= Solution
The <LOD score> is the base-ten log <likelihood> ratio against independent assortment. For the two families,
$$
\boxed{Z_{\max}=\log_{10}\frac{L_1(\widehat\theta)L_2(\widehat\theta)}{L_1(1/2)L_2(1/2)},\qquad0\leq\widehat\theta\leq\tfrac12.}
$$
For a known-phase first family this becomes
$$
Z_{\max}=r_1\log_{10}(2\widehat\theta)
+(m_1-r_1)\log_{10}(2(1-\widehat\theta))
+\log_{10}\frac{L_2(\widehat\theta)}{L_2(1/2)},
$$
with endpoint terms interpreted by limits, such as $0\log0=0$ when the corresponding count is zero. Positive values favour <genetic linkage>, and the <likelihood> ratio is $10^{Z_{\max}}$. A conventional large positive threshold such as 3 represents a <likelihood> ratio of 1000, not automatically a posterior <probability> or a universal 5% test.
For a test calibrated to this study, use the distribution of the maximized statistic under $\theta=1/2$, for example by simulating transmissions conditional on the same parental <genotype> and ascertainment scheme. Reject for sufficiently large values with the chosen significance threshold; phase uncertainty and the boundary null prevent assuming an unqualified ordinary interior Wilks reference. The missing pedigrees prevent the numerical value or their explicit <likelihood> polynomial from being supplied, but the likelihood-ratio definition and testing procedure apply once those data are recovered.
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