In recent decades, cultural consensus theory (CCT) models have been extensively applied across research domains to explore shared cultural knowledge and beliefs. These models are parameterized in terms of person-specific cognitive parameters such as abilities and guessing biases as well as item difficulties. Although psychometric test theory is also formalized in terms of abilities and item difficulties, a quality that clearly sets CCT models apart from other test theory models is their specification to operate on data in which the answer key is latent. In doing so, CCT models specify the answer key as parameters of the model, and also involved with this specification are procedures to verify the integrity of the answer key that is estimated. In this article, the authors develop the following methods to propagate the application of these CCT models in the field of social surveys: (1) by extending the underlying cognitive model to be able to account for uncertainty in decision making ("don’t know" responses), (2) by allowing covariate information to be entered in the analysis, and (3) by deriving statistical inference in the hierarchical Bayesian framework. The proposed model is fit to data describing knowledge on science and on aging to demonstrate the novel findings that can be achieved by the approach.