Electronic Supplementary Material for Oecologia

Structural diversity promotes productivity of mixed, uneven-aged forests in

southwestern Germany

Adrian Dănescu*1, Axel T Albrecht1, Jürgen Bauhus2

1Forest Research Institute of Baden-Württemberg, Wonnhaldestr. 4, 79100 Freiburg, Germany2Chair of Silviculture, Faculty of Environment and Natural Resources, Freiburg University,

Tennenbacherstr. 4, 79108 Freiburg, Germany

Corresponding author. Tel.: +49 7614 018 278; Fax:. +49 7614 018 333. E-mail:

adrian.danescu@forst.bwl.de

Table of contents

Appendix 1. Statistical modeling of tree height.......................................................................................2

Appendix 2. Additional information regarding growth predictors...........................................................3

Comparison of calibration and evaluation datasets..............................................................................3

Site quality variables............................................................................................................................3

Correlation of diversity indices............................................................................................................6

Appendix 3. Response variable transformation........................................................................................7

Appendix 4. Linear mixed-effects modeling............................................................................................9

Technical details...................................................................................................................................9

Detailed modeling results...................................................................................................................10

References...............................................................................................................................................15

1

Appendix 1. Statistical modeling of tree height.

Central to the calculation of stem and stand volumes is the relationship between tree height and

diameter.

Standard protocols for data collection and preliminary processing were employed for the long-term

forest plot network (Ehring et al. 1999). In contrast to measuring tree diameters, where a full inventory

takes place during each plot census, tree heights are measured for a representative subset of trees

during a census (i.e. a minimum of 30 trees per species is aimed for). In the office, height

measurements from the field are used to estimate the coefficients of a nonlinear diameter-height

function, which can then be employed to predict missing tree heights. For a given species, the

expected tree height in a certain plot and on a given census date is:

TH=1.3+a ∙eb /(DBH +c) (S

E1)

where a, b, c are model coefficients to be estimated and DBH is diameter at breast height.

From a total of 67,362 tree observations (i.e. after thinning) that required height information for

calculating structural diversity indices, approximately 33 % (21,998) were measured in the field. The

remaining 45,364 tree heights that had to be estimated with Eq.(SE1), were (i) derived based on

measurements of the same species in the same plot on the same survey date (95 %), (ii) derived based

on measurements of the same species in the same plot but from an earlier survey, or in a different plot

with similar characteristics (e.g. stocking and site conditions) (4 %), or (iii), for rare species, obtained

by using the height-diameter relationship of a different species from the same stand (1 %).

2

Appendix 2. Additional information regarding growth predictors

Comparison of calibration and evaluation datasets

The experimental plots used for this study are clustered in 16 sites spread across Baden-Württemberg,

a state in the southwestern part of Germany (Fig. SF1). In order to calibrate and evaluate the models

using independent data, we split the initial dataset at the site level (i.e. the highest hierarchical level in

our data). Additionally, we adjusted the selection of sites manually for each model (Table ST1) so as

to make sure that sample sizes and diversity gradients would be similar in the resulting pairs of

calibration and evaluation datasets.

Table ST1 Data structure and diversity characteristics of evaluation and calibration datasets. GCd is the Gini coefficient for tree diameters, LikeJ the closeness to a J-shaped diameter-frequency curve, and Hs is Shannon’s species diversity index

AttributeTree-level analysis Stand-level analysisBeech Fir Spruce

Calib. Eval. Calib. Eval. Calib. Eval. Calib. Eval.Tree-year obs. 1809 1836 16832 16885 7787 8073 - -Trees/Plots/Sites 423/19/7 318/10/7 2767/21/7 3730/28/8 1781/22/8 2284/28/8 -/27/8 -/25/8

LikeJAvg. 4.6 4.9 5.4 5.9 4.8 5.7 4.3 4.0Range 0–10 2–10 0–10 0–10 0–10 0–10 0–10 0–10

GCd Avg. 0.44 0.54 0.42 0.43 0.41 0.43 0.42 0.43Range 0.17–0.75 0.21–0.74 0.16–0.74 0.15–0.75 0.16–0.74 0.15–0.75 0.15–0.75 0.16–0.74

Hs Avg. 0.75 0.74 0.74 0.75 0.74 0.75 0.75 0.74Range 0.41–1.1 0.1–1.14 0.24–1.14 0.39–1.25 0–1.14 0.39–1.25 0.39–1.25 0–1.14

We also built scatterplots of the DBH – relBAL relationship across the calibration and evaluation

datasets, for the three species that we analyzed at the tree level (Fig. SF2). They show that for spruce

and fir the entire gradient of competitive pressure is represented in our data, whereas this is not the

case for beech. The latter species is clearly constrained on average to the understory, most likely due

to silvicultural interventions. The largest difference between evaluation and calibration datasets can be

seen for spruce, where a significant number of trees with small diameters occupy dominant positions

in the canopy.

