Table 4.

Variance and co-variance implied by ACE-β model (Eqs. 18–19)

MZ and DZ Twins:${\mathbf{V}}_P^{\text{MZ}}$ and ${\mathbf{V}}_P^{\text{DZ}}$
	Observed outcome (phenotype)
	x1j	y1j	x2j	y2j
x1j	V[1, 1]
y1j	βV[1, 1] + V[2, 1]	β2V[1, 1] + 2βV[2, 1] + V[2, 2]
x2j	V[3, 1]	βV[3, 1] + V[4, 1]	V[1, 1]
y2j	βV[3, 1] + V[4, 1]	β2V[3, 1] + 2βV[4, 1] + V[4, 2]	βV[1, 1] + V[2, 1]	β2V[1, 1] + 2βV[2, 1] + V[2, 2]

Note: V[l, k] refers to rowl and column k of the variance/covariance matrix for MZ or DZ twins in Eqs. (15–16) with e_yx = 0 (see also Table 2). The expected variance of the phenotypes, ${\sigma }_x^2$ for x and ${\sigma }_y^2$ for y is equal for MZ and DZ twins and can be obtained from the above table as ${\sigma }_x^2=a_{xx}^2+c_{xx}^2+e_{xx}^2$ and ${\sigma }_y^2={\beta }^2{({\sigma }_x^2)}^2+2\beta (a_{xx}{a}_{yx}+c_{xx}{c}_{yx})+(a_{yx}^2+a_{yy}^2+c_{yx}^2+c_{yy}^2+e_{yx}^2+e_{yy}^2)$.