Two algorithms for estimating test complexity levels

We study the problem of estimating the complexity levels of test problems and levels of preparation of the students that arises in learning management systems. To solve the problem, we propose two algorithms for processing test results. The first algorithm is based on the assumption that random answers of the test takers are described by a logistic distribution. To compute test problem complexities and levels of preparation of the students, we use the maximum likelihood method and the quasi-Newton Broyden–Fletcher–Goldfarb–Shanno optimization method, where the likelihood function is constructed in a special way based on Rasch's model. The second algorithm is heuristic and is based on recurrent recomputation of initial estimates obtained by adding up the positive answers of students separately by columns and rows of the matrix of answers, where columns correspond to answers of all students for a specific test, and rows correspond to answers of a specific student for all tests. We consider an example where we compare the results of applying the proposed algorithms.