Mapping question items to skills with non-negative matrix factorization

Intelligent learning environments need to assess the student skills to tailor course material, provide helpful hints, and in general provide some kind of personalized interaction. To perform this assessment, question items, exercises, and tasks are presented to the student. This assessment relies on a mapping of tasks to skills. However, the process of deciding which skills are involved in a given task is tedious and challenging. Means to automate it are highly desirable, even if only partial automation that provides supportive tools can be achieved. A recent technique based on Non-negative Matrix Factorization (NMF) was shown to offer valuable results, especially due to the fact that the resulting factorization allows a straightforward interpretation in terms of a Q-matrix. We investigate the factors and assumptions under which NMF can effectively derive the underlying high level skills behind assessment results. We demonstrate the use of different techniques to analyze and interpret the output of NMF. We propose a simple model to generate simulated data and to provide lower and upper bounds for quantifying skill effect. Using the simulated data, we show that, under the assumption of independent skills, the NMF technique is highly effective in deriving the Q-matrix. However, the NMF performance degrades under different ratios of variance between subject performance, item difficulty, and skill mastery. The results corroborates conclusions from previous work in that high level skills, corresponding to general topics like World History and Biology, seem to have no substantial effect on test performance, whereas other topics like Mathematics and French do. The analysis and visualization techniques of the NMF output, along with the simulation approach presented in this paper, should be useful for future investigations using NMF for Q-matrix induction from data.