1 Introduction and Description of Terminology

The principles of biometrical and quantitative genetics lie at the heart of virtually all of the statistical models examined in this book. Thus, an understanding of biometrical genetics is fundamental to our statistical approach to twin and family data. Biometrical models relate the ``latent,'' or unobserved, variables of our structural models to the functional effects of genes. It is these effects, based on the principles of Mendelian genetics, that give our structural models a degree of validity quite unusual in the social sciences. The purpose of this chapter is to provide a brief introduction to biometrical models. Extensive treatments of the subject have been provided by Mather and Jinks (1982) and Falconer (1990). Here we employ the notation of Mather and Jinks.

Before we begin our discussion of biometrical genetics, we must describe some of the terms that are encountered frequently in biometrical and classical genetic discourse. For the present purposes, we use the term gene in reference to a ``unit factor of inheritance'' that influences an observable trait or traits, following the earlier usage by Fuller and Thompson (1978). Observable characteristics are referred to as phenotypes. The site of a gene on a chromosome is known as the locus. Alleles are alternative forms of a gene that occupy the same locus on a chromosome. They often are symbolized as A and a or B and b or $A_1$ and $A_2$. The simplest system for a segregating locus involves only two alleles (A and a), but there also may be a large number of alleles in a system. For example, the HLA locus on chromosome 6 is known to have 18 alleles at the A locus, 41 alleles at the B locus, 8 at C, about 20 at DR, 3 at DQ, and 6 at DP (Bodmer 1987). Nevertheless, if one or two alleles are much more frequent than the others, a two-allele system provides a useful approximation and leads to an accurate account for the phenotypic variation and covariation with which we are concerned. The genotype is the chromosomal complement of alleles for an individual. At a single locus (with two alleles) the genotype may be symbolized AA, Aa, or aa; if we consider multiple loci the genotype of an individual may be written as AABB, AABb, AAbb, AaBB, AaBb, Aabb, aaBB, aaBb, or aabb, in the case of two loci, for example. Homozygosity refers to a state of identical alleles at corresponding loci on homologous chromosomes; for example, AA or aa for one locus, or AABB, aabb, AAbb, or aaBB for two loci. In contrast, heterozygosity refers to a state of unlike alleles at corresponding loci, Aa or AaBb, for example. When numeric or symbolic values are assigned to specific genotypes they are called genotypic values. The additive value of a gene is the sum of the average effects of the individual alleles. Dominance deviations refer to the extent to which genotypes differ from the additive genetic value. A system in which multiple loci are involved in the expression of a single trait is called polygenic (``many genes''). A pleiotropic system (``many growths'') is one in which the same gene or genes influence more than one trait.

Biometrical models are based on the measurable effects of different genotypes that arise at a segregating locus, which are summed across all of the loci that contribute to a continuously varying trait. The number of loci generally is not known, but it is usual to assume that a relatively large number of genes of equivalent effect are at work. In this way, the categories of Mendelian genetics that lead to binomial distributions for traits in the population tend toward continuous distributions such as the normal curve. Thus, the statistical parameters that describe this model are those of continuous distributions, including the first moment, or the mean; second moments, or variances, covariances, and correlation coefficients; and higher moments such as measures of skewness where these are appropriate. This polygenic model was originally developed by Sir Ronald Fisher in his classic paper ``The correlation between relatives on the supposition of Mendelian inheritance'' (Fisher, 1918), in which he reconciled Galtonian biometrics with Mendelian genetics. One interesting feature of the polygenic biometrical model is that it predicts normal distributions for traits when very many loci are involved and their effects are combined with a multitude of environmental influences. Since the vast majority of biological and behavioral traits approximate the normal distribution, it is an inherently plausible model for the geneticist to adopt. We might note, however, that although the normality expected for a polygenic system is statistically convenient as well as empirically appropriate, none of the biometrical expectations with which we shall be concerned depend on how many or how few genes are involved. The expectations are equally valid if there are are only one or two genes, or indeed no genes at all.

In the simplest two-allele system (A and a) there are two parameters that define the measurable effects of the three possible genotypes, AA, Aa, and aa. These parameters are $d$, which is twice the measured difference between the homozygotes AA and aa, and $h$, which defines the measured effect of the heterozygote Aa, insofar as it does not fall exactly between the homozygotes. The point between the two homozygotes is $m$, the mean effect of homozygous genotypes. We refer to the parameters $d$ and $h$ as genotypic effects. The scaling of the three genotypes is shown in Figure 3.1.

Figure 3.1: The $d$ and $h$ increments of the gene difference $A$ - $a$. $Aa$ may lie on either side of $m$ and the sign of $h$ will vary accordingly; in the case illustrated $h$ would be negative. (Adapted from Mather and Jinks, 1977, p. 32).

To make the simple two-allele model concrete, let us imagine that we are talking about genes that influence adult stature. Let us assume that the normal range of height for males is from 4 feet 10 inches to 6 feet 8 inches; that is, about 22 inches. And let us assume that each somatic chromosome has one gene of roughly equivalent effect. Then, roughly speaking, we are thinking in terms of loci for which the homozygotes contribute $\pm \frac{1}{2}$ inch (from the midpoint), depending on whether they are AA, the increasing homozygote, or aa, the decreasing homozygotes. In reality, although some loci may contribute greater effects than this, others will almost certainly contribute less; thus we are talking about the kind of model in which any particular polygene is having an effect that would be difficult to detect by the methods of classical genetics. Similarly, while the methods of linkage analysis may be appropriate for a number of quantitative loci, it seems unlikely that the majority of causes of genetic variation would be detectable by these means. The biometrical approach, being founded upon an assumption that inheritance may be polygenic, is designed to elucidate sources of genetic variation is these types of systems.