Site quality variables

In order to describe site quality in our models we initially considered a large pool of candidate growth

predictors. Many of them, however, were discarded due to their collinearity with other predictors or

because they were non-significant in the models.

3

We initially considered two classes of temperature and precipitation variables: (i) as mean annual

values (i.e. capturing annual fluctuations of atmospheric conditions), and (ii) as mean moving window

means of the previous 20 years (i.e. describing long-term climatic fluctuations).

Simple geographic variables like altitude and more complex ones, such as indices for diffuse, direct

and reflected potential solar radiation (Wilson and Gallant 2000) were also obtained at the plot level.

A soil quality index with two levels (i.e. high and low) was developed for this study based on site

classification data for the analyzed plots. Soil characteristics such as water holding capacity, humus

form and soil type were considered for its calculation.

Fig. SF 1 Distribution of the 16 experimental sites across the state of Baden-Württemberg. Darker tones of gray indicate

higher elevations. Inset at the top shows the general location within Germany

4

Fig. SF2 Scatterplots of DBH and relBAL for fir, spruce and beech, across the calibration and evaluation datasets

5

Correlation of diversity indices

In order to examine the correlation among the highest-performing diversity indices in our analysis, we

pooled all available plot data and generated a correlogram (Fig. SF3).

Fig. SF3 Correlogram for a subset of diversity indices used in the analysis: GCd, Gini coefficient of the DBH distribution;

Hh, Shannon index for tree height classes; Hs, Shannon index for species; LikeJ, closeness to a J-shaped DBH distribution;

Skew, skewness of the DBH distribution; VarD, coefficient of variation of the DBH distribution. The correlogram contains

three distinct regions: (i) the cells underneath the main diagonal of the matrix contain Spearman’s rho, as a non-parametric

correlation measure (larger fonts indicate a stronger correlation), (ii) the main diagonal shows a histogram for each diversity

index, and (iii) the cells above the main diagonal contain scatterplots with a red linear regression curve

6

Appendix 3. Response variable transformation

Initial modeling attempts with non-transformed or log-transformed response variables indicated

violations of the normality assumption (the latter transformation assumes that growth follows a power-

law function of DBH for the tree models or stand basal area for the stand model, and was implemented

by taking the natural logarithm of both response and predictor). Consequently, we decided to test the

Box-Cox transformation (Box and Cox 1964), a power transformation that has already proved its

merits in other studies (e.g. Fischer 2014). In our case, this transformation substantially reduced the

curvature in the loess fits of the residual plots (e.g. Robinson and Hamann 2010) compared to other

types of transformations (Fig. SF4).

The Box-Cox transformation has the following form:

y i(λ)={( y i

λ−1) /λ ,∧λ ≠ 0ln ( y i ) ,∧λ=0

(S

E2)

The optimal value for the transformation parameter λ was estimated iteratively by maximizing its

likelihood function, based on all available continuous predictors (i.e. a full model) (Box and Cox

1964). When λ ≠ 0 the back-transformation to the original scale was performed with the formula:

y i=( y¿¿ i( λ) ⋅ λ+1)1λ ¿

(S

E3)

Due to the presence of zero entries in the tree increment data, a prerequisite for performing

logarithmic or Box-Cox transformations was adding a positive constant ϕ to the response variables.

Similarly to other tree increment studies (e.g. Zhang et al. 2004; Pokharel and Dech 2012; Berrill and

O’Hara 2013), we selected ϕ=1 and subtracted this value when performing the retransformation.

7

Fig. SF4 Residual plots based on two types of variable transformation: Box-Cox transformation (BC) on the left column and

a log-log transformation (LL) on the right column. The first three rows illustrate residuals for the tree basal area increment

(BAI) models for spruce, fir and beech. The last row shows residuals for the relative stand basal area increment (relBAI).

