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<math> S </math> is the number of studies
<math> S </math> is the number of studies
<math> f_{i} </math> is the pooled allele frequency of <math>i^{th}</math> variant
<math> f_{i,k} </math> is the allele frequency of <math>i^{th}</math> variant in <math>k^{th}</math> study
<math> {\delta_{k}} </math> is the deviation of trait value of <math>k^{th}</math> 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 a gene.
<math> \mathbf{w^T} = (w_1,w_2,...,w_m)^T</math> is the vector of weights for <math>m</math> rare variants in a gene.
<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>.
<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>.
'''Optimized method for unbalanced studies (--useExact)''':
<math>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})}}</math>
<math>V_{meta_i}={\sigma^{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){\Omega_{k}}+n_{k}{\delta_{k}^{2}})}/(n-1)</math>
===BURDEN META ANALYSIS===
===BURDEN META ANALYSIS===
<math>T_{meta_{burden}}=\mathbf{w^TU_{meta}}\bigg/\sqrt{\mathbf{w^TV_{meta}w}} \sim\mathbf{N}(0,1)</math>,
<math>T_{meta_{burden}}=\mathbf{w^TU_{meta}}\bigg/\sqrt{\mathbf{w^TV_{meta}w}} \sim\mathbf{N}(0,1)</math>,
where <math>\mathbf{U_{meta}} = (U_{meta_1},U_{meta_2},...,U_{meta_m})</math> and <math> \mathbf{V_{meta}}=cov(\mathbf{U_{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.
where <math>\mathbf{U_{meta}} = (U_{meta_1},U_{meta_2},...,U_{meta_m})^T</math> and <math> \mathbf{V_{meta}}=cov(\mathbf{U_{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===
===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
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>,
<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
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>,
<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>.
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
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>,
<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>
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===
===SKAT META ANALYSIS===
<math>\mathbf{Q}\sim\sum_{i=1}^m{\lambda_i\chi_{1,i}^2}, </math> where <math>\left(\lambda_1,\lambda_2,\dots,\lambda_m\right)</math> are eigen values of <math>\mathbf{V_{meta}^\frac{1}{2}}\mathbf{W}\mathbf{V_{meta}^\frac{1}{2}}</math>.
<math>\mathbf{Q}\sim\sum_{i=1}^m{\lambda_i\chi_{1,i}^2}, </math> where <math>\left(\lambda_1,\lambda_2,\dots,\lambda_m\right)</math> are eigen values of <math>\mathbf{V_{meta}^\frac{1}{2}}\mathbf{W}\mathbf{V_{meta}^\frac{1}{2}}</math>.
[[Category:RAREMETAL]]
