Statistics and Its Interface

Volume 15 (2022)

Number 4

A Gibbs sampler for estimating the graded item response model with likert-scale data via the Pólya–Gamma distribution: a calculationally efficient data-augmentation scheme

Pages: 463 – 474



Zhaoyuan Zhang (School of Mathematics and Statistics, and Institute of Applied Mathematics, Yili Normal University, Yining, Xinjiang, China)

Jiwei Zhang (School of Mathematics and Statistics, Yunnan University, Kunming, Yunnan, China)

Jing Lu (School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin, China)


This paper reports the use of a highly effective Pólya–Gamma Gibbs sampling algorithm [32] based on auxiliary variables to estimate the parameters of the graded response model (GRM; [34]) that has been used widely in educational and psychological assessments. As its name suggests, the algorithm can be viewed as an extension of the traditional Gibbs sampling algorithm, overcoming the defect that the latter is ineffective for Bayesian non-conjugate models. By introducing auxiliary variables, non-conjugate models are transformed into conjugate ones, and posterior sampling is easier to implement with the help of the traditional Gibbs sampling algorithm. Also, the algorithm avoids the Metropolis–Hastings sampling algorithm’s tedious adjustment of tuning parameters to achieve an appropriate acceptance probability. Two simulation studies are conducted, and data from the Sexual Compulsivity Scale are subjected to detailed analysis to further illustrate the proposed methodology.


auxiliary variables, Bayesian estimation methods, graded response model, item response theory, Pólya–Gamma Gibbs sampling algorithm

The authors’ work was supported by the National Natural Science Foundation of China (Grant No. 12001091), by the Postdoctoral Science Foundations (Grant No. 2021M690587 and Grant No. 2021T140108), by the Fundamental Research Funds for the Central Universities of China (Grant No. 2412020QD025), and by the Yili Normal University 2021 Annual Research Project (Grant No. 2021YSBS012).

Received 25 May 2021

Accepted 18 November 2021

Published 4 March 2022