= Solution
The output describes <complete-linkage clustering>: the distance between two current groups is the largest original <Jaccard distance> between a member of one and a member of the other. Starting with singleton groups, <agglomerative hierarchical clustering> merges a closest pair at each step. In the merge array, a negative index denotes the original row of that number, and a positive index denotes the group created at that earlier merge. A printed height is the dissimilarity at which the new group forms.
Reading those references recursively gives the following <dendrogram> construction, with exact heights recovered from the binary profiles:
* Fred and Gbenga join at $0$; Graham and Tim join at $1/8$; Chad and Nicolas join at $1/6$.
* Frederic joins Chad–Nicolas at $1/3$, and John then joins that group at the same height $1/3$. These tied heights produce successive branches on the same horizontal level.
* Philip and Mark join at $3/8$; Garfield joins Fred–Gbenga at $2/5$.
* Graham–Tim joins Philip–Mark at $4/9$; Chad–Nicolas–Frederic–John joins Garfield–Fred–Gbenga at $1/2$.
* Sauli joins Graham–Tim–Philip–Mark at $5/8$, and Juliet joins that group at $3/4$.
* The two remaining groups join at $7/8$.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-42-dendrogram.png]
{title=Complete-linkage dendrogram reconstructed from exact binary-profile distances}
For example, a cut strictly between $1/2$ and $5/8$ gives four groups: Chad–Nicolas–Frederic–John–Garfield–Fred–Gbenga, Graham–Tim–Philip–Mark, Sauli, and Juliet. A cut strictly between $3/4$ and $7/8$ gives two larger groups. \b[The leaf order is not intrinsic; the merge memberships and heights are.] Reflecting branches leaves the <dendrogram> unchanged in meaning. The zero-height pair indicates identical recorded responses, while high joins indicate comparatively dissimilar profiles. These conclusions depend on the choice of <Jaccard distance>, which ignores shared negative answers; the <dendrogram> itself does not establish substantive demographic classes.
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