## Of Average Effects, Average Excesses and Additive Variance

Today’s post is a bit of a CYA (cover your a__) post.  I once got in a bit of a kerfuffle with Warren Ewens over my loose use of the concept of additive genetic variance.  It was a little like me bringing a knife to a gunfight.  I was hopelessly outclassed by his intellect, and he appropriately schooled me.  This is my attempt to make sure that people like Ewens understand that I now am aware of some of the technical issues surrounding additive genetic variance.

Sir Ronald Fisher defined the additive genetic variance to be the covariance between the average effect of allele and the average excess.  When mating is random, and everything is in linkage equilibrium this rather esoteric definition is equal to four times the similarity among half sibs, and it is predictive of the response to selection.  In other words, Fisher, very elegantly made a very precise definition of additive genetic variance that has many important implications for (his) real world.

So, it is worth briefly defining the average effect and the average excess.  The genotypic value is the average phenotype of a particular genotype averaged across all of the genetic and environmental variants it might experience.  Thus, any given individual will have a phenotype that is different than its genotypic value due to unexplained causes (usually called environment), but the members of a particular genotype will average out to the genotypic value.  Thus, for a one locus two allele system you could get A1A1 = g11, A1A2=g12, and A2A2=g22.

We can now define the average effect of a gene substitution to be the average change in phenotype of an individual if a random allele is replaced by the allele in question.  It is usually measured as a deviation from the population mean, but that doesn’t actually affect the calculations.  Thus, the average effect of the A1 allele can be calculated by taking the average of a substitution into all of the possible genotypes:

A1A1 => A1A1 = g11 – g11 = 0

A1A2 => ½ A1A2 + ½ A1A1 = ½ (g12 – g12) + ½ (g11 – g12) = ½ (g11 – g12)

A2A2 => A1A2 = g12 – g22

The average effect of the A1 allele is then just the weighted average of these changes.

On the other hand, the average excess is the average phenotype measured as a deviation from the population mean (and here it is important) of an individual that contains a particular allele.  Thus, the average excess of the A1 allele is the weighted average of two times the A1A1 genotype (because there are two A1 alleles), i.e., 2(g11 – M), and the A1A2 genotype, (g12 – M) where M is the population mean phenotype.

Oddly enough if the population is in multilocus HW equilibrium and there are no genotype environment interactions or correlations the average effect is exactly equal to the average effect.  Yea, I didn’t believe it either, but it does work.  So, in our perfect world the additive genetic variance equals the covariance between average excesses and the average effects, equals the variance in average effects, equals four times the variance among half sibs, equals the twice the covariance between parents and offspring, equals the response to selection.  It’s amazing.

The difference between the average effect of an allele and the average excess of an allele is that the average effect measures the effect of an allele averaged over all genotypes based on their expected frequency, whereas the average excess measures the effect of an allele in the associations it is actually found in.

The problem comes when mating is not random and there is linkage disequilibrium. In this case the average effect and the average excess are not equal, and the relationship among the different measures is broken.  Falconer tried to work this out and ultimately declared that when average excesses did not equal average effects efforts to measure the additive genetic variance were “doomed to failure” (Falconer 1985 Genetical Research Cambridge 46: 337-347).  In short, when the average excess and the average effect are not equal, which is probably always in the real world, the additive genetic variance is an unmeasurable statistic with no predictive value.  Good thing its central to quantitative genetic theory, huh?

So, what to do?  Ewens insists to me that since Fisher defined the additive genetic variance that is the correct definition, and we have to give it to him.  My thought is that we should do, perhaps with circular overtones, is to drop the “genetic” and define the “effective additive variance” (eVA) to be the variance that can contribute to a response to individual selection.

If we define effective eVA as the variance that can result in a response to selection just about everything heritable can be called “eVA”, including things that are decidedly NOT genetic.  One solution is to restrict the eVA to genetic components of the patterning node.  This takes care of perhaps one of the biggest problems with Fisher’s definition, which is that mating is not random.  There actually is one study that took this into account (Tonsor and Goodnight 1997.  Evolution 51: 1773-1784).  This study is interesting in that we did a standard half-sib mating design, however, instead of choosing mates from a uniform random distribution, we chose them randomly from an exponential distribution that had the same shape as the pollen flow distribution.  This modification had virtually no effect on the eVA, but that may have been the result of the choice of using a wide-spread weed.  However, it does not solve the problem that for any phenotypically based measure it will be virtually impossible to definitively distinguish “genetic” from “epigenetic” and “non-genetic”.  Thus, at this point we should probably punt, call it “effective additive variance” rather than “effective additive genetic variance”, and recognize that anything that contributes to the resemblance between parents and offspring will potentially get pooled into the effective additive variance.

This actually presents an interesting issue, since we know that different forms of inheritance have very different fidelities, for want of a better term.  That is the fidelity of nuclear DNA is tremendous, as evidenced by genetic diseases that can run in families for generations.  Conversely a single disruption of culture (such as a newborn getting adopted) can destroy a legacy of cultural inheritance.  Using this reasoning, it becomes obvious to me that it is still important to use clever experimental designs to divide the effective additive variance into components, such as effective nuclear genetic additive variance, effective maternal additive variance, effective epigenetic additive variance, and so on.