Extending Ordinal Regression with a Latent Zero-Augmented Beta Distribution

Abstract

Ecological abundance data are often recorded on an ordinal scale in which the lowest category represents species absence. One common example is when plant species cover is visually assessed within bounded quadrats and then assigned to pre-defined cover class categories. We present an ordinal beta hurdle model that directly models ordinal category probabilities with a biologically realistic beta-distributed latent variable. A hurdle-at-zero model allows ecologists to explore distribution (absence) and abundance processes in an integrated framework. This provides an alternative to cumulative link models when data are inconsistent with the assumption that the odds of moving into a higher category are the same for all categories (proportional odds). Graphical tools and a deviance information criterion were developed to assess whether a hurdle-at-zero model should be used for inferences rather than standard ordinal methods. Hurdle-at-zero and non-hurdle ordinal models fit to vegetation cover class data produced substantially different conclusions. The ordinal beta hurdle model yielded more precise parameter estimates than cumulative logit models, although out-of-sample predictions were similar. The ordinal beta hurdle model provides inferences directly on the latent biological variable of interest, percent cover, and supports exploration of more realistic ecological patterns and processes through the hurdle-at-zero or two-part specification. We provide JAGS code as an on-line supplement. Supplementary materials accompanying this paper appear on-line.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

References

  1. Agresti, A. (2010), Analysis of ordinal categorical data John Wiley and Sons, Hoboken, NJ, USA.

    Google Scholar 

  2. Agresti, A., and Kateri, M. (2014), Some Remarks on Latent Variable Models in Categorical Data Analysis. Communications in Statistics - Theory and Methods, 43, 801–814.

    MathSciNet  Article  MATH  Google Scholar 

  3. Albert, J. H., and Chib, S. (1993), Bayesian analysis of binary and polychotomous response data. Journal of the American Statistical Association, 88, 669–679.

    MathSciNet  Article  MATH  Google Scholar 

  4. Ananth, C. V., and Kleinbaum, D. G. (1997), Regression models for ordinal responses: a review of methods and applications. International Journal of Epidemiology, 26, 1323–1333.

    Article  Google Scholar 

  5. Andrewartha, H. G., and Birch, L. C. (1954), The distribution and abundance of animals. University of Chicago Press, Chicago, Illinois, USA.

    Google Scholar 

  6. Bonham, C. D. (1989), Measurements for terrestrial vegetation John Wiley and Sons, New York, NY.

    Google Scholar 

  7. Braun-Blanquet, J. (1932), Plant sociology. The study of plant communities. McGraw-Hill, New York, NY, US.

  8. Branscum, A. J., Johnson, W. O., and Thurmond, M. C. (2007), Bayesian beta regression: applications to household expenditure data and genetic distance between foot-and-mouth disease viruses. Australian New Zealand Journal of Statistics, 49, 287–301.

    MathSciNet  Article  MATH  Google Scholar 

  9. Chambers, J. C., Roundy, B. A., Blank, R. R., Meyer, S. E., and Whittaker, A. (2007), What makes Great Basin sagebrush ecosystems invasible by Bromus tectorum? Ecological Monographs, 77, 117–145.

    Article  Google Scholar 

  10. Chen, J., Shiyomi, M, Hori, Y., and Yamamura, Y. (2008a), Frequency distribution models for spatial patterns of vegetation abundance. Ecological Modelling, 211, 403–410.

    Article  Google Scholar 

  11. Chen, J., Shiyomi, M., Bonham, C. D., Yasuda, T., Hori, Y., and Yamamura, Y. (2008b), Plant cover estimation based on the beta distribution in grassland vegetation. Ecological Research, 23, 813–819.

    Article  Google Scholar 

  12. Christensen, R. H. B. (2014), Ordinal:Regression models for ordinal data. R package version 2014.12-22. Available at http://cran.r-project.org/web/packages/ordinal/index.html.

  13. Congdon, P. (2005), Bayesian models for categorical data. John Wiley and Sons, Hoboken, NJ, USA.

    Google Scholar 

  14. Coudun, C. and Gegout, J.-C. (2007). Quantitative prediction of the distribution and abundance of Vaccinium myrtillus with climatic and edaphic factors. Journal of Vegetation Science, 18, 517–524.

