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 γ-density-power-weight (Fujisawa and Eguchi, 2008; Windham, 1995), where γ 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 γ 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 γ-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 γ’s in average performance has similar capability to capture minimizer γ 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 γ procedure and other robust methods. This paper discusses some limitations of this method, and future work.

AdaTiSS: A Novel Data-Adaptive Robust Method for Quantifying Tissue Specificity Scores

“…Recently in a general linear regression problem, we developed a novel data-adaptive robust estimation based on density-power-weight under unknown outlier distribution and non-vanishing outlier proportion (Wang, et al, 2019). In the question of quantifying TS, we restrict the multivariable model analyzed in (Wang, et al, 2019) to a univariate model in the Gaussian population, and robustly estimate population information to get our data-adaptive quantitative TS in the form of robust z-score (AdaTiSS). Our AdaTiSS takes into account gene heterogeneities under various outlier proportions and magnitudes.…”

“…It achieves robustness and data-adaptiveness by selecting a turning parameter based on the data to optimize the population estimation. We summarize the procedure and the algorithm in the following subsections, and more statistical analysis can be found in (Wang, et al, 2019).…”

“…Applying the method in (Wang, et al, 2019) to estimate the null parameters, we take the technique of weighting the mixture density by the density power 0 ,where ≥ 0, to down weight the outliers and therefore purify the reweighted mixture model. The rational is that under a proper , the outliers go to the tails of the reweighted null density and thus the outliers do not contribute much to the population estimation.…”

“…The selected ̂ * varies from gene to gene, trying to optimize the estimation for each gene. (Wang, et al, 2019) provided more statistical analysis on the data-adaptive procedure. Here, we apply this new procedure to estimate the population distribution under the Gaussian-null mixture model formulated in (1) in the problem of quantifying TS.…”

“…We then formulated the question to robustly estimate the population distribution. In a linear regression problem, we developed a novel data-adaptive robust estimation based on density-power-weight under unknown outlier distribution and non-vanishing outlier proportion (Wang, et al, 2019). In the question of quantifying TS, we focused on the Gaussian-population mixture model.…”