RAREMETALWORKER METHOD

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Brief Introduction

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:

\mathbf{y} is the vector of observed quantitative trait

\mathbf{X} is the design matrix

\mathbf{G_i} is the genotype vector of the i^{th} variant

 \bar{\mathbf{G_i}} is the vector of average genotype of the i^{th} variant

\boldsymbol{\beta_c} is the vector of covariate effects

\beta_i is the scalar of fixed genetic effect of the i^{th} variant

\mathbf{g} is the random genetic effects

\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

We used the following model for the trait:

 \mathbf{y}=\mathbf{X}\boldsymbol{\beta_c}+\beta_i(\mathbf{G_i}-\bar{\mathbf{G_i}})+\mathbf{g}+\boldsymbol{\varepsilon} .

Here, the 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:

 U_i=(\mathbf{G_i}-\mathbf{\bar{G_i}} )^T \hat{\boldsymbol{\Omega}}^{-1}(\mathbf{y}-\mathbf{X}\boldsymbol{\beta})

We further derive the variance-covariance matrix of these statistics as

 \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}}) .

The score test statistic, T_i=(U_i^2)/V_{ii}, 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 (  U_i ) together with quality information of that marker, including HWE p-value, call rate, and allele counts.

RAREMETALWORKER also stores the covariance matrices ( \mathbf{V} ) 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:

\mathbf{y}=\mathbf{X}\boldsymbol{\beta} +\mathbf{g}+ \boldsymbol{\varepsilon}


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}}=\hat{\sigma_g^2}\mathbf{K}+\hat{\sigma_e^2}\mathbf{I}.

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}.