# RAREMETAL METHOD

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## 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:

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

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

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

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

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

${\displaystyle S}$ is the number of studies

${\displaystyle \mathbf {w^{T}} =(w_{1},w_{2},...,w_{m})^{T}}$ is the vector of weights for ${\displaystyle m}$ 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

${\displaystyle U_{meta_{i}}=\sum _{k=1}^{S}{U_{i,k}}}$

and its variance

${\displaystyle V_{meta_{i}}=\sum _{k=1}^{S}{V_{ii,k}}}$

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

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

### 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

${\displaystyle T_{meta_{burden}}=\mathbf {w^{T}U_{meta}} {\bigg /}{\sqrt {\mathbf {w^{T}V_{meta}w} }}\sim \mathbf {N} (0,1)}$.

### 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

${\displaystyle T_{meta_{VT}}=\max(T_{b\left(f_{1}\right)},T_{b\left(f_{2}\right)},\dots ,T_{b\left(f_{m}\right)})}$,

where $\displaystyle T_{b}_{f_{i}}$ is the burden test statistic under allele frequency threshold ${\displaystyle f_{i}}$, and can be constructed from single variant meta-analysis statistics using

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

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

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 ${\displaystyle \mathbf {0} }$ and covariance ${\displaystyle {\boldsymbol {\Omega }}}$, written as

${\displaystyle \left(T_{b\left(f_{1}\right)},T_{b\left(f_{2}\right)},\dots ,T_{b\left(f_{m}\right)}\right)}$${\displaystyle \sim \mathbf {MVN} \left(\mathbf {0} ,{\boldsymbol {\Omega }}\right)}$,

where ${\displaystyle {\boldsymbol {\Omega _{ij}}}={\frac {{\boldsymbol {\phi }}_{f_{i}}^{T}\mathbf {V_{meta}} {\boldsymbol {\phi }}_{f_{j}}}{{\sqrt {{\boldsymbol {\phi }}_{f_{i}}^{T}\mathbf {V_{meta}} {\boldsymbol {\phi }}_{f_{i}}}}{\sqrt {{\boldsymbol {\phi }}_{f_{j}}^{T}\mathbf {V_{meta}} {\boldsymbol {\phi }}_{f_{j}}}}}}}$

### 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

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

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

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

${\displaystyle \mathbf {Q} \sim \sum _{i=1}^{m}{\lambda _{i}\chi _{1,i}^{2}},}$ where ${\displaystyle \left(\lambda _{1},\lambda _{2},\dots ,\lambda _{m}\right)}$ are eigen values of ${\displaystyle \mathbf {V_{meta}^{\frac {1}{2}}} \mathbf {W} \mathbf {V_{meta}^{\frac {1}{2}}} }$.

Formulae for RAREMETAL
Test Statistics Null Distribution Notation
Single Variant ${\displaystyle T=\sum _{i=1}^{n}{U_{i}}{\bigg /}{\sqrt {\sum _{i=1}^{n}{V_{i}}}}}$ ${\displaystyle T\sim \mathbf {N} (0,1)}$ ${\displaystyle U_{i}{\text{ is the score statistic from study }}i;}$${\displaystyle V_{i}{\text{ is the variance of }}U_{i}.}$
un-weighted Burden ${\displaystyle T_{b}=\sum _{i=1}^{n}{\mathbf {U_{i}} }{\Big /}{\sqrt {\sum _{i=1}^{n}{\mathbf {V_{i}} }}}}$ ${\displaystyle T_{b}\sim \mathbf {N} (0,1)}$ ${\displaystyle \mathbf {U_{i}} {\text{ is the vector of score statistics from study }}i,or}$ ${\displaystyle \mathbf {U_{i}} =\{U_{i1},...,U_{im}\};}$ ${\displaystyle \mathbf {V_{i}} {\text{ is the covariance of }}\mathbf {U_{i}} .}$
Weighted Burden ${\displaystyle T_{wb}=\mathbf {w^{T}} \sum _{i=1}^{n}{\mathbf {U_{i}} }{\bigg /}{\sqrt {\mathbf {w^{T}} \left(\sum _{i=1}^{n}{\mathbf {V_{i}} }\right)\mathbf {w} }}}$ ${\displaystyle T_{wb}\sim \mathbf {N} (0,1)}$ ${\displaystyle \mathbf {w^{T}} =\{w_{1},w_{2},...,w_{m}\}^{T}{\text{ is the weight vector.}}}$
VT ${\displaystyle T_{VT}=\max(T_{b\left(f_{1}\right)},T_{b\left(f_{2}\right)},\dots ,T_{b\left(f_{m}\right)}),{\text{ where}}}$${\displaystyle 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}}}}}$ ${\displaystyle \left(T_{b\left(f_{1}\right)},T_{b\left(f_{2}\right)},\dots ,T_{b\left(f_{m}\right)}\right)}$${\displaystyle \sim \mathbf {MVN} \left(\mathbf {0} ,{\boldsymbol {\Omega }}\right){\text{,}}}$${\displaystyle {\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}}}}}}}$ ${\displaystyle {\boldsymbol {\phi }}_{f_{j}}{\text{ is a vector of }}0{\text{s and }}1{\text{s,}}}$ ${\displaystyle {\text{indicating the inclusion of a variant using threshold }}f_{j};}$
SKAT ${\displaystyle \mathbf {Q} =\left(\sum _{i=1}^{n}{\mathbf {U_{i}^{T}} }\right)\mathbf {W} \left(\sum _{i=1}^{n}{\mathbf {U_{i}} }\right)}$ ${\displaystyle \mathbf {Q} \sim \sum _{i=1}^{m}{\lambda _{i}\chi _{1,i}^{2}},{\text{ where}}}$ ${\displaystyle \left(\lambda _{1},\lambda _{2},\dots ,\lambda _{m}\right){\text{ are eigen values of}}}$${\displaystyle \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}}}$ ${\displaystyle \mathbf {W} {\text{ is a diagonal matrix of weights.}}}$