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[[Category:RAREMETALWORKER]]
==Useful Links==
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==
== Brief Introduction==
== Modeling Relatedness ==
[[RAREMETALWORKER]] generates single variant association test statistics for a single study prior to meta-analysis. This page provides a brief description of the statistics that
RAREMETALWORKER calculates, together with key formulae.
== Key Statistics for Analysis of Single Study ==
===NOTATIONS===
We use the following notations to describe our methods:
<math>\mathbf{y}</math> is the vector of observed quantitative trait
<math>\mathbf{X}</math> is the design matrix
<math>\mathbf{G_i}</math> is the genotype vector of the <math>i^{th}</math> variant
<math> \bar{\mathbf{G_i}}</math> is the vector of average genotype of the <math>i^{th}</math> variant
<math>\boldsymbol{\beta_c}</math> is the vector of covariate effects
<math>\beta_i</math> is the scalar of fixed genetic effect of the <math>i^{th}</math> variant
<math>\mathbf{g}</math> is the random genetic effects
<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.
===SINGLE VARIANT SCORE TEST===
We used the following model for the trait:
<math> \mathbf{y}=\mathbf{X}\boldsymbol{\beta_c}+\beta_i(\mathbf{G_i}-\bar{\mathbf{G_i}})+\mathbf{g}+\boldsymbol{\varepsilon} </math>.
Here, the 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:
<math> U_i=(\mathbf{G_i}-\mathbf{\bar{G_i}} )^T \hat{\boldsymbol{\Omega}}^{-1}(\mathbf{y}-\mathbf{X}\boldsymbol{\beta}) </math>
We further derive the variance-covariance matrix of these statistics as
<math> \mathbf{V}=(\mathbf{G}-\bar{\mathbf{G}})^T (\hat{\boldsymbol{\Omega}}^{-1}-\hat{\boldsymbol{\Omega}}^{-1} \mathbf{X}(\mathbf{X^T}\hat{\boldsymbol{\Omega}}^{-1}\mathbf{X})^{-1} \mathbf{X^T} \hat{\boldsymbol{\Omega}}^{-1})(\mathbf{G}-\bar{\mathbf{G}}) </math>.
The score test statistic, <math>T_i=(U_i^2)/V_{ii}</math>, is asymptotically distributed as chi-squared with one degree of freedom. The score test p-value is reported in RAREMETALWORKER.
===SUMMARY STATISTICS AND COVARIANCE MATRICES===
RAREMETALWORKER automatically stores the score statistics for each marker ( <math> U_i </math>) together with quality information of that marker, including HWE p-value, call rate, and allele counts.
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 ===
We use a variance component model to handle familial relationships. We estimate the variance components under the null model:
<math>\mathbf{y}=\mathbf{X}\boldsymbol{\beta} +\mathbf{g}+ \boldsymbol{\varepsilon}</math>
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>,
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 [http://www.nature.com/nmeth/journal/v8/n10/full/nmeth.1681.html 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>.
===ANALYZING MARKERS ON CHROMOSOME X===
== Modeling Relatedness ==
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>.
