Nonlinear regression is increasingly used to develop allometric equations for forest biomass estimation (i.e., as opposed to the traditional approach of log‐transformation followed by linear regression). Most statistical software packages, however, assume additive errors by default, violating a key assumption of allometric theory and possibly producing spurious models. Here, we show that such models may bias stand‐level biomass estimates by up to 100 percent in young forests, and we present an alternative nonlinear fitting approach that conforms with allometric theory

Hughes, R. Flint

Litton, Creighton M.

Mascaro, Joseph

Schnitzer, Stefan A.

Uowolo, Amanda

English

epublications@Marquette

Marquette University e-Publications@Marquette Biological Sciences Faculty Research and Publications Biological Sciences, Department of 11-2011 Minimizing Bias in Biomass Allometry: Model Selection and Log‐Transformation of Data Joseph Mascaro Creighton M. Litton R. Flint Hughes Amanda Uowolo Stefan A. Schnitzer Follow this and additional works at: https://epublications.marquette.edu/bio_fac  Part of the Biology Commons  Marquette University e-Publications@Marquette  Biological Sciences Faculty Research and Publications/College of Arts and Sciences  This paper is NOT THE PUBLISHED VERSION; but the author’s final, peer-reviewed manuscript. The published version may be accessed by following the link in the citation below.  Biotropica, Vol. 43, No. 6 (November 2011) : 649-653. DOI. This article is © The Association for Tropical Biology and Conservation and permission has been granted for this version to appear in e-Publications@Marquette. The Association for Tropical Biology and Conservation does not grant permission for this article to be further copied/distributed or hosted elsewhere without the express permission from The Association for Tropical Biology and Conservation. Minimizing Bias in Biomass Allometry: Model Selection and Log-Transformation of Data  Joseph Mascaro Department of Global Ecology, Carnegie Institution for Science, Stanford, California Smithsonian Tropical Research Institute, Apartado 2072, Balboa, Republic of Panama Department of Biological Sciences, University of Wisconsin – Milwaukee, Milwaukee, Wisconsin Creighton M. Litton Department of Natural Resources and Environmental Management, University of Hawai‘i at Mânoa, Honolulu, Hawaii R. Flint Hughes Institute of Pacific Islands Forestry, USDA Forest Service, Hilo, Hawaii Amanda Uowolo Institute of Pacific Islands Forestry, USDA Forest Service, Hilo, Hawaii Stefan A. Schnitzer Smithsonian Tropical Research Institute, Apartado 2072, Balboa, Republic of Panama Department of Biological Sciences, University of Wisconsin – Milwaukee, Milwaukee, Wisconsin  ABSTRACT Nonlinear regression is increasingly used to develop allometric equations for forest biomass estimation (i.e., as opposed to the traditional approach of log-transformation followed by linear regression). Most statistical software packages, however, assume additive errors by default, violating a key assumption of allometric theory and possibly producing spurious models. Here, we show that such models may bias stand-level biomass estimates by up to 100 percent in young forests, and we present an alternative nonlinear fitting approach that conforms with allometric theory. FOREST ECOSYSTEMS EXERT A LARGE INFLUENCE ON THE GLOBAL CARBON CYCLE through the flux and storage of carbon in plant biomass, and quantification of these carbon fluxes and stocks depends to a great degree on allometric models used to estimate aboveground tree biomass (Chave et al. 2005). The relationship between tree diameter and biomass is highly conserved, with idealized trees exhibiting a power-law relationship: aboveground biomass=a× diameterb where a and b are regression coefficients (Niklas 2006). Traditionally, linear models have been fit to log-transformed diameter and biomass data, but the increasing availability of advanced statistical packages has lead to greater use of nonlinear models fit directly to untransformed diameter and biomass data (e.g., Litton et al. 2006). This approach may be favored in part because its use avoids the need for transformation (and back-transformation; Baskerville 1972). In practice, however, the default nonlinear technique used by most statistical packages assumes homogeneity of errors, which cannot be safely assumed with most allometry data. For trees, in particular, ideal allometry data are strongly heteroscedastic, exhibiting increasing variation in biomass with increasing diameter (Chave et al. 2005). When nonlinear fitting techniques are applied without accounting for heteroscedasticity, the resulting models may include substantial biases even while maintaining high r2 and low mean square errors. Here, we contrast linear and nonlinear fitting approaches for estimating aboveground biomass using harvest data from six tree species of various habits, including one species with no previously published allometric model: Psidium cattleianum (strawberry guava). Psidium cattleianum is native to Brazil and has been widely introduced in Oceania, causing particularly dramatic alterations to the forest ecosystems of Hawai‘i where it is considered one of the state's most disruptive introduced plants (Little & Skolmen 1989, Wagner et al. 1999). The species has a high potential to alter ecosystem carbon storage by causing changes to forest structure and nutrient dynamics (Hughes & Denslow 2005, Asner et al. 2008), and thus its allometric characteristics are of considerable importance. METHODS Study species and harvest procedures.