AdaReg: Data Adaptive Robust Estimation in Linear Regression with Application in GTEx Gene Expressions

Abstract

With the development of high-throughput RNA sequencing (RNA-seq) technology, the Genotype Tissue-Expression (GTEx) project (Consortium et al., 2015) generated a valuable resource of gene expression data from more than 11,000 samples. The large-scale data set is a powerful resource for understanding the human transcriptome. However, the technical variation, sequencing background noise and unknown factors make the statistical analysis challenging. To eliminate the possibility that outliers might affect the estimation of population distribution, we need a more robust estimation method, a method that will adapt to heterogeneous genes and further optimize the estimate for each gene. We followed the approach of the robust estimation based on{gamma} -density-power-weight (Fujisawa and Eguchi, 2008; Windham, 1995), where{gamma} is the exponent of density weight which controls the balance between bias and variance. As far as we know, our work is the first to propose a procedure to tune the parameter{gamma} to balance the bias-variance trade-off under the mixture distributions. We constructed a robust likelihood criterion based on weighted densities in the mixture model of Gaussian population distribution mixed with unknown outlier distribution, and developed a data-adaptive{gamma} -selection procedure embedded into the robust estimation. We provided a heuristic analysis on the selection criterion and found that our practical selection trend under various{gamma} s in average performance has similar capability to capture minimizer{gamma} as the inestimable Mean Squared Error (MSE) trend from our simulation studies under a series of settings. Our data-adaptive robustifying procedure in the linear regression problem (AdaReg) shows a significant advantage in both simulation studies and real data application of heart samples from the GTEx project compared to the fixed{gamma} procedure and other robust methods. This paper discusses some limitations of this method, and future work.