Regularization Methods for High-Dimensional Instrumental Variables Regression With an Application to Genetical Genomics

Abstract

In genetical genomics studies, it is important to jointly analyze gene expression data and genetic variants in exploring their associations with complex traits, where the dimensionality of gene expressions and genetic variants can both be much larger than the sample size. Motivated by such modern applications, we consider the problem of variable selection and estimation in high-dimensional sparse instrumental variables models. To overcome the difficulty of high dimensionality and unknown optimal instruments, we propose a two-stage regularization framework for identifying and estimating important covariate effects while selecting and estimating optimal instruments. The methodology extends the classical two-stage least squares estimator to high dimensions by exploiting sparsity using sparsity-inducing penalty functions in both stages. The resulting procedure is efficiently implemented by coordinate descent optimization. For the representative L1 regularization and a class of concave regularization methods, we establish estimation, prediction, and model selection properties of the two-stage regularized estimators in the high-dimensional setting where the dimensionality of co-variates and instruments are both allowed to grow exponentially with the sample size. The practical performance of the proposed method is evaluated by simulation studies and its usefulness is illustrated by an analysis of mouse obesity data. Supplementary materials for this article are available online.

A directed learning strategy integrating multiple omic data improves genomic prediction

“…At first, we were confused with the above result because Lin et al . () reported that the predicted transcriptome obtained with genetic variants as IV significantly improved phenotype predictions. Soon afterwards, we noticed that only parts of transcripts were predictable with relatively high predictability (Figure a), and we then proposed to use the predicted values of ‘genetically predictable genes (GPGs)’ of the first layer, denoted by PT.1L.GPGs, to predict phenotypes.…”

“…Lin et al . () used a two‐stage least squares (2SLS) method to choose an optimal sparse subset of β 0 and Γ 0 . Through strict mathematical derivation and a large scale of simulation test, they claimed that the 2SR method was reliable and powerful for genomic prediction.…”

“…Here, we proposed such an integration strategy, that is multilayered least absolute shrinkage and selection operator (MLLASSO), for improving GP. The key idea of MLLASSO is to implement an innovative directed learning strategy that allows us to learn three layers of genetic features (we denote them as 'GFs.1L', 'GFs.2L' and 'GFs.3L' in the rest of this study) supervised by transcriptomic and metabolomic data using genetic variants as instrumental variables (IV), which has been proven to be an efficient statistical technique to select and estimate optimal instruments (Lin et al, 2015). Our approach is still GP because it only requires genomic markers as the input data, but it differs from the traditional GP in that it integrates transcriptomic and metabolic information into a single model and may capture higher order information of gene interactions.…”

Systems biology approaches to epidemiological studies of complex diseases

“…In contrast to the ordinary linear model regressing y on X , model (2) does not require that the covariate X and the error η be independent, thus substantially relaxing the assumptions of ordinary regression models and being more appealing in data analysis. Wei et al 15 developed two-stage penalized estimation procedure to estimate the parameters and to simultaneously identify the possible instruments and genes that are associated with the phenotype y .…”

Regularized estimation in sparse high-dimensional multivariate regression, with application to a DNA methylation study

“…With the development of modern technology for data collection, high-dimensional data have become increasingly common in many scientific research fields, e.g., genome-wide studies (Lin et al 2015), biomedical sciences (Mukherjee et al. 2015), economics and finance (Basu and Michailidis 2015).…”