 Solution

Under the original assumptions the extra affected child is impossible. A generation-3 woman can have at most one copy of the unique introduced disease allele, while an unrelated outside spouse has none. Their children cannot inherit two defective copies. Thus both linkage and no-linkage likelihoods of the enlarged data are zero, and the original single-entry model has no defined updated LOD score. At least one assumption must be relaxed: another introduced disease allele, a new mutation, a phenocopy, or the relationship of the outside spouse.

If the model is enlarged to allow an independent outside genetic carrier, the reported extra information contains disease status but no new genetic marker genotypes. In the simple extension with fixed outside heterozygous-carrier probability $c$, the old disease observations force both generation-2 family ancestors to be genetic carriers. A further generation-3 sister inherits the disease copy with probability $1/2$, independently of the existing offspring transmissions. Given that she and her spouse are genetic carriers, exactly one of their three children is affected with probabilityThe additional phenotype likelihood is therefore $27c/128$. It is the same at $\theta =0$ and $\theta =1/2$, so it cancels and the LOD remains unchanged in this phenotype-only, independent-carrier extension. If the identity of the affected child is specified, the factor is $9c/128$, which also cancels. No such conclusion applies automatically if new genetic marker data are supplied or the revised pedigree founder model couples this branch to the old genetic marker evidence.

In general an enlarged data set changes the score byafter choosing a coherent model with positive likelihood. The additional affected child alone does not justify assigning it the autozygosity evidence of an affected child of a cousin marriage.