Hostname: page-component-6898cfcb55-th4rj Total loading time: 0 Render date: 2026-11-11T14:48:30.257Z Has data issue: false hasContentIssue false
  • English
  • Français

Bayesian Comparison of Latent Variable Models: Conditional Versus Marginal Likelihoods

Published online by Cambridge University Press:  01 January 2025

Edgar C. MerkleOpen the ORCID record for Edgar C. Merkle[Opens in a new window],
Daniel Furr and
Sophia Rabe-Hesketh
Edgar C. Merkle*
Affiliation:
University of Missouri
Daniel Furr
Affiliation:
University of California, Berkeley
Sophia Rabe-Hesketh
Affiliation:
University of California, Berkeley
*
Correspondence should be made to Edgar C. Merkle, University of Missouri, Columbia, MO, USA.Email: merklee@missouri.edu
Get accessRights & Permissions [Opens in a new window]

Abstract

Typical Bayesian methods for models with latent variables (or random effects) involve directly sampling the latent variables along with the model parameters. In high-level software code for model definitions (using, e.g., BUGS, JAGS, Stan), the likelihood is therefore specified as conditional on the latent variables. This can lead researchers to perform model comparisons via conditional likelihoods, where the latent variables are considered model parameters. In other settings, however, typical model comparisons involve marginal likelihoods where the latent variables are integrated out. This distinction is often overlooked despite the fact that it can have a large impact on the comparisons of interest. In this paper, we clarify and illustrate these issues, focusing on the comparison of conditional and marginal Deviance Information Criteria (DICs) and Watanabe–Akaike Information Criteria (WAICs) in psychometric modeling. The conditional/marginal distinction corresponds to whether the model should be predictive for the clusters that are in the data or for new clusters (where “clusters” typically correspond to higher-level units like people or schools). Correspondingly, we show that marginal WAIC corresponds to leave-one-cluster out cross-validation, whereas conditional WAIC corresponds to leave-one-unit out. These results lead to recommendations on the general application of the criteria to models with latent variables.

Keywords

Bayesian information criteriaconditional likelihoodcross-validationDICIRTleave-one-cluster outmarginal likelihoodMCMCSEMWAIC

Information

Type
Original Paper
Information
Psychometrika , Volume 84 , Issue 3 , September 2019, pp. 802 - 829
Copyright
Copyright © 2019 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.)

Article purchase

Temporarily unavailable

Footnotes

Electronic supplementary material The online version of this article (https://doi.org/10.1007/s11336-019-09679-0) contains supplementary material, which is available to authorized users.

The R code and the real MGRM item parameters used in this paper are available online.

