Difference between revisions of "RAREMETAL METHOD"

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Revision as of 22:02, 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

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \mathbf{Q}=\mathbf{{U_{meta}}^T}\mathbf{W}\mathbf{U_{meta}} {| 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}} || 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.} |-style="height: 50pt;" | 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.} |}