An Insight to Estimated Item Response Matrix in Item Response Theory

This paper investigates the performance of item response theory based on distance criteria rather than likelihood criteria. For this purpose, the estimated item response matrix is introduced. This matrix is a reconstruction of the item response matrix using maximum likelihood estimates of the parameters in item response theory. Then the distance between the observed and estimated matrices can be determined using the Frobenius matrix norm. An approximated low-rank matrix can be generated from the observed item response matrix by singular value decomposition, and the distance between the observed and low-rank matrices can be obtained in the same way. By comparing these two distances, we can evaluate the performance of the estimated item response matrix comparable to the performance of an approximated low-rank matrix. Applying this comparison to actual examination data, it is found that the rank of the approximated low-rank matrix that is equivalent to the estimated item response matrix is very low when using matrices as training data. However, using test data, the predictive ability of item response theory seems high enough since the minimum distance between the approximated low-rank matrix and the observed item response matrix is approximately equal to or slightly less than the distance between the estimated item response matrix and the observed item response matrix. This fact has been first discovered by utilizing the estimated item response matrix defined here.