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.
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Agresti, A. (2010), Analysis of ordinal categorical data John Wiley and Sons, Hoboken, NJ, USA.
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.
Albert, J. H., and Chib, S. (1993), Bayesian analysis of binary and polychotomous response data. Journal of the American Statistical Association, 88, 669–679.
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.
Andrewartha, H. G., and Birch, L. C. (1954), The distribution and abundance of animals. University of Chicago Press, Chicago, Illinois, USA.
Bonham, C. D. (1989), Measurements for terrestrial vegetation John Wiley and Sons, New York, NY.
Braun-Blanquet, J. (1932), Plant sociology. The study of plant communities. McGraw-Hill, New York, NY, US.
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.
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.
Chen, J., Shiyomi, M, Hori, Y., and Yamamura, Y. (2008a), Frequency distribution models for spatial patterns of vegetation abundance. Ecological Modelling, 211, 403–410.
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.
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.
Congdon, P. (2005), Bayesian models for categorical data. John Wiley and Sons, Hoboken, NJ, USA.
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.
Damgaard, C. (2009), On the distribution of plant abundance data. Ecological Informatics, 4, 76–82.
Damgaard, C. (2012), Trend analyses of hierarchical pin-point cover data. Ecology, 93, 1269–1274.
Daubenmire, R. F. (1959), A canopy-coverage method. Northwest Science, 33, 43–64.
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.
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.
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.
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.
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.
Fahrmeier, L, and Tutz, G. (2001), Multivariate statistical modelling based on generalized linear models. Springer, Berlin.
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.
Ferrari, S. L. P. and Cribari-Neto, F. (2004), Beta regression for modelling rates and proportions. Journal of Applied Statistics, 31, 799–815.
Gelbard, J. L. and Belnap, J. (2003), Roads as conduits for exotic plant invasions in a semiarid landscape. Conservation Biology, 17, 420-432.
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.
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.
Guisan, A. and Harrell, F. E. (2000), Ordinal response regression models in Ecology. Journal of Vegetation Science, 11, 617–626.
Hall, D. B. (2000), Zero-inflated Poisson and Binomial Regression with Random Effects: A Case Study. Biometrics, 56, 1030–1039.
Heilbron, D. C. (1994), Zero-altered and other Regression Models for Count Data with Added Zeros. Biometrical Journal, 36, 531–547.
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.
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.
Holland, M. D. and Gray, B. R. (2011), Multinomial mixture model with heterogeneous classification probabilities. Ecological and Environmental Statistics, 18, 257–270.
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.
Ishwaran, H. (2000), Univariate and multirater ordinal cumulative link regression with covariate specific cutpoints. The Canadian Journal of Statistics, 28, 715–730.
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.
Kim, J-H. (2003), Assessing practical significance of the proportional odds assumption. Statistics and Probability Letters, 65, 233–239.
Kosmidis, I. and Firth, D. (2010), A generic algorithm for reducing bias in parametric estimation.Electronic Journal of Statistics, 4, 1097–1112.
Lachenbruch, P. A. (2002), Analysis of Data with Excess Zeros. Statistical Methods in Medical Research, 11, 297–302.
Lambert, D. (1992), Zero-Inflated Poisson Regression, with an Application to Defects in Manufacturing. Technometrics, 34, 1–14.
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.
Link, W. A. and Eaton, M. J. (2012) On thinning of chains in MCMC. Methods in Ecology and Evolution , 3, 112–115.
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.
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.
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.
Millar, R. B. (2009), Comparison of hierachical Bayesian models for overdispersed count data using DIC and Bayes’ factors. Biometrics, 65, 962–969.
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.
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.
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.
Ospina, R. and Ferrari, S. L. (2010), Inflated beta distributions. Statistical Papers, 51, 111–126.
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/
Plummer, M. (2008), Penalized loss functions for Bayesian model comparison. Biostatistics, 9, 523–539.
Plummer, M. (2015), JAGS Version 4.0.0 user manual. Available at https://sourceforge.net/projects/mcmc-jags/files/Manuals/4.x/.
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.
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.
Royle, J. A. and Link, W. A. (2005), A general class of multinomial mixture models for anuran calling survey data. Ecology, 86, 2505–2512.
Schabenberger, O. (1995), The use of ordinal response methodology in Forestry. Forest Science, 41, 321–336.
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.
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.
Stevens, D. L., and Olsen, A. R. (2004), Spatially balanced sampling of natural resources. Journal of the American Statistical Association, 99, 262–278.
Stroup, W. W. (2014), Rethinking the analysis of non-normal data in plant and soil science. Agronomy Journal, 106, 1–17.
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.
Venables, W. N., and Ripley, B. D. (2002), Modern Applied Statistics with S. Fourth Edition. Springer, New York. ISBN 0-387-95457-0
Wenger, S. J. and Freeman, M. C. (2008), Estimating species occurrence, abundance, and detection probability using zero-inflated distributions. Ecology, 89, 2953–2959.
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.
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.
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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
- Beta regression
- Cumulative link model
- Grouped continuous
- Hurdle model
- Midpoint regression
- Non-proportional odds
- Plant abundance
- Proportional odds model