) is equal to A quantitative character is a trait which exhibits continuous variation.
Even with a relatively small number of genes involved, the variation for a character will often appear continuous, due to measurement error and, more importantly, environmental effects.
(See Figure 6.2 in F+M).
* Most characters are continuously distributed.
* We do not fully understand the genetic basis of most traits, therefore we must describe them statistically.
* Even with better understanding of the genetic basis of a trait, a statistical description of that trait can be extremely useful.
* Quantitative genetic understanding has allowed substantial gains in agricultural output.
* QG concepts has allowed new understanding of evolutionary processes.
The CLT states that the sum of a large number of independent variables is distributed approximately by a normal distribution.
The Poisson distribution with large enough mean is closely approximated by the normal distribution.
If Xi is the ith value of X, then the mean of X (or ) is equal to
The mean is also sometimes called the expected value, and as such it is written as E[X].
The variance is the expected squared deviation form the mean:
Covariance is a generalization of the concept of variance. It defines the expected value of the product of two deviations from the mean:
The variance is also equal to the difference between the expected value of the square of the Xi's minus the square of the expected value of the Xi's:
The expected value of a sum of independent values is the sum of the expected values:
,
and the variance of a sum of independent values is the sum of the variances of those values.
For sums of values which are not independent, then the expected value involves the covariance:
The mean of Xi+1 is E[X]+1. The variance of Xi+1 is Var[X].
The mean of a constant (C) times Xi is C 
. The variance of CXi is C2Var[X].
The phenotype is a function of both the genotype and the environment:
The mean (M) in a randomly mating population with p A1 alleles and q=1-p A2 alleles is
When the effects of different loci combine additively, then the overall mean is equal to the sum of the contributions of each locus.
***Special Homework problem: What is mathematical relationship between d and h in the population genetic literature?
The average effect is the mean deviation from the population mean of individuals which received that allele from on parent, when the other allele is chosen at random from the population.
[See Table 7.2 in F+M]
The average effect of a gene substitution is slightly different: this is the difference which would be caused by changing one allele in an average individual into the other allele, so that:
The average effects are also derivable from a least squares regression of phenotype on allele copy number.
The value of an individual, as measured by the average value of its offspring, is called its breeding value.
This is twice the deviation of the offspring from the population mean (since the individual only contributes half of the alleles to its offspring).
This is also the sum of the average effects of the individual.
With random mating, the mean breeding value is zero.
The genetic effects (G) can be further partitioned.
Ignoring interactions among loci, G=A+D.
In this equation, the A refers to the additive effects, which is the sum of the breeding values, and the D refers to dominance deviations.
These dominance deviations refer to the deviation of diploid genotypic values from the sum of the average effects of those genotypes, due to the interaction between alleles at the same locus.
[See Figure 7.2 in F+M.]
If different loci interact to form the phenotype, then there is epistasis, and this interaction affects the composition of phenotypes in the population.
See the table 8.1 in F+M.
Heritability: the proportion of variance which is determined by genetics.
Broad-sense heritability: VG/VP.
Narrow-sense heritability: VA/VP. *** This is the important one, generally.
The partitioning of variance follows an ANOVA kind of logic. In fact ANOVA's were invented for exactly this reason by R. A. Fisher.
Additive variance is variance of the breeding values.
The variance of the dominance deviations is
See Fig 8.1 in F+M.
The interaction (or epistatic) variance can be a complicated function of interactions across several loci. See F+M.
Having the additive variance be greater than zero does not imply that alleles interact additively. Even with complete dominance, or with complete epistasis, there will usually be additive variance. {see figure 8.1 in F+M, figure 1 in Whitlock et al. 1995 ARES 26:601-629.}
G and E can be correlated (i.e. "good" offspring being provisioned more); more importantly, there can be interaction terms.
{Graph of bristle numbers}
Breeders v. evolutionists
Lab v. field
VE v. measurement error
repeatability: correlation of repeated measures of the same individual