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, 13:28, 20 May 2019
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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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| <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:
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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}(f_{i}-f_{i,k})}}</math> | + | '''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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| <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]] |