Improved Estimation of Correlation in Microarray Data Analysis

In the original work on clustering due to Eisen et al., in which they performed one of the most highly-re-analyzed microarray dataset of gene expressions, the authors claimed to have “found in the budding yeast Saccharomyces cerevisiae that clustering gene expression data groups together efficiently genes of known similar function.” However, they measured similarity between any pair of genes using a somewhat non-standard definition of correlation coefficient instead of Pearson’s correlation coefficient, an unbiased estimator. Eisen et al.’s paper remains mysteriously silent about how drastically the clusters of genes would change if one changed the definition of similarity back to Pearson’s correlation coefficient, or to any other from a larger family of estimators between Pearson’s and Eisen et al.’s, obtained by a “shrinkage coefficient” taking a value between 0 and 1. Their approach raised several issues: what would be the best shrinkage coefficient; how can one compute it and whether it can be computed quickly in a closed-form. Mishra and his students answered these questions in a recent paper, but left it for future research to understand if there is even a wider family of similarity metrics, even though such metrics may not be computable in closed form. We take up this problem in this paper and suggest how to compute a somewhat better similarity metric using an MCMC algorithm; how to define an intuitively clear Bayesian risk assessment and finally, how to interpret the empirical results obtained through simulation. ∗To whom correspondence should be addressed. E-mail: [email protected] 1 Problem Formulation In the post-genomic biology, clustering genes by their similarity has now occupied many biologists and statisticians for almost half a decade. Although the relevance of such a succinct representation in understanding fundamental principles of biology is yet to be firmly established, there is much less disagreement that the resulting data reorganization may add clarity to the subsequent bioinformatic analysis and experiment design, e.g., interpreting ChIP-Chip experiments, looking for cis-regulatory elements, etc. There is now a rapidly mushrooming body of genomics literature devoted to clustering, co-clustering, bi-clustering, etc., with random or designed sets of conditions and different definitions of similarity, and yet, there is much less attention paid to derive a statistically robust definition for similarity of genes. In the usual setting, starting with a series of expression microarray experimental data, one wishes to estimate similarity between the expression levels of a pair of genes because it is frequently indicative of functional relationships between them. Highly correlated transcriptomic behavior of a group of genes often suggests the presence of causal relationships, usually through common regulatory mechanisms. Identifying such potential relationships is of primary importance 1. In understanding and modeling microarray and other genetic data, and 2. In inferring functional relationships crucial to predictive and other kinds of inference. These identifications frequently arise from partitioning genes into closely related groups, called clusters. Traditionally, algorithms for cluster analysis of expression data are based on statistical properties of gene expressions and result in organizing genes according to similarities between their patterns (see [1]).