A Note on the Item Information Function of the Four-Parameter Logistic Model

Abstract

This article focuses on four-parameter logistic (4PL) model as an extension of the usual threeparameter logistic (3PL) model with an upper asymptote possibly different from 1. For a given item with fixed item parameters, Lord derived the value of the latent ability level that maximizes the item information function under the 3PL model. The purpose of this article is to extend this result to the 4PL model. A generic and algebraic method is developed for that purpose. The result is practically illustrated by an example and several potential applications of this result are outlined. Keywords item response theory, four-parameter logistic model, item information function, maximizationIn the field of dichotomous item response theory (IRT) models, the three-parameter logistic (3PL) model (Birnbaum, 1968) and the simpler one-and two-parameter logistic (1PL and 2PL) models have received most attention in the past decades. However, an extended version of the 3PL model was also suggested, by allowing an upper asymptote possibly smaller than 1. This four-parameter logistic (4PL) model, early proposed by Barton and Lord (1981) and barely mentioned by Hambleton and Swaminathan (1985), did not receive a lot of attention until recently. As Loken and Rulison (2010) pointed out, the strong dominancy of the 3PL model in the literature, the lack of consensus on its usefulness, and the technical difficulty in estimating the upper asymptote accurately, are strong arguments against the use of the 4PL model. Despite these conceptual drawbacks, the 4PL model was reconsidered recently in the literature. One reason is the recent improvement in computational power and resources, together with the development of accurate statistical modeling software. Some early arguments toward accurate estimation of the upper asymptote can be found in Linacre (2004) and Rupp (2003). Very recently, Loken and Rulison (2010) developed a Bayesian framework to calibrate the

Revisiting the 4-Parameter Item Response Model: Bayesian Estimation and Application

“…Several studies argued that d from the 4PL can be estimated to account for early careless errors to improve ability estimation (Liao et al, 2012;Rulison & Loken, 2009). Furthermore, for the 4PL, Magis (2013) derived an expression for the value of θ that maximizes item information and Ogasawara (2012) derived the asymptotic distribution ofθ. Additionally, researchers interested in measuring psychopathology traits have noted value in modeling lower and upper asymptotes of item response functions (Reise & Waller, 2003;Waller & Reise, 2010) in instances where subjects with higher levels of psychopathology may be reluctant to self-report attitudes, behaviors, and/or experiences.…”

Reducing outcome measures in mental health: a systematic review of the methods

“…When a second restriction is included (all items now have equal discrimination and guessing is not a factor) then the 2-PL is reduced to a 1-PL model. There is also a 4-PL model which has received less attention and this includes a factor which can be due to stress, tiredness or inattention for example (Magis, 2013).…”

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“…The thOpt procedure (Li and Schafer 2005;Magis 2013). In the thOpt rule, the item selected is the one belonging to the subset of administrable items of the bank (B) with minimum distance between the currently estimated trait levelθ and the value where the item achieves its maximum in the Fisher information function θ max…”