 Solution

The LOD score is the base-ten log likelihood ratio against independent assortment. For the two families,For a known-phase first family this becomeswith endpoint terms interpreted by limits, such as $0\log 0=0$ 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.