    Article  Google Scholar 

  15. Damgaard, C. (2009), On the distribution of plant abundance data. Ecological Informatics, 4, 76–82.

    Article  Google Scholar 

  16. Damgaard, C. (2012), Trend analyses of hierarchical pin-point cover data. Ecology, 93, 1269–1274.

    Article  Google Scholar 

  17. Daubenmire, R. F. (1959), A canopy-coverage method. Northwest Science, 33, 43–64.

    Google Scholar 

  18. Davies, K. W., Boyd, C. S., Beck, J. L., Bates, J. D., Svejcar, T. J., and Gregg, M. A. (2011), Saving the sagebrush sea: an ecosystem conservation plan for big sagebrush plant communities. Biological Conservation, 144, 2573–2584.

    Article  Google Scholar 

  19. Davies, K. W., Nafus, A. M., and Madsen, M. D. (2013), Medusahead invasion along unimproved roads, animal trails, and random transects. Western North American Naturalist, 73, 54–59.

    Article  Google Scholar 

  20. Duff, T. J., Bell, T. L., and York, A. (2011), Patterns of plant abundances in natural systems: is there value in modelling both species abundance and distribution?. Australian Journal of Botany, 59, 719–733.

    Google Scholar 

  21. Eskelson, N. I., Madsen, L., Hagar, J. C., and Temesgen, H. (2011), Estimating riparian understory vegetation cover with beta regression and copula models. Forest Science, 57, 212–221.

    Google Scholar 

  22. Esposito, D. M., Shanahan, E., and Rodhouse, T. J. (2016), UCBN and GRYN Sagebrush Steppe Vegetation Monitoring: Double Observer Study 2015. John Day Fossil Beds National Monument-Clarno Unit and City of Rocks National Reserve. Natural Resource Reporting Series NPS/UCBN/NRR-2016/1052. National Park Service, Fort Collins, Colorado.

  23. Fahrmeier, L, and Tutz, G. (2001), Multivariate statistical modelling based on generalized linear models. Springer, Berlin.

  24. Feng, X., Zhu, J., and Steen-Adams, M. M. (2015), On regression analysis of spatial proportional data with zero/one values. Spatial Statistics, 14, 452–471.

    MathSciNet  Article  Google Scholar 

  25. Ferrari, S. L. P. and Cribari-Neto, F. (2004), Beta regression for modelling rates and proportions. Journal of Applied Statistics, 31, 799–815.

    MathSciNet  Article  MATH  Google Scholar 

  26. Gelbard, J. L. and Belnap, J. (2003), Roads as conduits for exotic plant invasions in a semiarid landscape. Conservation Biology, 17, 420-432.

    Article  Google Scholar 

  27. Gelfand, A. E. and Smith, A. F. M. (1990), Sampling-based approaches to calculating marginal densities. Journal of the American Statistical Association, 85, 398–409.

    MathSciNet  Article  MATH  Google Scholar 

  28. Gruen, B., Kosmidis, I., and Zeileis, A. (2012), Extended Beta Regression in R: Shaken, Stirred, Mixed, and Partitioned. Journal of Statistical Software, 48, 1–25.

    Google Scholar 

  29. Guisan, A. and Harrell, F. E. (2000), Ordinal response regression models in Ecology. Journal of Vegetation Science, 11, 617–626.

    Article  Google Scholar 

  30. Hall, D. B. (2000), Zero-inflated Poisson and Binomial Regression with Random Effects: A Case Study. Biometrics, 56, 1030–1039.

    MathSciNet  Article  MATH  Google Scholar 

  31. Heilbron, D. C. (1994), Zero-altered and other Regression Models for Count Data with Added Zeros. Biometrical Journal, 36, 531–547.

    Article  MATH  Google Scholar 

  32. Higgs, M. D. and Hoeting, J. A. (2010), A clipped latent variable model for spatially correlated ordered categorical data. Computational Statistics and Data Analysis, 54, 1999–2011.

    MathSciNet  Article  MATH  Google Scholar 

  33. Higgs, M. D. and Ver Hoef, J. M. (2012), Discretized and Aggregated: modeling dive depth of Harbor Seals from Ordered Categorical data with temporal autocorrelation. Biometrics, 68, 965–974.

    MathSciNet  Article  MATH  Google Scholar 

  34. Holland, M. D. and Gray, B. R. (2011), Multinomial mixture model with heterogeneous classification probabilities. Ecological and Environmental Statistics, 18, 257–270.

    MathSciNet  Article  Google Scholar 

  35. Irvine, K. M. and Rodhouse, T. J. (2010), Power analysis for trend in ordinal cover classes: implications for long-term vegetation monitoring. Journal of Vegetation Science, 21, 1152–1161.

