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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 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 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.


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 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  .