# ABO allele frequencies: Why use the EM algorithm?

In textbooks and lecture notes and slides posted online, determining allele frequencies using blood type information (ABO), under the assumption of Hardy-Weinberg equilibrium, is accomplished using the EM algorithm. It seems to me, though, that this problem can be solved using basic algebra: If $$a$$, $$b$$, $$o$$ are the frequencies of the $$A$$, $$B$$, and $$O$$ alleles, respectively, and $$p_A$$, $$p_B$$, and $$p_O$$ are the proportions of the population in question displaying blood types $$A$$, $$B$$, and $$O$$, then

$$p_A = a^2 + 2ao,\quad p_B = b^2 + 2bo,\quad\text{and}\quad p_O = o^2.$$

Inverting the third equation gives $$o$$ in terms of $$p_O$$:

$$o = \sqrt{p_O}.$$

Plugging that into the first equation and rearranging, we get the quadratic equation.

$$a^2 + 2\sqrt{p_O}a - p_A = 0$$

Solving using the quadratic formula and throwing away the negative solution, we get

$$a = \sqrt{p_O + p_A} - \sqrt{p_O}.$$

Symmetrically,

$$b = \sqrt{p_O + p_B} - \sqrt{p_O}.$$

Thus, we've solved for the allele frequencies $$a$$, $$b$$, and $$o$$ with only basic algebra.

In the standard textbook case in which $$p_A = fa = 186/521$$, $$p_B = 38/521$$, and $$p_O = 284/521$$, this gives

$$a \approx 0.21,\quad b\approx0.05, \quad\text{and}\quad o\approx0.74,$$

which is close to what you get after a few iterations of the EM algorithm.

Question (finally): If the above calculation is correct (if not, please let me know!), what is a "simplest" non-synthetic (real data, from the literature) example of an allele frequency computation from phenotype data that actually requires a sophisticated technique like the EM algorithm?