  Cognitive diagnosis models (CDMs) are a family of discrete latent attribute
models that serve as statistical basis in educational and psychological
cognitive diagnosis assessments. CDMs aim to achieve fine-grained inference on
individuals' latent attributes, based on their observed responses to a set of
designed diagnostic items. In the literature, CDMs usually assume that items
require mastery of specific latent attributes and that each attribute is either
fully mastered or not mastered by a given subject. We propose a new class of
models, partial mastery CDMs (PM-CDMs), that generalizes CDMs by allowing for
partial mastery levels for each attribute of interest. We demonstrate that
PM-CDMs can be represented as restricted latent class models. Relying on the
latent class representation, we propose a Bayesian approach for estimation. We
present simulation studies to demonstrate parameter recovery, to investigate
the impact of model misspecification with respect to partial mastery, and to
develop diagnostic tools that could be used by practitioners to decide between
CDMs and PM-CDMs. We use two examples of real test data -- the fraction
subtraction and the English tests -- to demonstrate that employing PM-CDMs not
only improves model fit, compared to CDMs, but also can make substantial
difference in conclusions about attribute mastery. We conclude that PM-CDMs can
lead to more effective remediation programs by providing detailed
individual-level information about skills learned and skills that need to
study.