References

Celeux, G.Forbes, F.Robert, C. P., & Titterington, D. M. (2006). Deviance information criteria for missing data models. Bayesian Analysis, 1 (4), 651–673.CrossRefGoogle Scholar
daSilva, M. A.Bazán, J. L., & Huggins-Manley, A. C. (2019). Sensitivity analysis and choosing between alternative polytomous IRT models using Bayesian model comparison criteria. Communications in Statistics-Simulation and Computation, 48 (2), 601–620.CrossRefGoogle Scholar
De Boeck, P. (2008). Random item IRT models Random item IRT models. Psychometrika, 73, 533–559.CrossRefGoogle Scholar
Denwood, M. J. (2016). runjags: An R package providing interface utilities, model templates, parallel computing methods and additional distributions for MCMC models in JAGS. Journal of Statistical Software, 71 (9), 1–25. 10.18637/jss.v071.i09.CrossRefGoogle Scholar
Efron, B. (1986). How biased is the apparent error rate of a prediction rule?. Journal of the American Statistical Association, 81, 461–470.CrossRefGoogle Scholar
Fox, J. P. (2010). Bayesian item response modeling: Theory and applications, New York, NY: Springer.CrossRefGoogle Scholar
Furr, D. C. (2017). Bayesian and frequentist cross-validation methods for explanatory item response models. (Unpublished doctoral dissertation). University of California Berkeley, CA.Google Scholar
Gelfand, A. E.Sahu, S. K., & Carlin, B. P. (1995). Efficient parametrisations for normal linear mixed models. Biometrika, 82, 379–488.CrossRefGoogle Scholar
Gelman, A.Carlin, J. B.Stern, H. S.Dunson, D. B.Vehtari, A.Rubin, D. B.et.al (2013). Bayesian data analysis, 3 New York: Chapman & Hall/CRC.CrossRefGoogle Scholar
Gelman, A.Hwang, J., & Vehtari, A. (2014). Understanding predictive information criteria for Bayesian models. Statistics and Computing, 24, 997–1016.CrossRefGoogle Scholar
Gelman, A.Jakulin, A.Pittau, M. G., & Su, Y. S. (2008). A weakly informative default prior distribution for logistic and other regression models. The Annals of Applied Statistics, 2, 1360–1383.CrossRefGoogle Scholar
Gelman, A.Meng, X. L., & Stern, H. (1996). Posterior predictive assessment of model fitness via realized discrepancies. Statistica Sinica, 6, 733–807.Google Scholar
Gelman, A., & Rubin, D. B. (1992). Inference from iterative simulation using multiple sequences (with discussion). Statistical Science, 7, 457–511.CrossRefGoogle Scholar
Gronau, Q. F., & Wagenmakers, E. J. (2018). Limitations of Bayesian leave-one-out cross-validation for model selection. Computational Brain & Behavior, 2 (1), 1–11.CrossRefGoogle ScholarPubMed
Hoeting, J. A.Madigan, D.Raftery, A. E., & Volinsky, C. T. (1999). Bayesian model averaging: A tutorial. Statistical Science, 14, 382–417.Google Scholar
Kang, T.Cohen, A. S., & Sung, H. J. (2009). Model selection indices for polytomous items. Applied Psychological Medicine, 35, 499–518.CrossRefGoogle Scholar
Kaplan, D. (2014). Bayesian statistics for the social sciences, New York, NY: The Guildford Press.Google Scholar
Lancaster, T. (2000). The incidental parameter problem since 1948. Journal of Econometrics, 95, 391–413.CrossRefGoogle Scholar
Levy, R.Mislevy, R. J. (2016). Bayesian psychometric modeling, Boca Raton, FL: Chapman & Hall.Google Scholar
Li, F.Cohen, A. S.Kim, S. H., & Cho, S. J. (2009). Model selection methods for mixture dichotomous IRT models. Applied Psychological Measurement, 33, 353–373.CrossRefGoogle Scholar
Li, L.Qui, S., & Feng, C. X. (2016). Approximating cross-validatory predictive evaluation in Bayesian latent variable models with integrated IS and WAIC. Statistics and Computing, 26, 881–897.CrossRefGoogle Scholar
Lu, Z. H.Chow, S. M., & Loken, E. (2017). A comparison of Bayesian and frequentist model selection methods for factor analysis models. Psychological Methods, 22 (2), 361–381.CrossRefGoogle ScholarPubMed
Lunn, D.Jackson, C.Best, N.Thomas, A.Spiegelhalter, D. (2012). The BUGS book: A practical introduction to Bayesian analysis, New York, NY: Chapman & Hall/CRC.CrossRefGoogle Scholar
Lunn, D.Thomas, A.Best, N., & Spiegelhalter, D. (2000). WinBUGS—a Bayesian modelling framework: Concepts, structure, and extensibility. Statistics and Computing, 10, 325–337.CrossRefGoogle Scholar
Luo, U., & Al-Harbi, K. (2017). Performances of LOO and WAIC as IRT model selection methods. Psychological Test and Assessment Modeling, 59, 183–205.Google Scholar
Marshall, E. C., & Spiegelhalter, D. J. (2007). Identifying outliers in Bayesian hierarchical models: A simulation-based approach. Bayesian Analysis, 2 (2), 409–444.CrossRefGoogle Scholar
McElreath, R. (2015). Statistical rethinking: A Bayesian course with examples in R and Stan, New York, NY: Chapman & Hall/CRC.Google Scholar
Merkle, E. C., & Rosseel, Y. (2018). blavaan: Bayesian structural equation models via parameter expansion. Journal of Statistical Software, 85 (4), 1–30.CrossRefGoogle Scholar
Millar, R. B. (2009). Comparison of hierarchical Bayesian models for overdispersed count data using DIC and Bayes’ factors. Biometrics, 65, 962–969.CrossRefGoogle ScholarPubMed
Millar, R. B. (2018). Conditional vs. marginal estimation of predictive loss of hierarchical models using WAIC and cross-validation. Statistics and Computing, 28, 375–385.CrossRefGoogle Scholar
Mislevy, R. J. (1986). Bayes modal estimation in item response models. Psychometrika, 51, 177–195.CrossRefGoogle Scholar
Muthén, B., & Asparouhov, T. (2012). Bayesian structural equation modeling: A more flexible representation of substantive theory. Psychological Methods, 17, 313–335.CrossRefGoogle ScholarPubMed
