# 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

We use the following notations to describe our methods:

is the observed phenotype vector

is the design matrix

is the vector of covariate effects

is the scalar of fixed genetic effect of the variant

is the random genetic effects

is the non-shared environmental effects

## Single Variant Score Tests

We used the following model for the trait:

.

Here, [explain the formula].

In this model, is to measure the additive genetic effect of the variant. As usual, the score statistic for testing is:

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

.

Under the null, test statistics is asymptotically distributed as chi-squared with one degree of freedom.

## Summary Statistics and Covariance Matrices

RAREMETALWORKER automatically stores the score statistics for each marker ( ) together with quality information of that marker, including HWE p-value, call rate, and allele counts.

RAREMETALWORKER also stores the covariance matrices () of the score statistics of markers within a window.

## Modeling Relatedness

we use a variance component model to handle familial relationships. In a sample of n individuals, we model the observed phenotype vector () as a sum of covariate effects (specified by a design matrix and a vector of covariate effects ), additive genetic effects (modeled in vector ) and non-shared environmental effects (modeled in vector ). Thus the null model is:

We assume that genetic effects are normally distributed, with mean and covariance where the matrix summarizes kinship coefficients between sampled individuals and is a positive scalar describing the genetic contribution to the overall variance. We assume that non-shared environmental effects are normally distributed with mean and covariance , where is the identity matrix.

To estimate , we either use known pedigree structure to define or else use the empirical estimator , where is the count of variants, and are the genotype vector and estimated allele frequency for the variant, respectively. Each element in encodes the minor allele count for one individual. Model parameters , and , are estimated using maximum likelihood and the efficient algorithm described in Lippert et. al. For convenience, let the estimated covariance matrix of be .

## Chromosome X

To analyze markers on chromosome X, we fit an extra variance components , to model the variance explained by chromosome X. A kinship for chromosome 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 .