4 Summary

Table 3.3 replicates Table 3.2 employing genotypic frequencies appropriate to random mating and unequal gene frequencies. Using the table to calculate covariances among sibling pairs of the three types, MZ twins, DZ twins, and unrelated siblings, gives

Cov(MZ)	$=$	$2uv[d+(v-u)h]^2 + 4u^2v^2h^2$	$=$	$V_A + V_D$
Cov(DZ)	$=$	$uv[d+(v-u)h]^2 + u^2v^2h^2$	$=$	$\frac{1}{2}V_A + \frac{1}{4}V_D$
Cov(U)	$=$	$0$	$=$	$0$

Table 3.3: Genetic covariance components for MZ, DZ, and Unrelated Siblings with unequal gene frequencies at a single locus.

Genotype	Effect			Frequency
Pair	$x_{1i}$	$x_{2i}$		MZ	DZ	U
$AA,AA$	$d$	$d$		$u^2$	$u^4 + u^3v + \frac{1}{4}u^2v^2$	$u^4$
$AA,Aa$	$d$	$h$		-	$u^3v + \frac{1}{2}u^2v^2$	$2u^3v$
$AA,aa$	$d$	$-d$		--	$\frac{1}{4}u^2v^2$	$u^2v^2$
$Aa,AA$	$h$	$d$		--	$u^3v + \frac{1}{2}u^2v^2$	$2u^3v$
$Aa,Aa$	$h$	$h$		$2uv$	$u^3v + 3u^2v^2 + uv^3$	$4u^2v^2$
$Aa,aa$	$h$	$-d$		--	$\frac{1}{2}u^2v^2 + uv^3$	$2uv^3$
$aa,AA$	$-d$	$d$		--	$\frac{1}{4}u^2v^2$	$u^2v^2$
$aa,Aa$	$-d$	$h$		--	$\frac{1}{2}u^2v^2 + uv^3$	$2uv^3$
$aa,aa$	$-d$	$-d$		$u^4$	$\frac{1}{4}u^2v^2 + uv^3 + v^4$	$v^4$

By similar calculations, the expectations for half-siblings and for parents and their offspring may be shown to be $\frac{1}{4} V_A$ and $\frac{1}{2}V_A$, respectively. That is, these relationships do not reflect dominance effects. The MZ and DZ resemblances are the primary focus of this text, but all five relationships we have just discussed may be analyzed in the extended Mx approaches we discuss in Chapter . With more extensive genetical data, we can assess the effects of epistasis, or non-allelic interaction, since the biometrical model may be extended easily to include such genetic effects. Another important problem we have not considered is that of assortative mating, which one might have thought would introduce insuperable problems for the model. However, once we are working with genotypic values such as $A$ and $D$, the effects of assortment can be readily accommodated in the model by means of reverse path analysis (Wright, 1968) and the Pearson-Aitken treatment of selected variables (Aitken, 1934). Fulker (1988) describes this approach in the context of Fisher's (1918) model of assortment. In this chapter, we have given a brief introduction to the biometrical model that underlies the model fitting approach employed in this book, and we have indicated how additional genetic complexities may be accommodated in the model. However, in addition to genetic influences, we must consider the effects of the environment on any phenotype. These may be easily accommodated by defining environmental influences that are common to sib pairs and those that are unique to the individual. If these environmental effects are unrelated to the genotype, then the variances due to these influences simply add to the genetic variances we have just described. If they are not independent of genotype, as in the case of sibling interactions and cultural transmission, both of which are likely to occur in some behavioral phenotypes, then the Mx model may be suitably modified to account for these complexities, as we describe in Chapters 8 and .