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--- knowledge_assessment:q-matrix [2012/07/04 16:05]
jpetrovic created
+++ knowledge_assessment:q-matrix [2012/07/05 14:00]
jpetrovic
@@ Line 1: / Line 1: @@
 Q-matrix is a
    * "//method, which examines the inputs of many students to automatically extract relationships between questions and underlying concepts, and then uses those relationships in diagnosing and correcting student misconceptions.//"
+   * domain-independent knowledge model
+   * originally a binary matrix showing the relationship between test items and latent or underlying attributes, or concepts
+   * To build the q-matrix, experts constructed a relationship between test questions and concepts (referred to as attributes) and students taking the test were assigned knowledge states based on their test answers and the constructed q-matrix ((see Ham85 for a discussion of item-response theory))
+{{:knowledge_assessment:qm.jpg}}
+The goal of q-matrix construction is to extract underlying, or latent, variables, which account for studentsí differential performance on questions.
+Approaches:
+   * Hand construction of the q-matrix by experts' assigning concepts to questions and then comparing student answers to closest matrix responses. Problems: a q-matrix is a much more abstract measure of the relationships of questions to concepts. We might assume that the questions designed to test students are a more accurate reflection of the teaching objectives than an abstract construct which relates questions to underlying concepts.
+   * The alternative to this strategy is to design a method to extract a q-matrix, which explains student behavior, and reveals the underlying relationships between questions. Experts can examine the resulting q-matrix 25 to ensure that the extracted relationships seem to be valid, and then use that q-matrix to guide the generation of new problems.
+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.
+==== Q-matrix method ====
+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.

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