  We propose two probability-like measures of individual cluster-membership
certainty which can be applied to a hard partition of the sample such as that
obtained from the Partitioning Around Medoids (PAM) algorithm, hierarchical
clustering or k-means clustering. One measure extends the individual silhouette
widths and the other is obtained directly from the pairwise dissimilarities in
the sample. Unlike the classic silhouette, however, the measures behave like
probabilities and can be used to investigate an individual's tendency to belong
to a cluster. We also suggest two possible ways to evaluate the hard partition.
We evaluate the performance of both measures in individuals with ambiguous
cluster membership, using simulated binary datasets that have been partitioned
by the PAM algorithm or continuous datasets that have been partitioned by
hierarchical clustering and k-means clustering. For comparison, we also present
results from soft clustering algorithms such as soft analysis clustering
(FANNY) and two model-based clustering methods. Our proposed measures perform
comparably to the posterior-probability estimators from either FANNY or the
model-based clustering methods. We also illustrate the proposed measures by
applying them to Fisher's classic iris data set.
