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. Our method has been published in Liu et. al. 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

$f_{i}$ is the pooled allele frequency of $i^{th}$ variant

$f_{i,k}$ is the allele frequency of $i^{th}$ variant in $k^{th}$ study

${\delta _{k}}$ is the deviation of trait value of $k^{th}$ study

$\mathbf {w^{T}} =(w_{1},w_{2},...,w_{m})^{T}$ 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

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

Optimized method for unbalanced studies (--useExact):

$U_{meta_{i}}=\sum _{k=1}^{S}{U_{i,k}/{\hat {\Omega _{k}}}}-\sum _{k=1}^{S}{2n_{k}{\delta _{k}^{2}(f_{i}-f_{i,k})}}$

$V_{meta_{i}}={\sigma ^{2}}\sum _{k=1}^{S}{(V_{ii,k}{\Omega _{k}}-4n_{k}(ff'-f_{k}f_{k}'))}$

${\sigma ^{2}}=\sum _{k=1}^{S}{((n_{k}-1){\Omega _{k}}+n_{k}{\delta _{k}^{2}})}/(n-1)$

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 for a gene can be easily reconstructed as

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

where $\mathbf {U_{meta}} =(U_{meta_{1}},U_{meta_{2}},...,U_{meta_{m}})^{T}$ and $\mathbf {V_{meta}} =cov(\mathbf {U_{meta}} )$, representing a vector of single variant meta-analysis scores of $m$ variants in a gene and the covariance matrix of the scores across $m$ variants.

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

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

where $T_{b\left(f_{i}\right)}$ is the burden test statistic under allele frequency threshold $f_{i}$, and 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 ${\boldsymbol {\Omega }}$, written as

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

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

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

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 $\mathbf {V_{meta}^{\frac {1}{2}}} \mathbf {W} \mathbf {V_{meta}^{\frac {1}{2}}}$.