# Phenotypic variance

I’ve intermittently been reading through William Provine’s *The Origins of Theoretical Population Genetics*. It’s related to a project simmering on my back burner.

Meanwhile, last week I was talking with some students about the recent papers at the AAPA meetings about natural selection as assessed by quantitative traits. The students thought that some of these papers had omitted some basic details that seemed obvious from the point of view of quantitative genetics. Also, George Armelagos had mentioned Raymond Pearl, so I figured as long as I’m reading about Pearl, William Castle, R. A. Fisher and their attitudes toward quantitative genetics, I might as well note a few passages from Provine’s account.

Provine:

Fisher's express purpose in the paper was to interpret the well-established results of biometry in terms of Mendelian inheritance by ascertaining the biometrical properties of a Mendelian population.

I’ll just pause to note that Fisher’s formulation begins almost all textbooks in quantitative genetics and many in population genetics. The model that relates quantitative variation and genotypic variation is essential to all genetic analysis.

In particular, he wanted to show that Pearson was mistaken in concluding that the correlations between relatives in man contradicted the Mendelian scheme of inheritance. He began by defining a measure of the variability of a character in a population.

This is an essential step for any introduction to genetics also. I spend some time in all my courses talking about the relationship between genetic and phenotypic variation, using the measures of each as ways to talk about the ways they differ. We can analogize genetic variation to a digital readout – you have a genotype, or a set of genotypes, and the population’s variation has to do with the frequencies of those genotypes or the alleles that comprise them. So the variation is something that emerges from *counting genes*. You have **heterozygosity** (expected frequency of heterozygous genotypes), or **number of alleles**. At the sequence level, you count both alleles and the number of mutations that separate them – **average pairwise difference**, **number of segregating sites**.

Back to Provine:

Often the standard deviation σ was used for this purpose. But Fisher noted that

Now Provine gives a direct quote from Fisher 1918:

when there are two independent sources of variability capable of producing in an otherwise uniform population distributions with standard deviations σ_{1}and σ_{2}, it is found that the distribution, when both causes act together, has a standard deviation σ_{1}^{2}+ σ_{2}^{2}. It is therefore desirable in analysing the causes of variability to deal with the square of the standard deviation as the measure of variability. We shall term this quantity the Variance of the normal population to which it refers, and we may now ascribe to the constituent causes fractions or percentages of the total variance which they together produce (Fisher 1918:399).

I have always thought that this was a work of magic by Fisher. The additive quality of variance is such a useful characteristic for a measure of variation, it’s hard to imagine using anything else. Fisher continues:

For stature the coefficient of correlation between brothers is about .54, which we may interpret by saying that 54 per cent of their variance is accounted for by ancestry alone, and that 46 per cent must have some other explanation.

It is not sufficient to ascribe this last residue to the effects of environment. Numerous investigations by Galton and Pearson have shown that all measurable environment has much less effect on such measurements as stature. Further, the facts collected by Galton respecting identical twins show that in this case, where the essential nature is the same, the variance is far less. The simplest hypothesis, and the one which we shall examine, is that such features as stature are determined by a large number of Mendelian factors, and that the large variance among children of the same parents is due to the segregation of those factors in respect to which the parents are heterozygous. Upon this hypothesis we will attempt to determine how much more of the variance, in different measurable features, beyond that which is indicated by the fraternal correlation, is due to innate and heritable factors (Fisher 1918:400).

And that, in a nutshell, is why the *correlation* between relatives is not a measure of heritability. Fisher attempted to show that the segregation of Mendelian factors could account for a large fraction of the variance of stature, and substantially succeeded in showing that the environment had much less impact than had been assumed from the correlation between relatives.

Provine’s discussion continues along a different line, but he includes the characteristic line:

Fisher's 1918 paper was well received by the few geneticists who could understand his mathematics (147).