Journal of Advanced Statistics

Download PDF (551.9 KB) PP. 199 - 211 Pub. Date: December 1, 2016

DOI: 10.22606/jas.2016.14003

• Mehdi Razzaghi^**
  Mathematical and Digital Science, Bloomsburg University in Pennsylvania, 400 East 2nd Street, Bloomsburg, PA, United States
• Dong Zhang
  Mathematical and Digital Science, Bloomsburg University in Pennsylvania, 400 East 2nd Street, Bloomsburg, PA, United States

To determine the genes that are differentially expressed between samples in microarray experiments, traditionally the expression levels were assessed by taking the intensity levels at a spot on the array and flagging the gene if the magnitude of the fold change exceeded a threshold. Recently, however, there has been much effort to improve the methodology by incorporating the variability of the intensity ratios. While the Student’s t-test and several of its variants have been proposed by several authors, a methodology that has found widespread popularity is the application of the mixture model in a hierarchical approach whereby the mean of the distribution of the normalized log ratios is assumed to be a random variable having a mixture of two components. One component is a point mass distribution concentrated at zero to represent the non-differentially expressed genes and another component is a suitable distribution with zero mean to represent the differentially expressed genes. The normal and the Laplace distributions have been previously suggested for the differentially expressed genes component of the mixture. But, once again, the symmetry assumption can make these distributions unsuitable. Here, we take a more general approach and apply the beta-normal model to describe the distribution of the mean of the differentially expressed genes. The advantage of this approach is that we no longer assume symmetry for the distribution and let the data determine its shape. We show that our approach includes the earlier results based on normality assumption as a special case. Simulation results demonstrate that there are advantages in using the more general beta-normal distribution. An example with a microarray experimental data is utilized to provide further illustration.

Microarray experiments, differential expression, beta-normal, posterior odds

[1] M. Schena, D. Shalon, R. Davis, and P. Brown, “Quantitative monitoring of gene expression patterns in complementary dna microarray,” Science, vol. 270, pp. 467–470, 1995.

[2] M. Callow, S. Dudoit, E. Gong, T. Speed, and E. Rubin, “Microarray expression profiling identifies genes with altered expressionin hdl-deficient mice,” Genome Research, vol. 10, pp. 2022–2029, 2000.

[3] X. Cui and G. Churchill, “Statistical tests for differential expression in edna microarray experiments,” Genome Biology, vol. 4, p. 210, 2003.

[4] B. Efron, R. Tibshrani, V. Goss, and G. Chu, Microarrays and their use in a comparative experiment, Technical Report. Department of Health Research and Policy, Stanford University, Stanford, CA, 2000.

[5] V. Tusher, R. Tibshirani, and G. Chu, “Significance analysis of microarrays applied to the ionizing radiation response,” Proceedings of the National Academy of Sciences of the United States of America, vol. 98, no. 1, pp. 5116–5121, 2001.

[6] B. Allison, G. Gadbury, M. Heo, J. Fernandez, C. Lee, Y. Prolla, and R. Weindruch, “A mixture model approach for the analysis of microarray gene expression data,” Computational Statistics and Data Analysis, vol. 39, pp. 1–20, 2002.

[7] R. Delongchamp, T. Lee, and C. Velasco, “A method for computing the overall statistical significance of a treatment effect among a group of genes,” BMC Bioinformatics, vol. 11, 2006.

[8] I. Lonnstedt and T. Speed, “Replicated microarray data.” Statistica Sinica, vol. 12, pp. 31–46, 2002.

[9] D. Bhowmick, A. Davison, D. Goldstein, and Y. Ruffieux, “A laplace mixture model for identification of differential expression in microarray experiments,” Biostatistics, vol. 7, pp. 630–641, 2006.

[10] I. Jeffery, D. Higgins, and A. Culhane, “Comparison and evaluation of methods for generating differentially expressed gene lists from microarray data,” BMC Bioinformatics, vol. 7, p. 359, 2006.

[11] A. Hossain, J. Beyene, A. Willan, and P. Hu, “A flexible approximate likelihood ratio test for detecting differential expression in microarray data,” Computational Statistics and Data Analysis, vol. 53, pp. 3685–3695, 2009.

[12] N. Eugene, C. Lee, and F. Famoye, “Beta-normal distribution and its applications,” Communication in Statistics-Theory and Methods, vol. 31, pp. 497–512, 2002.

[13] A. K. Gupta and S. Nadarajah, “On the moments of the beta-normal distribution,” Communication in Statistics- Theory and Methods, vol. 33, pp. 1–13, 2004.

[14] F. Famoye, C. Lee, and N. Eugene, “Beta-normal distribution: Bimodality properties and applications,” Journal of Modern Applied Statistical Methods, vol. 3, pp. 85–103, 2004.

[15] M. Razzaghi, “Beta-normal distribution in dose-response modeling and risk assessment for quantitative responses,” Environmental and Ecological Statistics, vol. 16, pp. 25–36, 2009.

[16] C. Broyden, “Quasi-newton methods and their application to function minimisation.” Mathematics of Computation, vol. 21, pp. 368–381, 1967.

[17] F. Cartieaux, M. Thibaud, L. Zimmerli, P. Lessard, C. Sarrobert, P. David, A. Gherbaud, C. Robaglia, S. Somerville, and L. Nussaume, “Transcriptome analysis of arabidopsis colonized by a plant-growth promoting rhizobacterium reveals a general effect on disease resistance,” The Plant Journal, vol. 36, pp. 177– 190, 2003.

[18] B. Craig, M. Black, and R. Doerge, “Gene expression data: The technology and statistical analysis,” Journal of Agricultural, Biological, and Environmental Statistics, vol. 8, pp. 1–28, 2003.

[19] W. Pan, “A comparative review of statistical methods for discovering differentially expressed genes in replicated microarray experiments,” Bioinformatics, vol. 12, pp. 546–554, 2002.

[20] Y. Zhao and W. Pan, “Modified nonparametric approaches to detecting differentially expressed genes in replicated microarray experiments,” Bioinformatics, vol. 19, pp. 1046–1054, 2003.