    Article  Google Scholar 

  36. Ishwaran, H. (2000), Univariate and multirater ordinal cumulative link regression with covariate specific cutpoints. The Canadian Journal of Statistics, 28, 715–730.

  37. Kelley, M. E. and Anderson, S. J. (2008), Zero inflation in ordinal data: incorporating susceptibility to response through the use of a mixture model. Statistics in Medicine, 27, 3674–3688.

    MathSciNet  Article  Google Scholar 

  38. Kim, J-H. (2003), Assessing practical significance of the proportional odds assumption. Statistics and Probability Letters, 65, 233–239.

    MathSciNet  Article  MATH  Google Scholar 

  39. Kosmidis, I. and Firth, D. (2010), A generic algorithm for reducing bias in parametric estimation.Electronic Journal of Statistics, 4, 1097–1112.

    MathSciNet  Article  MATH  Google Scholar 

  40. Lachenbruch, P. A. (2002), Analysis of Data with Excess Zeros. Statistical Methods in Medical Research, 11, 297–302.

    Article  MATH  Google Scholar 

  41. Lambert, D. (1992), Zero-Inflated Poisson Regression, with an Application to Defects in Manufacturing. Technometrics, 34, 1–14.

    Article  MATH  Google Scholar 

  42. Larrabee, B., Scott, H. M., and Bello, N. M. (2014), Ordinary least squares regression of ordered categorical data: inferential implications in practice. Journal of Agricultural, Biological, and Environmental Statistics, 19, 373–386.

    MathSciNet  Article  MATH  Google Scholar 

  43. Link, W. A. and Eaton, M. J. (2012) On thinning of chains in MCMC. Methods in Ecology and Evolution , 3, 112–115.

    Article  Google Scholar 

  44. Mackenzie, D. I., Nichols, J. D., Royle, J. A., Pollock, K. H., Bailey, L. L., and Hines, J. E. (2006), Occupancy estimation and modeling:inferring patterns and dynamics of species occurrence. Elsevier Academic Press, Burlington, MA, USA.

    Google Scholar 

  45. Martin, T. G., Wintle, B. A., Rhodes, J. R., Kuhnert, P. M., Field, S. A., Low-Choy, S. J., Tyre, A. J., and Possingham, H. P. (2005), Zero tolerance ecology: improving ecological inference by modeling the source of zero observations. Ecology Letters, 8, 1235–1246.

    Article  Google Scholar 

  46. Milberg, P., Bergstedt, J., Fridman, J., Gunnar, O., Westerberg, L. (2008), Observer bias and random variation in vegetation monitoring data. Journal of Vegetation Science, 19, 633–644.

    Article  Google Scholar 

  47. Millar, R. B. (2009), Comparison of hierachical Bayesian models for overdispersed count data using DIC and Bayes’ factors. Biometrics, 65, 962–969.

    MathSciNet  Article  MATH  Google Scholar 

  48. Miller, R. F., Chambers, J. C., Pyke, D. A., Pierson, F. B., and Williams, C. J. (2013), A review of fire effects on vegetation and soils in the Great Basin region: response and ecological site characteristics. RMRS GTR-308. USDA Forest Service, Rocky Mountain Research Station, Fort Collins, Colorado, USA.

  49. Moulton, L. H. and Halsey, N. A. (1995), A Mixture Model with Detection Limits for Regression Analyses of Antibody Response to Vaccine. Biometrics, 51, 1570–1578.

    Article  MATH  Google Scholar 

  50. Neelon, B. H., O’Malley, A. J., and Normand, S-L T. (2010), A Bayesian model for repeated measures zero-inflated count data with application to outpatient psychiatric service use. Statistical Modelling, 10, 421–439.

    MathSciNet  Article  Google Scholar 

  51. Ospina, R. and Ferrari, S. L. (2010), Inflated beta distributions. Statistical Papers, 51, 111–126.

    MathSciNet  Article  MATH  Google Scholar 

  52. Plummer, M. (2003), JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling. In Proceedings of the 3rd International Workshop on Distributed Statistical Computing (DSC 2003), K. Hornik, F. Leisch, and A. Zeileis (eds.) Vienna, Austria. Available at: http://www.ci.tuwien.ac.at/Conferences/DSC-2003/

  53. Plummer, M. (2008), Penalized loss functions for Bayesian model comparison. Biostatistics, 9, 523–539.

    Article  MATH  Google Scholar 

  54. Plummer, M. (2015), JAGS Version 4.0.0 user manual. Available at https://sourceforge.net/projects/mcmc-jags/files/Manuals/4.x/.

