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 ===SINGLE VARIANT META ANALYSIS===   ===SINGLE VARIANT META ANALYSIS=== 
 Single variant metaanalysis score statistic can be reconstructed from score statistics and their variances generate by each study, assuming unrelated samples across studies, and is written   Single variant metaanalysis score statistic can be reconstructed from score statistics and their variances generate by each study, assuming unrelated samples across studies, and is written 
−  <math>T_i=\sum_{k=1}^S {U_{i,k}}\bigg/\sqrt{\sum_{k=1}^S{V_{ii,k}}}</math>  +  <math>T_i=\sum_{k=1}^S {U_{i,k}}\bigg/\sqrt{\sum_{k=1}^S{V_{ii,k}}}</math>. 
−  The score statistics asymptotically follows standard normal distribution, or <math>T\sim\mathbf{N}(0,1)</math>
 +  
 +  This score statistics asymptotically follows standard normal distribution, or <math>T\sim\mathbf{N}(0,1)</math> 
   
 ===BURDEN META ANALYSIS===   ===BURDEN META ANALYSIS=== 
Revision as of 21:10, 8 April 2014
INTRODUCTION
The key idea behind metaanalysis with RAREMETAL is that various genelevel 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 genelevel statistics can be derived and used to evaluate significance. Single variant statistics are calculated using the CochranMantelHaenszel 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
$S$ 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 metaanalysis score statistic can be reconstructed from score statistics and their variances generate by each study, assuming unrelated samples across studies, and is written
$T_{i}=\sum _{k=1}^{S}{U_{i,k}}{\bigg /}{\sqrt {\sum _{k=1}^{S}{V_{ii,k}}}}$.
This score statistics asymptotically follows standard normal distribution, or $T\sim \mathbf {N} (0,1)$
BURDEN META ANALYSIS
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}.$

unweighted 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}}$ 
$\mathbf {W} {\text{ is a diagonal matrix of weights.}}$
