A restricted latent class model is a family of latent variable models with broad applications in psychological and educational assessment, where the model is restricted via a latent matrix to reflect pre-specified assumptions on latent attributes. In this dissertation, I focus on the design and diagnosis of such models. First, the latent structure is often provided by experts and assumed to be correct upon construction, which may be subjective and misspecified. Recognizing this problem, I establish identifiability conditions that ensure the estimability of the structure matrix. With the theoretical development, a likelihood-based method is proposed to estimate and update the latent structure from the data. Second, it is usually assumed in cognitive diagnosis models, a group of such latent class models, that test items require mastery of specific skills – represented by latent attributes – and that each is either fully mastered or not by a subject. As a consequence, the concept of partial mastery may not be well accounted for. I propose a new class of models, partial mastery CDMs (PM-CDMs). This class generalizes both CDMs by allowing for partial mastery and mixed membership models by specifying mixed membership for each latent attribute dimension. I demonstrate that PM-CDMs can be represented as restricted latent class models and propose a Bayesian approach for estimation. Simulation studies show that the proposed method outperforms the existing approaches in latent structure estimation and the PM-CDM is able to investigate the impact of model misspecification with respect to partial mastery. I illustrate these methods through data in educational assessment. Latent structure estimation results provide interpretable results on the fraction subtraction data and the English test data demonstrate a case where PM-CDM improves the model-fit of the classical model settings.