Changes

<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 i^{th} variant

<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 i^{th} variant in k^{th} study

<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 k^{th} 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>.

<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>

 

<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>{\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>\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>.