β-empirical Bayes inference and model diagnosis of microarray data

Background: Microarray data enables the high-throughput survey of mRNA expression profiles at the genomic level; however, the data presents a challenging statistical problem because of the large number of transcripts with small sample sizes that are obtained. To reduce the dimensionality, various Bayesian or empirical Bayes hierarchical models have been developed. However, because of the complexity of the microarray data, no model can explain the data fully. It is generally difficult to scrutinize the irregular patterns of expression that are not expected by the usual statistical gene by gene models. Results: As an extension of empirical Bayes (EB) procedures, we have developed the beta-empirical Bayes (beta-EB) approach based on a beta-likelihood measure which can be regarded as an 'evidence-based' weighted (quasi-) likelihood inference. The weight of a transcript t is described as a power function of its likelihood, f(beta)(gamma(t)vertical bar theta). Genes with low likelihoods have unexpected expression patterns and low weights. By assigning low weights to outliers, the inference becomes robust. The value of beta, which controls the balance between the robustness and efficiency, is selected by maximizing the predictive beta(0)-likelihood by cross-validation. The proposed beta-EB approach identified six significant (p < 10(-5)) contaminated transcripts as differentially expressed (DE) in normal/tumor tissues from the head and neck of cancer patients. These six genes were all confirmed to be related to cancer; they were not identified as DE genes by the classical EB approach. When applied to the eQTL analysis of Arabidopsis thaliana, the proposed beta-EB approach identified some potential master regulators that were missed by the EB approach. Conclusions: The simulation data and real gene expression data showed that the proposed beta-EB method was robust against outliers. The distribution of the weights was used to scrutinize the irregular patterns of expression and diagnose the model statistically. When beta-weights outside the range of the predicted distribution were observed, a detailed inspection of the data was carried out. The beta-weights described here can be applied to other likelihood-based statistical models for diagnosis, and may serve as a useful tool for transcriptome and proteome studies.