1 Basic Genetic Model

Derivations of the expected variances and covariances of relatives under a simple univariate genetic model have been reviewed briefly in the chapters on biometrical genetics and path analysis (Chapters 3 and 5). In brief, from biometrical genetic theory we can write structural equations relating the phenotypes, $P$, of relatives $i$ and $j$ (e.g., BMI values of first and second members of twin pairs) to their underlying genotypes and environments which are latent variables whose influence we must infer. We may decompose the total genetic effect on a phenotype into contributions of:

• Additive effects of alleles at multiple loci (A),
• Dominance effects at multiple loci (D),
• Higher-order epistatic interactions between pairs of loci (additive $\times$ additive, additive $\times$ dominance, dominance $\times$ dominance: AA, AD, DD), and so on.

In practice even additive $\times$ dominance and dominance $\times$ dominance epistasis are confounded with dominance in studies of humans, and the power of resolving genetic dominance and additive $\times$ additive epistasis is very low. We shall therefore limit our consideration to additive and dominance genetic effects. Similarly, we may decompose the total environmental effect into that due to environmental influences shared by twins or sibling pairs reared in the same family (shared, common, or between-family environmental ($C$) effects), and that due to environmental effects that make family members differ from one another (within-family, specific, or random environmental ($E$) effects). Thus, the observed phenotypes, $P_{i}$ and $P_{j}$, will be linear functions of the underlying additive genetic deviations ($A_{i}$ and $A_{j}$), dominance genetic deviations ($D_{i}$ and $D_{j}$), shared environmental deviations ($C_{i}$ and $C_{j}$), and random environmental deviations ($E_{i}$ and $E_{j}$). Assuming all variables are scaled as deviations from zero, we have

$$P_{1} = e_{1} E_{1} + c_{1} C_{1} + a_{1} A_{1} + d_{1} D_{1}$$

$$P_{2} = e_{2} E_{2} + c_{2} C_{2} + a_{2} A_{2} + d_{2} D_{2}$$ (43)

In most models we do not expect the magnitude of genetic effects, or the environmental effects, to differ between first and second twins, so we set $e_{1} = e_{2} = e, c_{1} = c_{2} = c, a_{1} = a_{2} = a, d_{1} = d_{2} = d$. Likewise, we do not expect the values of $e, c, a$, and $d$ to vary as a function of relationship. In other words, the effects of genotype and environment on the phenotype are the same regardless of whether one is an MZ twin, a DZ twin, or not a twin at all. In matrix form, we may write

$$\left( \begin{array}{r} P_1 \ P_2 \end{array} \right) = \... ...C_1\ E_1\ D_1\ A_2\ C_2\ E_2\ D_2 \end{array} \right)$$

As shown in Chapter 5, this model generates a predicted covariance matrix ($\Sigma$) which is equal to

$$\left [ \begin{array}{cc} a^2+c^2+e^e+d^2 & a^2+c^2+d^2 \\ a^2+c^2+d^2 & a^2+c^2+e^e+d^2 \end{array} \right ]$$

Unless two or more waves of measurement are used, or several variables index the phenotype under study, residual effects (such as measurement error) will form part of the random environmental component, and are not explicitly included in the model. To obtain estimates for the genetic and environmental effects in this model, we must also specify the variances and covariances among the latent genetic and environmental factors. Two alternative parameterizations are possible: 1) the variance components approach (Chapter 3), or 2) the path coefficients model (Chapter 5). The variance components approach becomes cumbersome for designs involving more complex pedigree structures than pairs of relatives, but it does have some numerical advantages (see Chapter , p. ). In the variance components approach we estimate variances of the latent non-shared and shared environmental and additive and dominance genetic variables, $V_E$, $V_C$, $V_A$, or $V_D$, and fix $a = c = e = d =1$. Thus, the phenotypic variance is simply the sum of the four variance components. In the path coefficients approach we standardize the variances of the latent variables to unity ($V_E$ = $V_C$ = $V_A$ = $V_D$ =1) and estimate a combination of $a, c, e$, and $d$ as free parameters. Thus, the phenotypic variance is a weighted sum of standardized variables. In this volume we will often refer to models that have particular combinations of free parameters in the general path coefficients model. Specifically, we refer to an ACE model as one having only additive genetic, common environment, and random environment effects; an ADE model as one having additive genetic, dominance, and random environment effects; an AE model as one having additive genetic and random environment effects, and so on. Figures 5.3a and 5.3b in Chapter 5 represent path diagrams for the two alternative parameterizations of the full basic genetic model, illustrated for the case of pairs of monozygotic twins (MZ) or dizygotic twins (DZ), who may be reared together (MZT, DZT) or reared apart (MZA, DZA). For simplicity, we make certain strong assumptions in this chapter, which are implied by the way we have drawn the path diagrams in Figure 5.3:

1. No genotype-environment correlation, i.e., latent genetic variables $A$ are uncorrelated with latent environmental variables $C$ and $E$;
2. No genotype $\times$ environment interaction, so that the observed phenotypes are a linear function of the underlying genetic and environmental variables;
3. Random mating, i.e., no tendency for like to marry like, an assumption which is implied by fixing the covariance of the additive genetic deviations of DZ twins or full sibs to $0.5V_A$;
4. Random placement of adoptees, so that the rearing environments of separated twin pairs are uncorrelated.

We discuss ways in which these assumptions may be relaxed in subsequent chapters, particularly Chapter 9 and Chapter .