RAREMETALWORKER METHOD

Brief Introduction

RAREMETALWORKER(RMW) generates single variant association results from score test, together with summary statistics and covariance matrices of the score statistics.

In the following sections, we will go through the methods behind RWM including statistic model, handling sample relatedness, and definitions of statistics in the output.

Modeling Relatedness

we use a variance component model to handle familial relationships. In a sample of n individuals, we model the observed phenotype vector ($\mathbf{𝐲}$) as a sum of covariate effects (specified by a design matrix $\mathbf{𝐗}$ and a vector of covariate effects $\mathit{\beta }$), additive genetic effects (modeled in vector $\mathbf{𝐠}$) and non-shared environmental effects (modeled in vector $\mathit{\epsilon }$). Thus 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 $\mathrm{\Omega }=2{\sigma }_g^2\mathbf{𝐊}+{\sigma }_e^2\mathbf{𝐈}$.

Single Variant Score Tests

Our single variant association test is the score test using linear mixed model, treating single variants as fixed effects. The alternative model is:

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

In this model, the scalar parameter ${\gamma }_i$ is to measure the additive genetic effect of the $i^{th}$ variant. As usual, the score statistic for testing $H_0:{\gamma }_i=0$ is:

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

And the variance-covariance matrix of these statistics is (see Appendix A for details):

                       V=(G-G ̅ )^T (Ω ̂^(-1)-Ω ̂^(-1) X(X^T Ω ̂^(-1) X)^(-1) X^T Ω ̂^(-1) )(G-G ̅).

Under the null, test statistics T_i=(U_i^2)/V_ii is asymptotically distributed as chi-squared with one degree of freedom.