Sparse Linear Mixed Model Research Articles

Variable Selection in Heterogeneous Datasets: A Truncated-rank Sparse Linear Mixed Model with Applications to Genome-wide Association Studies.

A fundamental and important challenge in modern datasets of ever increasing dimensionality is variable selection, which has taken on renewed interest recently due to the growth of biological and medical datasets with complex, non-i.i.d. structures. Naïvely applying classical variable selection methods such as the Lasso to such datasets may lead to a large number of false discoveries. Motivated by genome-wide association studies in genetics, we study the problem of variable selection for datasets arising from multiple subpopulations, when this underlying population structure is unknown to the researcher. We propose a unified framework for sparse variable selection that adaptively corrects for population structure via a low-rank linear mixed model. Most importantly, the proposed method does not require prior knowledge of individual relationships in the data and adaptively selects a covariance structure of the correct complexity. Through extensive experiments, we illustrate the effectiveness of this framework over existing methods. Further, we test our method on three different genomic datasets from plants, mice, and humans, and discuss the knowledge we discover with our model.

Genetic Decoupling of Thermal Hardiness across Metamorphosis in Drosophila melanogaster.

As organisms age the environment fluctuates, exerting differential selection across ontogeny. In particular, highly seasonal environments expose life stages to often drastically different thermal environments. This developmental variation is particularly striking in organisms with complex life cycles, wherein life history stages also exhibit distinct morphologies, physiologies, and behaviors. Genes acting pleiotropically on thermal responses may produce genetic correlations across ontogeny, constraining the independent evolution of each life stage to their respective thermal environments. To investigate whether developmental genetic correlations constrain the evolution thermal hardiness of the fly Drosophila melanogaster, we applied quantitative genetic analyses to cold hardiness measured in both larvae and adults from isogenic lines of the Drosophila Genetic Reference Panel (DGRP), using survival at stressful low temperatures as the phenotypic metric. Using full genome resequencing data for the DGRP, we also implemented genome-wide association (GWA) analysis using Bayesian Sparse Linear Mixed Models (BSLMMs) to estimate associations between naturally segregating variation and cold hardiness for both larvae and adults. Quantitative genetic analyses revealed no significant genetic correlation for cold hardiness between life stages, suggesting complete genetic decoupling of thermal hardiness across the metamorphic boundary. Both quantitative genetic and GWA analyses suggested that polygenic variation underlies cold hardiness in both stages, and that associated loci largely affected one stage or the other, but not both. However, reciprocal enrichment tests and correlations between BSLMM parameters for each life stage support some shared physiological mechanisms that may reflect common cellular thermal response pathways. Overall, these results suggest no developmental genetic constraints on cold hardiness across metamorphosis in D. melanogaster, an important consideration in evolutionary models of responses to changing climates. Genetic correlations for environmental sensitivity across ontogeny remains largely unexplored in other organisms, thus assessing the generality of genetic decoupling will require further quantitative or population genetic analysis in additional species.

Prediction of gene expression with cis-SNPs using mixed models and regularization methods

BackgroundIt has been shown that gene expression in human tissues is heritable, thus predicting gene expression using only SNPs becomes possible. The prediction of gene expression can offer important implications on the genetic architecture of individual functional associated SNPs and further interpretations of the molecular basis underlying human diseases.MethodsWe compared three types of methods for predicting gene expression using only cis-SNPs, including the polygenic model, i.e. linear mixed model (LMM), two sparse models, i.e. Lasso and elastic net (ENET), and the hybrid of LMM and sparse model, i.e. Bayesian sparse linear mixed model (BSLMM). The three kinds of prediction methods have very different assumptions of underlying genetic architectures. These methods were evaluated using simulations under various scenarios, and were applied to the Geuvadis gene expression data.ResultsThe simulations showed that these four prediction methods (i.e. Lasso, ENET, LMM and BSLMM) behaved best when their respective modeling assumptions were satisfied, but BSLMM had a robust performance across a range of scenarios. According to R2 of these models in the Geuvadis data, the four methods performed quite similarly. We did not observe any clustering or enrichment of predictive genes (defined as genes with R2 ≥ 0.05) across the chromosomes, and also did not see there was any clear relationship between the proportion of the predictive genes and the proportion of genes in each chromosome. However, an interesting finding in the Geuvadis data was that highly predictive genes (e.g. R2 ≥ 0.30) may have sparse genetic architectures since Lasso, ENET and BSLMM outperformed LMM for these genes; and this observation was validated in another gene expression data. We further showed that the predictive genes were enriched in approximately independent LD blocks.ConclusionsGene expression can be predicted with only cis-SNPs using well-developed prediction models and these predictive genes were enriched in some approximately independent LD blocks. The prediction of gene expression can shed some light on the functional interpretation for identified SNPs in GWASs.

Survey of the Heritability and Sparse Architecture of Gene Expression Traits across Human Tissues.

