# The Variation Driven by Differences between Species and between Sites in Allometric Biomass Models

^{1}

^{2}

## Abstract

**:**

_{species}= 42.56%, SE = 6.10% for Dataset 1 and VPC

_{species}= 47.54%, SE = 6.07% for Dataset 2) was larger than that explained by site (VPC

_{site}= 20.08%, SE = 3.35% for Dataset 1 and VPC

_{site}= 8.27%, SE = 1.38% for Dataset 2). The proportion of variance explained by site decreased by 24%–44% and the proportion of variance explained by species changed only slightly, when height is included in the allometric biomass models (i.e., models based on diameter at breast height alone, compared to models based on diameter at breast height and tree height). Conclusions: Allometric biomass models were more species-specific than they were site-specific. Therefore, the species (i.e., differences between species) seems to be a more important driver of variability in allometric models compared to site (i.e., differences between sites). Including height in allometric biomass models helped reduce the dependency of these models, on sites only.

## 1. Introduction

^{β}) [4] is a widely accepted form of allometric biomass model. Despite the allometric scaling theory, which support an invariant allometric scaling (the scaling exponent β = 8/3) [5,6], there is little support from empirical evidence for this theory [7,8].

_{2}) were also reported to alter biomass allocation among tree organs [2,18,19]. Additionally, when in different stages of development, the trees were shown to have different allometries [20,21]. Competition between trees modifies the H-D ratio and crown size [22] with direct consequences on aboveground biomass allometry. This was confirmed by a study on allometry of dominant and supressed trees [23]. As trees do not respond very quickly to changes in competition, it is not only the current tree competition that is important, but also previous competition [10]. The type of mixture, exposing different levels of interspecific competition were shown to affect aboveground biomass allometry [24]. Furthermore, competition (intraspecific and interspecific) can be adjusted through measures of forest management. For example, thinning reduces competition between trees being proven to reduce the ratio between fine roots and leaf biomass [25]. Also, in even-aged stands, the patterns of tree competition are different compared to uneven-aged stands, where greater structural diversity may result in more diverse tree allometries. However, all these factors affecting allometry usually interact with each other, and also with the genotype. Therefore, including such factors as a fixed effect variable in allometric models, often give poor results. By considering the site only, as a vector of all these effects (main effects and their interactions) seems to be a more practical approach [2].

