# Changes

## RAREMETAL METHOD

, 13:28, 20 May 2019
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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. Our method has been published in [http://www.ncbi.nlm.nih.gov/pubmed/24336170 '''Liu et. al''']. The main formulae are tabulated in the following:
==KEY FORMULAE==
$S$ 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.
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 $T_{meta_{burden}}=\mathbf{w^TU_{meta}}\bigg/\sqrt{\mathbf{w^TV_{meta}w}} \sim\mathbf{N}(0,1)$,
where $T_\mathbf{meta_U_{burdenmeta}}=\mathbf(U_{w^TU_meta_1},U_{metameta_2},...,U_{meta_m}\bigg)^T</\sqrt{math> and [itex] \mathbf{w^TV_V_{meta}w}} \sim=cov(\mathbf{NU_{meta}(0})$,1)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
$T_{VT}=\max(T_{b\left(f_1\right)},T_{b\left(f_2\right)},\dots,T_{b\left(f_m\right)})$,
$T_{meta_{VT}}=\max(T_{b\left(f_1\right)},T_{b\left(f_2\right)},\dots,T_{b\left(f_m\right)})$, where $T_{b\left(f_i\right)}$ is the burden test statistic under any 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 $\boldsymbol{\Omega}$
As described by [http://www.ncbi.nlm.nih.gov/pubmed/21885029 '''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)$$\sim\mathbf{MVN}\left(\mathbf{0},\boldsymbol{\Omega}\right)$,
where $\boldsymbol{\Omega_{ij}}=\frac{\boldsymbol{\phi}_{f_i}^T\left(\sum_{i=1}^nmathbf{\mathbfV_{V_imeta}}\right)\boldsymbol{\phi}_{f_j}}{\sqrt{\boldsymbol{\phi}_{f_i}^T\left(\sum_{i=1}^nmathbf{\mathbfV_{V_imeta}}\right)\boldsymbol{\phi}_{f_i}}\sqrt{\boldsymbol{\phi}_{f_j}^T\left(\sum_{i=1}^nmathbf{\mathbfV_{V_imeta}}\right)\boldsymbol{\phi}_{f_j}}}$.
===SKAT META ANALYSIS===
{| border="1" cellpadding="5" cellspacing="0" align="center"|+'''Formulae for [[Category:RAREMETAL'''! scope="col" width="120pt" | Test! scope="col" width="50pt" | Statistics! scope="col" width="225pt" | Null Distribution! scope="col" width="225pt" | Notation|-| Single Variant || $T=\sum_{i=1}^n {U_i}\bigg/\sqrt{\sum_{i=1}^n{V_i}}$ || $T\sim\mathbf{N}(0,1)$ ||$U_i \text{ is the score statistic from study }i;$$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 || $T_{wb}=\mathbf{w^T}\sum_{i=1}^n{\mathbf{U_i}}\bigg/\sqrt{\mathbf{w^T}\left(\sum_{i=1}^n{\mathbf{V_i}}\right)\mathbf{w}}$ || $T_{wb}\sim\mathbf{N}(0,1)$ || $\mathbf{w^T}=\{w_1,w_2,...,w_m\}^T \text{ is the weight vector.}$|-style="height: 50pt;"| 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}$ || $\mathbf{W}\text{ is a diagonal matrix of weights.}$|}]]
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