Estimating Riparian Understory Vegetation Cover with Beta Regression and Copula Models

Understory vegetation communities are critical components of forest ecosystems. As a result, the importance of modeling understory vegetation characteristics in forested landscapes has become more apparent. Abundance measures such as shrub cover are bounded between 0 and 1, exhibit heteroscedastic error variance, and are often subject to spatial dependence. These distributional features tend to be ignored when shrub cover data are analyzed. The beta distribution has been used successfully to describe the frequency distribution of vegetation cover. Beta regression models ignoring spatial dependence (BR) and accounting for spatial depen- dence (BRdep) were used to estimate percent shrub cover as a function of topographic conditions and overstory vegetation structure in riparian zones in western Oregon. The BR models showed poor explanatory power (pseudo-R 2 0.34) but outperformed ordinary least-squares (OLS) and generalized least-squares (GLS) regression models with logit-transformed response in terms of mean square prediction error and absolute bias. We introduce a copula (COP) model that is based on the beta distribution and accounts for spatial dependence. A simulation study was designed to illustrate the effects of incorrectly assuming normality, equal variance, and spatial independence. It showed that BR, BRdep, and COP models provide unbiased parameter estimates, whereas OLS and GLS models result in slightly biased estimates for two of the three parameters. On the basis of the simulation study, 93-97% of the GLS, BRdep, and COP confidence intervals covered the true parameters, whereas OLS and BR only resulted in 84-88% coverage, which demonstrated the superiority of GLS, BRdep, and COP over OLS and BR models in providing standard errors for the parameter estimates in the presence of spatial dependence. FOR .S CI. 57(3):212-221.

[1]  A. Hudak,et al.  Mapping snags and understory shrubs for a LiDAR-based assessment of wildlife habitat suitability , 2009 .

[2]  P. McCullagh,et al.  Generalized Linear Models , 1972, Predictive Analytics.

[3]  Theresa Marquardt.  Accuracy and suitability of several stand sampling methods in riparian zones , 2010 .

[4]  Bill Ravens,et al.  An Introduction to Copulas , 2000, Technometrics.

[5]  G. Casella,et al.  Statistical Inference , 2003, Encyclopedia of Social Network Analysis and Mining.

[6]  R. Nelsen.  An Introduction to Copulas , 1998 .

[7]  B. McCullough,et al.  Regression analysis of variates observed on (0, 1): percentages, proportions and fractions , 2003 .

[8]  P. Anderson,et al.  Variable density management in Riparian Reserves: lessons learned from an operational study in managed forests of western Oregon, USA , 2004 .

[9]  Paul D. Anderson,et al.  BLM Density Management and Riparian Buffer Study: Establishment Report and Study Plan , 2006 .

[10]  Janet L. Ohmann,et al.  Evaluation and prediction of shrub cover in coastal Oregon forests (USA) , 2004 .

[11]  H. Bugmann,et al.  Predicting Understorey Vegetation Cover from Overstorey Attributes in Two Temperate Mountain Forests , 2003, Forstwissenschaftliches Centralblatt vereinigt mit Tharandter forstliches Jahrbuch.

[12]  S. Ferrari,et al.  Beta Regression for Modelling Rates and Proportions , 2004 .

[13]  Jun Chen,et al.  Distribution model and spatial variation of cover in grassland vegetation , 2006 .

[14]  Meike J. Wittmann,et al.  Mathematical Ecology , 2006 .

[15]  Christian Damgaard,et al.  On the distribution of plant abundance data , 2009, Ecol. Informatics.

[16]  Satishs Iyengar,et al.  Multivariate Models and Dependence Concepts , 1998 .

[17]  J. Hagar.  Wildlife species associated with non-coniferous vegetation in Pacific Northwest conifer forests: A review , 2007 .

[18]  A. Zeileis,et al.  Beta Regression in R , 2010 .

[19]  H. Joe.  Multivariate models and dependence concepts , 1998 .

[20]  S. Ferrari,et al.  On beta regression residuals , 2008 .

[21]  Jun Chen,et al.  Plant cover estimation based on the beta distribution in grassland vegetation , 2008, Ecological Research.

[22]  Rf Daubenmire,et al.  Canopy coverage method of vegetation analysis , 1959 .

[23]  Y. Hori,et al.  Frequency distribution models for spatial patterns of vegetation abundance , 2008 .

[24]  A. Raftery.  Bayesian Model Selection in Social Research , 1995 .

[25]  Michael Smithson,et al.  A better lemon squeezer? Maximum-likelihood regression with beta-distributed dependent variables. , 2006, Psychological methods.

[26]  N. Crookston,et al.  Understory cover and biomass indices predictions for forest ecosystems of the Northwestern United States , 2010 .

[27]  Bruce D. McCullough,et al.  Regression analysis of proportions in finance with self selection , 2008 .

[28]  Miina Rautiainen,et al.  Local Models for Forest Canopy Cover with Beta Regression , 2007 .

[29]  Eric R. Ziegel,et al.  Generalized Linear Models , 2002, Technometrics.

[30]  Charles E. Courchene,et al.  NATIONAL COUNCIL OF THE PAPER INDUSTRY FOR AIR AND STREAM IMPROVEMENT , 1993 .

[31]  Charles B. Halpern,et al.  Plant Species Diversity in Natural and Managed Forests of the Pacific Northwest , 1995 .

[32]  L. Madsen,et al.  Simulating correlated count data , 2007, Environmental and Ecological Statistics.

[33]  Scott Gates,et al.  Donut Shops and Speed Traps: Evaluating Models of Supervision on Police Behavior , 1993 .

[34]  J. Braun-Blanquet,et al.  Pflanzensoziologie: Grundzuge der Vegetationskunde. , 1967 .