## 2. Materials and Methods

#### 2.1. Biomass Data

#### 2.2. Methods

#### 2.2.1. The Rationale for the Proposed Methodology

#### 2.2.2. VPC of nested ANOVA

- (a)
- A multilevel model (random intercept and slope model) was fitted to log-log transformed data (for each dataset),$$\mathrm{ln}{\left(\mathrm{AGB}\right)}_{\mathrm{ijk}}={\mathsf{\beta}}_{0}+{\mathsf{\delta}}_{\mathrm{j}0}+{\mathsf{\gamma}}_{\mathrm{k}0}+{\mathsf{\beta}}_{1}\times \mathrm{ln}{\left(\mathrm{D}\right)}_{\mathrm{ijk}}+{\mathsf{\delta}}_{\mathrm{j}1}+{\mathsf{\gamma}}_{\mathrm{k}1}+{\mathsf{\beta}}_{2}\times \mathrm{ln}{\left(\mathrm{H}\right)}_{\mathrm{ijk}}+{\mathsf{\delta}}_{\mathrm{j}2}+{\mathsf{\gamma}}_{\mathrm{k}2}+{\mathsf{\epsilon}}_{\mathrm{ijk}}$$
_{ijk}is the log of AGB for tree i from site j and species k; β_{0}is the fixed part of the intercept; β_{1}is the fixed part of the ln(D) slope; β_{2}is the fixed part of the ln(H) slope; δ_{j0}is the random part of the intercept attributable to differences between sites; δ_{j1}is the random part of the ln(D) slope attributable to differences between sites; δ_{j2}is the random part of the ln(H) slope attributable to differences between sites; γ_{k0}is the random part of the intercept attributable to differences between species; γ_{k1}is the random part of the ln(D) slope attributable to differences species; γ_{k2}is the random part of the ln(H) slope attributable to the differences between species; ε_{ijk}is the error term of tree i from site j and species k, ε_{ijk}~N(0,σ^{2}). - (b)
- A back-transformed nonlinear model was used to calculate the predicted AGB. As the error distribution becomes lognormal in original scale, a back transformation correction factor (exp(σ
^{2}/2)) [41,42] was used, where σ is the residual standard error of the model in log-log scale. The back-transformed model that predicts AGB as a function of D and H is:$${\widehat{\mathrm{AGB}}}_{\mathrm{ijk}}=\mathrm{exp}\left({\mathsf{\beta}}_{0}\right)\times {\mathrm{D}}_{\mathrm{ijk}}{}^{{\mathsf{\beta}}_{1}}\cdot {\mathrm{H}}_{\mathrm{ijk}}{}^{{\mathsf{\beta}}_{2}}\times \mathrm{exp}\left(\frac{{\mathsf{\sigma}}^{2}}{2}\right)$$_{0}is the fixed part of the intercept (Equation (1)); β_{1}is the fixed part of ln(D) slope (Equation (1)); β_{2}is the fixed part of ln(H) slope (Equation (1)); D and H are the diameter at breast height and tree height; σ^{2}is the residual variance in Equation (1). - (c)
- The relative residual for each tree was calculated as,$${\mathrm{P}}_{\mathrm{ijk}}=\frac{{\mathrm{AGB}}_{\mathrm{ijk}}-{\widehat{\mathrm{AGB}}}_{\mathrm{ijk}}}{{\widehat{\mathrm{AGB}}}_{\mathrm{ijk}}}\times 100$$
_{ijk}is the observed AGB of tree i, from site j and species k; ${\widehat{AGB}}_{ijk}$ is the predicted AGB of tree i, from site j and species k (Equation (2)).

_{ijk}is the relative residual (Equation (3)) of tree i from site j and species k; μ is the overall mean of relative residuals; δ

_{j}is the random effect attributable to the differences between sites, δ

_{j}~N(μ, σ

_{site}

^{2}); γ

_{k}is the random effect attributable to differences between species, γ

_{k}~N(μ, σ

_{species}

^{2}); ε

_{ijk}is the error term of tree i from site j and species k, ε

_{ijk}~N(0, σ

_{ε}

^{2}).

_{species}

^{2}= var(γ

_{k}) is the variance explained by species in Equation (4), σ

_{site}

^{2}= var(δ

_{j}) is the variance explained by site in Equation (4) and σ

_{ε}

^{2}= var(ε

_{ijk}) is the residual variance in Equation (4).

#### 2.2.3. Bootstrap Analysis

_{species}and VPC

_{site}, Equations (5) and (6) resulted 100,000 VPC values. These values were used further to calculate the standard errors (SE) of VPC and to plot the density of VPC values, when comparing datasets and VPC sources.

#### 2.2.4. The effect of Including H as Predictor in Allometric Biomass Models

#### 2.2.5. Analysis of Random Effects

_{k}), and so has each site (i.e., δ

_{j}). Therefore, based on these species-specific means, we can evaluate how similar is allometry among species, and how the species can be grouped by their allometry. Two species with similar γ

_{k}values, show that trees of similar D and H share also similar aboveground biomass (AGB). To ease the interpretation of γ

_{k}, the γ

_{k}values were centred to zero, by subtracting the overall multispecies mean of relative residuals (μ, Equation (4)) from γ

_{k}. As a result, γ

_{k}> 0 means that trees of similar D and H from species k show greater AGB compared to the overall multispecies mean. The opposite is for γ

_{k}< 0. To show how the species are grouped by their allometry, a dendrogram was developed. The variable (i.e., γ

_{k}) was first standardized, Euclidean distances were calculated and then the Ward error sum of squares hierarchical clustering method (‘ward.D2’) was used [45].

