasing the dream: plant EST microarrays. Curr. Opin. Plant Biol., 3, 108-116.
44. SAS Institute (2001) SAS/IML Users Guide (Release 8.2). Cary: SAS Institute.
45. Satterthwaite, F. E. (1949) An approximate distribution of estimates of variance components. Biometr. Bull., 2, 110-114.
46. Saw, J. G., Yang, M. C. K. and Mo, T. C. (1984) Chebyshev inequality with estimated mean and variance. Am. Statistn, 38, 130-132.
47. Scheffé, H. (1959) The Analysis of Variance. New York: Wiley.
48. Šidák, Z. (1967) Rectangular confidence regions for the means of multivariate normal distributions. J. Am. Statist. Ass., 62, 626-633.
49. Tusher, V. G., Tibshirani, R. and Chu, G. (2001) Significance analysis of microarrays applied to the ionizing radiation response. Proc. Natn. Acad. Sci. USA, 98, 5116-5121.
50. Wilcox, R. R. (1993) Robustness in ANOVA. In Applied Analysis of Variance in the Behavioral Sciences (ed. E. Edwards), pp. 345-374. New York: Dekker.
51. Wilcox, R. R. (1997) Introduction to Robust Estimation and Hypothesis Testing. San Diego: Academic Press.
52. Wu, T. D. (2001) Analysing gene expression data from DNA microarrays to identify candidate genes. J. Pathol., 195, 53-65.
53. Zimmerman, D. W. (1996) A note on homogeneity of variance of scores and ranks. J. Exper. Educ., 64, 351-362.