Fold-Change Estimation of Differentially Expressed Genes using Mixture Mixed-Model

Microarray experiments produce expression measurements for thousands of genes simultaneously, though usually for a small number of RNA samples. The most common problem is the identification of genes that are differentially expressed between different groups of samples or biological conditions. As the number of genes far exceeds the number of RNA samples, the inherent multiplicity poses a severe problem in both hypothesis testing and effect estimation. While much of the recent literature is focused on the hypothesis aspects, we concentrate in this paper on effect estimation as a tool for the identification of differentially expressed genes. We propose a linear mixed model where the random effects are assumed to follow a mixture distribution, and study in detail the case of three normals, corresponding to genes that are down-, up- or non regulated. Our approach leads to a new type of non-linear shrinkage estimation, where a proportion of estimates is shrunk to zero, while the rest follows standard linear shrinkage. This allows us to estimate the log fold-change of the genes involved and to identify those that are differentially expressed within the same model framework. We investigate the operating characteristics of our method using simulation and spike-in studies, and illustrate its application to real data using a breast-cancer dataset.