# 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==
$V_{ij,k}$ 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#SINGLE_VARIANT_SCORE_TEST|'''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
$U_f_{i}$ is the pooled allele frequency of $i^{th}$ variant $f_{i,k}$ and is the allele frequency of $i^{th}$ variant in $k^{th}$ study $V_{ij,\delta_{k}}$ is the deviation of trait value of $k^{th}$ are described in detail study $\mathbf{w^T} = (w_1,w_2,...,w_m)^T$ is the vector of weights for $m$ rare variants in [[RAREMETALWORKER_method#SINGLE_VARIANT_SCORE_TEST|'''RAREMETALWORKER method''']]a gene.
===SINGLE VARIANT META ANALYSIS===
Single variant meta-analysis score statistic can be reconstructed from score statistics and their variances generate generated by each study, assuming that samples are unrelated across studies, and asymptotically follows standard normal distribution . Define meta-analysis score statistics as
$T_iU_{meta_i}=\sum_{k=1}^S {U_{i,k}}$ 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

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

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 [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 $\boldsymbol{\Omega_{ij}}=\frac{\boldsymbol{\phi}_{f_i}^T\mathbf{V_{meta}}\boldsymbol{\phi}_{f_j}}{\sqrt{\boldsymbol{\phi}_{f_i}^T\mathbf{V_{meta}}\boldsymbol{\phi}_{f_i}}\sqrt{\boldsymbol{\phi}_{f_j}^T\mathbf{V_{meta}}\boldsymbol{\phi}_{f_j}}}$.
===SKAT META ANALYSIS===
{| border="1" cellpadding="5" cellspacing="0" align="center"|+'''Formulae for RAREMETAL'''! scope="col" width="120pt" | Test! scope="col" width="50pt" | Statistics! scope="col" width="225pt" | Null Distribution! scope="col" width="225pt" | NotationSKAT 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|-| Single Variant || $T=\sum_mathbf{i=1Q}^n {U_i}\bigg/\sqrt{\sum_{i=1}^n{V_i}}$ || $T\sim\mathbf{N}(0,1)$ ||$U_i \text{ is the score statistic from study }i;$$V_i \textU_{ is the variance of meta} U_i.$|-| un-weighted Burden || $T_b=\sum_{i=1}^n{\mathbf{U_i}}\Big/\sqrt{\sum_{i=1T}^n{\mathbf{V_i}}W}$ || $T_b\sim\mathbf{N}(0,1)$ ||$\mathbfU_{U_imeta}\text{ is the vector of score statistics from study }i, or$ , where $\mathbf{U_iW}=\{U_{i1},...,U_{im}\};$ $\mathbf{V_i} \text{ is the covariance a diagonal matrix of weights of } \mathbf{U_i}rare variants included in a gene.$|-| Weighted Burden || $T_{wb}=\mathbf{w^T}\sum_{i=1}^n{\mathbf{U_i}}\biggAs shown in [http:/\sqrt{\mathbf{w^T}\left(\sum_{i=1}^n{\mathbf{V_i}}\right)\mathbf{w}}</math> || [itex]T_{wb}\sim\mathbf{N}(0,1)$ || $\mathbf{w^T}=\{w_1,w_2,www.ncbi.nlm.,w_m\}^T \text{ is the weight vectornih.}$|-style="height: 50pt;"| VT || $T_{VT}=\max(T_{b\left(f_1\right)},T_{b\left(f_2\right)},\dots,T_{b\left(f_m\right)}),\text{ where}$$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}} <gov/math> ||[itex] \left(T_{b\left(f_1\right)},T_{b\left(f_2\right)},\dots,T_{b\left(f_m\right)}\right)<pubmed/math>[itex]\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}}}$ || 21737059 $\boldsymbol{\phi}_{f_j}\text{ is a vector of } 0 \text{s and } 1\text{s'''Wu et. al'''],}$ $\text{indicating the inclusion null distribution of a variant using threshold }f_j;$ |-| SKAT || the $\mathbf{Q}=\left(\sum_{i=1}^n{\mathbf{U_i^T}}\right) \mathbf{W}\left(\sum_{i=1}^n{\mathbf{U_i}}\right)$ ||statistic follows a mixture chi-sqaured distribution described as $\mathbf{Q}\sim\sum_{i=1}^m{\lambda_i\chi_{1,i}^2},\text{ where}$ where $\left(\lambda_1,\lambda_2,\dots,\lambda_m\right)\text{$ are eigen values of}[/itex]$\left(\sum_mathbf{i=1}^n{\mathbfV_{V_imeta}}\right)^\frac{1}{2}}\mathbf{W}\left(\sum_mathbf{i=1}^nV_{\mathbf{V_imeta}}\right)^\frac{1}{2}}$ || $\mathbf{W}\text{ is a diagonal matrix of weights.}$|} [[Category:RAREMETAL]]
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