The Annals of Applied Statistics

Multi-rubric models for ordinal spatial data with application to online ratings data

Antonio R. Linero, Jonathan R. Bradley, and Apurva Desai

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Abstract

Interest in online rating data has increased in recent years in which ordinal ratings of products or local businesses are provided by users of a website, such as Yelp! or Amazon. One source of heterogeneity in ratings is that users apply different standards when supplying their ratings; even if two users benefit from a product the same amount, they may translate their benefit into ratings in different ways. In this article we propose an ordinal data model, which we refer to as a multi-rubric model, which treats the criteria used to convert a latent utility into a rating as user-specific random effects, with the distribution of these random effects being modeled nonparametrically. We demonstrate that this approach is capable of accounting for this type of variability in addition to usual sources of heterogeneity due to item quality, user biases, interactions between items and users and the spatial structure of the users and items. We apply the model developed here to publicly available data from the website Yelp! and demonstrate that it produces interpretable clusterings of users according to their rating behavior, in addition to providing better predictions of ratings and better summaries of overall item quality.

Article information

Source
Ann. Appl. Stat., Volume 12, Number 4 (2018), 2054-2074.

Dates
Received: September 2017
Revised: December 2017
First available in Project Euclid: 13 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1542078036

Digital Object Identifier
doi:10.1214/18-AOAS1143

Mathematical Reviews number (MathSciNet)
MR3875692

Keywords
Bayesian hierarchichal model data augmentation nonparametric Bayes ordinal data spatial prediction

Citation

Linero, Antonio R.; Bradley, Jonathan R.; Desai, Apurva. Multi-rubric models for ordinal spatial data with application to online ratings data. Ann. Appl. Stat. 12 (2018), no. 4, 2054--2074. doi:10.1214/18-AOAS1143. https://projecteuclid.org/euclid.aoas/1542078036


