Optimal Estimation of Genetic Relatedness in High-Dimensional Linear Models

ABSTRACT

Estimating the genetic relatedness between two traits based on the genome-wide association data is an important problem in genetics research. In the framework of high-dimensional linear models, we introduce two measures of genetic relatedness and develop optimal estimators for them. One is genetic covariance, which is defined to be the inner product of the two regression vectors, and another is genetic correlation, which is a normalized inner product by their lengths. We propose functional de-biased estimators (FDEs), which consist of an initial estimation step with the plug-in scaled Lasso estimator, and a further bias correction step. We also develop estimators of the quadratic functionals of the regression vectors, which can be used to estimate the heritability of each trait. The estimators are shown to be minimax rate-optimal and can be efficiently implemented. Simulation results show that FDEs provide better estimates of the genetic relatedness than simple plug-in estimates. FDE is also applied to an analysis of a yeast segregant dataset with multiple traits to estimate the genetic relatedness among these traits. Supplementary materials for this article are available online.

KEYWORDS:

• Genetic correlations
• Genome-wide association studies
• Inner product
• Minimax rate of convergence
• Quadratic functional

Supplementary Material

The supplementary materials present extended simulations in Section A. In Section B, we prove Theorem 2. In Section C, we prove (28) and (31) in Theorem 3. We also provide detailed proofs of extra lemmas in Section D. In Section E, we present detailed results of real data analysis.

Acknowledgment

The authors thank Alexandre Tsybakov for helpful discussion on Section 3.2, and the reviewer and AE for helpful comments.

Funding

The research of Wanjie Wang and Hongzhe Li was supported in part by NIH grants CA127334 and GM097505.