Nonlinear penalized estimation of true Q-matrix in cognitive diagnostic models

A key issue of cognitive diagnostic models (CDMs) is the correct identification of Q-matrix which indicates the relationship between attributes and test items. Previous CDMs typically assumed a known Q-matrix provided by domain experts such as those who developed the questions. However, misspecifications of Q-matrix had been discovered in the past studies. The primary purpose of this research is to set up a mathematical framework to estimate the true Q-matrix based on item response data. The model considers all Q-matrix elements as parameters and estimates them through EM algorithm. Two simulation designs are conducted to evaluate the feasibility and performance of the model. An empirical study is addressed to compare the estimated Q-matrix with the one designed by experts. The results show that the model performs well and is able to identify 60% to 90% of correct elements of Q-matrix. The model also indicates possible misspecifications of the designed Q-matrix in the fraction subtraction test.