  Cognitive diagnosis models (CDMs) are a popular tool for assessing students'
mastery of sets of skills. Given a set of $K$ skills tested on an assessment,
students are classified into one of $2^K$ latent skill set profiles that
represent whether they have mastered each skill or not. Traditional approaches
to estimating these profiles are computationally intensive and become
infeasible on large datasets. Instead, proxy skill estimates can be generated
from the observed responses and then clustered, and these clusters can be
assigned to different profiles. Building on previous work, we consider how to
optimally perform this clustering when not all $2^K$ profiles are possible,
e.g. because of hierarchical relationships among the skills, and when not all
possible profiles are present in the population. We compare hierarchical
clustering and several k-means variants, including semisupervised clustering
using simulated student responses. The empty k-means algorithm paired with a
novel method for generating starting centers yields the best overall
performance.
