# Changes

We assume that genetic effects are normally distributed, with mean $\mathbf{0}$ and covariance $\mathbf{K}\sigma_g^2$ where the matrix $\mathbf{K}$ summarizes kinship coefficients between sampled individuals and $\sigma_g^2$ is a positive scalar describing the genetic contribution to the overall variance. We assume that non-shared environmental effects are normally distributed with mean $\mathbf{0}$ and covariance $\mathbf{I}\sigma_e^2$, where $\mathbf{I}$ is the identity matrix.
To estimate $\mathbf{K}$, we either use known pedigree structure to define $\mathbf{K}$ or else use the empirical estimator  $\mathbf{K}=\frac{1}{l}\sum_{i=1}^l{(G_i-2f_i\mathbf{1})(G_i-2f_i\mathbf{1})\over 4f_i(1-f_i)}$,
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 $\hat{\boldsymbol{\beta}}$, $\hat{\sigma_g^2}$ and $\hat{\sigma_e^2}$, are estimated using maximum likelihood and the efficient algorithm described in Lippert et. al. For convenience, let the estimated covariance matrix of $\mathbf{y}$ be $\hat{\boldsymbol{\Omega}}=2\hat{\sigma_g^2}\mathbf{K}+\hat{\sigma_e^2}\mathbf{I}$.