# Changes

## RAREMETALWORKER METHOD

, 17:49, 16 March 2018
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[[Category:RAREMETALWORKER]]

Here are some useful links to key pages:
* The [[RAREMETALWORKER | '''RAREMETALWORKER documentation''']]
* The [[RAREMETALWORKER_command_reference | '''RAREMETALWORKER command reference''']]
* The [[RAREMETALWORKER_SPECIAL_TOPICS | '''RAREMETALWORKER special topics''']]
* The [[Tutorial:_RAREMETAL | '''RAREMETALWORKER quick start tutorial''']]
* The [[RAREMETAL_method | '''RAREMETAL method''']]
* The [[RAREMETAL_FAQ | '''FAQ''']]

== Brief Introduction==
We use the following notations to describe our methods:
$\mathbf{y}$ is the observed phenotype vector $\hat{\boldsymbol{\Omega}}$ estimated covariance matrix of $\mathbf{y}$observed quantitative trait
$\mathbf{X}$ is the design matrix
$\boldsymbol{\varepsilon}$ is the non-shared environmental effects

$\hat{\boldsymbol{\Omega}}$ is the estimated covariance matrix of $\mathbf{y}$

$\mathbf{K}$ is the kinship matrix

$\mathbf{K_X}$ is the kinship matrix of Chromosome X

$\sigma_g^2$ is the genetic component

${{\sigma_g}_X}^2$ is the genetic component for markers on chromosome X

$\sigma_e^2$ is the non-shared-environment component.
===SINGLE VARIANT SCORE TEST===
$\mathbf{y}=\mathbf{X}\boldsymbol{\beta_c}+\beta_i(\mathbf{G_i}-\bar{\mathbf{G_i}})+\mathbf{g}+\boldsymbol{\varepsilon}$.
Here, [explain the formula]quantitive trait for an individual is a sum of covariate effects, additive genetic effect from the $i^{th}$ variant and the polygenic background effects together with non-shared environmental effect.
In this model, $\beta_i$ is to measure the additive genetic effect of the $i^{th}$ variant. As usual, the score statistic for testing $H_0:\beta_i=0$ is:
=== MODELING RELATEDNESS ===
we We use a variance component model to handle familial relationships. The We estimate the variance components under the null model is:
$\mathbf{y}=\mathbf{X}\boldsymbol{\beta} +\mathbf{g}+ \boldsymbol{\varepsilon}$
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 $\hat{\boldsymbol{\Omega}}=\hat{\sigma_g^2}\mathbf{K}+\hat{\sigma_e^2}\mathbf{I}$.
==Chromosome =ANALYZING MARKERS ON 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 $\hat{\boldsymbol{\Omega}}=\hat{\sigma_g^2}\mathbf{K}+\hat{{\sigma_g}_X^2}\mathbf{K_X}+\hat{\sigma_e^2}\mathbf{I}$.
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