10 Sampling error

The model is able to handle most forms of sampling error. Typically, only a proportion of individuals are sampled in each subpopulation. The most realistic assumption is to assume that the number of copies of, say allele $A$, conditional on the allele frequency $p_i$ in subpopulation $i$ follow a hypergeometric distribution. If we have hypergeometric sampling of $N_i^{(s)}$ individuals from a finite population of $N_i^{(h)}$ individuals, then the conditional variance in the sampled gene frequency is

$$\operatorname{Var}(p_i'\vert p_i)=\frac{2N_i^{(h)}-2N_i^{(s)}}{2N_i^{(s)}(2N_i^{(h)}-1)}p_i(1-p_i).$$ (5)

The unconditional standardized variance of the sampled gene frequency is then

$$\operatorname{Var}(p_i) = \frac{1}{2N_i^*}+(1-\frac1{2N_i^*})c_{ii},$$ (6)

where

$$\frac{1}{2N_i^*} = \frac{2N_i^{(h)}-2N_i^{(s)}}{2N_i^{(s)}(2N_i^{(h)}-1)}.$$ (7)

This simplifies to the more standard binomial model when $N_i^{(h)} \rightarrow \infty$. Also, as $N_i^{(s)} \rightarrow \infty$ there sampling variance tends to zero.

To set up the model with sampling error, the two vectors Ns and Nh containing the appropriate sample sizes in each subpopulation (corresponding to $N_i^{(s)}$ and $N_i^{(h)}$, respectively) should be defined globally. The diagonal elements of the computed covariance matrix are then adjusted according to (6), before evaluating the likelihood of the data. In addition, in the simulate procedure, sampling error is added at the end of each simulated realization of the process. Sample sizes differing between loci is currently not handled. Assigning the value Inf to all elements of Nh results in binomial sampling. The model is set up with no sampling error by assigning a NA to Ns or by assigning Inf to some (or all) of the elements.