Hostname: page-component-5f745c7db-xx4dx Total loading time: 0 Render date: 2025-01-06T07:54:46.613Z Has data issue: true hasContentIssue false
  • English
  • Français

Gibbs Samplers for Logistic Item Response Models via the Pólya–Gamma Distribution: A Computationally Efficient Data-Augmentation Strategy

Published online by Cambridge University Press:  01 January 2025

Zhehan Jiang Open the ORCID record for Zhehan Jiang [Opens in a new window]  and
Jonathan Templin
Zhehan Jiang*
Affiliation:
The University of Alabama
Jonathan Templin
Affiliation:
The University of Kansas
*
Correspondence should be made to Zhehan Jiang, 309DE LB Gorgas Library, University Libraries, The University of Alabama, Tuscaloosa, USA. Email: zjiang17@ua.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Fully Bayesian estimation of item response theory models with logistic link functions suffers from low computational efficiency due to posterior density functions that do not have known forms. To improve algorithmic computational efficiency, this paper proposes a Bayesian estimation method by adopting a new data-augmentation strategy in uni- and multidimensional IRT models. The strategy is based on the Pólya–Gamma family of distributions which provides a closed-form posterior distribution for logistic-based models. In this paper, an overview of Pólya–Gamma distributions is described within a logistic regression framework. In addition, we provide details about deriving conditional distributions of IRT, incorporating Pólya–Gamma distributions into the conditional distributions for Bayesian samplers’ construction, and random drawing from the samplers such that a faster convergence can be achieved. Simulation studies and applications to real datasets were conducted to demonstrate the efficiency and utility of the proposed method.

