Testing Hardy-Weinberg

One approach to testing the hypothesis that genotypes are in Hardy-Weinberg proportions is quite simple. We can simply do a $\chi^2$ or $G$-test for goodness of fit between observed and predicted genotype (or phenotype) frequencies, where the predicted genotype frequencies are derived from our estimates of the allele frequencies in the population.¹ There's only one problem. To do either of these tests we have to know how many degrees of freedom are associated with the test. How do we figure that out? In general, the formula is

$$\begin{eqnarray*} \hbox{d.f.} = && (\hbox{\char93 of categories in the data -1 ... ... (\hbox{\char93 number of parameters estimated from the data}) \end{eqnarray*}$$

For this problem we have

$$\begin{eqnarray*} \hbox{d.f.} = && (\hbox{\char93 of phenotype categories in th... ... (\hbox{\char93 of allele frequencies estimated from the data}) \end{eqnarray*}$$

In the ABO blood group we have 4 phenotype categories, and 3 allele frequencies. That means that a test of whether a particular data set has genotypes in Hardy-Weinberg proportions will have $(4-1)-(3-1) = 1$ degrees of freedom for the test. Notice that this also means that if you have completely dominant markers, like RAPDs or AFLPs, you can't determine whether genotypes are in Hardy-Weinberg proportions because you have 0 degrees of freedom available for the test.