RAREMETALWORKER METHOD

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 observed phenotype vector

$\mathbf{𝐗}$ is the design matrix

${\mathbf{G}}_{\mathbf{𝐢}}$ is the genotype vector of the $i^{th}$ variant

$\overline{{\mathbf{G}}_{\mathbf{𝐢}}}$ is the vector of average genotype of the $i^{th}$ variant

${\mathit{\beta }}_{\mathit{c}}$ 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

$\mathit{\epsilon }$ is the non-shared environmental effects

SINGLE VARIANT SCORE TEST

We used the following model for the trait:

$\mathbf{𝐲}=\mathbf{𝐗}{\mathit{\beta }}_{\mathit{c}}+{\beta }_i({\mathbf{G}}_{\mathbf{𝐢}}-\overline{{\mathbf{G}}_{\mathbf{𝐢}}})+\mathbf{𝐠}+\mathit{\epsilon }$.

Here, [explain the formula].

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:

$U_i=({\mathbf{G}}_{\mathbf{𝐢}}-\overline{{\mathbf{G}}_{\mathbf{i}}}{)}^T{\hat{\mathrm{\Omega }}}^{-1}(\mathbf{𝐲}-\mathbf{𝐗}\mathit{\beta })$

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

$\mathbf{𝐕}=(\mathbf{𝐆}-\overline{\mathbf{𝐆}}{)}^T({\hat{\mathrm{\Omega }}}^{-1}-{\hat{\mathrm{\Omega }}}^{-1}\mathbf{𝐗}({\mathbf{𝐗}}^{\mathbf{𝐓}}{\hat{\mathrm{\Omega }}}^{-1}\mathbf{𝐗}{)}^{-1}{\mathbf{𝐗}}^{\mathbf{𝐓}}{\hat{\mathrm{\Omega }}}^{-1})(\mathbf{𝐆}-\overline{\mathbf{𝐆}})$.

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. The null model is:

$\mathbf{𝐲}=\mathbf{𝐗}\mathit{\beta }+\mathbf{𝐠}+\mathit{\epsilon }$

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

$\mathbf{𝐊}=\frac{1}{l}\sum _{i=1}^l\frac{(G_i-2f_i\mathbf{𝟏})(G_i-2f_i\mathbf{𝟏})}{4f_i(1-f_i)}$,

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 $\hat{\mathit{\beta }}$, $\widehat{{\sigma }_g^2}$ and $\widehat{{\sigma }_e^2}$, 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 $\hat{\mathrm{\Omega }}=\widehat{{\sigma }_g^2}\mathbf{𝐊}+\widehat{{\sigma }_e^2}\mathbf{𝐈}$.

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, ${\mathit{K}}_{\mathit{X}}$, can be estimated either from a pedigree, or from genotypes of marker from chromosome X. Then the estimated covariance matrix can be written as $\hat{\mathrm{\Omega }}=\widehat{{\sigma }_g^2}\mathbf{𝐊}+\widehat{{{\sigma }_g}_X^2}{\mathbf{K}}_{\mathbf{𝐗}}+\widehat{{\sigma }_e^2}\mathbf{𝐈}$.