Keywords

Bayesiandata augmentationIRTGibbsestimationPólya–Gamma
Type
Original Paper
Information
Psychometrika , Volume 84 , Issue 2 , June 2019 , pp. 358 - 374
Copyright
Copyright © 2018 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Albert, J. H., Chib, S. (1993). Bayesian analysis of binary and polychotomous response data. Journal of the American statistical Association, 88(422), 669–679.CrossRefGoogle Scholar
Aruoba, S. B., & Fernández-Villaverde, J. (2014). A comparison of programming languages in economics. NBER working paper no. w20263. Cambridge, MA: National Bureau of Economic Research.CrossRefGoogle Scholar
Baker, F. B., Kim, S. H. (2004). Item response theory: Parameter estimation techniques, Boca Raton: CRC Press.CrossRefGoogle Scholar
Biane, P., Pitman, J., Yor, M. (2001). Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions. Bulletin of the American Mathematical Society, 38(4), 435–465.CrossRefGoogle Scholar
Bock, R. D., Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46(4), 443–459.CrossRefGoogle Scholar
Brooks, S., Gelman, A., Jones, G., Meng, X. L. (2011). Handbook of Markov chain Monte Carlo, Boca Raton: CRC Press.CrossRefGoogle Scholar
Cai, L. (2010). High-dimensional exploratory item factor analysis by a Metropolis–Hastings Robbins–Monro algorithm. Psychometrika, 75(1), 33–57.CrossRefGoogle Scholar
Carlin, B. P., Polson, N. G., Stoffer, D. S. (1992). A Monte Carlo approach to nonnormal and nonlinear state-space modeling. Journal of the American Statistical Association, 87(418), 493–500.CrossRefGoogle Scholar
Chalmers, R. P. (2012). mirt: A multidimensional item response theory package for the R environment. Journal of Statistical Software, 48(6), 1–29.CrossRefGoogle Scholar
Chib, S., Greenberg, E., & Chen, Y. (1998). MCMC methods for fitting and comparing multinomial response models. NBER working paper no. 19802001. Cambridge, MA: National Bureau of Economic Research.Google Scholar
DeMars, C. E. (2016). Partially compensatory multidimensional item response theory models: Two alternate model forms. Educational and Psychological Measurement, 76(2), 231–257.CrossRefGoogle ScholarPubMed
Devroye, L. (2002). Simulating Bessel random variables. Statistics and Probability Letters, 57(3), 249–257.CrossRefGoogle Scholar
Dobra, A., Tebaldi, C., West, M. (2006). Data augmentation in multi-way contingency tables with fixed marginal totals. Journal of Statistical Planning and Inference, 136(2), 355–372.CrossRefGoogle Scholar
Duane, S., Kennedy, A. D., Pendleton, B. J., Roweth, D. (1987). Hybrid Monte Carlo. Physics Letters B, 195(2), 216–222.CrossRefGoogle Scholar
Eckes, T., Baghaei, P. (2015). Using testlet response theory to examine local dependence in C-tests. Applied Measurement in Education, 28(2), 85–98.CrossRefGoogle Scholar
Edwards, M. C. (2010). A Markov chain Monte Carlo approach to confirmatory item factor analysis. Psychometrika, 75(3), 474–497.CrossRefGoogle Scholar
Embretson, S. E., Reise, S. P. (2000). Item response theory for psychologists, Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
Feinberg, R. A., Rubright, J. D. (2016). Conducting simulation studies in psychometrics. Educational Measurement: Issues and Practice, 35(2), 36–49.CrossRefGoogle Scholar
Forster, J. J., Skene, A. M. (1994). Calculation of marginal densities for parameters of multinomial distributions. Statistics and Computing, 4(4), 279–286.CrossRefGoogle Scholar
Fox, J. P., Glas, C. A. (2001). Bayesian estimation of a multilevel IRT model using Gibbs sampling. Psychometrika, 66(2), 271–288.CrossRefGoogle Scholar
Frühwirth-Schnatter, S., Frühwirth, R. (2010). Data augmentation and MCMC for binary and multinomial logit models. In Kneib, T., Tutz, G. (Eds), Statistical modelling and regression structures, Heidelberg: Physica-Verlag HD 111–132.CrossRefGoogle Scholar
Gamerman, D. (1997). Sampling from the posterior distribution in generalized linear mixed models. Statistics and Computing, 7(1), 57–68.CrossRefGoogle Scholar
Geman, S., Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721–741.CrossRefGoogle ScholarPubMed
Girolami, M., Calderhead, B. (2011). Riemann manifold langevin and hamiltonian monte carlo methods. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73(2), 123–214.CrossRefGoogle Scholar
Harwell, M. R., Baker, F. B. (1991). The use of prior distributions in marginalized Bayesian item parameter estimation: A didactic. Applied Psychological Measurement, 15(4), 375–389.CrossRefGoogle Scholar
Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57(1), 97–109.CrossRefGoogle Scholar
Hoffman, M. D., Gelman, A. (2014). The No-U-turn sampler: Adaptively setting path lengths in Hamiltonian Monte Carlo. Journal of Machine Learning Research, 15(1), 1593–1623.Google Scholar
