Changes

We assume that genetic effects are normally distributed, with mean <math>\mathbf{0}</math> and covariance <math>\mathbf{K}\sigma_g^2</math> where the matrix <math>\mathbf{K}</math> summarizes kinship coefficients between sampled individuals and  <math>\sigma_g^2</math> is a positive scalar describing the genetic contribution to the overall variance. We assume that non-shared environmental effects are normally distributed with mean <math>\mathbf{0}</math> and covariance <math>\mathbf{I}\sigma_e^2</math>, where <math>\mathbf{I}</math> is the identity matrix.

We assume that genetic effects are normally distributed, with mean <math>\mathbf{0}</math> and covariance <math>\mathbf{K}\sigma_g^2</math> where the matrix <math>\mathbf{K}</math> summarizes kinship coefficients between sampled individuals and  <math>\sigma_g^2</math> is a positive scalar describing the genetic contribution to the overall variance. We assume that non-shared environmental effects are normally distributed with mean <math>\mathbf{0}</math> and covariance <math>\mathbf{I}\sigma_e^2</math>, where <math>\mathbf{I}</math> is the identity matrix.

To estimate <math>\mathbf{K}</math>, we either use known pedigree structure to define <math>\mathbf{K}</math> or else use the empirical estimator <math>\mathbf{K}={1\over l} ∑_(i=1)^l{(G_i-2f_i\mathbf{1})(G_i-2f_i\mathbf{1})\over 4f_i(1-f_i)} </math>,  

To estimate <math>\mathbf{K}</math>, we either use known pedigree structure to define <math>\mathbf{K}</math> or else use the empirical estimator <math>\mathbf{K}=1/l </math> ∑_(i=1)^l{(G_i-2f_i\mathbf{1})(G_i-2f_i\mathbf{1})\over 4f_i(1-f_i)} </math>,  

where l is the count of variants, G_i and f_i are the genotype vector and estimated allele frequency for the i^(th) variant, respectively. Each element in G_i encodes the minor allele count for one individual. Model parameters β ̂, (σ_g^2 ) ̂ and (σ_e^2 ) ̂, are estimated using maximum likelihood and the efficient algorithm described in Lippert et. al34. For convenience, let the estimated covariance matrix of y be Ω ̂=2(σ_g^2 ) ̂K ̂+(σ_e^2 ) ̂I.

where l is the count of variants, G_i and f_i are the genotype vector and estimated allele frequency for the i^(th) variant, respectively. Each element in G_i encodes the minor allele count for one individual. Model parameters β ̂, (σ_g^2 ) ̂ and (σ_e^2 ) ̂, are estimated using maximum likelihood and the efficient algorithm described in Lippert et. al34. For convenience, let the estimated covariance matrix of y be Ω ̂=2(σ_g^2 ) ̂K ̂+(σ_e^2 ) ̂I.