  Cognitive Diagnosis Models (CDMs) are a special family of discrete latent
variable models that are widely used in modern educational, psychological,
social and biological sciences. A key component of CDMs is a binary $Q$-matrix
characterizing the dependence structure between the items and the latent
attributes. Additionally, researchers also assume in many applications certain
hierarchical structures among the latent attributes to characterize their
dependence. In most CDM applications, the attribute-attribute hierarchical
structures, the item-attribute $Q$-matrix, the item-level diagnostic model, as
well as the number of latent attributes, need to be fully or partially
pre-specified, which however may be subjective and misspecified as noted by
many recent studies. This paper considers the problem of jointly learning these
latent and hierarchical structures in CDMs from observed data with minimal
model assumptions. Specifically, a penalized likelihood approach is proposed to
select the number of attributes and estimate the latent and hierarchical
structures simultaneously. An efficient expectation-maximization (EM) algorithm
and a latent structure recovery algorithm are developed, and statistical
consistency theory is also established under mild conditions. The good
performance of the proposed method is illustrated by simulation studies and a
real data application in educational assessment.
