A Paradox by Another Name Is Good Estimation

Abstract

This chapter describes the property of estimates of points in a multidimensional space that is labeled by some as paradoxical, shows when this property of the estimates is present, and also shows that the paradoxical result is not flaw in estimation because estimates improve with additional information even when the paradox occurs. The paradox is that when a correct response to a test item is added to the string of responses for an examinee to previous items, at least one of the coordinates of the new estimated θ-point decreases compared to the estimate based on the initial string of responses. The information presented in the chapter shows that this can occur whenever the likelihood function for the estimates has a particular form. This form is present in many cases when the item responses for a test can not be described by simple structure. Results are presented to show that the additional response improves the estimate of the θ-point even though the paradoxical result occurs.