Holmes, C. C., Held, L. (2006). Bayesian auxiliary variable models for binary and multinomial regression. Bayesian Analysis, 1(1), 145–168.Google Scholar
Imai, K., van Dyk, D. A. (2005). A Bayesian analysis of the multinomial probit model using marginal data augmentation. Journal of Econometrics, 124(2), 311–334.CrossRefGoogle Scholar
Jiang, Z., & Raymond, M. (2018). The use of multivariate generalizability theory to evaluate the quality of subscores. Applied Psychological Measurement. https://doi.org/10.1177/0146621618758698.CrossRefGoogle Scholar
Junker, B. W., Patz, R. J., VanHoudnos, N. M. (2016). Markov chain Monte Carlo for item response models. Handbook of Item Response Theory, Volume Two: Statistical Tools, 21, 271–325.Google Scholar
Kuo, T. C., Sheng, Y. (2015). Bayesian estimation of a multi-unidimensional graded response IRT model. Behaviormetrika, 42(2), 79–94.CrossRefGoogle Scholar
Lawley, D. N., Maxwell, A. E. (1971). Factor analysis as a statistical method, New York: Macmillan.Google Scholar
Lenk, P. J., DeSarbo, W. S. (2000). Bayesian inference for finite mixtures of generalized linear models with random effects. Psychometrika, 65(1), 93–119.CrossRefGoogle Scholar
Lord, F. M. (1980). Applications of item response theory to practical testing problems, London: Routledge.Google Scholar
Lunn, D. J., Thomas, A., Best, N., Spiegelhalter, D. (2000). WinBUGS-a Bayesian modelling framework: Concepts, structure, and extensibility. Statistics and Computing, 10(4), 325–337.CrossRefGoogle Scholar
Lynch, S. M. (2010). Introduction to applied Bayesian statistics and estimation for social scientists, New York: Springer.Google Scholar
Matlock, K. L., Turner, R. C., Gitchel, W. D. (2016). A study of reverse-worded matched item pairs using the generalized partial credit and nominal response models. Educational and Psychological Measurement, 78, 103–127.CrossRefGoogle ScholarPubMed
McDonald, R. P. (1999). Test theory: A unified treatment, London: Erlbaum.Google Scholar
Mislevy, R. J., Stocking, M. L. (1989). A consumer’s guide to LOGIST and BILOG. Applied Psychological Measurement, 13, 57–75.CrossRefGoogle Scholar
Monroe, S., Cai, L. (2014). Estimation of a Ramsay-curve item response theory model by the Metropolis–Hastings Robbins–Monro algorithm. Educational and Psychological Measurement, 74(2), 343–369.CrossRefGoogle Scholar
Niederreiter, H. (1978). Quasi-Monte Carlo methods and pseudo-random numbers. Bulletin of the American Mathematical Society, 84(6), 957–1041.CrossRefGoogle Scholar
Patz, R. J., Junker, B. W. (1999). A straightforward approach to Markov chain Monte Carlo methods for item response models. Journal of Educational and Behavioral Statistics, 24(2), 146–178.CrossRefGoogle Scholar
Polson, N. G., Scott, J. G., Windle, J. (2013). Bayesian inference for logistic models using Pólya–Gamma latent variables. Journal of the American statistical Association, 108(504), 1339–1349.CrossRefGoogle Scholar
R Core Team. (2018). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. Retrieved from http://www.Rproject.org/.Google Scholar
Robert, C. P., Casella, G. (2004). Monte Carlo statistical methods, New York: Springer.CrossRefGoogle Scholar
Schilling, S., Bock, R. D. (2005). High-dimensional maximum marginal likelihood item factor analysis by adaptive quadrature. Psychometrika, 70(3), 533–555.Google Scholar
Shaby, B., Wells, M. T. (2010). Exploring an adaptive Metropolis algorithm. Currently Under Review, 1, 1–17.Google Scholar
Sinharay, S. (2003). Assessing convergence of the Markov chain Monte Carlo algorithms: A review. ETS Research Report Series, 2003(1), i52.CrossRefGoogle Scholar
Skene, A. M., Wakefield, J. C. (1990). Hierarchical models for multicentre binary response studies. Statistics in Medicine, 9(8), 919–929.CrossRefGoogle ScholarPubMed
Spiegelhalter, D. J., Thomas, A., Best, N., & Lunn, D. (2003). WinBUGS user manual. Cambridge, UK: MRC Biostatistics Unit. Retrieved from http://www.mrc-bsu.cam.ac.uk/bugs.Google Scholar
Talhouk, A., Doucet, A., Murphy, K. (2012). Efficient Bayesian inference for multivariate probit models with sparse inverse correlation matrices. Journal of Computational and Graphical Statistics, 21(3), 739–757.CrossRefGoogle Scholar
Thissen, D., Wainer, H. (2001). Test scoring, Hillsdale, NJ: Lawrence Erlbaum Associates.CrossRefGoogle Scholar
van Der Linden, W. J., Hambleton, R. K. (1997). Item response theory: Brief history, common models, and extensions. In van der Linden, W. J., Hambleton, R. K. (Eds), Handbook of modern item response theory, New York: Springer 1–28.CrossRefGoogle Scholar
Zeithammer, R., Lenk, P. (2006). Bayesian estimation of multivariate-normal models when dimensions are absent. Quantitative Marketing and Economics, 4(3), 241–265.CrossRefGoogle Scholar