# RAREMETAL METHOD

## INTRODUCTION

The key idea behind meta-analysis with RAREMETAL is that various gene-level test statistics can be reconstructed from single variant score statistics and that, when the linkage disequilibrium relationships between variants are known, the distribution of these gene-level statistics can be derived and used to evaluate signifi-cance. Single variant statistics are calculated using the Cochran-Mantel-Haenszel method. The main formulae are tabulated in the following:

## KEY FORMULAE

### NOTATIONS

We denote the following to describe our methods:

$U_{i,k}$ is the score statistic for the $i^{th}$ variant from the $k^{th}$ study

$V_{ij,k}$ is the covariance of the score statistics between the $i^{th}$ and the $j^{th}$ variant from the $k^{th}$ study

$U_{i,k}$ and $V_{ij,k}$ are described in detail in RAREMETALWORKER method.

$\mathbf{U_k}$ is the vector of score statistics of rare variants in a gene from the $k^{th}$ study.

$\mathbf{V_k}$ is the variance-covariance matrix of score statistics of rare variants in a gene from the $k^{th}$ study, or $\mathbf{V_k} = cov(\mathbf{U_k})$

$S$ is the number of studies

$\mathbf{w^T} = (w_1,w_2,...,w_m)^T$ is the vector of weights for $m$ rare variants in a gene.

### SINGLE VARIANT META ANALYSIS

Single variant meta-analysis score statistic can be reconstructed from score statistics and their variances generated by each study, assuming that samples are unrelated across studies. Define meta-analysis score statistics as

$U_{meta_i}=\sum_{k=1}^S {U_{i,k}}$

and its variance

$V_{meta_i}=\sum_{k=1}^S{V_{ii,k}}$

Then the score test statistics for the $i^{th}$ variant $T_{meta_i}$ asymptotically follows standard normal distribution

$T_{meta_i}=U_{meta_i}\bigg/\sqrt{V_{meta_i}}=\sum_{k=1}^S {U_{i,k}}\bigg/\sqrt{\sum_{k=1}^S{V_{ii,k}}} \sim\mathbf{N}(0,1)$

### BURDEN META ANALYSIS

Burden test has been shown to be powerful detecting a group of rare variants that are unidirectional in effects. Once single variant meta analysis statistics are constructed, burden test score statistic can be easily reconstructed as

$T_{meta_{burden}}=\mathbf{w^TU_{meta}}\bigg/\sqrt{\mathbf{w^TV_{meta}w}} \sim\mathbf{N}(0,1)$.

### VT META ANALYSIS

Including variants that are not associated to phenotype can hurt power. Variable threshold test is designed to choose the optimal allele frequency threshold amongst rare variants in a gene, to gain power. The test statistic is defined as the maximum burden score statistic calculated using every possible frequency threshold

$T_{VT}=\max(T_{b\left(f_1\right)},T_{b\left(f_2\right)},\dots,T_{b\left(f_m\right)})$,

where the burden test statistic under any allele frequency threshold can be constructed from single variant meta-analysis statistics using

$T_{b\left(f_j\right)}=\boldsymbol{\phi}_{f_j}^\mathbf{T}\mathbf{U_{meta}}\bigg/\sqrt{\boldsymbol{\phi}_{f_j}^\mathbf{T}\mathbf{V_{meta}}\boldsymbol{\phi}_{f_j}}$,

where $j$ represents any allele frequency in a group of rare variants, $\boldsymbol{\phi}_{f_j}$ is a vector of 0 and 1, indicating if a variant is included in the analysis using frequency threshold $f_i$.

As described by Lin et. al, the p-value of this test can be calculated analytically using the fact that the burden test statistics together follow a multivariate normal distribution with mean $\mathbf{0}$ and covariance $\boldsymbol{\Omega}$

$\left(T_{b\left(f_1\right)},T_{b\left(f_2\right)},\dots,T_{b\left(f_m\right)}\right)$$\sim\mathbf{MVN}\left(\mathbf{0},\boldsymbol{\Omega}\right)$,

where $\boldsymbol{\Omega_{ij}}=\frac{\boldsymbol{\phi}_{f_i}^T\left(\sum_{i=1}^n{\mathbf{V_i}}\right)\boldsymbol{\phi}_{f_j}}{\sqrt{\boldsymbol{\phi}_{f_i}^T\left(\sum_{i=1}^n{\mathbf{V_i}}\right)\boldsymbol{\phi}_{f_i}}\sqrt{\boldsymbol{\phi}_{f_j}^T\left(\sum_{i=1}^n{\mathbf{V_i}}\right)\boldsymbol{\phi}_{f_j}}}$

### SKAT META ANALYSIS

SKAT is most powerful when detecting genes with rare variants having opposite directions in effect sizes. Meta-analysis statistic can also be re-constructed using single variant meta-analysis scores and their covariances

$\mathbf{Q}=\mathbf{{U_{meta}}^T}\mathbf{W}\mathbf{U_{meta}}$,

where $\mathbf{W}$ is a diagonal matrix of weights of rare variants included in a gene.

