# RAREMETALWORKER METHOD

Here are some useful links to key pages:

## 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 $\hat{\boldsymbol{\Omega}}=\hat{\sigma_g^2}\mathbf{K}+\hat{{\sigma_g}_X^2}\mathbf{K_X}+\hat{\sigma_e^2}\mathbf{I}$.