Posterior predictive model checking for multidimensionality in item response theory and Bayesian networks

If data exhibit a dimensional structure more complex than what is assumed, key conditional independence assumptions of the hypothesized model do not hold. The current work pursues posterior predictive model checking, a flexible family of Bayesian model checking procedures, as a tool for criticizing models in light of inadequately modeled dimensional structure. Factors hypothesized to influence dimensionality and dimensionality assessment are couched in conditional covariance theory and conveyed via geometric representations of multidimensionality. These factors and their hypothesized effects motivate a simulation study that investigates posterior predictive model checking in the context of item response theory for dichotomous observables. A unidimensional model is fit to data that follow compensatory or conjunctive multidimensional item response models to assess the utility of conducting posterior predictive model checking. Discrepancy measures are formulated at the level of individual items and pairs of items. A second study draws from the results of the first study and investigates the model checking techniques in the context of multidimensional Bayesian networks with inhibitory effects. Key findings include support for the hypothesized effects of the manipulated factors with regard to their influence on dimensionality assessment and the superiority of certain discrepancy measures for conducting posterior predictive model checking on dimensionality assessment. The application of these techniques to models both familiar to assessment and those that have not yet become standard practice speaks to the generality of the procedures and its potentially broad applicability.