# 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>. | ||

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+ | {| 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 23:57, 8 April 2014

## Contents

## 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:

is the score statistic for the variant from the study

is the covariance of the score statistics between the and the variant from the study

and are described in detail in **RAREMETALWORKER method**.

is the vector of score statistics of rare variants in a gene from the study.

is the variance-covariance matrix of score statistics of rare variants in a gene from the study, or

is the number of studies

is the vector of weights for 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

and its variance

Then the score test statistics for the variant asymptotically follows standard normal distribution

### 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 can be easily reconstructed as

.

### 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

,

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

,

where represents any allele frequency in a group of rare variants, is a vector of 0 and 1, indicating if a variant is included in the analysis using frequency threshold .

### 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

,

where is a diagonal matrix of weights of rare variants included in a gene.

As shown in **Wu et. al**, the null distribution of the statistic follows a mixture chi-sqaured distribution described as

where are eigen values of .

Test | Statistics | Null Distribution | Notation |
---|---|---|---|

Single Variant | |||

un-weighted Burden | |||

Weighted Burden | |||

VT | |||

SKAT |