Estimating relative viability

To estimate absolute viabilities, we have to be able to identify genotypes non-destructively. That's fine if we happen to be dealing with an experimental situation where we can do controlled crosses to establish known genotypes or if we happen to be studying an organism and a trait where we can identify the genotype from the phenotype of a zygote (or at least a very young individual) and from surviving adults.² What do we do when we can't follow the survival of individuals with known genotype? Give up?³

Remember that to make inferences about how selection will act, we only need to know relative viabilities, not absolute viabilities.⁴ We still need to know something about the genotypic composition of the population before selection, but it turns out that if we're only interested in relative viabilities, we don't need to follow individuals. All we need to be able to do is to score genotypes and estimate genotype frequencies before and after selection. Our data looks like this:

Genotype	$A_1A_1$	$A_1A_2$	$A_2A_2$
Frequency in zygotes	$x_{11}^{(z)}$	$x_{12}^{(z)}$	$x_{22}^{(z)}$
Frequency in adults	$x_{11}^{(a)}$	$x_{12}^{(a)}$	$x_{22}^{(a)}$

We also know that

$$\begin{eqnarray*} x_{11}^{(a)} &=& w_{11}x_{11}^{(z)}/\bar w \\ x_{12}^{(a)} &=... ...)}/\bar w \\ x_{22}^{(a)} &=& w_{22}x_{22}^{(z)}/\bar w \quad . \end{eqnarray*}$$

Suppose we now divide all three equations by the middle one:

$$\begin{eqnarray*} x_{11}^{(a)}/x_{12}^{(a)} &=& w_{11}x_{11}^{(z)}/w_{12}x_{12}^... ...}/x_{12}^{(a)} &=& w_{22}x_{22}^{(z)}/w_{12}x_{12}^{(z)} \quad , \end{eqnarray*}$$

or, rearranging a bit

$$\begin{eqnarray*} \frac{w_{11}}{w_{12}} &=& \left(\frac{x_{11}^{(a)}}{x_{12}^{(a... ...\right) \left(\frac{x_{12}^{(z)}}{x_{22}^{(z)}}\right) \quad . \end{eqnarray*}$$

This gives us a complete set of relative viabilities.

Genotype	$A_1A_1$	$A_1A_2$	$A_2A_2$
Relative viability	$\frac{w_{11}}{w_{12}}$	1	$\frac{w_{22}}{w_{12}}$

If we use the maximum-likelihood estimates for genotype frequencies before and after selection, we obtain maximum likelihood estimates for the relative viabilities.⁵ If we use Bayesian methods to estimate genotype frequencies (including the uncertainty around those estimates), we can use these formulas to get Bayesian estimates of the relative viabilities (and the uncertainty around them).