Maximum likelihood estimation based DINA model and Q-matrix learning

Cognitive diagnosis models (CDMs) are of growing interest in test development and in the measurement of human performance. The DINA (deterministic input, noisy, and gate) model has been widely used in cognitive diagnosis tests and in the process of test development. Central to many such models, the well-known Q-matrix [1], which specifies the item-attribute relationships and two noise parameters in DINA model related to item response functions are termed as slip and guessing. Slip parameter indicates a student with mastery of all of the attributes that an item requires fails to answer the item correctly. In contrast, guessing parameter indicates a student lacks the attributes that are required by an item but succeeds to answer the item correctly. In this paper, we developed a new method and presented an alternate recursive algorithm to learn Q-matrix and uncertainty variables slip and guessing based on Boolean Matrix Factorization (BMF) and Maximum Likelihood Estimation (MLE) respectively for DINA model of CDM. In particularly, we spontaneously transferred the deterministic Q-matrix learning problem into BMF problem. Because BMF is an NP-hard problem [2], we proposed an alternate recursive method to find approximate solution by adding one dimension in attribute latent space in each step. On the other hand, we analytically estimated the slip and guessing parameters through the maximum likelihood of uncertainty variables. The optimum process is alternate recursive between latent attribute space and uncertainty variable space. Simulation results show that the MLE Q-matrix learning algorithm has fast convergence to the optimal solution under suitable initial values of Q_init and A_init. This is extremely important and applicable when the method is extended to big data.