A unified linear mixed model for familial relatedness and population structure in genetic association studies

Abstract

Familial relatedness (FR) and population structure (PS) are two major sources for genetic correlation. In the human population, both FR and PS can further break down into additive and dominant components to account for potential additive and dominant genetic effects. In this study, besides the classical additive genomic relationship matrix, a dominant genomic relationship matrix is introduced. A link between the additive/dominant genomic relationship matrices and the coancestry (or kinship)/double coancestry coefficients is also established. In addition, a way to separate the FR and PS correlations based on the estimates of coancestry and double coancestry coefficients from the genomic relationship matrices is proposed. A unified linear mixed model is also developed, which can account for both the additive and dominance effects of FR and PS correlations as well as their possible random interactions. Finally, this unified linear mixed model is applied to analyze two study cohorts from UK Biobank.

On asymptotic distributions of several test statistics for familial relatedness in linear mixed models

“…It should be pointed out that this study has some limitations. In general, the FR correlation could present both an additive and a dominant random effect on a quantitative trait 15 . In this study, we only focused on hypothesis test of the additive FR effect, while the dominant effect of the FR correlation was ignored.…”

“…In general, the FR correlation could present both an additive and a dominant random effect on a quantitative trait. 15 In this study, we only focused on hypothesis test of the additive FR effect, while the dominant effect of the FR correlation was ignored. Our asymptotic results were restricted to a special LMM where the random effects for the expected FR correlation and residual errors were assumed to be normally distributed.…”

“…This can be done via a spectral decomposition on $$$ \boldsymbol{\Sigma} =\boldsymbol{VS}{\boldsymbol{V}}^{\mathsf{T}} $$$, where $$$ \boldsymbol{S}=\operatorname{diag}\left({\xi}_1,\dots, {\xi}_q\right) $$$ with its diagonal elements being the $$$ q $$$ positive eigenvalues of $$$ \boldsymbol{\Sigma} $$$, $$$ q=\mathit{\operatorname{rank}}\left(\boldsymbol{\Sigma} \right)\le n $$$, and $$$ \boldsymbol{V}=\left({\mathbf{v}}_1,\dots, {\mathbf{v}}_q\right) $$$ is a $$$ n\times q $$$ matrix with its $$$ j $$$th column vector $$$ {\mathbf{v}}_j $$$ being an eigenvector of $$$ \boldsymbol{\Sigma} $$$ corresponding to $$$ {\xi}_j $$$ for $$$ j=1,\dots, q $$$. Then, model () can be re‐written as the following 2,15 …”

“…Within each family, the expected FR correlation matrix $$$ 2\boldsymbol{\Phi} $$$ was extracted from the genomic relationship matrix by equating the expected coancestry coefficients being 0, $$$ 1/32 $$$, 1/16, 1/8, 1/4, 1/2, or 1, whichever is the closest. The PS correlation matrix $$$ 2{\boldsymbol{\Phi}}_p $$$ was then derived from the genomic relationship matrix and the expected FR correlation matrix $$$ 2\boldsymbol{\Phi} $$$ using the method proposed by DeVogel et al 15 To account for the PS correlation, the top 10 leading PCs from the PS correlation matrix $$$ 2{\boldsymbol{\Phi}}_p $$$ were calculated and adjusted as fixed covariates. Together with age (continuous), sex and BMI (continuous), the total number of fixed covariates including an intercept is $$$ p=14 $$$.…”

“…On the other hand, as the PS correlation is usually much weaker than the FR correlation, the expected FR correlation can be extracted from the genomic relationship matrix based on some pre-specified cutoff values. 15 Typically, the expected FR correlation can capture the leading part of the actual FR correlation between individuals across the whole genome. Just like the kinship coefficients, families with the same familial relationships (or family type) often share exactly the same expected FR correlation structure.…”