The difficulty of multiple-choice questions (MCQs) is a crucial factor for educational assessments. Predicting MCQ difficulty is challenging since it requires understanding both the complexity of reaching the correct option and the plausibility of distractors, i.e., incorrect options. In this paper, we propose a novel, two-stage method to predict the difficulty of MCQs. First, to better estimate the complexity of each MCQ, we use large language models (LLMs) to augment the reasoning steps required to reach each option. We use not just the MCQ itself but also these reasoning steps as input to predict the difficulty. Second, to capture the plausibility of distractors, we sample knowledge levels from a distribution to account for variation among students responding to the MCQ. This setup, inspired by item response theory (IRT), enable us to estimate the likelihood of students selecting each (both correct and incorrect) option. We align these predictions with their ground truth values, using a Kullback-Leibler (KL) divergence-based regularization objective, and use estimated likelihoods to predict MCQ difficulty. We evaluate our method on two real-world \emph{math} MCQ and response datasets with ground truth difficulty values estimated using IRT. Experimental results show that our method outperforms all baselines, up to a 28.3\% reduction in mean squared error and a 34.6\% improvement in the coefficient of determination. We also qualitatively discuss how our novel method results in higher accuracy in predicting MCQ difficulty.

Computerized adaptive testing (CAT) is a form of personalized testing that accurately measures students' knowledge levels while reducing test length. Bilevel optimization-based CAT (BOBCAT) is a recent framework that learns a data-driven question selection algorithm to effectively reduce test length and improve test accuracy. However, it suffers from high question exposure and test overlap rates, which potentially affects test security. This paper introduces a constrained version of BOBCAT to address these problems by changing its optimization setup and enabling us to trade off test accuracy for question exposure and test overlap rates. We show that C-BOBCAT is effective through extensive experiments on two real-world adult testing datasets.