Partitioning the phenotypic variance

Before we worry about how to estimate any of those variance components I just mentioned, we first have to understand what they are. So let's start with some definitions (Table 1).⁷

Table 1: Fundamental parameter definitions for quantitative genetics with one locus and two alleles.

Genotype	$A_1A_1$	$A_1A_2$	$A_2A_2$
Frequency	$p^2$	$2pq$	$q^2$
Genotypic value	$x_{11}$	$x_{12}$	$x_{22}$
Additive genotypic value	$2\alpha_1$	$\alpha_1 + \alpha_2$	$2\alpha_2$

You should notice something rather strange about Table 1 when you look at it. I motivated the entire discussion of quantitative genetics by talking about the need to deal with variation at many loci, and what I've presented involves only two alleles at a single locus. I do this for two reasons:

1. It's not too difficult to do the algebra with multiple alleles at one locus instead of only two, but it get's a little messy, and I'd rather avoid the mess.

2. Doing the algebra with multiple loci involves a lot of assumptions, which I'll mention when we get to applications, and the algebra is even worse than with multiple alleles at a single locus.

Fortunately, the basic principles extend with little modification to multiple loci, so we can see all of the underlying logic by focusing on one locus with two alleles where we have a chance of understanding what the different variance components mean.

Two terms in Table 1 will almost certainly be unfamiliar to you: genotypic value and additive genotypic value. Of the two, genotypic value is the easiest to understand (Figure 1). It simply refers to the average phenotype associated with a given genotype.⁸ The additive genotypic value refers to the average phenotype associated with a given genotype, as would be inferred from the additive effect of the alleles of which it is composed. That didn't help much, did it? That's because I now need to tell you what we mean by the additive effect of an allele.⁹

Figure 1: The phenotype distribution in a population in which the three genotypes at a single locus with two alleles occur in equal frequency. The $A_1A_1$ genotype has a mean trait value of 1, the $A_1A_2$ genotype has a mean trait value of 2, and the $A_2A_2$ genotype has a mean trait value of 3, but each genotype can produce a range of phenotypes with the standard deviation of the distribution being 0.25 in each case.