A Bayesian Semiparametric Item Response Model with Dirichlet Process Priors

Abstract

In Item Response Theory (IRT), item characteristic curves (ICCs) are illustrated through logistic models or normal ogive models, and the probability that examinees give the correct answer is usually a monotonically increasing function of their ability parameters. However, since only limited patterns of shapes can be obtained from logistic models or normal ogive models, there is a possibility that the model applied does not fit the data. As a result, the existing method can be rejected because it cannot deal with various item response patterns.To overcome these problems, we propose a new semiparametric IRT model using a Dirichlet process mixture logistic distribution. Our method does not rely on assumptions but only requires that the ICCs be a monotonically nondecreasing function; that is, our method can deal with more types of item response patterns than the existing methods, such as the one-parameter normal ogive models or the two-or threeparameter logistic models.We conducted two simulation studies whose results indicate that the proposed method can express more patterns of shapes for ICCs and can estimate the ability parameters more accurately than the existing parametric and nonparametric methods. The proposed method has also been applied to Facial Expression Recognition data with noteworthy results.

On the Bayesian Nonparametric Generalization of IRT-Type Models

“…In the Bayesian context, Dirichlet processes (DP), mixtures of DP (MDP), Dirichlet process mixtures (DPM), Pólya trees (PT), and mixtures of PT (MPT) have been used to define nonparametric models for random effects distributions (Bush & MacEachern, 1996;Mukhopadhyay & Gelfand, 1997;Müller & Rosner, 1997;Walker & Mallick, 1997;Kleinman & Ibrahim, 1998a, 1998bHanson, 2006;Jara, Hanson, & Lesaffre, 2009). Bayesian nonparametric methods in IRT-type models have been considered in the discussion of Ishwaran (1998) to Roberts and Rosenthal (1998), in Duncan and MacEachern (2008), and also by Karabatsos and Walker (2009) and Miyazaki and Hoshino (2009).…”

Maximum Marginal Likelihood Estimation of a Monotonic Polynomial Generalized Partial Credit Model with Applications to Multiple Group Analysis

“…While we do not rule out the possibility of constructing GPC functions of other types of polynomials (e.g., splines), such an approach will likely not offer the same level of parsimony as GPC-MP, and it is unclear whether this would have any advantage over existing approaches (e.g., monotone splines; Ramsay & Winsberg, 1991). Finally, our approach has not yet been compared with other promising, though computationally intensive Bayesian non-parametric approaches (e.g., Duncan & MacEachern, 2008; Miyazaki & Hoshino, 2009). 4 …”

“…For instance, if all non-standard IRFs tend to follow the same shape, researchers may suspect that a non-normally distributed latent trait is the culprit and choose to model the distribution using an empirical histogram approach (Mislevy, 1984; Woods, 2007a), Ramsay curves (Woods & Thissen, 2006; Woods, 2006, 2007b, 2008) or Davidian curves (Woods & Lin, 2008). Alternatively, non-standard IRFs can be modeled using techniques such as non-parametric methods (Ramsay, 1991; Rossi, Wang, & Ramsay, 2002; Samejima, 1977, 1979, 1984; Sijtsma, Debets, & Molenaar, 1990), semi-parametric methods (Liang, 2007; Ramsay & Winsberg, 1991), or by using Bayesian non-parametric estimation (Duncan & MacEachern, 2008, 2013; Miyazaki & Hoshino, 2009; Qin, 1998). …”

Semiparametric Item Response Functions in the Context of Guessing

“…Although the LMPA does not completely overlap with the mixture IRF, the two are almost indistinguishable. Thus, while some approaches do more closely represent the mixing of IRFs or their parameters to achieve flexible IRF shapes (e.g., Duncan & MacEachern, ; Miyazaki & Hoshino, ) or explicit identification of latent classes (e.g., Rost, ), the LMPA approach may provide a reasonable alternative solution that can accommodate nonstandard IRF shapes. Importantly, it may also serve as a leading indicator followed by more detailed and computationally demanding analysis.…”

“…Nonparametric methods may quickly estimate IRFs with good recovery, including smoothed isotonic regression (SISO; Lee, , ) and kernel smoothing (Ramsay, )—the latter of which can result in IRFs that are not monotonically increasing. Bayesian nonparametric methods may also be employed (Duncan & MacEachern, ; Miyazaki & Hoshino, ; Qin, ), but may be slow to estimate (Liang, ).…”