6 Path Models for the Classical Twin Study

To introduce genetic models and to further illustrate the tracing rules both for standardized variables and unstandardized variables, we examine some simple genetic models of resemblance. The classical twin study, in which MZ twins and DZ twins are reared together in the same home is one of the most powerful designs for detecting genetic and shared environmental effects. Once we have collected such data, they may be summarized as observed covariance matrices (Chapter 2), but in order to test hypotheses we need to derive expected covariance matrices from the model. We first digress briefly to review the biometrical principles outlined in Chapter 3, in order to express the ideas in a path-analytic context.

In contrast to the regression models considered in previous sections, many genetic analyses of family data postulate independent variables (genotypes and environments) as latent rather than manifest variables. In other words, the genotypes and environments are not measured directly but their influence is inferred through their effects on the covariances of relatives. However, we can represent these models as path diagrams in just the same way as the regression models. The brief introduction to path-analytic genetic models we give here will be treated in greater detail in Chapter 6, and thereafter.

From quantitative genetic theory (see Chapter 3), we can write equations relating the phenotypes $P_i$ and $P_j$ of relatives $i$ and $j$ (e.g., systolic blood pressures of first and second members of a twin pair), to their underlying genotypes and environments. We may decompose the total genetic effect on a phenotype into that due to the additive effects of alleles at multiple loci, that due to the dominance effects at multiple loci, and that due to the epistatic interactions between loci (Mather and Jinks, 1982). 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 effects), and that due to environmental effects which make family members differ from one another (`random', `specific', or `within-family' environmental effects). Thus, the observed phenotypes, $P_i$ and $P_j$, are assumed to be linear functions of the underlying additive genetic variance ($A_i$ and $A_j$), dominance variance ($D_i$ and $D_j$), shared environmental variance ($C_i$ and $C_j$) and random environmental variance ($E_i$ and $E_j$). In quantitative genetic studies of human populations, epistatic genetic effects are usually confounded with dominance genetic effects, and so will not be considered further here. Assuming all variables are scaled as deviations from zero, we have

$$P_1 = e_1 E_1 + c_1 C_1 + a_1 A_1 + d_1 D_1$$

and

$$P_2 = e_2 E_2 + c_2 C_2 + a_2 A_2 + d_2 D_2$$

Particularly for pairs of twins, we do not expect the magnitude of genetic or environmental effects to vary as a function of relationship so we set $e_1 = e_2 = e$, $c_1 = c_2 = c$, $a_1 = a_2 = a$, and $d_1 = d_2 = d$. In matrix form, we write

$$\left( \begin{array}{c} P_1 \\ P_2 \end{array}\right) = \le... ...\\ A_1 \\ D_1 \\ E_2 \\ C_2 \\ A_2 \\ D_2 \end{array}\right) .$$

Unless two or more waves of measurement are used, or several observed variables index the phenotype under study, residual effects are included in the random environmental component, and are not separately specified in the model.

Figures 5.4a and 5.4b represent two alternative parameterizations of the 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). In Figure 5.4a, the traditional path coefficients model, the variances of the latent variables $A_1, C_1, E_1, D_1$ and $A_2, C_2, E_2, D_2$ are standardized ( $V_E = V_C = V_A = V_D = 1$, and the path coefficients $e, c, a$, or $d$ -- quantifying the paths from the latent variables to the observed variable, measured on both twins, $P_1$ and $P_2$ -- are free parameters to be estimated. Figure 5.4b is called a variance components model because it fixes $e = c = a = d = 1$, and estimates separate random environmental, shared environmental, additive genetic and dominance genetic variances instead.

The traditional path model illustrates tracing rules for standardized variables, and is straightforward to generalize to multivariate problems; the variance components model illustrates an unstandardized path model. Provided all parameter estimates are non-negative, tracing the paths in either parameterization will give the same solution, with $V_A = a^2$, $V_D = d^2$, $V_C = c^2$ and $V_E = e^2$.

Figure 5.4: Alternative representations of the basic genetic model: a) variance components model, and b) traditional path coefficients model.