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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?

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