— We reanalyzed five published models for predicting total aboveground biomass, four of which were fit using nonlinear regression by ordinary least squares on untransformed biomass and diameter data (diameter at breast height [dbh] at 1.3 m from the ground, or basal diameter): (1) Metrosideros polymorpha (ohi‘a); (2) Diospyros sandwicensis (lama); (3) Psydrax odorata (alahe‘e); and (4) Dodonaea viscosa (a‘ali‘i) (Litton et al. 2006, Litton & Kauffman 2008), and a fifth using linear regression by ordinary least squares on ln-transformed biomass and dbh data: (5) Rhamnus cathartica (common buckthorn) (Mascaro & Schnitzer 2011). We also analyzed one unpublished dataset for (6) P. cattleianum (strawberry guava). The species selected grow as shrubs (1, 4–6), small trees (1–6), or large trees (1–2), and occur across a variety of habitats, including tropical dry forests (1–4), tropical mesic and wet forests (1–4, 6), and temperate to subtropical forests (5). For each dataset, 15–34 trees for each species were cut at ground level and all aboveground parts (i.e., wood, twigs, leaves, fruit) were separated and weighed with spring scales (Table 1). Subsamples of each tissue type were collected and dried to constant mass in a forced air oven at 60°–90°C (temperatures were consistent within a given species) to correct for moisture content and determine the total aboveground dry weight of each tree (Appendix S1). Psidium cattleianum individuals were harvested from the Laupahoehoe Unit of the Hilo Forest Reserve on Hawai‘i Island in 2008. Individuals with branching below 1.3 m on the main stem were excluded, and thus the proposed model here should be applied to shrub-form Psidium with caution. Details on the published models for the other five species are available in the original publications (Litton et al. 2006, Litton & Kauffman 2008, Mascaro & Schnitzer 2011). Table 1. Summary of harvest data for six tree allometry datasets. Diameter (cm) is diameter at breast height (dbh, 1.3 m from the ground), except for Dodonaea viscosa, which is basal diameter. Species N Minimum diameter Maximum diameter References Metrosideros polymorpha 30 0.3 33.1 Litton and Kauffman (2008) Diospyros sandwicensis 25 1.8 20.2 Litton et al. (2006) Psydrax odorata 34 0.5 4.6 Litton et al. (2006) Dodonaea viscosa 20 0.5a 2.9a Litton and Kauffman (2008) Psidium cattleianum 26 0.2 18.2 This study Rhamnus cathartica 15 1.2 24.7 Mascaro and Schnitzer (2011) a Basal diameter (cm). Statistical analyses.— We used three techniques to fit allometric models to biomass and diameter data. First, we applied the traditional allometric approach (sensuBaskerville 1972) by fitting a linear model to each dataset of the form ln(𝑦𝑦)  =  ln(𝑎𝑎)  +  𝑏𝑏 ×  𝑙𝑙𝑙𝑙(𝑥𝑥) + 𝜀𝜀 (1a) where 𝑦𝑦 is the dependent variable (total aboveground biomass), 𝑥𝑥 is the independent variable (diameter), 𝑎𝑎 and 𝑏𝑏 are regression coefficients and is regression error. Each linear model was then back-transformed to a power function of the form: 𝑦𝑦 =  𝑎𝑎𝑥𝑥𝑏𝑏  ×  CF (1b) where CF is a correction factor computed as CF = 𝑒𝑒(MSE/2) (1c) and MSE is the mean square error of the regression. The CF accounts for the back-transformation of the regression error, and is a requisite step in the use of linear models and ln-transformed data in allometry (Baskerville 1972). Second, we applied nonlinear regression to fit a power model to each dataset of the form 𝑦𝑦 =  𝑎𝑎𝑥𝑥𝑏𝑏  ×  ε (2) In the case of four of the five published models, these fits served to replicate the published results of Litton and Kauffman (2008) and Litton et al. (2006). Additionally, model 2 replicates the ‘default’ power-law fit for commonly used statistical packages (e.g., SigmaPlot, JMP, and R – specifically the nlm function). Finally, we applied a variant of model 2 with an additional term to account for heteroscedasticity that is common in allometric models. Specifically, we allowed regression error to scale as a function of tree diameter by introducing an additional constant (k) 𝑦𝑦 = 𝑎𝑎𝑥𝑥𝑏𝑏 + 𝑥𝑥𝑘𝑘  × 𝜀𝜀𝑒𝑒 (3) All model parameters were estimated using maximum likelihood estimation in SAS (2008). For comparative purposes, r2 and MSE were assessed in the power-law form for all models (i.e., equation 1b for model 1). Corrected (i.e., for sample size) Akaike Information Criterion scores (AICc) were assessed for all three models (follows Draper & Smith 1998). RESULTS AND DISCUSSION All three model fits were significant for all six species(i.e., r2 >0.73, P<0.0001) (Table S1; Fig. 1). For all species, however, very high ΔAICc scores suggested that model 2 was the least effective of the three in predicting biomass (Table S1). In four of the six species, model 2 introduced a substantial and consistent bias for smaller diameter stems compared with harvested biomass. For M. polymorpha, the error averaged 223 percent for individuals <8 cm dbh, and reached 400 percent error within that span (Fig. 2A). Similar biases were found for D. viscosa (80% for stems <1.5 cm basal diameter), P. cattleianum (51% for stems <14 cm dbh), and R. cathartica (74% for stems <18 cm dbh). Thus, although model 2 is the ideal form for tree allometry (Niklas 2006), fitting this model without controlling for variant error structure over the range of diameters sampled (a near ubiquitous feature of allometric datasets) may result in spurious equations that are not applicable across the entire size range used in generating the models.  