There are T traits, which may be binary or quantitative. We standardize each trait and the genotype for each single-nucleotide polymorphism (SNP) j so that they all have mean zero and variance one. The length-T vector of marginal (i.e., not controlling for other SNPs), true effects of SNP j on each of the traits is denoted βj. We assume that these are random effects with mean 0 and variance-covariance matrix Ω that is the same across j. The mean is zero because we treat the choice of reference allele as arbitrary. We make the common assumption14,21,22 that the βj’s are identically distributed across j. The assumption implies that the expected amount of phenotypic variance explained is equal for each SNP, regardless of SNP characteristics such as allele frequency.