_{j}in Equation (4)) would also tell how the allometry of trees vary between sites. However, the site, as defined above, unlike the species, has a precise spatial delimitation. As a consequence, in order to find whether the effects of site on biomass allometry depend on geographical gradients, the correlations between site random effect (i.e., δ

_{j}) and the geographical gradients (e.g., latitude, longitude, altitude) were presented. Because altitude was not provided for all locations, the altitude was extracted as a function of latitude and longitude, using package ‘elevatr’ in R [46].

#### 2.2.6. Data Processing

## 3. Results

#### 3.1. The Species Explained a Larger Proportion of Variance in Allometric Models Compared to Sites

_{species}= 42.56% (SE = 6.10%; 95% confidence interval: 29.56%–53.41%) for Dataset 1 and VPC

_{species}= 47.54% (SE = 6.07%; 95% confidence interval: 34.44%–58.15%) for Dataset 2. By contrast, the proportion of variance that was attributable to differences between sites was VPC

_{site}= 20.08% (SE = 3.35%; 95% confidence interval: 14.16%–27.26%) for Dataset 1 and VPC

_{site}= 8.27% (SE = 1.38%; 95% confidence interval: 5.89%–11.29%) for Dataset 2. The distributions of VPC values resulted from bootstrap analysis, show a clearer differentiation between VPC

_{species}and VPC

_{site}for Dataset 2, where the was no overlapping of distributions (Figure 2). However, the difference between VPC

_{species}and VPC

_{site}was significant for both datasets (p < 0.0001).

#### 3.2. Including H as Predictor in Allometric Models Reduced The Proportion of Variance Attributable to Differences between Sites, but Had Marginal Effect on That Attributable to Differences between Species

_{site}(Figure 3). When both D and H were used to predict AGB, instead of D alone, the proportion of variance explained by site in allometric biomass models was reduced by 24%–44%, from 26.32 to 20.08% (for Dataset 1) and from 14.63% to 8.27% (for Dataset 2). By contrast, the VPC

_{species}increased slightly, from 40.77% to 42.56% for Dataset 1 and from 46.66% to 47.54% for Dataset 2. Furthermore, the distributions of VPC values resulted from bootstrap analysis showed an overlap of 25% for Dataset 1 and only 4% for Dataset 2 (Figure 3). Nevertheless, the distributions of VPC

_{species}showed a much greater overlap, of 80% for Dataset 1 and 89% for Dataset 2 (Figure 3a1,a2). This suggests that, including H in allometric models helps reduce site-specificity, with only a marginal effect on species dependency.

#### 3.3. The Species and Site Random Effects

_{k}in Equation (4)), shows how the species were grouped by their allometry. A large and positive γ

_{k}value shows that, on average, the trees of similar D and H exhibit greater AGB for species k than the multispecies average. For example, the species Quercus mongolica Fisch. ex Ledeb. (Mongolian oak) stands out in Dataset 1, showing the largest AGB (on average, 43.2% larger than the multispecies mean), whereas the species Populus nigra L. (Black poplar) stands out as the species producing the lowest AGB for trees of similar D and H (on average, 33.2% less than the multispecies mean) (Figure 4). There are many similarities between the two datasets. For example, Quercus spp., Acer spp., Fagus sylvatica L., showed larger AGB than the multispecies mean in both datasets; Populus sp. showed lower AGB than the multispecies average. Also, the species that were common to both datasets showed relatively similar γ

_{k}values (e.g., Pseudotsuga menziesii: 1.5% vs. 4.7%; Quercus rubra: 20.5% vs. 30.1%; Fagus sylvatica: 30.6% vs. 36.9%). However, as can be observed in Figure 4, the allometry of species within the same genus or family can vary greatly. Therefore, using a species-specific model to another species within the same genus or family (being justified that species within the same genus of family share a larger proportion of genotype) can risk producing large prediction bias.

_{j}(i.e., site random effect in Equation (4), taking one value for each site) was significantly correlated with altitude (r = −0.140, p = 0.031), latitude (r = −0.213, p < 0.001) and longitude (r = 0.189, p = 0.003). Therefore, in Canada, the trees (of given D and H), located at lower altitudes, tend to have greater AGB than those located at higher altitudes; the trees located in the South tend to have greater AGB than those located in the North; the trees located in the West tend to have greater AGB than those located in the East. The latter may be a side effect caused by the Rocky Mountains range in western Canada which creates conditions across the country for a gradient in tree biomass allometry. However, for Dataset 1, the correlations between δ

_{j}and geographical coordinates were not significant (Altitude: r = 0.027, p = 0.749; Latitude: r = −0.096, p = 0.271; Longitude: r = −0.119, p = 0.172).

