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| <math> S </math> is the number of studies | | <math> S </math> is the number of studies |
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| + | <math> f_{i} </math> is the pooled allele frequency of <math>i^{th}</math> variant |
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| + | <math> f_{i,k} </math> is the allele frequency of <math>i^{th}</math> variant in <math>k^{th}</math> study |
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| + | <math> {\delta_{k}} </math> is the deviation of trait value of <math>k^{th}</math> study |
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| <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. |
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| and its variance | | and its variance |
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− | <math>V_{meta_i}=\sum_{k=1}^S{V_{ii,k}}</math> | + | <math>V_{meta_i}=\sum_{k=1}^S{V_{ii,k}}</math>. |
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| Then the score test statistics for the <math>i^{th}</math> variant <math>T_{meta_i}</math> asymptotically follows standard normal distribution | | Then the score test statistics for the <math>i^{th}</math> variant <math>T_{meta_i}</math> asymptotically follows standard normal distribution |
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− | <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>. |
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| + | '''Optimized method for unbalanced studies (--useExact)''': |
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| + | <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> |
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| + | <math>V_{meta_i}={\sigma^{2}}\sum_{k=1}^S{(V_{ii,k}{\Omega_{k}}-4n_{k}(ff'-f_{k}f_{k}'))}</math> |
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| + | <math>{\sigma^{2}}=\sum_{k=1}^S{((n_{k}-1){\Omega_{k}}+n_{k}{\delta_{k}^{2}})}/(n-1)</math> |
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| ===BURDEN META ANALYSIS=== | | ===BURDEN META ANALYSIS=== |
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− | 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 can be easily reconstructed as | + | 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 |
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− | <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>, |
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| + | 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. |
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| ===VT META ANALYSIS=== | | ===VT META ANALYSIS=== |
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| 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 |
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| <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>, |
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| 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 |
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| <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>, |
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| 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>. |
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| 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 |
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| <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>, |
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− | 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>. |
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| ===SKAT META ANALYSIS=== | | ===SKAT META ANALYSIS=== |
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| <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>. |
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| + | [[Category:RAREMETAL]] |