Changes

<math>\mathbf{y}=\mathbf{X}\beta +\mathbf{g}+ \boldsymbol{\varepsilon}</math>

<math>\mathbf{y}=\mathbf{X}\beta +\mathbf{g}+ \boldsymbol{\varepsilon}</math>

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 '''K''' 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.

 

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.

== Single Variant Score Tests ==

== Single Variant Score Tests ==