Understanding the genetic architecture of gene expression traits is key to elucidating the underlying mechanisms of complex traits. Here, for the first time, we perform a systematic survey of the heritability and the distribution of effect sizes across all representative tissues in the human body. We find that local h2 can be relatively well characterized with 59% of expressed genes showing significant h2 (FDR < 0.1) in the DGN whole blood cohort. However, current sample sizes (n ≤ 922) do not allow us to compute distal h2. Bayesian Sparse Linear Mixed Model (BSLMM) analysis provides strong evidence that the genetic contribution to local expression traits is dominated by a handful of genetic variants rather than by the collective contribution of a large number of variants each of modest size. In other words, the local architecture of gene expression traits is sparse rather than polygenic across all 40 tissues (from DGN and GTEx) examined. This result is confirmed by the sparsity of optimal performing gene expression predictors via elastic net modeling. To further explore the tissue context specificity, we decompose the expression traits into cross-tissue and tissue-specific components using a novel Orthogonal Tissue Decomposition (OTD) approach. Through a series of simulations we show that the cross-tissue and tissue-specific components are identifiable via OTD. Heritability and sparsity estimates of these derived expression phenotypes show similar characteristics to the original traits. Consistent properties relative to prior GTEx multi-tissue analysis results suggest that these traits reflect the expected biology. Finally, we apply this knowledge to develop prediction models of gene expression traits for all tissues. The prediction models, heritability, and prediction performance R2 for original and decomposed expression phenotypes are made publicly available (https://github.com/hakyimlab/PrediXcan).

Multilocus approaches for the measurement of selection on correlated genetic loci.

The study of ecological speciation is inherently linked to the study of selection. Methods for estimating phenotypic selection within a generation based on associations between trait values and fitness (e.g. survival) of individuals are established. These methods attempt to disentangle selection acting directly on a trait from indirect selection caused by correlations with other traits via multivariate statistical approaches (i.e. inference of selection gradients). The estimation of selection on genotypic or genomic variation could also benefit from disentangling direct and indirect selection on genetic loci. However, achieving this goal is difficult with genomic data because the number of potentially correlated genetic loci (p) is very large relative to the number of individuals sampled (n). In other words, the number of model parameters exceeds the number of observations (p≫n). We present simulations examining the utility of whole-genome regression approaches (i.e. Bayesian sparse linear mixed models) for quantifying direct selection in cases where p≫n. Such models have been used for genome-wide association mapping and are common in artificial breeding. Our results show they hold promise for studies of natural selection in the wild and thus of ecological speciation. But we also demonstrate important limitations to the approach and discuss study designs required for more robust inferences.

Heritability estimation in high dimensional sparse linear mixed models

Motivated by applications in genetic fields, we propose to estimate the heritability in high-dimensional sparse linear mixed models. The heritability determines how the variance is shared between the different random components of a linear mixed model. The main novelty of our approach is to consider that the random effects can be sparse, that is may contain null components, but we do not know either their proportion or their positions. The estimator that we consider is strongly inspired by the one proposed by Pirinen, Donnelly and Spencer (2013), and is based on a maximum likelihood approach. We also study the theoretical properties of our estimator, namely we establish that our estimator of the heritability is $\sqrt{n}$-consistent when both the number of observations $n$ and the number of random effects $N$ tend to infinity under mild assumptions. We also prove that our estimator of the heritability satisfies a central limit theorem which gives as a byproduct a confidence interval for the heritability. Some Monte-Carlo experiments are also conducted in order to show the finite sample performances of our estimator.

Poly-omic prediction of complex traits: OmicKriging.

High-confidence prediction of complex traits such as disease risk or drug response is an ultimate goal of personalized medicine. Although genome-wide association studies have discovered thousands of well-replicated polymorphisms associated with a broad spectrum of complex traits, the combined predictive power of these associations for any given trait is generally too low to be of clinical relevance. We propose a novel systems approach to complex trait prediction, which leverages and integrates similarity in genetic, transcriptomic, or other omics-level data. We translate the omic similarity into phenotypic similarity using a method called Kriging, commonly used in geostatistics and machine learning. Our method called OmicKriging emphasizes the use of a wide variety of systems-level data, such as those increasingly made available by comprehensive surveys of the genome, transcriptome, and epigenome, for complex trait prediction. Furthermore, our OmicKriging framework allows easy integration of prior information on the function of subsets of omics-level data from heterogeneous sources without the sometimes heavy computational burden of Bayesian approaches. Using seven disease datasets from the Wellcome Trust Case Control Consortium (WTCCC), we show that OmicKriging allows simple integration of sparse and highly polygenic components yielding comparable performance at a fraction of the computing time of a recently published Bayesian sparse linear mixed model method. Using a cellular growth phenotype, we show that integrating mRNA and microRNA expression data substantially increases performance over either dataset alone. Using clinical statin response, we show improved prediction over existing methods. We provide an R package to implement OmicKriging (http://www.scandb.org/newinterface/tools/OmicKriging.html).

Polygenic Modeling with Bayesian Sparse Linear Mixed Models

Both linear mixed models (LMMs) and sparse regression models are widely used in genetics applications, including, recently, polygenic modeling in genome-wide association studies. These two approaches make very different assumptions, so are expected to perform well in different situations. However, in practice, for a given dataset one typically does not know which assumptions will be more accurate. Motivated by this, we consider a hybrid of the two, which we refer to as a “Bayesian sparse linear mixed model” (BSLMM) that includes both these models as special cases. We address several key computational and statistical issues that arise when applying BSLMM, including appropriate prior specification for the hyper-parameters and a novel Markov chain Monte Carlo algorithm for posterior inference. We apply BSLMM and compare it with other methods for two polygenic modeling applications: estimating the proportion of variance in phenotypes explained (PVE) by available genotypes, and phenotype (or breeding value) prediction. For PVE estimation, we demonstrate that BSLMM combines the advantages of both standard LMMs and sparse regression modeling. For phenotype prediction it considerably outperforms either of the other two methods, as well as several other large-scale regression methods previously suggested for this problem. Software implementing our method is freely available from http://stephenslab.uchicago.edu/software.html.