  Cognitive Diagnosis Models (CDMs) are a special family of discrete latent
variable models widely used in educational, psychological and social sciences.
In many applications of CDMs, certain hierarchical structures among the latent
attributes are assumed by researchers to characterize their dependence
structure. Specifically, a directed acyclic graph is used to specify
hierarchical constraints on the allowable configurations of the discrete latent
attributes. In this paper, we consider the important yet unaddressed problem of
testing the existence of latent hierarchical structures in CDMs. We first
introduce the concept of testability of hierarchical structures in CDMs and
present sufficient conditions. Then we study the asymptotic behaviors of the
likelihood ratio test (LRT) statistic, which is widely used for testing nested
models. Due to the irregularity of the problem, the asymptotic distribution of
LRT becomes nonstandard and tends to provide unsatisfactory finite sample
performance under practical conditions. We provide statistical insights on such
failures, and propose to use parametric bootstrap to perform the testing. We
also demonstrate the effectiveness and superiority of parametric bootstrap for
testing the latent hierarchies over non-parametric bootstrap and the na\"ive
Chi-squared test through comprehensive simulations and an educational
assessment dataset.
