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Q-matrix is a matrix describing relations of questions and concepts required for their understanding. It is a domain-independent model of knowledge represented by a binary matrix showing the relationship between test items and latent or underlying attributes, or concepts.
Q-matrix is a MxN matrix, where M equals the number of questions in an assessment, and N equals the total number of concepts required for understanding all questions. The matrix element A[i,j] equals 1 if the i-th concept is required for correctly answering j-th question and 0 if the i-th concept is NOT required for correctly answering j-th question. Alternatively, matrix values can be not just {0,1}, but real numbers from the interval [0,1], describing the probability that a student who knows i-th concept will correctly answer j-th question.
The goal of q-matrix construction is to extract underlying, or latent, variables, which account for studentsí differential performance on questions.
Approaches:
Factor analysis: How to automatically determine concepts? Using covariance matrix. Number of concepts should be smaller than number of questions. Still, this methos has proven to be less fault tollerant.
The q-matrix method is a simple hill-climbing algorithm that creates a matrix representing relationships between concepts and questions directly. The algorithm varies c, the number of concepts, and the values in the q-matrix, minimizing the total error for all students for a given set of n questions. To avoid of local minima, each hill-climbing search is seeded with different random Q-matrices and the best of these is kept.
When forming a correlation matrix, we lose individual student data in favor of calculating average relationships between questions. The q-matrix method is optimized to assign each student the most appropriate knowledge state, using all available response data for each student.
As Sellers found in her research, the results obtained through q-matrix analysis seem to describe relationships among variables in interpretable ways. Factor analysis and principal components analysis, on the other hand, do not readily offer interpretable results.
Later researchers found that, although the q-matrix model was a good way to compare student data to a concept model, expert-constructed q-matrices did not correspond to student data any better than random q-matrices did.
the findings in Brewer's previous research, which found that the factor analysis method performed poorly in comparison with the q-matrix method when fewer observations were available.