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.

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

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

$S$ is the number of studies

${\mathbf{𝐰}}^{\mathbf{𝐓}}=(w_1,w_2,...,w_m{)}^T$ 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

$U_{met{a}_i}=\sum _{k=1}^S{U}_{i,k}$

and its variance

$V_{met{a}_i}=\sum _{k=1}^S{V}_{ii,k}$

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

$T_{met{a}_i}=U_{met{a}_i}/\sqrt{V_{met{a}_i}}=\sum _{k=1}^S{U}_{i,k}/\sqrt{\sum _{k=1}^S{V}_{ii,k}}\sim \mathbf{𝐍}(0,1)$

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

$T_{met{a}_{burden}}={\mathbf{𝐰}}^{\mathbf{𝐓}}{\mathbf{U}}_{\mathbf{𝐦}\mathbf{𝐞}\mathbf{𝐭}\mathbf{𝐚}}/\sqrt{{\mathbf{𝐰}}^{\mathbf{𝐓}}{\mathbf{V}}_{\mathbf{𝐦}\mathbf{𝐞}\mathbf{𝐭}\mathbf{𝐚}}\mathbf{𝐰}}\sim \mathbf{𝐍}(0,1)$.

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

$T_{b\left(f_j\right)}={\mathit{\varphi }}_{f_j}^{\mathbf{𝐓}}{\mathbf{U}}_{\mathbf{𝐦}\mathbf{𝐞}\mathbf{𝐭}\mathbf{𝐚}}/\sqrt{{\mathit{\varphi }}_{f_j}^{\mathbf{𝐓}}{\mathbf{V}}_{\mathbf{𝐦}\mathbf{𝐞}\mathbf{𝐭}\mathbf{𝐚}}{\mathit{\varphi }}_{f_j}}$,

where $j$ represents any allele frequency in a group of rare variants, ${\mathit{\varphi }}_{f_j}$ 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 $\mathrm{\Omega }$, written as

$(T_{b\left(f_1\right)},T_{b\left(f_2\right)},\dots ,T_{b\left(f_m\right)})$$\sim \mathbf{𝐌}\mathbf{𝐕}\mathbf{𝐍}(\mathbf{𝟎},\mathrm{\Omega })$,

where ${\mathrm{\Omega }}_{\mathit{i}\mathit{j}}=\frac{{\mathit{\varphi }}_{f_i}^T{\mathbf{V}}_{\mathbf{𝐦}\mathbf{𝐞}\mathbf{𝐭}\mathbf{𝐚}}{\mathit{\varphi }}_{f_j}}{\sqrt{{\mathit{\varphi }}_{f_i}^T{\mathbf{V}}_{\mathbf{𝐦}\mathbf{𝐞}\mathbf{𝐭}\mathbf{𝐚}}{\mathit{\varphi }}_{f_i}}\sqrt{{\mathit{\varphi }}_{f_j}^T{\mathbf{V}}_{\mathbf{𝐦}\mathbf{𝐞}\mathbf{𝐭}\mathbf{𝐚}}{\mathit{\varphi }}_{f_j}}}$

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

$\mathbf{𝐐}={{\mathbf{U}}_{\mathbf{𝐦}\mathbf{𝐞}\mathbf{𝐭}\mathbf{𝐚}}}^{\mathbf{𝐓}}\mathbf{𝐖}{\mathbf{U}}_{\mathbf{𝐦}\mathbf{𝐞}\mathbf{𝐭}\mathbf{𝐚}}$,

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 $({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_m)$ are eigen values of ${𝐕}_{\mathbf{𝐦}\mathbf{𝐞}\mathbf{𝐭}\mathbf{𝐚}}^{\frac{1}{2}}\mathbf{𝐖}{𝐕}_{\mathbf{𝐦}\mathbf{𝐞}\mathbf{𝐭}\mathbf{𝐚}}^{\frac{1}{2}}$.