Dashed black lines represent loess fits to the residuals and illustrate the amount of curvature in the plots.

8

Appendix 4. Linear mixed-effects modeling

Technical details

The linear mixed-effects models for tree increment had the following form (Pinheiro and Bates 2000):

y ijkt=X ijkt ∙ β+Z i , jkt ∙b i+Z ij ,kt ∙ bij+Zijk ,t ∙ bijk+εijkt (S

E4)

where y ijkt is the observation vector of the response variable for the tth measurement of the kth tree,

located within the jth plot, within the ith experimental site. X ijkt is the design matrix containing the

predictor variables for fixed effects and β is the parameter vector for the same model component.

Considering that site (i), plot (ij) and the tree within a plot (ijk) were modeled as random effects, Zi , jkt

is the random effects design matrix for site, Zij , kt is the random effects design matrix for plots nested

within sites and Zijk ,t is the random effects design matrix for trees nested within plots. Moreover,

b i , bij , bijk are the random effect parameter vectors for site, plot within site and tree within plot, and

ε ijkt is the error term. It is assumed that the random effects b i , bij , bijk and the error ε ijkt are normally

distributed with mean zero and variances σ i2 , σ ij

2 , σ ijk2 and σ ijkt

2 , respectively.

The linear mixed-effects model for stand basal area increment took the following form:

y ijt=X ijt ∙ β+Z i , jt ∙ bi+Z ij ,t ∙ bij+εijt (S

E5)

Where y ijt is the observation vector of the response variable for the tth survey of the jth plot, located

within the ith experimental site. X ijt is the design matrix for fixed effects and β is the parameter vector

for the same model component. Zi , jt and Zij ,t are the design matrices for random effects site (i) and

plot (ij). Similarly to the tree-level models, it is assumed that the random effects parameters for site

and plot (b i and b ij) and the error term ε ijt are normally distributed with mean zero and variances σ i2,

σ ij2 and σ ijt

2 , respectively.

The first order autoregressive covariance structure, AR(1), which imposes smaller correlations with

increasing time lag based on a single correlation parameter (φ), was introduced to reduce the temporal

autocorrelation of the within-tree residuals (Pinheiro and Bates 2000).

The power (Eq.(SE6)) and exponential (Eq.(SE7)) variance functions were alternatively tested, with

the aim of explicitly addressing the heteroscedasticity present in our models (Pinheiro and Bates

2000).

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Var (εl )=σ2 ∙|v l|2 ∙ δ (S

E6)

Var (εl )=σ2 ∙ exp (2∙ δ ∙ v l ) (S

E7)

whereε l are residuals within the innermost level of grouping (i.e. tree for the tree-level models and

stand for the stand-level models); δ is the variance parameter to be estimated;σ is the initial variance

of the innermost residuals; vl is the variance covariate (DBH in the tree-level models and stand basal

area in the stand-level model).

The last step of model selection consisted in ranking models with significant diversity indices

according to their AIC values. The Akaike Information Criterion (Burnham and Anderson 2002) takes

the following form:

AIC = −2∙log(£) + 2∙K (S

E8)

where log(£) is the log-likelihood of the model and K the number of parameters in the model.

We also calculated Akaike weights (w i), i.e. the relative likelihood of model i given the data and the

set of R models (Burnham and Anderson 2002):

w i=exp(−1

2∙ Δi)

∑j=1

R

exp(−12

∙ Δ j)(S

E9)

where j is the set of R alternative models, Δi = AICi – min(AIC).

Detailed modeling results

As expected, after obtaining the initial optimal models, altering the diversity component in the second

step affected the parameter estimates and the standard errors of the other fixed effects, as well as the

variances of the random effects. However, none of the estimates of the other fixed effects changed

their sign or lost their statistical significance during this process. The results of gradually testing the

available diversity indices in each of the models are shown in Table ST2 (i.e. only models with

statistically significant diversity indices and models omitting diversity indices), and the models are

sorted ascendingly by AIC. Tables ST3 and ST4 contain parameter estimates only for the best models

(i.e. the highest ranking models according to AIC in Table ST2; see also the simplified Eq.(SE10)-

(SE14)).

