Paradoxical Results and Item Bundles

Hooker, Finkelman, and Schwartzman (Psychometrika, 2009, in press) defined a paradoxical result as the attainment of a higher test score by changing answers from correct to incorrect and demonstrated that such results are unavoidable for maximum likelihood estimates in multidimensional item response theory. The potential for these results to occur leads to the undesirable possibility of a subject’s best answer being detrimental to them. This paper considers the existence of paradoxical results in tests composed of item bundles when compensatory models are used. We demonstrate that paradoxical results can occur when bundle effects are modeled as nuisance parameters for each subject. However, when these nuisance parameters are modeled as random effects, or used in a Bayesian analysis, it is possible to design tests comprised of many short bundles that avoid paradoxical results and we provide an algorithm for doing so. We also examine alternative models for handling dependence between item bundles and show that using fixed dependency effects is always guaranteed to avoid paradoxical results.