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

Take all components to be centered and let their marginal variances be $V_A,V_C,V_E$. A linear structural equation model iswhere $C_i,E_{i1},E_{i2}$ are mutually independent and the $E_{ij}$ are identically distributed. The causal directed acyclic graph has arrows

For monozygotic twins, set $A_{i1}=A_{i2}=G_i$ with $\mathrm{Var}(G_i)=V_A$; the genetic cause is completely shared. For dizygotic twins, one explicit construction iswhere $G_i,G_{i1},G_{i2}$ 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.

The common marginal trait variance isFor monozygotic pairs, shared genes and shared environment giveFor dizygotic pairs, the genetic covariance is halved while the common environment is unchanged:Subtracting cancels the common-environment variance and proves Falconer's formulathe heritability under the ACE model.

Suppose assortative mating raises the dizygotic genetic correlation from $1/2$ to $\rho >1/2$. Thenso the formula returnsThus using the nominal one-half genetic correlation produces downward bias in estimated heritability, assuming the other ACE assumptions remain valid.