= Solution
Take the affected sib pairs from independent families with the same disease model. Conditional on their marker sharing states, multiply the <recessive affected-sib-pair likelihoods>:
$$
\mathcal L(\theta)=L_0^{n_0}L_1^{n_1}L_2^{n_2}.
$$
Write $x=\sqrt{L_0}=\theta(1-\theta)$, so that $\sqrt{L_2}=1/2-x$ and $L_1=x(1/2-x)$. Consequently
$$
\boxed{\mathcal L(\theta)=x^{\,2n_0+n_1}\left(\frac12-x\right)^{2n_2+n_1},\qquad 0\le x\le\frac14.}
$$
For an unordered count sample there can also be a <multinomial likelihood> coefficient, and the Mendelian <probabilities> of the observed marker sharing states supply further factors. All these factors are independent of $\theta$.
If <genetic-study ascertainment> explicitly conditions on both children being affected, the normalizing disease <probability> is constant:
$$
P(\text{both affected})=\frac14L_0+\frac12L_1+\frac14L_2=\frac{(x+1/2-x)^2}{4}=\frac1{16}.
$$
Thus the conditional sharing <likelihood> differs from the displayed <likelihood function> only by factors independent of the <recombination fraction>. Multiple pairs drawn from the same larger sibship would not, in general, justify treating these pair <likelihoods> as independent.
Back to article page
