Robust Linear Models for Cis-eQTL Analysis

<div><p>Expression Quantitative Trait Loci (eQTL) analysis enables characterisation of functional genetic variation influencing expression levels of individual genes. In outbread populations, including humans, eQTLs are commonly analysed using the conventional linear model, adjusting for relevant covariates, assuming an allelic dosage model and a Gaussian error term. However, gene expression data generally have noise that induces heavy-tailed errors relative to the Gaussian distribution and often include atypical observations, or outliers. Such departures from modelling assumptions can lead to an increased rate of type II errors (false negatives), and to some extent also type I errors (false positives). Careful model checking can reduce the risk of type-I errors but often not type II errors, since it is generally too time-consuming to carefully check all models with a non-significant effect in large-scale and genome-wide studies. Here we propose the application of a robust linear model for eQTL analysis to reduce adverse effects of deviations from the assumption of Gaussian residuals. We present results from a simulation study as well as results from the analysis of real eQTL data sets. Our findings suggest that in many situations robust models have the potential to provide more reliable eQTL results compared to conventional linear models, particularly in respect to reducing type II errors due to non-Gaussian noise. Post-genomic data, such as that generated in genome-wide eQTL studies, are often noisy and frequently contain atypical observations. Robust statistical models have the potential to provide more reliable results and increased statistical power under non-Gaussian conditions. The results presented here suggest that robust models should be considered routinely alongside other commonly used methodologies for eQTL analysis.</p></div

P-value correspondence in Myers <i>et al</i>. data set [25].

<p>Scatter plot of −<i>log</i>₁₀(p-values) from Myers <i>et al</i>. data set [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0127882#pone.0127882.ref025" target="_blank">25</a>]. (Key: green = significant in both models, red = significant in the conventional model only, blue = significant in the robust model only, data from points marked with black squares are shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0127882#pone.0127882.g005" target="_blank">Fig 5</a>)</p

Results from comparative analysis of Myers <i>et al</i>. data set [25].

<p>SNP effect size estimates and standard errors for eQTLs significant in both models (A, D), in the robust model only (B, E), and in the linear model only (C, F).</p

P-value correspondence in Grundberg <i>et al</i>. data set [26].

<p>Scatter plot of −<i>log</i>₁₀(p-values) from MuTHER data set [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0127882#pone.0127882.ref026" target="_blank">26</a>]. (Key: green = significant in both models, red = significant in the conventional model only, blue = significant in the robust model only, data from points marked with black squares are shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0127882#pone.0127882.s002" target="_blank">S1 Fig</a>).</p

Power analysis results (empirical residuals from robust model fit).

<p>A) Residuals from a random sample of eQTL models. B) Residuals from a random sample from models found to be significant only in the robust eQTL model. (‘cont’ = residual from robust model fit of Myers <i>et al</i>. [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0127882#pone.0127882.ref025" target="_blank">25</a>] data set; ‘no cont’ = Gaussian residuals.)</p

Power analysis results (mixture contamination model).

<p>A) Power as a function of contamination proportion. B) Power as a function of study size. C) Power as a function of the genetic effect size. (Simulation parameters: 10000 samples; A, B and D: N = 100; B, C and D:<i>π</i> = 0.95)</p

Concordance (number and proportion of mRNAs with at least one eQTL SNP) between the conventional and robust models (Myers et al. data set [25].

<p>Concordance (number and proportion of mRNAs with at least one eQTL SNP) between the conventional and robust models (Myers et al. data set [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0127882#pone.0127882.ref025" target="_blank">25</a>].</p

Power analysis results (heavy-tailed).

<p>A) Power as a function of degrees of freedom in the student t-distribution. B) Power as a function of study size. C) Power as a function of the genetic effect size. (Simulation parameters: 10000 samples, A-B, D: N = 100, B-D:<i>df</i> = 4)</p

Comparison of miRNA profiles across tissues.

<p>The left panel (A) shows the single-linkage hierarchical clustering of inter-tissue profile correlations. In the right panel (B) the top 10 most tissue specific islet miRNAs are displayed in descending order. The colors indicate the normalized expression levels of these miRNAs across the different profiles used in the analysis.</p

Association results with <i>FTO</i> SNP rs9939609 and <i>NEGR1</i> SNP rs2815752 in categories of BMI, compared with normal-weight controls.

<p>β, effect size; SE, standard error.</p