Recall equation 1 in which a pair of traits is expressed as a linear combination of genetic effects and an error term that includes environmental and unmodelled genetic effects:where is a matrix of coefficients, is a -vector of coded genetic markers, and is a pair of random errors that are independent of . Assume that the markers are independent and standardised. In the usual case of single nucleotide polymorphism (SNP) genotypes under Hardy-Weinberg Equilibrium, where is the number of minor alleles and is the minor allele frequency at SNP . The genetic effects are regarded as fixed across samples but random over with , and . Then the variance-covariance matrix of is written asFor continuous traits, assume without loss of generality that and are standardised so that and are the proportions of variation of each trait explained by . These quantities will be called the explained genetic variances, and are bounded above by the heritabilities.
