Difference between revisions of "RAREMETAL METHOD"

From Genome Analysis Wiki
Jump to: navigation, search
(→‎KEY FORMULAE)
(→‎NOTATIONS)
Line 6: Line 6:
 
===NOTATIONS===
 
===NOTATIONS===
 
We denote the following to describe our methods:
 
We denote the following to describe our methods:
−
<math>\mathbf{U_i,k} </math> is the score statistic for the <math>i^{th} </math> variant from the <math> k^{th} </math> study
+
<math>\mathbf{U_{i,k}} </math> is the score statistic for the <math>i^{th} </math> variant from the <math> k^{th} </math> study
  
−
<math>\mathbf{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>\mathbf{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> S </math> is the number of studies
 
<math> S </math> is the number of studies

Revision as of 14:28, 1 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: \mathbf{U_{i,k}} is the score statistic for the i^{th} variant from the k^{th} study

\mathbf{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 


===SINGLE VARIANT META ANALYSIS===

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.
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.}
Retrieved from "https://genome.sph.umich.edu/w/index.php?title=RAREMETAL_METHOD&oldid=9854"