Minimum Distance Estimation of Multidimensional Diffusion-Based Item Response Theory Models

Abstract

Diffusion-based item response theory models are models for responses and response times on psychological tests, which can be used as measurement models in the same way as standard item response theory models (Tuerlinckx, Molenaar, & van der Maas, 2016). Their range of application, however, is narrowed by the fact that multidimensional versions of the model are not easy to fit. Marginal maximum likelihood estimation (e.g., Molenaar, Tuerlinckx, & van der Maas, 2015a) is computationally intensive and infeasible for multidimensional versions. The weighted least squares estimator of Ranger, Kuhn, and Szardenings (2016) is inefficient. Here, we propose an alternative estimator that is more efficient than the least squares estimator and less demanding than the maximum likelihood estimator. The estimator is based on minimum distance estimation and consists in modeling the sample quantiles and sample covariances. The performance of the estimator is investigated in a simulation study. The simulation study corroborates that the estimator performs well. The application of the estimator is demonstrated with real data.

Keywords:

• Diffusion model
• item response theory
• minimum distance estimation

Article information

Conflict of interest disclosures: Each author signed a form for disclosure of potential conflicts of interest. No authors reported any financial or other conflicts of interest in relation to the work described.

Ethical principles: The authors affirm having followed professional ethical guidelines in preparing this work. These guidelines include obtaining informed consent from human participants, maintaining ethical treatment and respect for the rights of human or animal participants, and ensuring the privacy of participants and their data, such as ensuring that individual participants cannot be identified in reported results or from publicly available original or archival data.

Funding: This work was not supported.

Role of the funders/sponsors: None of the funders or sponsors of this research had any role in the design and conduct of the study; collection, management, analysis, and interpretation of data; preparation, review, or approval of the manuscript; or decision to submit the manuscript for publication.

Acknowledgments: We would like to thank the Editor Prof. Molenaar, the Associate Editor Prof. Edwards and two referees for their patience and their helpful and constructive comments, which led to many improvements. We are also indebted to Prof. Ferrando and Prof. Lorenzo-Seva for their generosity to share the data. Without this support, this publication would not have been possible.

The ideas and opinions expressed herein are those of the authors alone, and endorsement by the author’s institutions is not intended and should not be inferred.

Notes

1 The following discussion is limited to diffusion models with latent traits; estimation of the standard diffusion model with parameter variability is discussed in Ratcliff and Tuerlinckx (2002), Voss and Voss (2008), Voss et al. (2013) and Ratcliff and Childers (2015).

2 We use the term classical minimum distance estimation in order to distinguish the estimation approach from minimum distance estimation with density-based distances (Basu, Harris, & Basu, 1997).

3 We chose the D-diffusion model as this model is the preferred model for data from personality tests. The proposed estimation approach, however, could easily be adapted to other variants of the diffusion-based item response theory model.

4 In limited information estimation of item response models, one estimates the model’s parameters by modeling the joint solution probabilities in all item pairs (Maydeu-Olivares & Joe, 2005). These statistics are the raw moments of the responses of order two. Here, we use the covariances which are the centered moments of order two. Both quantities are closely related.