With the growing amount and diversity of intermediate omics data complementary to genomics (e.g., DNA methylation, gene expression, and protein abundance), there is a need to develop methods to incorporate intermediate omics data into conventional genomic evaluation. The omics data helps decode the multiple layers of regulation from genotypes to phenotypes, thus forms a connected multi-layer network naturally. We developed a new method named NN-LMM to model the multiple layers of regulation from genotypes to intermediate omics features, then to phenotypes, by extending conventional linear mixed models (“LMM”) to multi-layer artificial neural networks (“NN”). NN-LMM incorporates intermediate omics features by adding middle layers between genotypes and phenotypes. Linear mixed models (e.g., pedigree-based BLUP, GBLUP, Bayesian Alphabet, single-step GBLUP, or single-step Bayesian Alphabet) can be used to sample marker effects or genetic values on intermediate omics features, and activation functions in neural networks are used to capture the nonlinear relationships between intermediate omics features and phenotypes. NN-LMM had significantly better prediction performance than the recently proposed single-step approach for genomic prediction with intermediate omics data. Compared to the single-step approach, NN-LMM can handle various patterns of missing omics measures, and allows nonlinear relationships between intermediate omics features and phenotypes. NN-LMM has been implemented in an open-source package called “JWAS”.
In conventional linear models for whole-genome prediction and genome-wide association studies (GWAS), it is usually assumed that the relationship between genotypes and phenotypes is linear. Bayesian neural networks have been used to account for non-linearity such as complex genetic architectures. Here, we introduce a method named NN-Bayes, where” NN” stands for neural networks, and” Bayes” stands for Bayesian Alphabet models, including a collection of Bayesian regression models such as BayesA, BayesB, BayesC, and Bayesian LASSO. NN-Bayes incorporates Bayesian Alphabet models into non-linear neural networks via hidden layers between SNPs and observed traits. Thus, NN-Bayes attempts to improve the performance of genome-wide prediction and GWAS by accommodating non-linear relationships between the hidden nodes and the observed trait, while maintaining genomic interpretability through the Bayesian regression models that connect the SNPs to the hidden nodes. For genomic interpretability, the posterior distribution of marker effects in NN-Bayes is inferred by Markov chain Monte Carlo (MCMC) approaches and used for inference of association through posterior inclusion probabilities (PIPs) and window posterior probability of association (WPPA). In simulation studies with dominance and epistatic effects, performance of NN-Bayes was significantly better than conventional linear models for both GWAS and whole-genome prediction, and the differences on prediction accuracy were substantial in magnitude. In real data analyses, for the soy dataset, NN-Bayes achieved significantly higher prediction accuracies than conventional linear models, and results from other four different species showed that NN-Bayes had similar prediction performance to linear models, which is potentially due to the small sample size. Our NN-Bayes is optimized for high-dimensional genomic data and implemented in an open-source package called” JWAS”. NN-Bayes can lead to greater use of Bayesian neural networks to account for non-linear relationships due to its interpretability and computational performance.
Background: Bayesian regression models are widely used in genomic prediction, where the effects of all markers are estimated simultaneously by combining the information from the phenotypic data with priors for the marker effects and other parameters such as variance components or membership probabilities. Inferences from most Bayesian regression models are based on Markov chain Monte Carlo methods, where statistics are computed from a Markov chain constructed to have a stationary distribution that is equal to the posterior distribution of the unknown parameters. In practice, chains of tens of thousands steps are typically used in whole-genome Bayesian analyses, which is computationally intensive. Methods: In this paper, we propose a fast parallelized algorithm for Bayesian regression models called independent intensive Bayesian regression models (BayesXII, "X" stands for Bayesian alphabet methods and "II" stands for "parallel") and show how the sampling of each marker effect can be made independent of samples for other marker effects within each step of the chain. This is done by augmenting the marker covariate matrix by adding p (the number of markers) new rows such that columns of the augmented marker covariate matrix are orthogonal. Ideally, the computations at each step of the MCMC chain can be accelerated by k times, where k is the number of computer processors, up to p times, where p is the number of markers. Results: We demonstrate the BayesXII algorithm using the prior for BayesCπ , a Bayesian variable selection regression method, which is applied to simulated data with 50,000 individuals and a medium-density marker panel (∼ 50,000 markers). To reach about the same accuracy as the conventional samplers for BayesCπ required less than 30 min using the BayesXII algorithm on 24 nodes (computer used as a server) with 24 cores on each node. In this case, the BayesXII algorithm required one tenth of the computation time of conventional samplers for BayesCπ. Addressing the heavy computational burden associated with Bayesian methods by parallel computing will lead to greater use of these methods.
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