Formulae for RAREMETAL

Test	Statistics	Null Distribution	Notation
Single Variant	$T=\sum _{i=1}^n{U}_i/\sqrt{\sum _{i=1}^n{V}_i}$	$T\sim \mathbf{𝐍}(0,1)$	$U_i\text{ is the score statistic from study }i;$$V_i\text{ is the variance of }{U}_i.$
un-weighted Burden	$T_b=\sum _{i=1}^n{\mathbf{U}}_{\mathbf{𝐢}}/\sqrt{\sum _{i=1}^n{\mathbf{V}}_{\mathbf{𝐢}}}$	$T_b\sim \mathbf{𝐍}(0,1)$	${\mathbf{U}}_{\mathbf{𝐢}}\text{ is the vector of score statistics from study }i,or$ ${\mathbf{U}}_{\mathbf{𝐢}}=\{U_{i1},...,U_{im}\};$ ${\mathbf{V}}_{\mathbf{𝐢}}\text{ is the covariance of }{\mathbf{U}}_{\mathbf{𝐢}}.$
Weighted Burden	$T_{wb}={\mathbf{𝐰}}^{\mathbf{𝐓}}\sum _{i=1}^n{\mathbf{U}}_{\mathbf{𝐢}}/\sqrt{{\mathbf{𝐰}}^{\mathbf{𝐓}}\left(\sum _{i=1}^n{\mathbf{V}}_{\mathbf{𝐢}}\right)\mathbf{𝐰}}$	$T_{wb}\sim \mathbf{𝐍}(0,1)$	${\mathbf{𝐰}}^{\mathbf{𝐓}}=\{w_1,w_2,...,w_m{\}}^T\text{ is the weight vector.}$
VT	$T_{VT}=\max (T_{b\left(f_1\right)},T_{b\left(f_2\right)},\dots ,T_{b\left(f_m\right)}),\text{ where}$$T_{b\left(f_j\right)}={\mathit{\varphi }}_{f_j}^{\mathbf{𝐓}}\sum _{i=1}^n{\mathbf{U}}_{\mathbf{𝐢}}/\sqrt{{\mathit{\varphi }}_{f_j}^{\mathbf{𝐓}}\left(\sum _{i=1}^n{\mathbf{V}}_{\mathbf{𝐢}}\right){\mathit{\varphi }}_{f_j}}$	$(T_{b\left(f_1\right)},T_{b\left(f_2\right)},\dots ,T_{b\left(f_m\right)})$$\sim \mathbf{𝐌}\mathbf{𝐕}\mathbf{𝐍}(\mathbf{𝟎},\mathrm{\Omega })\text{,}$$\text{where }{\mathrm{\Omega }}_{\mathit{i}\mathit{j}}=\frac{{\mathit{\varphi }}_{f_i}^T\left(\sum _{i=1}^n{\mathbf{V}}_{\mathbf{𝐢}}\right){\mathit{\varphi }}_{f_j}}{\sqrt{{\mathit{\varphi }}_{f_i}^T\left(\sum _{i=1}^n{\mathbf{V}}_{\mathbf{𝐢}}\right){\mathit{\varphi }}_{f_i}}\sqrt{{\mathit{\varphi }}_{f_j}^T\left(\sum _{i=1}^n{\mathbf{V}}_{\mathbf{𝐢}}\right){\mathit{\varphi }}_{f_j}}}$	${\mathit{\varphi }}_{f_j}\text{ is a vector of }0\text{s and }1\text{s,}$ $\text{indicating the inclusion of a variant using threshold }{f}_j;$
SKAT	$\mathbf{𝐐}=\left(\sum _{i=1}^n{𝐔}_{\mathbf{𝐢}}^{\mathbf{𝐓}}\right)\mathbf{𝐖}\left(\sum _{i=1}^n{\mathbf{U}}_{\mathbf{𝐢}}\right)$	$\mathbf{𝐐}\sim \sum _{i=1}^m{\lambda }_i{\chi }_{1,i}^2,\text{ where}$ $({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_m)\text{ are eigen values of}$${\left(\sum _{i=1}^n{\mathbf{V}}_{\mathbf{𝐢}}\right)}^{\frac{1}{2}}\mathbf{𝐖}{\left(\sum _{i=1}^n{\mathbf{V}}_{\mathbf{𝐢}}\right)}^{\frac{1}{2}}$	$\mathbf{𝐖}\text{ is a diagonal matrix of weights.}$