## 4. Discussion

_{site}= 23.61% (for model based on D, H and wood density to predict AGB). This value is larger than VPC

_{site}values reported here (i.e., VPC

_{site}= 20.08% for Dataset 1 and VPC

_{site}= 8.27% for Dataset 2). Therefore, because of the larger VPC value (23.61%), it can be speculated that wood density may not have explained entirely the variance that arose from differences between species.

_{k}(Equation (4)) parameters (meaning greater AGB for trees of similar D and H), whereas species with generally low wood density (e.g., Populus spp., Salix spp., Pinus spp., Picea spp.) have shown generally negative values of γ

_{k}(Figure 4).

_{H-D}= 22.68%, SE = 3.75%; Dataset 2: VPC

_{H-D}= 20.77%, SE = 2.12%) was greater than that explained by species (Dataset 1: VPC

_{H-D}= 11.96%, SE = 3.05%; Dataset 2: VPC

_{H-D}= 16.17%, SE = 3.60%), for both datasets. Therefore, these results validate the assumption that, the reason for the reduction in site dependency of allometric models when H is included as a predictor, is related to the fact that H-D ratio varies more between sites than between species.

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The number of trees per species (

**a**), and the number of trees per site (

**b**) presented in base 10 log scale.

**Figure 2.**The Variance Partition Coefficient (VPC) density, for Dataset 1 (

**a**) and Dataset 2 (

**b**). Note: The density curves are based on VPC values resulted from bootstrap analysis; the ‘overlap’ shows the percentage of the overlapping regions of the two distributions.

**Figure 3.**The Variance Partition Coefficient (VPC) density for models based on D only (AGB = f(D)) and for models based on both D and H (AGB = f(D, H)). Note: The density curves are based on VPC values resulted from bootstrap analysis; the ‘overlap’ shows the percentage of the overlapping regions of the two distributions.

**Figure 4.**The dendrograms showing the grouping of species by their allometry, for Dataset 1 and Dataset 2. Note: (1) The γ

_{k}values (Equation (4)) are presented in parenthesis for each species (γ

_{k}was centred to zero by subtracting the mean μ from each γ

_{k}value) and the sample size for each species. (2) In different colours are the species that were grouped by cutting the dendrogram at a distance of 1.0 (vertical dashed red line).

**Figure 5.**The VPC

_{H-D}, showing the proportion of variance caused by differences between species and differences between sites in the variability of H-D ratio (ratio between height and diameter at breast height).

Characteristic | Dataset 1 | Dataset 2 |
---|---|---|

Region | Europe and Asia | Canada |

Sample size | 4921 | 5199 |

D range (cm) | 5.0–72.9 | 5.0–74.3 |

H range (m) | 3.1–42.8 | 2.5–40.8 |

AGB range (kg) | 2.2–4291.3 | 2.2–2951.4 |

Number of tree species | 56 | 38 |

Number of trees per species (range) | 5–1656 | 11–704 |

Number of sites | 133 | 237 |

Number of trees per site (range) | 5–278 | 5–164 |

References | [34] | [35,36] |

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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Dutcă, I.
The Variation Driven by Differences between Species and between Sites in Allometric Biomass Models. *Forests* **2019**, *10*, 976.
https://doi.org/10.3390/f10110976

**AMA Style**

Dutcă I.
The Variation Driven by Differences between Species and between Sites in Allometric Biomass Models. *Forests*. 2019; 10(11):976.
https://doi.org/10.3390/f10110976

**Chicago/Turabian Style**

Dutcă, Ioan.
2019. "The Variation Driven by Differences between Species and between Sites in Allometric Biomass Models" *Forests* 10, no. 11: 976.
https://doi.org/10.3390/f10110976