Conditional dependence among items in the DINA model: Application of the multivariate probit model

Abstract

The deterministic input, noisy ``and'' gate (DINA) model is a tractable and interpretable cognitive diagnosis model that can be used to identify students' mastery and nonmastery of skills in a subject domain of interest. We introduce the multivariate probit DINA (DINA-MP) model, which is a re-formulation of the DINA model that can account for potential relationships between test items that may remain even after conditioning on the students' underlying skills. A computationally efficient parameter-expanded Monte Carlo EM (PX-MCEM) algorithm is outlined for maximum likelihood estimation of the parameters of the proposed model. A simulation study is conducted to determine the extent to which ignoring between-item conditional dependence may yield biased estimates with understated standard errors, thus yielding confidence intervals that are misleadingly narrow and tests with inflated Type I errors. Finally, fraction subtraction data are analyzed to examine the practical viability of the DINA-MP model and the corresponding PX-MCEM algorithm.