A Restricted Four-Parameter IRT Model: The Dyad Four-Parameter Normal Ogive (Dyad-4PNO) Model
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THEORY AND METHODS A RESTRICTED FOUR-PARAMETER IRT MODEL: THE DYAD FOUR-PARAMETER NORMAL OGIVE (DYAD-4PNO) MODEL
Justin L. Kern DEPARTMENT OF EDUCATIONAL PSYCHOLOGY, UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
Steven Andrew Culpepper DEPARTMENT OF STATISTICS, UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
Recently, there has been a renewed interest in the four-parameter item response theory model as a way to capture guessing and slipping behaviors in responses. Research has shown, however, that the nested three-parameter model suffers from issues of unidentifiability (San Martín et al. in Psychometrika 80:450– 467, 2015), which places concern on the identifiability of the four-parameter model. Borrowing from recent advances in the identification of cognitive diagnostic models, in particular, the DINA model (Gu and Xu in Stat Sin https://doi.org/10.5705/ss.202018.0420, 2019), a new model is proposed with restrictions inspired by this new literature to help with the identification issue. Specifically, we show conditions under which the four-parameter model is strictly and generically identified. These conditions inform the presentation of a new exploratory model, which we call the dyad four-parameter normal ogive (Dyad-4PNO) model. This model is developed by placing a hierarchical structure on the DINA model and imposing equality constraints on a priori unknown dyads of items. We present a Bayesian formulation of this model, and show that model parameters can be accurately recovered. Finally, we apply the model to a real dataset. Key words: four-parameter model, Bayesian statistics, slipping, identification, hierarchical DINA model.
1. Introduction Within the past decade, there has been a revival of interest in Barton and Lord’s (1981) four-parameter model (4PM) first proposed nearly 40 years ago (Feuerstahler and Waller 2014; Waller and Feuerstahler 2017; Culpepper 2016). This model is a simple generalization of the three-parameter model in that it includes both a lower asymptote and an upper asymptote for the probability of a correct item response as follows: P(Y j = 1|θ, α j , β j , g j , s j ) = g j + (1 − s j − g j )(α j θ − β j )
(1)
where (·) denotes an arbitrary link function; Barton and Lord (1981) introduced the fourparameter logistic distribution (4PL) with a logit link and the four-parameter normal ogive model (4PNO) uses a probit link, . Here, θ is the examinee latent trait level; and β j , α j , g j , and 1 − s j are the jth item’s threshold, slope, lower asymptote, and upper asymptote parameters, respectively. Specifically, g j and 1 − s j represent the minimum and maximum probabilities for a correct response, respectively. Correspondence should be made to Steven Andrew Culpepper, Department of Statistics, University of Illinois at Urbana-Champaign, 725 South Wright Street, Champaign, IL61820, USA. Email: [email protected]
© 2020 The Psychometric Society
PSYCHOMETRIKA
Current research has shown that the presence of an upper asymptote may be tenable in the cases of psychological assessment (
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