Difference between revisions of "RAREMETAL METHOD"

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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>.
{| border="1" cellpadding="5" cellspacing="0" align="center"
|+'''Formulae for RAREMETAL'''
! scope="col" width="120pt" | Test
! scope="col" width="50pt" | Statistics
! scope="col" width="225pt" | Null Distribution
! scope="col" width="225pt" | Notation
| 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>
| 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>
| 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>
|-style="height: 50pt;"
| 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>
| 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>

Revision as of 23:54, 8 April 2014


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:



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 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


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 test has been shown to be powerful detecting a group of rare variants that are unidirectional in effects. 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).


Including variants that are not associated to phenotype can hurt power. Variable threshold test is designed to choose the optimal allele frequency threshold amongst rare variants in a gene, to gain power. The test statistic is defined as the maximum burden score statistic calculated using every possible frequency threshold


where the burden test statistic under any allele frequency threshold can be constructed from single variant meta-analysis statistics using

T_{b\left(f_j\right)}=\boldsymbol{\phi}_{f_j}^\mathbf{T}\mathbf{U_{meta}}\bigg/\sqrt{\boldsymbol{\phi}_{f_j}^\mathbf{T}\mathbf{V_{meta}}\boldsymbol{\phi}_{f_j}} ,

where j represents any allele frequency in a group of rare variants, \boldsymbol{\phi}_{f_j} is a vector of 0 and 1, indicating if a variant is included in the analysis using frequency threshold f_i.


SKAT is most powerful when detecting genes with rare variants having opposite directions in effect sizes. Meta-analysis statistic can also be re-constructed using single variant meta-analysis scores and their covariances


where \mathbf{W} is the diagonal matrix of weights of rare variants included in a gene.

As shown in Wu et. al, the null distribution of the  \mathbf{Q} statistic follows a mixture chi-sqaured distribution described as

\mathbf{Q}\sim\sum_{i=1}^m{\lambda_i\chi_{1,i}^2}, where \left(\lambda_1,\lambda_2,\dots,\lambda_m\right) are eigen values of \mathbf{V_{meta}^\frac{1}{2}}\mathbf{W}\mathbf{V_{meta}^\frac{1}{2}}.