As shown in Wu et. al, the null distribution of the $\mathbf{Q}$ statistic follows a mixture chi-sqaured distribution described as

$\mathbf{Q}\sim\sum_{i=1}^m{\lambda_i\chi_{1,i}^2},$ where $\left(\lambda_1,\lambda_2,\dots,\lambda_m\right)$ are eigen values of $\mathbf{V_{meta}^\frac{1}{2}}\mathbf{W}\mathbf{V_{meta}^\frac{1}{2}}$.

Formulae for RAREMETAL
Test Statistics Null Distribution Notation
Single Variant $T=\sum_{i=1}^n {U_i}\bigg/\sqrt{\sum_{i=1}^n{V_i}}$ $T\sim\mathbf{N}(0,1)$ $U_i \text{ is the score statistic from study }i;$$V_i \text{ is the variance of } U_i.$
un-weighted Burden $T_b=\sum_{i=1}^n{\mathbf{U_i}}\Big/\sqrt{\sum_{i=1}^n{\mathbf{V_i}}}$ $T_b\sim\mathbf{N}(0,1)$ $\mathbf{U_i}\text{ is the vector of score statistics from study }i, or$ $\mathbf{U_i}=\{U_{i1},...,U_{im}\};$ $\mathbf{V_i} \text{ is the covariance of } \mathbf{U_i}.$
Weighted Burden $T_{wb}=\mathbf{w^T}\sum_{i=1}^n{\mathbf{U_i}}\bigg/\sqrt{\mathbf{w^T}\left(\sum_{i=1}^n{\mathbf{V_i}}\right)\mathbf{w}}$ $T_{wb}\sim\mathbf{N}(0,1)$ $\mathbf{w^T}=\{w_1,w_2,...,w_m\}^T \text{ is the weight vector.}$
VT $T_{VT}=\max(T_{b\left(f_1\right)},T_{b\left(f_2\right)},\dots,T_{b\left(f_m\right)}),\text{ where}$$T_{b\left(f_j\right)}=\boldsymbol{\phi}_{f_j}^\mathbf{T}\sum_{i=1}^n{\mathbf{U_i}}\bigg/\sqrt{\boldsymbol{\phi}_{f_j}^\mathbf{T}\left(\sum_{i=1}^n{\mathbf{V_i}}\right)\boldsymbol{\phi}_{f_j}}$ $\left(T_{b\left(f_1\right)},T_{b\left(f_2\right)},\dots,T_{b\left(f_m\right)}\right)$$\sim\mathbf{MVN}\left(\mathbf{0},\boldsymbol{\Omega}\right)\text{,}$$\text{where }\boldsymbol{\Omega_{ij}}=\frac{\boldsymbol{\phi}_{f_i}^T\left(\sum_{i=1}^n{\mathbf{V_i}}\right)\boldsymbol{\phi}_{f_j}}{\sqrt{\boldsymbol{\phi}_{f_i}^T\left(\sum_{i=1}^n{\mathbf{V_i}}\right)\boldsymbol{\phi}_{f_i}}\sqrt{\boldsymbol{\phi}_{f_j}^T\left(\sum_{i=1}^n{\mathbf{V_i}}\right)\boldsymbol{\phi}_{f_j}}}$ $\boldsymbol{\phi}_{f_j}\text{ is a vector of } 0 \text{s and } 1\text{s,}$ $\text{indicating the inclusion of a variant using threshold }f_j;$
SKAT $\mathbf{Q}=\left(\sum_{i=1}^n{\mathbf{U_i^T}}\right) \mathbf{W}\left(\sum_{i=1}^n{\mathbf{U_i}}\right)$ $\mathbf{Q}\sim\sum_{i=1}^m{\lambda_i\chi_{1,i}^2},\text{ where}$ $\left(\lambda_1,\lambda_2,\dots,\lambda_m\right)\text{ are eigen values of}$$\left(\sum_{i=1}^n{\mathbf{V_i}}\right)^\frac{1}{2}\mathbf{W}\left(\sum_{i=1}^n{\mathbf{V_i}}\right)^\frac{1}{2}$ $\mathbf{W}\text{ is a diagonal matrix of weights.}$