The length-T vector of GWAS estimates is denoted β^j, which is equal to the true effect vector plus estimation error, βj+εj. The estimation error is the sum of sampling variation and biases (such as population stratification or technical artifacts). With any standard GWAS estimator (such as OLS or logistic regression), sampling variation is uncorrelated with βj. We assume that the biases are also uncorrelated with βj. The variance-covariance matrix of εj, denoted ∑j, may differ across SNPs j due to differences in the SNPs’ sample sizes per trait and the SNPs’ sample overlap between traits, although we only account for the former in our estimation of ∑j.
