MOTIVATION: The functions of genes in networks are typically correlated due to their functional connectivity. Variable selection methods have been developed to select important genes associated with a trait while incorporating network graphical information. However, no method has been proposed to quantify the uncertainty of individual genes under such settings. RESULTS: In this paper, we construct confidence intervals (CIs) and provide P-values for parameters of a high-dimensional linear model incorporating graphical structures where the number of variables p diverges with the number of observations. For combining the graphical information, we propose a graph-constrained desparsified LASSO (least absolute shrinkage and selection operator) (GCDL) estimator, which reduces dramatically the influence of high correlation of predictors and enjoys the advantage of faster computation and higher accuracy compared with the desparsified LASSO. Theoretical results show that the GCDL estimator achieves asymptotic normality. The asymptotic property of the uniform convergence is established, with which an explicit expression of the uniform CI can be derived. Extensive numerical results indicate that the GCDL estimator and its (uniform) CI perform well even when predictors are highly correlated. AVAILABILITY AND IMPLEMENTATION: An R package implementing the proposed method is available at https://github.com/XiaoZhangryy/gcdl.

We consider parsimonious modeling of high-dimensional multivariate additive models using regression splines, with or without sparsity assumptions. The approach is based on treating the coefficients in the spline expansions as a third-order tensor. Note the data does not have tensor predictors or tensor responses, which distinguishes our study from the existing ones. A Tucker decomposition is used to reduce the number of parameters in the tensor. We also combined the Tucker decomposition with penalization to enable variable selection. The proposed method can avoid the statistical inefficiency caused by estimating a large number of nonparametric functions. We provide sufficient conditions under which the proposed tensor-based estimators achieve the optimal rate of convergence for the nonparametric regression components. We conduct simulation studies to demonstrate the effectiveness of the proposed novel approach in fitting high-dimensional multivariate additive models and illustrate its application on a breast cancer copy number variation and gene expression data set.

MOTIVATION: The joint analysis of multiple phenotypes is important in many biological studies, such as plant and animal breeding. The heritability estimation for a linear combination of phenotypes is designed to account for correlation information. Existing methods for estimating heritability mainly focus on single phenotypes under random-effect models. These methods also require some stringent conditions, which calls for a more flexible and interpretable method for estimating heritability. Fixed-effect models emerge as a useful alternative. RESULTS: In this article, we propose a novel heritability estimator based on multivariate ridge regression for linear combinations of phenotypes, yielding accurate estimates in both sparse and dense cases. Under mild conditions in the high-dimensional setting, the proposed estimator appears to be consistent and asymptotically normally distributed. Simulation studies show that the proposed estimator is promising under different scenarios. Compared with independently combined heritability estimates in the case of multiple phenotypes, the proposed method significantly improves the performance by considering correlations among those phenotypes. We further demonstrate its application in heritability estimation and correlation analysis for the Oryza sativa rice dataset. AVAILABILITY AND IMPLEMENTATION: An R package implementing the proposed method is available at https://github.com/xg-SUFE1/MultiRidgeVar, where covariance estimates are also given together with heritability estimates. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.