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References

  • Albert, J. H. and Chib, S. (1993). Bayesian analysis of binary and polychotomous response data. J. Amer. Statist. Assoc. 88 669–679.
  • Albert, J. H. and Chib, S. (1997). Bayesian methods for cumulative, sequential and two-step ordinal data regression models. Technical report, Dept. Mathematics and Statistics, Bowling Green State Univ., Bowling Green, OH.
  • Banerjee, S., Gelfand, A. E., Finley, A. O. and Sang, H. (2008). Gaussian predictive process models for large spatial data sets. J. R. Stat. Soc. Ser. B. Stat. Methodol. 70 825–848.
  • Bao, J. and Hanson, T. E. (2015). Bayesian nonparametric multivariate ordinal regression. Canad. J. Statist. 43 337–357.
  • Berrett, C. and Calder, C. A. (2012). Data augmentation strategies for the Bayesian spatial probit regression model. Comput. Statist. Data Anal. 56 478–490.
  • Binder, D. A. (1978). Bayesian cluster analysis. Biometrika 65 31–38.
  • Bobadilla, J., Ortega, F., Hernando, A. and Gutiérrez, A. (2013). Recommender systems survey. Knowl.-Based Syst. 46 109–132.
  • Bradley, J. R., Holan, S. H. and Wikle, C. K. (2015). Multivariate spatio-temporal models for high-dimensional areal data with application to longitudinal employer-household dynamics. Ann. Appl. Stat. 9 1761–1791.
  • Bradley, J. R., Wikle, C. K. and Holan, S. H. (2016). Bayesian spatial change of support for count-valued survey data with application to the American community survey. J. Amer. Statist. Assoc. 111 472–487.
  • Cargnoni, C., Müller, P. and West, M. (1997). Bayesian forecasting of multinomial time series through conditionally Gaussian dynamic models. J. Amer. Statist. Assoc. 92 640–647.
  • Carlin, B. P. and Polson, N. G. (1992). Monte Carlo Bayesian methods for discrete regression models and categorical time series. In Bayesian Statistics, 4 (Peñíscola, 1991) (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. Smith, eds.) 109–125. Oxford Univ. Press, New York.
  • Chen, M.-H. and Dey, D. K. (2000). A unified Bayesian approach for analyzing correlated ordinal response data. Braz. J. Probab. Stat. 14 87–111.
  • Cowles, M. K. (1996). Accelerating Monte Carlo Markov chain convergence for cumulative-link generalized linear models. Stat. Comput. 6 101–111.
  • Cressie, N. A. C. (2015). Statistics for Spatial Data, revised ed. Wiley, New York.
  • Cressie, N. and Johannesson, G. (2008). Fixed rank kriging for very large spatial data sets. J. R. Stat. Soc. Ser. B. Stat. Methodol. 70 209–226.
  • De Oliveira, V. (2000). Bayesian prediction of clipped Gaussian random fields. Comput. Statist. Data Anal. 34 299–314.
  • De Oliveira, V. (2004). A simple model for spatial rainfall fields. Stoch. Environ. Res. Risk Assess. 18 131–140.
  • DeYoreo, M. and Kottas, A. (2014). Bayesian nonparametric modeling for multivariate ordinal regression. Preprint. Available at arXiv:1408.1027.
  • Escobar, M. D. and West, M. (1995). Bayesian density estimation and inference using mixtures. J. Amer. Statist. Assoc. 90 577–588.
  • Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 209–230.
  • Fritsch, A. and Ickstadt, K. (2009). Improved criteria for clustering based on the posterior similarity matrix. Bayesian Anal. 4 367–391.
  • Gelman, A. (2006). Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper). Bayesian Anal. 1 515–533.
  • Gill, J. and Casella, G. (2009). Nonparametric priors for ordinal Bayesian social science models: Specification and estimation. J. Amer. Statist. Assoc. 104 453–464.
  • Härdle, W. and Müller, M. (2000). Multivariate and semiparametric kernel regression. In Smoothing and Regression: Approaches, Computation, and Application (M. G. Schimek, ed.). Wiley, New York.
  • Higgs, M. D. and Hoeting, J. A. (2010). A clipped latent variable model for spatially correlated ordered categorical data. Comput. Statist. Data Anal. 54 1999–2011.
  • Hoffman, M. D., Blei, D. M., Wang, C. and Paisley, J. (2013). Stochastic variational inference. J. Mach. Learn. Res. 14 1303–1347.
  • Houlsby, N., Hernández-Lobato, J. M. and Ghahramani, Z. (2014). Cold-start active learning with robust ordinal matrix factorization. In International Conference on Machine Learning 766–774.
  • Ishwaran, H. and Zarepour, M. (2002). Dirichlet prior sieves in finite normal mixtures. Statist. Sinica 12 941–963.
  • Knorr-Held, L. (1995). Dynamic cumulative probit models for ordinal panel-data; a Bayesian analysis by Gibbs sampling. Technical report, Ludwig-Maximilians-Univ., Munich.
  • Koren, Y. and Sill, J. (2011). OrdRec: An ordinal model for predicting personalized item rating distributions. In Proceedings of the Fifth ACM Conference on Recommender Systems 117–124. ACM, New York.
  • Kottas, A., Müller, P. and Quintana, F. (2005). Nonparametric Bayesian modeling for multivariate ordinal data. J. Comput. Graph. Statist. 14 610–625.
  • Linero, A. R. (2018). Bayesian regression trees for high dimensional prediction and variable selection. J. Amer. Statist. Assoc. 113 626–636.
  • Linero, A. R., Bradley, J. R. and Desai, A. (2018). Supplement to “Multi-rubric models for ordinal spatial data with application to online ratings data”. DOI:10.1214/18-AOAS1143SUPP.
  • Marlin, B. M. and Zemel, R. S. (2009). Collaborative prediction and ranking with non-random missing data. In Proceedings of the Third ACM Conference on Recommender Systems 5–12. ACM, New York.
  • Marlin, B. M., Zemel, R. S., Roweis, S. and Slaney, M. (2007). Collaborative filtering and the missing at random assumption. In Proceedings of the Twenty-Third Conference on Uncertainty in Artificial Intelligence 267–275. AUAI Press, Corvallis, OR.
  • Mnih, A. and Salakhutdinov, R. R. (2008). Probabilistic matrix factorization. In Advances in Neural Information Processing Systems 20 1257–1264. Curran Associates, Red Hook, NY.
  • Neal, R. M. (2003). Slice sampling. Ann. Statist. 31 705–767.
  • Paquet, U., Thomson, B. and Winther, O. (2012). A hierarchical model for ordinal matrix factorization. Stat. Comput. 22 945–957.
  • Rasmussen, C. E. and Williams, C. K. I. (2006). Gaussian Processes for Machine Learning. MIT Press, Cambridge, MA.
  • Sang, H. and Huang, J. Z. (2012). A full scale approximation of covariance functions for large spatial data sets. J. R. Stat. Soc. Ser. B. Stat. Methodol. 74 111–132.
  • Schliep, E. M. and Hoeting, J. A. (2015). Data augmentation and parameter expansion for independent or spatially correlated ordinal data. Comput. Statist. Data Anal. 90 1–14.
  • Simpson, D., Rue, H., Riebler, A., Martins, T. G. and Sørbye, S. H. (2017). Penalising model component complexity: A principled, practical approach to constructing priors. Statist. Sci. 32 1–28.
  • Teh, Y. W., Jordan, M. I., Beal, M. J. and Blei, D. M. (2006). Hierarchical Dirichlet processes. J. Amer. Statist. Assoc. 101 1566–1581.
  • Velozo, P. L., Alves, M. B. and Schmidt, A. M. (2014). Modelling categorized levels of precipitation. Braz. J. Probab. Stat. 28 190–208.

Supplemental materials

  • Identifiability of model parameters. In this supplementary material we discuss identifiability of the model parameters; we give empirical evidence that the latent variables are identified up to orthogonal transformations.