Take all components to be centered and let their marginal variances be $V_A,V_C,V_E$. A <linear structural equation model> is
$$
C_{i1}=C_{i2}=C_i,
\qquad
Y_{ij}=A_{ij}+C_i+E_{ij},
$$
where $C_i,E_{i1},E_{i2}$ are mutually independent and the $E_{ij}$ are identically distributed. The <causal directed acyclic graph> has arrows
$$
A_{i1}\to Y_{i1}\leftarrow C_i\to Y_{i2}\leftarrow A_{i2},
\qquad
E_{i1}\to Y_{i1},
\qquad
E_{i2}\to Y_{i2}.
$$

For <monozygotic twins>, set $A_{i1}=A_{i2}=G_i$ with $\operatorname{Var}(G_i)=V_A$; the genetic cause is completely shared. For <dizygotic twins>, one explicit construction is
$$
A_{i1}=G_i+G_{i1},
\qquad
A_{i2}=G_i+G_{i2},
$$
where $G_i,G_{i1},G_{i2}$ are independent with variance $V_A/2$. Then both additive genetic terms have variance $V_A$, while their covariance is $V_A/2$ and their <correlation coefficient> is $1/2$. In a fuller graph, the shared $G_i$ points to both genetic components and the unique $G_{ij}$ points only to its own component.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 221 / 2 / i / Solution

{title2=DZ twin}
{wiki=Twin#Dizygotic_(fraternal)_twins}

Dizygotic twins develop from two zygotes and, absent assortative mating and other complications, share one half of additive genetic variation on average.


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