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RAREMETAL METHOD

1,333 bytes added, 12:28, 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==
<math>V_{ij,k} </math> is the covariance of the score statistics between the <math>i^{th} </math> and the <math>j^{th} </math> variant from the <math> k^{th} </math> study
 
<math>U_{i,k} </math> and <math>V_{ij,k} </math> are described in detail in [[RAREMETALWORKER_method#SINGLE_VARIANT_SCORE_TEST|'''RAREMETALWORKER method''']].
<math>\mathbf{U_k}</math> is the vector of score statistics of rare variants in a gene from the <math> k^{th} </math> study.
<math> S </math> is the number of studies
<math>U_f_{i} </math> is the pooled allele frequency of <math>i^{th}</math> variant <math> f_{i,k} </math> and is the allele frequency of <math>i^{th}</math> variant in <math>k^{th}</math> study <math>V_{ij,\delta_{k}} </math> is the deviation of trait value of <math>k^{th} </math> are described in detail study <math> \mathbf{w^T} = (w_1,w_2,...,w_m)^T</math> is the vector of weights for <math>m</math> 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. Define meta-analysis score statistics as
<math>U_{meta,imeta_i}=\sum_{k=1}^S {U_{i,k}}</math>
and its variance
<math>V_{metameta_i}=\sum_{k=1}^S{V_{ii,k}}</math>. Then the score test statistics for the <math>i^{th}</math> variant <math>T_{meta_i}</math> asymptotically follows standard normal distribution  <math>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)</math>.
Then the score test statistics for the <math>i^{th}</math> variant <math>T_{meta,i}</math> asymptotically follows standard normal distribution
'''Optimized method for unbalanced studies (--useExact)''': <math>T_U_{meta,imeta_i}=\sum_{k=1}^S {U_{metai,ik}\bigg/\sqrthat{V_\Omega_{meta,ik}}}=-\sum_{k=1}^S {U_2n_{k}{\delta_{k}^{2}(f_{i}-f_{i,k})}\bigg}</math> <math>V_{meta_i}={\sqrtsigma^{2}}\sum_{k=1}^S{(V_{ii,k}{\Omega_{k}}-4n_{k}(ff'-f_{k}f_{k}'))}</math> <math>{\sigma^{2}}=\sum_{k=1}^S{((n_{k} -1){\simOmega_{k}}+n_{k}{\mathbfdelta_{Nk}^{2}})}/(0,n-1)</math>
===BURDEN META ANALYSIS===
Meta-analysis burden score test statistics for a gene can be reconstructed
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 <math>T_{meta,bmeta_{burden}}=\mathbf{w^T}\sum_TU_{k=1}^S{\mathbf{U_kmeta}}\Bigbigg/\sqrt{\mathbf{w^TTV_{meta}w}}\leftsim\sum_mathbf{kN}(0,1)</math>, where <math>\mathbf{U_{meta}} =1(U_{meta_1},U_{meta_2},...,U_{meta_m})^S{T</math> and <math> \mathbf{V_kV_{meta}}\right=cov(\mathbf{wU_{meta}})</math>, representing a vector of single variant meta-analysis scores of <math>m</math> variants in a gene and the covariance matrix of the scores across <math>m</math>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
 
 
<math>T_{meta_{VT}}=\max(T_{b\left(f_1\right)},T_{b\left(f_2\right)},\dots,T_{b\left(f_m\right)})</math>,
 
where <math>T_{b\left(f_i\right)}</math> is the burden test statistic under allele frequency threshold <math>f_i</math>, and can be constructed from single variant meta-analysis statistics using
 
 
<math>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}} </math>,
 
 
where <math>j</math> represents any allele frequency in a group of rare variants, <math>\boldsymbol{\phi}_{f_j}</math> is a vector of 0 and 1, indicating if a variant is included in the analysis using frequency threshold <math>f_i</math>.
 
 
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 <math>\mathbf{0}</math> and covariance <math>\boldsymbol{\Omega}</math>, written as
 
 
<math> \left(T_{b\left(f_1\right)},T_{b\left(f_2\right)},\dots,T_{b\left(f_m\right)}\right)</math><math>\sim\mathbf{MVN}\left(\mathbf{0},\boldsymbol{\Omega}\right) </math>,
 
 
where <math>\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}}}</math>.
===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 || <math>T=\sum_mathbf{i=1Q}^n {U_i}\bigg/\sqrt{\sum_{i=1}^n{V_i}}</math> || <math>T\sim\mathbf{N}(0,1)</math> ||<math> U_i \text{ is the score statistic from study }i;</math><math> V_i \textU_{ is the variance of meta} U_i.</math>|-| un-weighted Burden || <math>T_b=\sum_{i=1}^n{\mathbf{U_i}}\Big/\sqrt{\sum_{i=1T}^n{\mathbf{V_i}}W}</math> || <math>T_b\sim\mathbf{N}(0,1)</math> ||<math> \mathbfU_{U_imeta}\text{ is the vector of score statistics from study }i, or </math> , where <math> \mathbf{U_iW}=\{U_{i1},...,U_{im}\};</math> <math>\mathbf{V_i} \text{ is the covariance a diagonal matrix of weights of } \mathbf{U_i}rare variants included in a gene.</math>|-| Weighted Burden || <math>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> || <math>T_{wb}\sim\mathbf{N}(0,1)</math> || <math> \mathbf{w^T}=\{w_1,w_2,www.ncbi.nlm.,w_m\}^T \text{ is the weight vectornih.}</math>|-style="height: 50pt;"| VT || <math>T_{VT}=\max(T_{b\left(f_1\right)},T_{b\left(f_2\right)},\dots,T_{b\left(f_m\right)}),\text{ where}</math><math>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> ||<math> \left(T_{b\left(f_1\right)},T_{b\left(f_2\right)},\dots,T_{b\left(f_m\right)}\right)<pubmed/math><math>\sim\mathbf{MVN}\left(\mathbf{0},\boldsymbol{\Omega}\right)\text{,} </math><math>\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}}}</math> || 21737059 <math> \boldsymbol{\phi}_{f_j}\text{ is a vector of } 0 \text{s and } 1\text{s'''Wu et. al'''],} </math> <math>\text{indicating the inclusion null distribution of a variant using threshold }f_j; </math> |-| SKAT || the <math>\mathbf{Q}=\left(\sum_{i=1}^n{\mathbf{U_i^T}}\right) \mathbf{W}\left(\sum_{i=1}^n{\mathbf{U_i}}\right)</math> ||statistic follows a mixture chi-sqaured distribution described as <math>\mathbf{Q}\sim\sum_{i=1}^m{\lambda_i\chi_{1,i}^2},\text{ where}</math> where <math>\left(\lambda_1,\lambda_2,\dots,\lambda_m\right)\text{ </math> are eigen values of}</math><math>\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}}</math> || <math>\mathbf{W}\text{ is a diagonal matrix of weights.}</math>|} [[Category:RAREMETAL]]
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