On Separable Tests, Correlated Priors and Paradoxical Results in Multidimensional Item Response Theory

This note considers the problem of paradoxical results in multidimensional item response theory. This is the possibility that a deliberately incorrect response could be beneficial to an examinee. An example is created that demonstrates that, when three or more ability dimensions are considered, paradoxical results can occur for Bayesian estimates even for separable tests when all the dimensions have positive prior correlation. A discussion of the mathematics behind the result is given and a computationally feasible means to check for the existence of paradoxical results is developed.