# Changes

,  13:29, 1 April 2014
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$\mathbf{K}=\frac{1}{l}\sum_{i=1}^l{(G_i-2f_i\mathbf{1})(G_i-2f_i\mathbf{1})\over 4f_i(1-f_i)}$,

$\mathbf{K}=\frac{1}{l}\sum_{i=1}^l{(G_i-2f_i\mathbf{1})(G_i-2f_i\mathbf{1})\over 4f_i(1-f_i)}$,
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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 $\hat{\boldsymbol{\Omega}}=2\hat{\sigma_g^2}\mathbf{K}+\hat{\sigma_e^2}\mathbf{I}$.
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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 [http://www.nature.com/nmeth/journal/v8/n10/full/nmeth.1681.html Lippert et. al]. For convenience, let the estimated covariance matrix of $\mathbf{y}$ be $\hat{\boldsymbol{\Omega}}=2\hat{\sigma_g^2}\mathbf{K}+\hat{\sigma_e^2}\mathbf{I}$.

==Chromosome X==

==Chromosome X==

To analyze markers on chromosome X, we fit an extra variance components ${{\sigma_g}_X}^2$, to model the variance explained by chromosome X. A kinship for chromosome X, $\boldsymbol{K_X}$, can be estimated either from a pedigree, or from genotypes of marker from chromosome X. Then the estimated covariance matrix can be written as $\hat{\boldsymbol{\Omega}}=2\hat{\sigma_g^2}\mathbf{K}+2\hat{{\sigma_g}_X^2}\mathbf{K_X}+\hat{\sigma_e^2}\mathbf{I}$.

To analyze markers on chromosome X, we fit an extra variance components ${{\sigma_g}_X}^2$, to model the variance explained by chromosome X. A kinship for chromosome X, $\boldsymbol{K_X}$, can be estimated either from a pedigree, or from genotypes of marker from chromosome X. Then the estimated covariance matrix can be written as $\hat{\boldsymbol{\Omega}}=2\hat{\sigma_g^2}\mathbf{K}+2\hat{{\sigma_g}_X^2}\mathbf{K_X}+\hat{\sigma_e^2}\mathbf{I}$.
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