# Difference between revisions of "RAREMETAL METHOD"

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<math> S </math> is the number of studies | <math> S </math> is the number of studies | ||

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+ | <math> f_{i} </math> is the pooled allele frequency of <math>i^{th}</math> variant | ||

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+ | <math> f_{i,k} </math> is the allele frequency of <math>i^{th}</math> variant in <math>k^{th}</math> study | ||

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+ | <math> {\delta_{k}} </math> is the deviation of trait value of <math>k^{th}</math> study | ||

<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. | <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. | ||

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and its variance | and its variance | ||

− | <math>V_{meta_i}=\sum_{k=1}^S{V_{ii,k}}</math> | + | <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 | 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> | + | <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>. |

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+ | '''Optimized method for unbalanced studies (--useExact)''': | ||

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+ | <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> | ||

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+ | <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 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 | + | 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>, | ||

− | <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=== | ===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 | 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>, | <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 | 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>, | <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>. | 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 | 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>, | <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> | + | |

+ | 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 META ANALYSIS=== | ||

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## Latest revision as of 12:28, 20 May 2019

## 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. Our method has been published in **Liu et. al**. 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 pooled allele frequency of variant

is the allele frequency of variant in study

is the deviation of trait value of study

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 generated 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

.

**Optimized method for unbalanced studies (--useExact)**:

### 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

,

where and , representing a vector of single variant meta-analysis scores of variants in a gene and the covariance matrix of the scores across 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

,

where is the burden test statistic under allele frequency threshold , and 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 .

As described by **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 and covariance , written as

,

where .

### 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 .