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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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|+'''Formulae for RAREMETAL'''
! scope="col" width="120pt" | Test
! scope="col" width="50pt" | Statistics
! scope="col" width="225pt" | Null Distribution
! scope="col" width="225pt" | Notation
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| Single Variant || <math>T=\sum_{i=1}^n {U_i}\bigg/\sqrt{\sum_{i=1}^n{V_i}}</math> || <math>T\sim\mathbf{N}(0,1)</math> ||<math> U_i \text{ is the score statistic from study }i;</math><math> V_i \text{ is the variance of } U_i.</math>
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| un-weighted Burden || <math>T_b=\sum_{i=1}^n{\mathbf{U_i}}\Big/\sqrt{\sum_{i=1}^n{\mathbf{V_i}}}</math> || <math>T_b\sim\mathbf{N}(0,1)</math> ||<math> \mathbf{U_i}\text{ is the vector of score statistics from study }i, or </math> <math> \mathbf{U_i}=\{U_{i1},...,U_{im}\};</math> <math>\mathbf{V_i} \text{ is the covariance of } \mathbf{U_i}.</math>
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| Weighted Burden || <math>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}}</math> || <math>T_{wb}\sim\mathbf{N}(0,1)</math> || <math> \mathbf{w^T}=\{w_1,w_2,...,w_m\}^T \text{ is the weight vector.}</math>
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| 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> || <math>\mathbf{W}\text{ is a diagonal matrix of weights.}</math>
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