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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:
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 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:
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
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 .