Estimation of a four‐parameter item response theory model

Abstract

We explore the justification and formulation of a four-parameter item response theory model (4PM) and employ a Bayesian approach to recover successfully parameter estimates for items and respondents. For data generated using a 4PM item response model, overall fit is improved when using the 4PM rather than the 3PM or the 2PM. Furthermore, although estimated trait scores under the various models correlate almost perfectly, inferences at the high and low ends of the trait continuum are compromised, with poorer coverage of the confidence intervals when the wrong model is used. We also show in an empirical example that the 4PM can yield new insights into the properties of a widely used delinquency scale. We discuss the implications for building appropriate measurement models in education and psychology to model more accurately the underlying response process.

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

“…Lowering the upper asymptote for these items also corresponded to increasing the item slopes to fit the data (see additional discussion in Waller & Reise, 2010). Note that Loken and Rulison (2010) reported similar findings in an application to delinquency data where several item slopes were three times larger for the four-parameter model.…”

“…Recent research renewed interest in Barton and Lord's (1981) four-parameter logistic (4PL) item response theory (IRT) model (Feuerstahler & Waller, 2014;Liao, Ho, Yen, & Cheng, 2012, Loken & Rulison, 2010Ogasawara, 2012;Rulison & Loken, 2009;Waller & Reise, 2010). The 4PL is a generalization of the three-parameter model and includes an upper asymptote for the probability of a correct response,…”

“…One consistent discussion point regarding the 4PL pertains to the difficulty in estimating item parameters. Prior research considered Bayesian model formulations with Metropolis-Hastings (MH) sampling (Loken & Rulison, 2010) and marginal maximum likelihood (Feuerstahler & Waller, 2014) to estimate 4PL parameters. This paper extends prior research on Bayesian estimation of IRT models (e.g., Albert, 1992;Béguin & Glas 2001;Patz & Junker, 1999a, b;Sahu, 2002).…”

“…That is, one identification restriction is that the upper asymptote exceeds the lower asymptote, d > c. Existing Bayesian algorithms for the 4PL do not incorporate the monotonicity restriction and instead employ highly informative prior distributions that avoid violating the condition with a higher probability. For example, Loken and Rulison (2010) used priors of c ∼ Beta (20, 80) and d ∼ Beta (80,20), which are fairly concentrated in the regions of 0.2 and 0.8, respectively. In contrast, the Bayesian formulation discussed below demonstrates accurate parameter recovery with less informative priors for the lower and upper asymptotes.…”

Item Factor Analysis: A Primer and New Open-Source Implementation

“…Refer to Thissen and Steinberg (1986) for an item model taxonomy. Fortunately, the most popular item models can be represented with just three flexible item models: the 4PL (Loken & Rulison, 2010), the graded response model (Samejima, 1969), and the nominal model (Thissen, Cai, & Bock, 2010). Each model will be described in turn.…”