Changes

| VT || <math>T_{VT}=\max(T_{b\left(f_1\right)},T_{b\left(f_2\right)},\dots,T_{b\left(f_m\right)}),\text{ where}</math><math>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}} </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)\text{,} </math><math>\text{where }\boldsymbol{\Omega_{ij}}=\frac{\boldsymbol{\phi}_{f_i}^T\left(\sum_{i=1}^n{\mathbf{V_i}}\right)\boldsymbol{\phi}_{f_j}}{\sqrt{\boldsymbol{\phi}_{f_i}^T\left(\sum_{i=1}^n{\mathbf{V_i}}\right)\boldsymbol{\phi}_{f_i}}\sqrt{\boldsymbol{\phi}_{f_j}^T\left(\sum_{i=1}^n{\mathbf{V_i}}\right)\boldsymbol{\phi}_{f_j}}}</math> ||  <math> \boldsymbol{\phi}_{f_j}\text{ is a vector of } 0 \text{s and } 1\text{s,} </math> <math>\text{indicating the inclusion of a variant using threshold }f_j; </math>  

| VT || <math>T_{VT}=\max(T_{b\left(f_1\right)},T_{b\left(f_2\right)},\dots,T_{b\left(f_m\right)}),\text{ where}</math><math>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}} </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)\text{,} </math><math>\text{where }\boldsymbol{\Omega_{ij}}=\frac{\boldsymbol{\phi}_{f_i}^T\left(\sum_{i=1}^n{\mathbf{V_i}}\right)\boldsymbol{\phi}_{f_j}}{\sqrt{\boldsymbol{\phi}_{f_i}^T\left(\sum_{i=1}^n{\mathbf{V_i}}\right)\boldsymbol{\phi}_{f_i}}\sqrt{\boldsymbol{\phi}_{f_j}^T\left(\sum_{i=1}^n{\mathbf{V_i}}\right)\boldsymbol{\phi}_{f_j}}}</math> ||  <math> \boldsymbol{\phi}_{f_j}\text{ is a vector of } 0 \text{s and } 1\text{s,} </math> <math>\text{indicating the inclusion of a variant using threshold }f_j; </math>  

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| SKAT || <math>\mathbf{Q}=\left(\sum_{i=1}^n{\mathbf{U_i^T}}\right) \mathbf{W}\left(\sum_{i=1}^n{\mathbf{U_i}}\right)</math> ||<math>\mathbf{Q}\sim\sum_{i=1}^m{\lambda_i\chi_{1,i}^2},\text{ where}</math> <math>\left(\lambda_1,\lambda_2,\dots,\lambda_m\right)\text{ are eigen values of}</math><math>\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}</math>

| SKAT || <math>\mathbf{Q}=\left(\sum_{i=1}^n{\mathbf{U_i^T}}\right) \mathbf{W}\left(\sum_{i=1}^n{\mathbf{U_i}}\right)</math> ||<math>\mathbf{Q}\sim\sum_{i=1}^m{\lambda_i\chi_{1,i}^2},\text{ where}</math> <math>\left(\lambda_1,\lambda_2,\dots,\lambda_m\right)\text{ are eigen values of}</math><math>\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}</math> || \mathbf{W}\text{ is a diagonal matrix where diagonal elements are weights.}

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