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The Dirichlet Dual Response Model: An Item Response Model for Continuous Bounded Interval Responses

Author

Listed:
  • Matthias Kloft

    (University of Marburg)

  • Raphael Hartmann

    (University of Marburg)

  • Andreas Voss

    (Heidelberg University)

  • Daniel W. Heck

    (University of Marburg)

Abstract

Standard response formats such as rating or visual analogue scales require respondents to condense distributions of latent states or behaviors into a single value. Whereas this is suitable to measure central tendency, it neglects the variance of distributions. As a remedy, variability may be measured using interval-response formats, more specifically the dual-range slider (RS2). Given the lack of an appropriate item response model for the RS2, we develop the Dirichlet dual response model (DDRM), an extension of the beta response model (BRM; Noel & Dauvier in Appl Psychol Meas, 31:47–73, 2007). We evaluate the DDRM’s performance by assessing parameter recovery in a simulation study. Results indicate overall good parameter recovery, although parameters concerning interval width (which reflect variability in behavior or states) perform worse than parameters concerning central tendency. We also test the model empirically by jointly fitting the BRM and the DDRM to single-range slider (RS1) and RS2 responses for two Extraversion scales. While the DDRM has an acceptable fit, it shows some misfit regarding the RS2 interval widths. Nonetheless, the model indicates substantial differences between respondents concerning variability in behavior. High correlations between person parameters of the BRM and DDRM suggest convergent validity between the RS1 and the RS2 interval location. Both the simulation and the empirical study demonstrate that the latent parameter space of the DDRM addresses an important issue of the RS2 response format, namely, the scale-inherent interdependence of interval location and interval width (i.e., intervals at the boundaries are necessarily smaller).

Suggested Citation

  • Matthias Kloft & Raphael Hartmann & Andreas Voss & Daniel W. Heck, 2023. "The Dirichlet Dual Response Model: An Item Response Model for Continuous Bounded Interval Responses," Psychometrika, Springer;The Psychometric Society, vol. 88(3), pages 888-916, September.
  • Handle: RePEc:spr:psycho:v:88:y:2023:i:3:d:10.1007_s11336-023-09924-7
    DOI: 10.1007/s11336-023-09924-7
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    References listed on IDEAS

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    1. Fumiko Samejima, 1973. "Homogeneous case of the continuous response model," Psychometrika, Springer;The Psychometric Society, vol. 38(2), pages 203-219, June.
    2. Chalmers, R. Philip, 2012. "mirt: A Multidimensional Item Response Theory Package for the R Environment," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 48(i06).
    3. Lewandowski, Daniel & Kurowicka, Dorota & Joe, Harry, 2009. "Generating random correlation matrices based on vines and extended onion method," Journal of Multivariate Analysis, Elsevier, vol. 100(9), pages 1989-2001, October.
    4. Yvonnick Noel, 2014. "A Beta Unfolding Model for Continuous Bounded Responses," Psychometrika, Springer;The Psychometric Society, vol. 79(4), pages 647-674, October.
    5. Jonah Gabry & Daniel Simpson & Aki Vehtari & Michael Betancourt & Andrew Gelman, 2019. "Visualization in Bayesian workflow," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 182(2), pages 389-402, February.
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