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==Single Variant Meta Analysis ==
==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. Our method has been published in [http://www.ncbi.nlm.nih.gov/pubmed/24336170 '''Liu et. al''']. The main formulae are tabulated in the following:
==KEY FORMULAE==
== Gene-level Meta Analysis ==
===NOTATIONS===
We denote the following to describe our methods:
<math>U_{i,k} </math> is the score statistic for the <math>i^{th} </math> variant from the <math> k^{th} </math> study
<math>V_{ij,k} </math> is the covariance of the score statistics between the <math>i^{th} </math> and the <math>j^{th} </math> variant from the <math> k^{th} </math> study
<math>U_{i,k} </math> and <math>V_{ij,k} </math> are described in detail in [[RAREMETALWORKER_method#SINGLE_VARIANT_SCORE_TEST|'''RAREMETALWORKER method''']].
<math>\mathbf{U_k}</math> is the vector of score statistics of rare variants in a gene from the <math> k^{th} </math> study.
<math>\mathbf{V_k}</math> is the variance-covariance matrix of score statistics of rare variants in a gene from the <math> k^{th} </math> study, or <math>\mathbf{V_k} = cov(\mathbf{U_k})</math>
<math> S </math> is the number of studies
<math> f_{i} </math> is the pooled allele frequency of <math>i^{th}</math> variant
<math> f_{i,k} </math> is the allele frequency of <math>i^{th}</math> variant in <math>k^{th}</math> study
<math> {\delta_{k}} </math> is the deviation of trait value of <math>k^{th}</math> study
== Conditional Analysis ==
<math> \mathbf{w^T} = (w_1,w_2,...,w_m)^T</math> is the vector of weights for <math>m</math> rare variants in a gene.
===SINGLE VARIANT META ANALYSIS===
Single variant meta-analysis score statistic can be reconstructed from score statistics and their variances generated by each study, assuming that samples are unrelated across studies. Define meta-analysis score statistics as
<math>U_{meta_i}=\sum_{k=1}^S {U_{i,k}}</math>
and its variance
<math>V_{meta_i}=\sum_{k=1}^S{V_{ii,k}}</math>.
Then the score test statistics for the <math>i^{th}</math> variant <math>T_{meta_i}</math> asymptotically follows standard normal distribution
<math>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)</math>.
'''Optimized method for unbalanced studies (--useExact)''':
<math>U_{meta_i}=\sum_{k=1}^S {U_{i,k}/\hat{\Omega_{k}}}-\sum_{k=1}^S{2n_{k}{\delta_{k}^{2}(f_{i}-f_{i,k})}}</math>
<math>V_{meta_i}={\sigma^{2}}\sum_{k=1}^S{(V_{ii,k}{\Omega_{k}}-4n_{k}(ff'-f_{k}f_{k}'))}</math>
<math>{\sigma^{2}}=\sum_{k=1}^S{((n_{k}-1){\Omega_{k}}+n_{k}{\delta_{k}^{2}})}/(n-1)</math>
===BURDEN META ANALYSIS===
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 for a gene 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>,
where <math>\mathbf{U_{meta}} = (U_{meta_1},U_{meta_2},...,U_{meta_m})^T</math> and <math> \mathbf{V_{meta}}=cov(\mathbf{U_{meta}})</math>, representing a vector of single variant meta-analysis scores of <math>m</math> variants in a gene and the covariance matrix of the scores across <math>m</math> variants.
===VT META ANALYSIS===
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
<math>T_{meta_{VT}}=\max(T_{b\left(f_1\right)},T_{b\left(f_2\right)},\dots,T_{b\left(f_m\right)})</math>,
where <math>T_{b\left(f_i\right)}</math> is the burden test statistic under allele frequency threshold <math>f_i</math>, and can be constructed from single variant meta-analysis statistics using
<math>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}} </math>,
where <math>j</math> represents any allele frequency in a group of rare variants, <math>\boldsymbol{\phi}_{f_j}</math> is a vector of 0 and 1, indicating if a variant is included in the analysis using frequency threshold <math>f_i</math>.
As described by [http://www.ncbi.nlm.nih.gov/pubmed/21885029 '''Lin et. al'''], the p-value of this test can be calculated analytically using the fact that the burden test statistics together follow a multivariate normal distribution with mean <math>\mathbf{0}</math> and covariance <math>\boldsymbol{\Omega}</math>, written as
<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) </math>,
where <math>\boldsymbol{\Omega_{ij}}=\frac{\boldsymbol{\phi}_{f_i}^T\mathbf{V_{meta}}\boldsymbol{\phi}_{f_j}}{\sqrt{\boldsymbol{\phi}_{f_i}^T\mathbf{V_{meta}}\boldsymbol{\phi}_{f_i}}\sqrt{\boldsymbol{\phi}_{f_j}^T\mathbf{V_{meta}}\boldsymbol{\phi}_{f_j}}}</math>.
===SKAT META ANALYSIS===
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
<math>\mathbf{Q}=\mathbf{{U_{meta}}^T}\mathbf{W}\mathbf{U_{meta}}</math>,
where <math>\mathbf{W}</math> is a diagonal matrix of weights of rare variants included in a gene.
As shown in [http://www.ncbi.nlm.nih.gov/pubmed/21737059 '''Wu et. al'''], the null distribution of the <math> \mathbf{Q} </math> statistic follows a mixture chi-sqaured distribution described as
<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>.
[[Category:RAREMETAL]]