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The Site, Plot and Tree random effects organized a significant amount of variance in the tree-level

models, with the exception of the beech model, where Site was not significant (Table ST3). Similarly,

the Site and Plot random effects significantly improved the stand-level models (Table ST4). The

exponential variance function (Eq.(SE7)) with DBH as variance covariate best stabilized the variance

in the tree-level models (Table ST3), whereas the same effect was achieved in the stand-level model

using the power variance function (Eq.(SE6)) with stand basal area as variance covariate (Table ST4).

The first order autoregressive covariance structure (AR(1)) significantly improved the tree-level

models (Table ST3), yet its utility was not supported at the stand level (Table ST4).

The model selection process resulted in similar fixed structures for the tree-level models (Eq.(SE10)-

(SE12) and Table ST3). Overall, the final models for fir and spruce shared most of their predictors,

with small differences only in their site quality components: for spruce, tree growth was only

influenced by the total growing season precipitation (GSP) whereas for fir both mean annual

temperature (MAT) and GSP significantly influenced growth. For both fir and spruce, the diameter at

the beginning of the growth period (DBH) and its quadratic form accounted for tree size effects, and

the relative BAL and the stand basal area (BA) described competition. Similarly, in the beech model,

tree size was accounted for by DBH and its quadratic form, whereas competition was represented by

BA (i.e. a proxy for the effect of stand density). In the beech model, thinning-based release effects

were accounted for by the d/D ratio. From the tested site quality predictors, only MAT was

significantly related to the basal area increment of beech trees.

Fir: (BAI ¿¿ tree+1)λ−1λ

=β0+β1 ∙ DBH 2+β2 ∙ DBH+β3 ∙ relBAL+β4 ∙BA+β5 ∙ MAT +β6 ∙GSP+β7 ∙ Skew ¿(SE10)

Spruce: (BAI ¿¿ tree+1)λ−1λ

=β0+β1 ∙ DBH 2+β2 ∙ DBH+β3 ∙ relBAL+β4 ∙BA+β5 ∙GSP+β6 ∙ VarD ¿(SE11)

Beech: (BAI ¿¿ tree+1)λ−1λ

=β0+β1 ∙ DBH 2+β2 ∙DBH+β3 ∙MAT +β4 ∙ d / D+β5∙ BA +β6 ∙ Hh¿(SE12)

Stand: (relBAI ¿¿ stand)λ−1λ

=β0+β1 ∙BA+β2 ∙ MAT +β3 ∙ LikeJ ¿(SE13)

(relBAI ¿¿ stand)λ−1λ

=β0+β1 ∙BA+β2 ∙ MAT +β3 ∙ Hs ¿(SE14)

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Table ST2 Goodness-of-fit statistics for models with significant diversity indices and models omitting diversity indices. The models are sorted ascendingly by AIC. See Eq.(SE8), (SE9) for details regarding AIC, Δi and wi

Response variable Diversity

index

Parameter sign AICi Δi wi

BAItree (Fir) Skew + 33127.5 0.0 1.0GCd + 33193.6 66.1 0.0Hh + 33213.1 85.6 0.0VarH + 33214.2 86.7 0.0VarD + 33216.6 89.1 0.0Ed + 33220.3 92.8 0.0GCh + 33221.3 93.8 0.0Hd + 33292.3 164.8 0.0Eh + 33304.1 176.6 0.0LikeJ + 33305.6 178.1 0.0Omitted 33374.8 247.3 0.0

BAItree (Spruce) VarD + 25665.5 0.0 1.0GCd + 25673.4 8.0 0.0VarH + 25700.5 35.1 0.0GCh + 25707.9 42.5 0.0Skew + 25791.6 126.1 0.0Hd + 25865.7 200.3 0.0Hh + 25910.6 245.1 0.0Ed − 25921.3 255.8 0.0Omitted 25950.9 285.4 0.0

BAItree (Beech) Hh + 4090.2 0.0 1.0Omitted 4102 11.8 0.0

relBAIstand LikeJ + −90.3 0.0 0.7Hs + −89.2 1.1 0.1GCd + −86.4 3.9 0.1GCh + −85.9 4.4 0.0VarH + −76.8 13.5 0.0VarD + −76.3 14.0 0.0Omitted −72.7 17.6 0.0Skew + −70.8 19.5 0.0

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Table ST3 Parameter estimates with their standard errors (S.E.) for the best tree-level models (Eq.(SE10)-(SE12)). Parameter estimates rely on the restricted maximum likelihood method

Parameters Fir Spruce BeechEstimate S.E. Estimate S.E. Estimate S.E.

