Item response prediction for incomplete response matrix using the EM-type item response theory with application to adaptive online ability evaluation system

The item response theory (IRT) gives us the valuable information about the difficulties of problems as well as the abilities of students, whereas the classical test method provides only the abilities of students with pre-determined scores to each problem. To enhance the use of the IRT, we have developed a concise IRT evaluation Web system via the drag-and-drop Excel file in which 0/1 scores of the test result are stored. In addition, we have introduced an online adaptive IRT system to assess the students' abilities more accurately with fewer problems. In such a system, the item bank is pre-stored and the problem difficulties are determined in advance. However, as the number of online adaptive examinees becomes large, the calibration for parameters to problems, incorporating the new examinees' results for problem difficulties, may be needed. For the calibration, parameter estimation methods of problem difficulties and students' abilities for incomplete response matrices are required. In this paper, we propose a new method to estimate the problem difficulties and students' abilities for incomplete item response matrices via the LIRT, which is based on the item response theory and the EM-type algorithm. Then, we show a calibration procedure expressing the problem difficulties and students' abilities to some online adaptive system. We have found the estimates for discrimination parameters vary to some extent from the beginning to the end. However, the estimates for the difficulty parameters do not vary much, which corresponds to that the estimates for the ability parameters do not vary much.