Navarro, D. (2018). Between the devil and the deep blue sea: Tensions between scientific judgement and statistical model selection. Computational Brain & Behavior, 2 (1), 28–34.CrossRefGoogle Scholar
Naylor, J. C., & Smith, A. F. (1982). Applications of a method for the efficient computation of posterior distributions. Journal of the Royal Statistical Society C (Applied Statistics), 31, 214–225.Google Scholar
Neyman, J., & Scott, E. L. (1948). Consistent estimates based on partially consistent observations. Econometrica, 16, 1–32.CrossRefGoogle Scholar
O’Hagan, A. (1976). On posterior joint and marginal modes. Biometrika, 63, 329–333.CrossRefGoogle Scholar
Piironen, J., & Vehtari, A. (2017). Comparison of Bayesian predictive methods for model selection. Statistics and Computing, 27, 711–735.CrossRefGoogle Scholar
Pinheiro, J. C., & Bates, D. M. (1995). Approximations to the log-likelihood function in the nonlinear mixed-effects model. Journal of Computational Graphics and Statistics, 4, 12–35.CrossRefGoogle Scholar
Plummer, M. (2003). JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling. In K. Hornik, Leisch, F. & Zeileis, A. (Eds.), Proceedings of the 3rd international workshop on distributed statistical computing.Google Scholar
Plummer, M. (2008). Penalized loss functions for Bayesian model comparison. Biostatistics, 9 (3), 523–539.CrossRefGoogle ScholarPubMed
Rabe-Hesketh, S.Skrondal, A., & Pickles, A. (2005). Maximum likelihood estimation of limited and discrete dependent variable models with nested random effects. Journal of Econometrics, 128 (2), 301–323.CrossRefGoogle Scholar
Raftery, A. E.Lewis, S. M. (1995). The number of iterations, convergence diagnostics, and generic Metropolis algorithms, London: Chapman and Hall.Google Scholar
Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical linear models: Applications and data analysis methods, 2 Thousand Oaks, CA: Sage.Google Scholar
Rosseel, Y. (2012). lavaan: An R package for structural equation modeling. Journal of Statistical Software, 48 (2), 1–36.CrossRefGoogle Scholar
Song, X. Y., & Lee, S. Y. (2012). Basic and advanced Bayesian structural equation modeling: With applications in the medical and behavioral sciences, Chichester, UK: Wiley.CrossRefGoogle Scholar
Spiegelhalter, D. J.Best, N. G.Carlin, B. P., & van der Linde, A. (2002). Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society Series, B64, 583–639.CrossRefGoogle Scholar
Spielberger, C. (1988). State-trait anger expression inventory research edition [Computer software manual]. FL: Odessa.Google Scholar
Stan Development Team. (2014). Stan modeling language users guide and reference manual, version 2.5.0 [Computer software manual]. http://mc-stan.org/.Google Scholar
Trevisani, M., & Gelfand, A. E. (2003). Inequalities between expected marginal log-likelihoods, with implications for likelihood-based model complexity and comparison measures. The Canadian Journal of Statistics, 31, 239–250.CrossRefGoogle Scholar
Vansteelandt, K. (2000). Formal models for contextualized personality psychology (Unpublished doctoral dissertation), Belgium: University of Leuven Leuven.Google Scholar
Vehtari, A., Gelman, A., & Gabry, J. (2016). loo: Efficient leave-one-out cross-validation and WAIC for Bayesian models. R package version 0.1.6. https://github.com/stan-dev/loo.Google Scholar
Vehtari, A.Gelman, A., & Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing, 27, 1413–1432.CrossRefGoogle Scholar
Vehtari, A.Mononen, T.Tolvanen, V.Sivula, T., & Winther, O. (2016). Bayesian leave-one-out cross-validation approximations for Gaussian latent variable models. Journal of Machine Learning Research, 17, 1–38.Google Scholar
Vehtari, A.Simpson, D. P.Yao, Y., & Gelman, A. (2018). Limitations of "Limitations of Bayesian leave-one-out cross-validation for model selection". Computational Brain & Behavior, 2 (1), 22–27.CrossRefGoogle Scholar
Watanabe, S. (2010). Asymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory. Journal of Machine Learning Research, 11, 3571–3594.Google Scholar
White, I. R. (2010). simsum: Analyses of simulation studies including Monte Carlo error. The Stata Journal, 10, 369–385.CrossRefGoogle Scholar
Wicherts, J. M.Dolan, C. V., & Hessen, D. J. (2005). Stereotype threat and group differences in test performance: A question of measurement invariance. Journal of Personality and Social Psychology, 89 (5), 696–716.CrossRefGoogle ScholarPubMed
Yao, Y.Vehtari, A.Simpson, D., & Gelman, A. (2018). Using stacking to average Bayesian predictive distributions (with discussion). Bayesian Analysis, 13, 917–1007. https://10.1214/17-BA1091.CrossRefGoogle Scholar
Zhang, X.Tao, J.Wang, C., & Shi, N. Z. (2019). Bayesian model selection methods for multilevel IRT models: A comparison of five DIC-based indices. Journal of Educational Measurement, 56, 3–27.CrossRefGoogle Scholar
Zhao, Z., & Severini, T. A. (2017). Integrated likelihood computation methods. Computational Statistics, 32, 281–313.CrossRefGoogle Scholar
Zhu, X., & Stone, C. A. (2012). Bayesian comparison of alternative graded response models for performance assessment applications. Educational and Psychological Measurement, 7 (2), 5774–799.Google Scholar
Supplementary material: File

Merkle et al. supplementary material

Merkle et al. supplementary material
Download Merkle et al. supplementary material(File)
File 7.4 KB