# Difference between revisions of "RAREMETAL METHOD"

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<math>\mathbf{Q}=\mathbf{{U_{meta}}^T}\mathbf{W}\mathbf{U_{meta}}</math>, | <math>\mathbf{Q}=\mathbf{{U_{meta}}^T}\mathbf{W}\mathbf{U_{meta}}</math>, | ||

− | where <math>\mathbf{W}</math> is | + | 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 | 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>. | <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>. |

## Revision as of 23:55, 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:

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 .