Hostname: page-component-669899f699-chc8l Total loading time: 0 Render date: 2025-04-26T05:41:29.092Z Has data issue: false hasContentIssue false
  • English
  • Français

Ordered Beta Regression: A Parsimonious, Well-Fitting Model for Continuous Data with Lower and Upper Bounds

Published online by Cambridge University Press:  27 July 2022

Robert Kubinec Open the ORCID record for Robert Kubinec [Opens in a new window]
Robert Kubinec*
Affiliation:
Division of Social Sciences, New York University Abu Dhabi, Abu Dhabi, United Arab Emirates. E-mail: rmk7@nyu.edu
*
Corresponding author Robert Kubinec
Get access Rights & Permissions [Opens in a new window]

Abstract

I propose a new model, ordered Beta regression, for continuous distributions with both lower and upper bounds, such as data arising from survey slider scales, visual analog scales, and dose–response relationships. This model employs the cut point technique popularized by ordered logit to fit a single linear model to both continuous (0,1) and degenerate [0,1] responses. The model can be estimated with or without observations at the bounds, and as such is a general solution for these types of data. Employing a Monte Carlo simulation, I show that the model is noticeably more efficient than ordinary least squares regression, zero-and-one-inflated Beta regression, rescaled Beta regression, and fractional logit while fully capturing nuances in the outcome. I apply the model to a replication of the Aidt and Jensen (2014, European Economic Review 72, 52–75) study of suffrage extensions in Europe. The model can be fit with the R package ordbetareg to facilitate hierarchical, dynamic, and multivariate modeling.

Keywords

Bayesian statisticslimited dependent variablesregression modeling
Type
Article
Information
Political Analysis , Volume 31 , Issue 4 , October 2023 , pp. 519 - 536
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of the Society for Political Methodology

