# Changes

## RAREMETAL Documentation

, 11:30, 26 August 2013
Approach
| Weighted Burden || $T_{wb}=\mathbf{w^T}\sum_{i=1}^n{\mathbf{U_i}}\bigg/\sqrt{\mathbf{w^T}\left(\sum_{i=1}^n{\mathbf{V_i}}\right)\mathbf{w}}$ || $T_{wb}\sim\mathbf{N}(0,1)$ || $\mathbf{w^T}=\{w_1,w_2,...,w_m\}^T \text{ is the weight vector.}$
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| VT || $T_{VT}=\max(T_{b\left(f_1\right)},T_{b\left(f_2\right)},\dots,T_{b\left(f_m\right)}),\text{ where}$$T_{b\left(f_j\right)}=\boldsymbol{\phi}_{f_j}^\mathbf{T}\sum_{i=1}^n{\mathbf{U_i}}\bigg/\sqrt{\boldsymbol{\phi}_{f_j}^\mathbf{T}\left(\sum_{i=1}^n{\mathbf{V_i}}\right)\boldsymbol{\phi}_{f_j}}$ ||$\left(T_{b\left(f_1\right)},T_{b\left(f_2\right)},\dots,T_{b\left(f_m\right)}\right)$$\sim\mathbf{MVN}\left(\mathbf{0},\boldsymbol{\Omega}\right)\text{, where }\boldsymbol{\Omega_{ij}}=\mathbf{\phi_i^T}\sum_{i=1}^n{\mathbf{V_i}}\mathbf{\phi_j}\bigg/\sqrt{\mathbf{\phi_i^T}\sum_{i=1}^n{\mathbf{V_i}}\mathbf{\phi_i}}\sqrt{\mathbf{\phi_j^T}\left(\sum_{i=1}^n{\mathbf{V_i}}\mathbf{\phi_j}}$ || $\boldsymbol{\phi}_{f_j}\text{ is a vector of } 0 \text{s and } 1\text{s,}$ $\text{indicating whether the variant is included using threshold }f_j;$ $\boldsymbol{\Phi}= \{\phi_{f_1},\phi_{f_2},...\phi_{f_m}\},\text{ is a matrix of indicators.}$
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| SKAT || $\mathbf{Q}=\left(\sum_{i=1}^n{\mathbf{U_i^T}}\right) \mathbf{W}\left(\sum_{i=1}^n{\mathbf{U_i}}\right)$ ||$\mathbf{Q}\sim\sum_{i=1}^m{\lambda_i\chi_{1,i}^2},\text{ where}$ $\left(\lambda_1,\lambda_2,\dots,\lambda_m\right)\text{ are eigen values of}$$\left(\sum_{i=1}^n{\mathbf{V_i}}\right)^\frac{1}{2}\mathbf{W}\left(\sum_{i=1}^n{\mathbf{V_i}}\right)^\frac{1}{2}$
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