# Changes

## RAREMETAL METHOD

, 13:28, 20 May 2019
Tag category
$S$ is the number of studies
$f_{i}$ is the pooled allele frequency of $i^{th}$ variant
$f_{i,k}$ is the allele frequency of $i^{th}$ variant in $k^{th}$ study
${\delta_{k}}$ is the deviation of trait value of $k^{th}$ study
$\mathbf{w^T} = (w_1,w_2,...,w_m)^T$ is the vector of weights for $m$ rare variants in a gene.
$T_{meta_i}=U_{meta_i}\bigg/\sqrt{V_{meta_i}}=\sum_{k=1}^S {U_{i,k}}\bigg/\sqrt{\sum_{k=1}^S{V_{ii,k}}} \sim\mathbf{N}(0,1)$.
Optimized method for unbalanced studies:
'''Optimized method for unbalanced studies (--useExact)''': $U_{meta_i}=\sum_{k=1}^S {U_{i,k}/\hat{\Omega_{k}}}-\sum_{k=1}^S{2n_{k}{\delta_{k}^{2}(f_{i}-f_{i,k})}}$ $V_{meta_i}={\sigma^{2}}\sum_{k=1}^S{(V_{ii,k}{\Omega_{k}}-4n_{k}(ff'-f_{k}f_{k}'))}$ ${\sigma^{2}}=\sum_{k=1}^S{((n_{k}-1){\Omega_{k}}+n_{k}{\delta_{k}^{2}})}/(n-1)$
===BURDEN META ANALYSIS===
$\mathbf{Q}\sim\sum_{i=1}^m{\lambda_i\chi_{1,i}^2},$ where $\left(\lambda_1,\lambda_2,\dots,\lambda_m\right)$ are eigen values of $\mathbf{V_{meta}^\frac{1}{2}}\mathbf{W}\mathbf{V_{meta}^\frac{1}{2}}$.

[[Category:RAREMETAL]]
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