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==Useful Links==
Here are some useful links to key pages:
* The [[RAREMETALWORKER | '''RAREMETALWORKER documentation''']]
* The [[RAREMETALWORKER_command_reference | '''RAREMETALWORKER command reference''']]
* 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:
<math>\mathbf{y}</math> is the vector of observed phenotype vectorquantitative trait
<math>\mathbf{X}</math> is the design matrix
<math>\boldsymbol{\varepsilon}</math> is the non-shared environmental effects
<math> \hat{\boldsymbol{\Omega}} </math> is the estimated covariance matrix of <math>\mathbf{y}</math>
<math>\mathbf{K}</math> is the kinship matrix
<math>\mathbf{K_X}</math> is the kinship matrix of Chromosome X
<math> \sigma_g^2 </math> is the genetic component
<math> {{\sigma_g}_X}^2 </math> is the genetic component for markers on chromosome X
<math>\sigma_e^2 </math> is the non-shared-environment component.
<math> \mathbf{y}=\mathbf{X}\boldsymbol{\beta_c}+\beta_i(\mathbf{G_i}-\bar{\mathbf{G_i}})+\mathbf{g}+\boldsymbol{\varepsilon} </math>.
Here, [explain the formula]quantitive trait for an individual is a sum of covariate effects, additive genetic effect from the <math> i^{th} </math> variant and the polygenic background effects together with non-shared environmental effect.
In this model, <math>\beta_i</math> is to measure the additive genetic effect of the <math>i^{th}</math> variant. As usual, the score statistic for testing <math>H_0:\beta_i=0</math> is:
RAREMETALWORKER also stores the covariance matrices (<math> \mathbf{V} </math>) of the score statistics of markers within a window, size of which can be specified through command line.
== Modeling Relatedness = MODELING RELATEDNESS ===we We use a variance component model to handle familial relationships. The We estimate the variance components under the null model is:
<math>\mathbf{y}=\mathbf{X}\boldsymbol{\beta} +\mathbf{g}+ \boldsymbol{\varepsilon}</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}}=\hat{\sigma_g^2}\mathbf{K}+\hat{\sigma_e^2}\mathbf{I}</math>.
To analyze markers on chromosome X, we fit an extra variance components <math> {{\sigma_g}_X}^2 </math>, to model the variance explained by chromosome X. A kinship for chromosome X, <math> \boldsymbol{K_X} </math>, can be estimated either from a pedigree, or from genotypes of marker from chromosome X. Then the estimated covariance matrix can be written as <math>\hat{\boldsymbol{\Omega}}=\hat{\sigma_g^2}\mathbf{K}+\hat{{\sigma_g}_X^2}\mathbf{K_X}+\hat{\sigma_e^2}\mathbf{I}</math>.

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