Figure 1 Allometric models predicting aboveground biomass based on stem diameter for six woody species. Diameter (cm) is diameter at breast height (dbh, 1.3 m from the ground), except for Dodonaea viscosa, which is basal diameter.  Figure 2 Percent error in allometric models predicting aboveground biomass based on stem diameter for six woody species. Diameter (cm) is diameter at breast height (dbh, 1.3 m from the ground), except for Dodonaea viscosa, which is basal diameter. The equation biases described almost exclusively affect the smaller stems in the datasets. However, in practice we have found that such biases can result in substantial error in estimating plot-level aboveground biomass in forests that contain many small trees (e.g., young, regenerating stands), and early successional stands dominated by small diameter individuals are an increasingly important component of forested landscapes across the globe (Swanson et al. 2011). Importantly, three of the species for which the biases were detected are capable of forming large, monotypic stands of small individuals with very high stem densities (Mueller-Dombois & Fosberg 1998, Mascaro & Schnitzer 2007, Asner et al. 2008). We applied all three M. polymorpha models to plot inventory data from young, open forests on Hawai‘i Island (stand density=1670 stems/ha; dbh range=2–12 cm; mean dbh=3.8 cm; trees in stand >8 cm dbh=5%; J. Mascaro & F. Hughes unpubl. data), and found that use of model 2 resulted in a ∼100 percent overestimation of stand-level aboveground biomass compared to the other models. However, when applied to an older stand with trees of varying size (stand density=1140 stems/ha; dbh range=3.2–32.7 cm; mean dbh=15.4 cm; trees in stand >8 cm dbh=93%; C.M. Litton & J.B. Kauffman unpubl. data), all three models produced similar stand totals, highlighting that the bias outlined here becomes less important as mean tree size increases. To fit an appropriate model, a multiplicative—rather than additive—error term is required. The simplest way to account for multiplicative errors is to perform a log-transformation on the diameter and biomass data, thus normalizing the biomass error structure along the range of diameter values as in model 1 (Baskerville 1972). Note that the additive error term (ɛ) in model 1 (eq 1a) represents multiplicative errors on the original scale. An alternative can be accomplished by introducing an additional regression constant that allows errors to scale with diameter (i.e., model 3). Here, model 3 performed better than model 1 (AICc) for three of six species examined, and for two of these the ΔAICc scores from model 3 to 1 were >10. By contrast, for the three species where model 1 performed better, the ΔAICc from model 1 to 3 was always <5. Thus, the few examples considered here suggest that model 3 may be more reliable, although we emphasize that models 1 and 3 were very similar overall. Our results highlight that the decision to log-transform raw data in allometry is more than one of statistical convenience. Packard and Boardman (2008) argued that fitting linear models on log-transformed data leads to results that are ‘biased and misleading’ because such models operate in geometric rather than arithmetic space, and that analyses should be performed on the original scale. However, Kerkhoff and Enquist (2009) note that many allometric characteristics of organisms are ‘multiplicative by nature’ and thus fitting models to log-transformed data is perfectly acceptable because accounting for proportional rather than absolute variation is most important. Our results support the latter view. Models that assumed multiplicative errors (1 and 3) described the data very well, although the results of model 3 indicate that log-transformation is not the only way to satisfy this assumption. In conclusion, fitting nonlinear biomass allometry models while assuming additive errors can produce systematic biases in estimates for smaller diameter trees. This bias can lead to large errors in landscape-scale biomass estimation from stand-level datasets that are dominated by small trees. In light of these results, we believe that added care should be used to determine whether an allometric model is appropriate for the error structure of the data. As a solution, we propose that published allometric models should be accompanied by: (1) the raw data used to generate the regression equations (e.g., as an appendix); and (2) plots of the residuals. Such an approach would allow individual investigators to judge whether the model is adequate for their needs as published, or if alternative approaches are more appropriate for estimating biomass for a given tree or stand. ACKNOWLEDGMENTS Jim Baldwin (USFS-PSW) provided assistance fitting models. J. Mascaro was supported by a grant from Applied Ecological Services Inc., an NSF Graduate Research Fellowship, and a UWM Golda Meir Library Scholar Award. Additional support was provided by a grant from the National Science Foundation to C.M. Litton (DEB-0816486). Comments from four anonymous reviewers improved this manuscript. LITERATURE CITED Asner, G. P., R. F.Hughes, P. M.Vitousek, D. E.Knapp, T.Kennedy-Bowdoin, J.Boardman, R. E.Martin, M.Eastwood, and R. O.Green. 2008. 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Minimizing Bias in Biomass Allometry: Model Selection and Log‐Transformation of Data

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Minimizing Bias in Biomass Allometry: Model Selection and Log‐Transformation of Data

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