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}=\frac{1}{l}\sum_{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}=\frac{1}{l}\sum_{i=1}^l{(G_i-2f_i\mathbf{1})(G_i-2f_i\mathbf{1})\over 4f_i(1-f_i)} </math>,  

 

where <math>l</math> is the count of variants, <math>G_i</math> and <math>f_i</math> are the genotype vector and estimated allele frequency for the <math>i^{th}</math> variant, respectively. Each element in <math>G_i</math> encodes the minor allele count for one individual. Model parameters <math>\hat{\boldsymbol{\beta}}</math>, <math>\hat{\sigma_g^2}</math> and <math>\hat{\sigma_e^2}</math>, are estimated using maximum likelihood and the efficient algorithm described in Lippert et. al. For convenience, let the estimated covariance matrix of <math>\mathbf{y}</math> be <math>\hat{\boldsymbol{\Omega}}=2\hat{\sigma_g^2}\mathbf{K}+\hat{\sigma_e^2}\mathbf{I}</math>.

where <math>l</math> is the count of variants, <math>G_i</math> and <math>f_i</math> are the genotype vector and estimated allele frequency for the <math>i^{th}</math> variant, respectively. Each element in <math>G_i</math> encodes the minor allele count for one individual. Model parameters <math>\hat{\boldsymbol{\beta}}</math>, <math>\hat{\sigma_g^2}</math> and <math>\hat{\sigma_e^2}</math>, are estimated using maximum likelihood and the efficient algorithm described in Lippert et. al. For convenience, let the estimated covariance matrix of <math>\mathbf{y}</math> be <math>\hat{\boldsymbol{\Omega}}=2\hat{\sigma_g^2}\mathbf{K}+\hat{\sigma_e^2}\mathbf{I}</math>.