Phenotypic Values

Phenotypic values are calculated from the genotypic values for each individual for each trait. Each individual's phenotypic value is calculated from its genotypic value with an environmental effect determined by the heritability $h^2$. The individual's genotypic value is based on the alleles it inherited at the quantitative trait loci. To calculate genetic values, we use Cockerham's general genetic model [CockerhamCockerham1954].

$$G_i = \sum_{r=1}^m a_r x_{ir} + \sum_{r=1}^m d_r z_{ir} + \sum_{r\neq s} b^{AA}_{rs} x_{ir} x_{is} + \sum_{r\neq s} b^{AD}_{rs} x_{ir} z_{is}$$

$$\mbox{} + \sum_{r\neq s} b^{DA}_{rs} z_{ir} x_{is} + \sum_{r\neq s} b^{DD}_{rs} z_{ir} z_{is}$$ (2.8)

The parameters $a_r, d_r$ are the additive and dominance effects of QTL $r$. The $b$'s are epistatic interactions. The superscripts on the $b$'s are for the type of interaction: We distinguish between additive by additive (AA), additive by dominance (AD), dominance by additive (DA) and dominance by dominance (DD) interactions. The $x$ and $z$ are coded variables denoting the genotype of the QTL. The $x_{ir}$ take on values $(1, 0, -1)$ for QTL genotypes $(QQ, Qq, qq)$, while the $z_{ir}$ are $1/2$ for heterozygotes and $-1/2$ for homozygotes.

This results in a vector of genotypic values, one entry per individual in the simulated data set. The genetic variance is the sample variance of this vector of genotypic values. Call it $\sigma^2_g$. The environmental variance, $\sigma^2_e$ is defined by

$$\sigma^2_e =\sigma^2_g ( \frac{1}{h^2} - 1)$$ (2.9)

where $h^2$ is the heritability of the trait. The extra environmental effect is taken from a normal distribution with mean 0 and variance $\sigma^2_e$. If the environmental variance is specified, the heritability is ignored and the environmental variance is used directly. For each individual in the data set, a random variable with mean zero and variance $\sigma^2_e$ is generated and added to the genotypic value. This is the phenotypic value of that individual, and is printed in the output file.