# RAREMETAL METHOD

Single Variant $T=\sum_{i=1}^n {U_i}\bigg/\sqrt{\sum_{i=1}^n{V_i}}$ $T\sim\mathbf{N}(0,1)$ $U_i \text{ is the score statistic from study }i;$$V_i \text{ is the variance of } U_i.$
un-weighted Burden $T_b=\sum_{i=1}^n{\mathbf{U_i}}\Big/\sqrt{\sum_{i=1}^n{\mathbf{V_i}}}$ $T_b\sim\mathbf{N}(0,1)$ $\mathbf{U_i}\text{ is the vector of score statistics from study }i, or$ $\mathbf{U_i}=\{U_{i1},...,U_{im}\};$ $\mathbf{V_i} \text{ is the covariance of } \mathbf{U_i}.$
Weighted Burden $T_{wb}=\mathbf{w^T}\sum_{i=1}^n{\mathbf{U_i}}\bigg/\sqrt{\mathbf{w^T}\left(\sum_{i=1}^n{\mathbf{V_i}}\right)\mathbf{w}}$ $T_{wb}\sim\mathbf{N}(0,1)$ $\mathbf{w^T}=\{w_1,w_2,...,w_m\}^T \text{ is the weight vector.}$
VT $T_{VT}=\max(T_{b\left(f_1\right)},T_{b\left(f_2\right)},\dots,T_{b\left(f_m\right)}),\text{ where}$$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)$$\sim\mathbf{MVN}\left(\mathbf{0},\boldsymbol{\Omega}\right)\text{,}$$\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}}}$ $\boldsymbol{\phi}_{f_j}\text{ is a vector of } 0 \text{s and } 1\text{s,}$ $\text{indicating the inclusion of a variant using threshold }f_j;$
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}$ $\mathbf{W}\text{ is a diagonal matrix of weights.}$