Box-Cox transformationλ 0.0932 0.3415 0.3012γ 1.1902 1.0862 1.0773

Fixed effects

β01.2887 0.2314 2.1644 0.4912 2.3334 0.5473

β1−0.0012 <0.0001 −0.002 0.0001 −0.0022 0.0003

β20.0844 0.0018 0.0844 0.0037 0.1271 0.0046

β3−0.9078 0.0951 −2.6347 0.1757 0.1526 0.0502

β4−0.0282 0.0017 −0.0248 0.0043 0.1327 0.0321

β50.2283 0.0167 0.0049 0.0002 −0.0348 0.0057

β60.0016 0.0001 0.049 0.0027 0.6717 0.163

β70.441 0.0272 - - - -

Variance components

σ i2 0.15 1.059 n.s.

σ ij2 0.0411 0.1735 0.3817

σ ijk2 0.1385 0.49 0.1516

σ ε2 0.4816 1.3838 0.6145

δ 0.014 0.0203 0.0206φ 0.5019 0.2647 0.5246

Model evaluationē 3.3855 −2.9680 −2.7931

ē% 15.8 −16.4 −23.2

Note: λ, transformation parameter; γ, bias correction multiplier; δ, variance parameter; φ, autocorrelation parameter (AR(1));

σ ε2 residual variance; σ i

2, σ ij2, σ ijk

2 , variances for the site, plot and tree random effects; ē and ē% mean bias and mean relative

bias; n.s., not significant.

13

Table ST4 Parameter estimates with their standard errors (S.E.) for the two highest ranking stand-level models (Eq.(SE13) and (SE14)). Parameter estimates are based on the restricted maximum likelihood method

Parameters Structural diversity Species diversityEstimate S.E. Estimate S.E.

Box-Cox transformationλ −0.4845 −0.4845γ 1.092 1.092

Fixed effects

β00.7953 0.0473 0.8108 0.0534

β1−0.0185 0.0016 −0.0179 0.0018

β20.1299 0.0242 0.1338 0.0254

β30.0323 0.0079 0.327 0.1233

Variance components

σ i2 0.0153 0.0199

σ ij2 0.0023 0.0033

σ ε2 0.0185 0.0186

δ 0.0701 0.0797φ n.s. n.s.

Model evaluationē −0.1693 −0.2601

ē% −5.5 −8.5

Note: λ, transformation parameter; γ, bias correction multiplier; δ, variance parameter; φ, autocorrelation parameter (AR(1));

σ ε2 residual variance; σ i

2, σ ij2, variances for site and plot random effects; ē and ē% mean bias and mean relative bias; n.s., not

significant.

14

References

Berrill J-P, O’Hara KL (2013) Estimating site productivity in irregular stand structures by indexing the

basal area or volume increment of the dominant species. Can J For Res 44:92-100 doi:10.1139/cjfr-

2013-0230

Box GE, Cox DR (1964) An analysis of transformations. J R Stat Soc Series B Stat Methodol 26:211-

252

Burnham KP, Anderson DR (2002) Model selection and multimodel inference: A practical

information-theoretic approach. 2nd ed. Springer, New York

Ehring A, Klädtke J, Yue C (1999) Ein interaktives Programm zur Erstellung von

Bestandeshöhenkurven. Centralblatt für das gesamte Forstwesen 116:47-52

Fischer C (2014) Zur Variablentransformation in Durchmesserzuwachsmodellen. In: Kohnle U,

Klädtke J (eds) Sektion Ertragskunde Jahrestagung 2014. Deutscher Verband Forstlicher

Forschungsanstalten, Lenzen a.d. Elbe, pp 1-15

Pinheiro JC, Bates DM (2000) Mixed-effects models in S and S-PLUS. Springer, New York

Pokharel B, Dech JP (2012) Mixed-effects basal area increment models for tree species in the boreal

forest of Ontario, Canada using an ecological land classification approach to incorporate site

effects. Forestry 85:255-270 doi:10.1093/forestry/cpr070

Robinson AP, Hamann JD (2010) Forest analytics with R: an introduction. Springer, New York

Wilson JP, Gallant JC (2000) Terrain analysis: principles and applications. Wiley, New York

Zhang L, Peng C, Dang Q (2004) Individual-tree basal area growth models for jack pine and black

spruce in northern Ontario. The Forestry Chronicle 80:366-374

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