Extending Ordinal Regression with a Latent Zero-Augmented Beta Distribution
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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.
KeywordsBeta regression Cumulative link model Grouped continuous Hurdle model Midpoint regression Non-proportional odds Plant abundance Proportional odds model
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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