An example

Let's see how this works with some real data from Dobzhansky's work on chromosome inversion polymorphisms in Drosophila pseudoobscura.⁶

Genotype	$ST/ST$	$ST/CH$	$CH/CH$	Total
Number in larvae	41	82	27	150
Number in adults	57	169	29	255

You may be wondering how the sample of adults can be larger than the sample of larvae. That's because to score an individual's inversion type, Dobzhansky had to kill it. The numbers in larvae are based on a sample of the population, and the adults that survived were not genotyped as larvae. As a result, all we can do is to estimate the relative viabilities.

$$\begin{eqnarray*} \frac{w_{11}}{w_{12}} &=& \left(\frac{x_{11}^{(a)}}{x_{12}^{(a... ...9/255}\right) \left(\frac{82/150}{27/150}\right) = 0.52 \quad . \end{eqnarray*}$$

So it looks as if we have balancing selection, i.e., the fitness of the heterozygote exceeds that of either homozygote.

We can check to see whether this conclusion is statistically justified by comparing the observed number of individuals in each genotype category in adults with what we'd expect if all genotypes were equally likely to survive.

Genotype	$ST/ST$	$ST/CH$	$CH/CH$
Expected	$\left(\frac{41}{150}\right)255$	$\left(\frac{82}{150}\right)255$	$\left(\frac{27}{150}\right)255$
	69.7	139.4	45.9
Observed	57	169	29
$\chi^2_2 = 14.82$, $P < 0.001$

So we have strong evidence that genotypes differ in their probability of survival.

We can also use our knowledge of how selection works to predict the genotype frequencies at equilibrium:

$$\begin{eqnarray*} \frac{w_{11}}{w_{12}} &=& 1 - s_1 \\ \frac{w_{22}}{w_{12}} &=& 1 - s_2 \quad . \end{eqnarray*}$$

So $s_1 = 0.33$, $s_2 = 0.48$, and the predicted equilibrium frequency of the $ST$ chromosome is $s_2/(s_1+s_2) = 0.59$.

Now all of those estimates are maximum-likelihood estimates. Doing these estimates in a Bayesian context is relatively straightforward and the details will be left as an excerise.⁷ In outline we simply

1. Estimate the gentoype frequencies before and after selection as samples from a multinomial.

2. Apply the formulas above to calculate relative viabilities and selection coefficients.

3. Determine whether the 95% credible intervals for $s_1$ or $s_2$ overlap 0.⁸

4. Calculate the equilibrium frequency from $s_2/(s_1+s_2)$, if $s_1 > 0$ and $s_2 > 0$. Otherwise, determine which fixation state will be approached.

In the end you then have not only viability estimates and their associated uncertainties, but a prediction about the ultimate composition of the population, associated with an accompanying level of uncertainty.