The bivariate model represented for an individual for two phenotypes (P1 and P2):

P1 = G1 + NG1 = A1+ D1 + H1 + E1

P2 = G2 + NG2 = A2+ D2 + H2 + E2

Var (P1) = Var(G1) + Var (NG1)

Var (P2) = Var(G2) + Var (NG2)

Broad-sense heritability (P1) = Var(G1)/Var (P1) = H2

Narrow-sense heritability (P1) = Var(A1)/Var (P1) = h2

Covar (P1, P2) = Covar (G1, G2) + Covar (NG1, NG2)

Phenotypic correlation: r (P1, P2) = Covar (P1, P2)/SD(P1) × SD(P2)

Genetic correlation: r (G1, G2) = Covar (G1, G2)/SD(G1) × SD(G2);

where P is an individual’s phenotypic value (possibly a residual after correction for fixed effects of, e.g., age and sex), G is genotypic value and NG stands for non-genetic value. Var(P) is the variance of the phenotype (or the phenotypic residual); var(G) and var(NG) stand for genetic and non-genetic variance components (assuming no covariance of G and NG). G can be decomposed into additive genetic (A) and non-additive (dominance; D) values; non-genetic influences can be distinguished into those that are common to members from the same household (called household effects (H) in Mendel) and all other (unique; E) environmental effects.

The covariance between two phenotypes, here labeled P1 and P2 (e.g., MDD and smoking) likewise can be decomposed into genetic and non-genetic covariance. The correlation of P1 and P2 is obtained by scaling the phenotypic covariance by the product of the standard deviations of P1 and P2. Likewise, the genetic correlation is obtained by dividing the genetic covariance by the standard deviations of G1 and G2.