Difference between revisions of "RAREMETAL METHOD"

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(BURDEN META ANALYSIS)
(BURDEN META ANALYSIS)
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Once single variant meta analysis statistics are constructed, burden test score statistic can be easily reconstructed as
 
Once single variant meta analysis statistics are constructed, burden test score statistic can be easily reconstructed as
  
<math>T_{meta_{burden}}=\mathbf{w^TU_{meta}}\bigg/\sqrt{\mathbf{w^TV_{meta}w}} \sim\mathbf{N}(0,1)</math>
+
<math>T_{meta_{burden}}=\mathbf{w^TU_{meta}}\bigg/\sqrt{\mathbf{w^TV_{meta}w}} \sim\mathbf{N}(0,1)</math>.
  
 
===VT META ANALYSIS===
 
===VT META ANALYSIS===

Revision as of 22:48, 8 April 2014

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

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

\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

 \mathbf{w^T} = (w_1,w_2,...,w_m)^T is the vector of weights for m rare variants in a gene.

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

Once single variant meta analysis statistics are constructed, burden test score statistic can be easily reconstructed as

T_{meta_{burden}}=\mathbf{w^TU_{meta}}\bigg/\sqrt{\mathbf{w^TV_{meta}w}} \sim\mathbf{N}(0,1).

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.}