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

Edited by Jeff Gill

References

Aidt, T. S., and Jensen, P. S.. 2014. “Workers of the World, Unite! Franchise Extensions and the Threat of Revolution in Europe, 1820–1938.” European Economic Review 72: 52–75. https://doi.org/10.1016/j.euroecorev.2014.08.001 CrossRefGoogle Scholar
Betancourt, M. 2019. “Ordinal Regression.” Case Study 1. https://betanalpha.github.io/assets/case_studies/ordinal_regression.html.Google Scholar
Bürkner, P.-C. 2017. “brms: An R Package for Bayesian Multilevel Models Using Stan.” Journal of Statistical Software 80 (1): 128. https://doi.org/10.18637/jss.v080.i01 CrossRefGoogle Scholar
Carpenter, B., et al. 2017. “Stan: A Probabilistic Programming Language.” Journal of Statistical Software 76: 1–32.CrossRefGoogle Scholar
Cooper, D. J., and Kagel, J. H.. 2016. “Other-Regarding Preferences: A Selective Survey of Experimental Results.” In The Handbook of Experimental Economics, edited by J. H. Kagel and A. E. Roth, Vol. II, 217–289. Princeton, NJ: Princeton University Press.Google Scholar
Ferrari, S., and Cribari-Neto, F.. 2004. “Beta Regression for Modelling Rates and Proportions.” Journal of Applied Statistics 31 (7): 799–815.CrossRefGoogle Scholar
Fisher, R., Ricardo, G., and Fox, D.. 2020. “Bayesian Concentration-Response Modelling Using jagsNEC.” Zenodo. https://doi.org/10.5281/zenodo.3966864 CrossRefGoogle Scholar
Gelman, A., and Carlin, J.. 2014. “Beyond Power Calculations: Assessing Type S (Sign) and Type M (Magnitude) Errors.” Perspectives on Psychological Science 9 (6): 641–651.CrossRefGoogle ScholarPubMed
Horrace, W. C., and Oaxaca, R. L.. 2006. “Results on the Bias and Inconsistency of Ordinary Least Squares for the Linear Probability Model.” Economics Letters 90 (3): 321–327. https://doi.org/10.1016/j.econlet.2005.08.024 CrossRefGoogle Scholar
Jaynes, E. T. 2003. Probability Theory: The Logic of Science. Cambridge, United Kingdom: Cambridge University Press.CrossRefGoogle Scholar
Kubinec, R. 2022. “Replication Data for Ordered Beta Regression: A Parsimonious, Well-Fitting Model for Continuous Data with Lower and Upper Bounds.” https://doi.org/10.7910/DVN/5XYO7O, Harvard Dataverse, V1, UNF:6:LGP00JRsQf1X+9SAoOLJxQ== [fileUNF].CrossRefGoogle Scholar
Lee, K. A., Hicks, G., and Nino-Murcia, G.. 1991. “Validity and Reliability of a Scale to Assess Fatigue.” Psychiatry Research 36 (3): 291–298.CrossRefGoogle ScholarPubMed
Leeper, T. J. 2021. “Interpreting Regression Results Using Average Marginal Effects with r’s Margins.” Package Vignette, January. https://cran.r-project.org/web/packages/margins/vignettes/TechnicalDetails.pdf.Google Scholar
Liu, F., and Kong, Y.. 2015. “Zoib: An R Package for Bayesian Inference for Beta Regression and Zero/One Inflated Beta Regression.” The R Journal 7 (2): 34–51.CrossRefGoogle Scholar
Liu, M., and Wang, Y.. 2015. “Data Collection Mode Effect on Feeling Thermometer Questions: A Comparison of Face-to-Face and Web Surveys.” Computers in Human Behavior 48: 212–218. https://doi.org/10.1016/j.chb.2015.01.057 CrossRefGoogle Scholar
McCullagh, P. 1980. “Regression Models for Ordinal Data.” Journal of the Royal Statistical Society 42 (2): 109–142.Google Scholar
Monk, T. H. 1989. “A Visual Analogue Scale Technique to Measure Global Vigor and Affect.” Psychiatry Research 27: 89–99.CrossRefGoogle ScholarPubMed
Myles, P. S., Myles, D. B., Wendy Galagher, D., Boyd, C. C., MacDonald, N., and Dennis, A.. 2017. “Measuring Acute Postoperative Pain Using the Visual Analog Scale: The Minimal Clinically Important Difference and Patient Acceptable Symptom State.” British Journal of Anaesthesia 118 (3): 424–429.CrossRefGoogle ScholarPubMed
Nelson, S. C. 2008. “Feeling Thermometer.” In Encyclopedia of Survey Research Methods, edited by Editor Paul J. Lavrakas, 277–277. Newbury Park, CA: SAGE Publishing.Google Scholar
Ospina, R., and Ferrari, S. L. P.. 2012. “A General Class of Zero-or-One Inflated Beta Regression Models.” Computational Statistics and Data Analysis 56 (6): 1609–1623.CrossRefGoogle Scholar
Papke, L. E., and Wooldridge, J. M.. 1996. “Econometric Methods for Fractional Response Variables with an Application to 401(k) Plan Participation Rates.” Journal of Applied Econometrics 11 (6): 619–632. https://doi.org/10.1002/(SICI)1099-1255(199611)11:6%3C619::AID-JAE418%3E3.0.CO;2-1 3.0.CO;2-1>CrossRefGoogle Scholar
Prentice, R. L. 1976. “A Generalization of the Probit and Logit Methods for Dose Response Curves.” Biometrics 32 (4): 761–768. https://doi.org/10.2307/2529262 CrossRefGoogle ScholarPubMed
Ritz, C., Baty, F., Streibig, J. C., and Gerhard, D.. 2015. “Dose-Response Analysis Using R.” PLoS One 10 (12): e0146021. https://doi.org/10.1371/journal.pone.0146021 CrossRefGoogle ScholarPubMed
Roster, C. A., Lucianetti, L., and Albaum, G.. 2015. “Exploring Slider vs. Categorical Response Formats in Web-Based Surveys.” Journal of Research Practice 11 (1): D1.Google Scholar
Samejima, F. 1997. “Graded Response Model.” In Handbook of Modern Item Response Theory, edited by van der Linden, W. J. and Hambleton, R. K., 85–100. Berlin, Germany: Springer.CrossRefGoogle Scholar
Smithson, M., and Verkuilen, J.. 2006. “A Better Lemon Squeezer? Maximum-Likelihood Regression with Beta-Distributed Independent Variables.” Psychological Methods 11 (1): 54–71.CrossRefGoogle Scholar
Stan Development Team. 2016. “Stan Modeling Language Users Guide and Reference Manual.” Manual, Stan Development Team. http://mc-stan.org.Google Scholar
Swearingen, C. J., Melguizo, M. S., and Bursac, Z.. 2012. “Inflated Beta Regression: Zero, One, and Everything in Between.” SAS Global Forum, 11. https://support.sas.com/resources/papers/proceedings12/325-2012.pdf.Google Scholar
Vehtari, A., Gelman, A., and Gabry, J.. 2016. “Practical Bayesian Model Evaluation Using Leave-One-Outcross-Validation and WAIC.” Statistical Computing 27: 1413–1432.CrossRefGoogle Scholar
Zorn, C. J. W. 1998. “An Analytic and Empirical Examination of Zero-Inflated and Hurdle Poisson Specifications.” Sociological Methods & Research 26 (3): 368–400.CrossRefGoogle Scholar
Supplementary material: Link

Kubinec Dataset

Dataset

https://doi.org/10.7910/DVN/5XYO7O
Link
Supplementary material: PDF

Kubinec supplementary material

Appendix

Download Kubinec supplementary material(PDF)
PDF 4.9 MB