  55. Reisner, M. D., Grace, J. B., Pyke, D. A., and Doescher, P. S. (2013), Conditions favouring Bromus tectorum dominance of endangered sagebrush steppe ecosystems. Journal of Applied Ecology, 50, 1039–1049.

    Article  Google Scholar 

  56. Rodhouse, T. J., Irvine, K. M., Sheley, R. L., Smith, B. S., Hoh, S., Esposito, D. M., and Mata-Gonzalez, R. (2014), Predicting foundation bunchgrass species abundances: model-assisted decision-making in protected-area sagebrush-steppe. Ecosphere, 5, art208.

  57. Royle, J. A. and Link, W. A. (2005), A general class of multinomial mixture models for anuran calling survey data. Ecology, 86, 2505–2512.

    Article  Google Scholar 

  58. Schabenberger, O. (1995), The use of ordinal response methodology in Forestry. Forest Science, 41, 321–336.

    Google Scholar 

  59. Schliep, E. M. and Hoeting, J. A. (2013), Multilevel latent Gaussian process model for mixed disrete and continuous multivariate response data. Journal of Agricultural, Biological, and Environmental Statistics, 18, 492–513.

    MathSciNet  Article  MATH  Google Scholar 

  60. Spiegelhalter, D., Best, N., Carlin, B., and van der Linde, A. (2002), Bayesian measures of model complexity and fit (with discussion). Journal of the Royal Statistical Society B, 64, 583–639.

  61. Stevens, D. L., and Olsen, A. R. (2004), Spatially balanced sampling of natural resources. Journal of the American Statistical Association, 99, 262–278.

    MathSciNet  Article  MATH  Google Scholar 

  62. Stroup, W. W. (2014), Rethinking the analysis of non-normal data in plant and soil science. Agronomy Journal, 106, 1–17.

    Article  Google Scholar 

  63. Tamhane, A. C., Ankenman, B. E., and Yang, Y. (2002), The beta distribution as a latent response model for ordinal data (I): estimation of location and dispersion parameters. Journal of Statistical Computation and Simulation, 72, 473–494.

    MathSciNet  Article  MATH  Google Scholar 

  64. Venables, W. N., and Ripley, B. D. (2002), Modern Applied Statistics with S. Fourth Edition. Springer, New York. ISBN 0-387-95457-0

    Google Scholar 

  65. Wenger, S. J. and Freeman, M. C. (2008), Estimating species occurrence, abundance, and detection probability using zero-inflated distributions. Ecology, 89, 2953–2959.

    Article  Google Scholar 

  66. Yeo, J. J., Rodhouse, T. J., Dicus, G. H., Irvine, K. M., and Garrett, L. K. (2009), Sagebrush steppe vegetation monitoring protocol. Upper Columbia Basin Network. Version 1.0. Natural Resource Report NPS/UCBN/NRR–2009/142. National Park Service, Fort Collins, CO, USA.

Download references

Acknowledgments

We thank Dr. Megan D. Higgs for early discussions on this work and her assistance with WinBUGS code for clipping latent distributions. Dr. Brian Gray provided encouragement and interest in this work and we are appreciative. We also thank Dr. Andrew Hoegh, two anonymous reviewers’, and our associate editor’s comments and suggestion for revising our paper. The work by K. M. Irvine was funded through an Interagency Agreement P12PG70586 with the National Park Service. T. J. Rodhouse was funded by Upper Columbia Basin Network Inventory and Monitoring Program of the National Park Service. I. N. Keren’s participation was secured by an interagency agreement with Montana State’s Institute on Ecosystems with funding by North Central Climate Science Center. Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the U.S. Government.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Kathryn M. Irvine.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 237 KB)

Supplementary material 2 (csv 99 KB)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Irvine, K.M., Rodhouse, T.J. & Keren, I.N. Extending Ordinal Regression with a Latent Zero-Augmented Beta Distribution. JABES 21, 619–640 (2016). https://doi.org/10.1007/s13253-016-0265-2

Download citation

Keywords

  • Beta regression
  • Cumulative link model
  • Grouped continuous
  • Hurdle model
  • Midpoint regression
  • Non-proportional odds
  • Plant abundance
  • Proportional odds model