RAREMETAL METHOD

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Revision as of 21:39, 8 April 2014 by Shuang Feng (talk | contribs) (BURDEN META ANALYSIS)
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INTRODUCTION

The key idea behind meta-analysis with RAREMETAL is that various gene-level test statistics can be reconstructed from single variant score statistics and that, when the linkage disequilibrium relationships between variants are known, the distribution of these gene-level statistics can be derived and used to evaluate signifi-cance. Single variant statistics are calculated using the Cochran-Mantel-Haenszel method. The main formulae are tabulated in the following:

KEY FORMULAE

NOTATIONS

We denote the following to describe our methods:

U_{i,k} is the score statistic for the i^{th} variant from the  k^{th} study

V_{ij,k} is the covariance of the score statistics between the i^{th} and the j^{th} variant from the  k^{th} study

\mathbf{U_k} is the vector of score statistics of rare variants in a gene from the  k^{th} study.

\mathbf{V_k} is the variance-covariance matrix of score statistics of rare variants in a gene from the  k^{th} study, or \mathbf{V_k} = cov(\mathbf{U_k})

 S is the number of studies

U_{i,k} and V_{ij,k} are described in detail in RAREMETALWORKER method.

SINGLE VARIANT META ANALYSIS

Single variant meta-analysis score statistic can be reconstructed from score statistics and their variances generate by each study, assuming that samples are unrelated across studies. Define meta-analysis score statistics as

U_{meta,i}=\sum_{k=1}^S {U_{i,k}}

and its variance

V_{meta,i}=\sum_{k=1}^S{V_{ii,k}}

Then the score test statistics for the i^{th} variant T_{meta,i} asymptotically follows standard normal distribution

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)

BURDEN META ANALYSIS

Meta-analysis burden score test statistics for a gene can be reconstructed

T_meta,b=\sum_{k=1}^S{\mathbf{U_k}}\Big/\sqrt{\sum_{k=1}^S{\mathbf{V_k}}}

VT META ANALYSIS

SKAT META ANALYSIS

Formulae for RAREMETAL
Test Statistics Null Distribution Notation
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.}