[Paper Review] Optimal Estimation of Co-heritability in High-dimensional Linear Models

Zijian Guo, Wanjie Wang+2 more|arXiv (Cornell University)|May 24, 2016

Genetic Mapping and Diversity in Plants and Animals6 references3 citations

TL;DR

This paper proposes functional de-biased estimators (FDEs) for co-heritability in high-dimensional linear models, focusing on the inner product and normalized inner product of regression vectors from GWAS data. The method achieves minimax optimal rates of convergence and significantly outperforms naive plug-in estimators in simulations and real yeast data analysis.

ABSTRACT

Co-heritability is an important concept that characterizes the genetic associations within pairs of quantitative traits. There has been significant recent interest in estimating the co-heritability based on data from the genome-wide association studies (GWAS). This paper introduces two measures of co-heritability in the high-dimensional linear model framework, including the inner product of the two regression vectors and a normalized inner product by their lengths. Functional de-biased estimators (FDEs) are developed to estimate these two co-heritability measures. In addition, estimators of quadratic functionals of the regression vectors are proposed. Both theoretical and numerical properties of the estimators are investigated. In particular, minimax rates of convergence are established and the proposed estimators of the inner product, the quadratic functionals and the normalized inner product are shown to be rate-optimal. Simulation results show that the FDEs significantly outperform the naive plug-in estimates. The FDEs are also applied to analyze a yeast segregant data set with multiple traits to estimate heritability and co-heritability among the traits.

Motivation & Objective

• To develop optimal estimation methods for co-heritability between pairs of quantitative traits using high-dimensional linear models.
• To define co-heritability via the inner product and normalized inner product of regression vectors, avoiding reliance on mixed-effects models.
• To establish minimax optimal rates of convergence for estimators of inner products, quadratic functionals, and normalized inner products.
• To overcome bias from irrelevant SNPs by focusing on relevant genetic variants through functional de-biasing.
• To validate the method using simulation studies and real yeast segregant data with multiple traits.

Proposed method

• Develops functional de-biased estimators (FDEs) to correct bias in high-dimensional regression coefficient estimation.
• Defines co-heritability as the inner product and normalized inner product of two regression vectors β and γ.
• Applies de-biasing techniques to estimate β and γ under sub-Gaussian design matrices with high-dimensional p and sample size n.
• Uses a two-step procedure: first, estimate β and γ via regularized methods (e.g., Lasso); second, de-bias using a dual formulation to construct asymptotically normal estimators.
• Establishes theoretical guarantees via concentration inequalities and high-probability bounds under sparsity and sub-Gaussian assumptions.
• Derives minimax lower bounds and proves the FDEs achieve the optimal rate of convergence.

Experimental results

Research questions

• RQ1 Can co-heritability between two quantitative traits be optimally estimated in high-dimensional linear models without assuming knowledge of causal SNPs?
• RQ2 What are the minimax optimal rates of convergence for estimating co-heritability measures such as inner product and normalized inner product?
• RQ3 How do functional de-biased estimators (FDEs) compare to naive plug-in estimators in finite samples under high-dimensional settings?
• RQ4 Can the proposed method effectively estimate co-heritability in real GWAS data with multiple correlated traits?
• RQ5 What is the theoretical justification for the optimality of the FDEs in terms of minimax risk?

Key findings

• The FDEs for the inner product, quadratic functionals, and normalized inner product achieve the minimax optimal rate of convergence under the given high-dimensional model.
• The FDEs significantly outperform naive plug-in estimators in simulation studies, especially in high-dimensional and sparse settings.
• Theoretical analysis confirms that the FDEs are rate-optimal, with convergence rates of order O(√(k log p / n)) under sparsity and sub-Gaussian design.
• The method successfully estimates co-heritability in a real yeast segregant data set, demonstrating practical utility for multi-trait genetic analysis.
• Theoretical bounds show that the FDEs achieve high-probability control of estimation error under sparsity and sub-Gaussian design assumptions.
• The method avoids the bias introduced by using all genotyped SNPs by focusing on relevant variants through de-biasing, improving accuracy over existing approaches.