Negative Information and the Three-Parameter Logistic Model

Abstract

The 3-parameter logistic model is commonly used to describe the relationship among an unobserved latent trait (ability), unobserved item properties, and an observed binary outcome. We show that for certain values of the item properties and latent ability, the observed information about ability contained in the binary response is negative. This result has implications for maximization procedures, such as Newton-Raphson; approximate sampling methods, such as the Metropolis-Hastings algorithm; and Bayesian adaptive testing. All of these typically utilize the observed information. This result is contrasted with the fact that observed negative information does not occur in the limiting case with no guessing (2-parameter logistic model). The probability of negative information is expressed by a simple formula. This research extends the work of Samejima (1973) and Yen, Burket, and Sykes (1991). In many educational tests, dichotomously scored multiple-choice items are given to examinees in order to obtain estimates of their true underlying ability. A widely adopted tenet is that a test with more items must yield estimates of ability with greater precision than the corresponding test with some items deleted. We show that not to be the case for the three-parameter logistic (3PL) model when the standard errors of ability estimates are based on the total observed information function. We focus on this issue by computing the additional information about a single examinee's ability from a single binary item response (generalizations to multiple responses are straightfor