@@ -9,7 +9,7 @@ This document gives an update on the analyses completed during and after the wor

**Workshop participants:** David A. Coomes, Daniel Falster, Rob Kooyman, Daniel Laughlin, Lourens Poorter, Mark Vanderwel, Ghislain Vieilledent, Mark Westoby, Joe Wright

**Data contributors:** J. Caspersen, H. Zeng, S. Gourlet-Fleury, B. Herault, G. Ståhl, J. Thompson, S. Richardson, P. Ruiz, I-F. Sun, N. Swenson, M Uriarte, M. Zavala, N. E. Zimmermann, M. Hanewinkel, J. Zimmerman, Yusuke Onoda,, Hiroko Kurokawa, Masahiro AIBA, MRN Québec and other

**Other participants and Data contributors:** J. Caspersen, H. Zeng, S. Gourlet-Fleury, B. Herault, G. Ståhl, J. Thompson, S. Richardson, P. Ruiz, I-F. Sun, N. Swenson, M Uriarte, M. Zavala, N. E. Zimmermann, M. Hanewinkel, J. Zimmerman, Yusuke Onoda, Hiroko Kurokawa, Masahiro Aiba, MRN Québec and other

\newpage

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@@ -70,7 +70,7 @@ where:

an individual $i$ from species $f$ growing in plot plot $p$ in census $t$,

- $D_i$ is the diameter of the individual $i$,

- $B_{n}$ is the the basal area of neighborhood tree of species $n$,

- $G\textrm{max}_{f,p,i}$ is the maximum growth rate of the focal species $f$ on the plot $p$ for the individual $i$\color{red}(is the i necessary here?)\color{black}

- $G\textrm{max}_{f,p,i}$ is the maximum growth rate of the focal species $f$ on the plot $p$ for the individual $i$,

- $s$ and $g$ are functions representing the size and the competition effect respectively, and

- $\lambda_{n,f}$ is a parameter representing the growth reduction for a

unit of neighborhood basal area increase of species $n$ on species

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@@ -96,8 +96,7 @@ The central part of the analysis involves comparing alternative models for $\lam

\lambda_{n,f} = a +b \times (t_{n} - t_{f}).

\end{equation}

During the first day of the workshop we discussed the logic behind the hierarchical trait distance model, noting a fundamental similarity in this model to one based on competitive effect and competitive tolerance (where traits conferring a

high competitive effect also confer a high competition tolerance[^compreponse]). We also discussed the possibility of including a model with separate

The logic behind the hierarchical trait distance model, can be understand through a decomposition of competition in competitive effect and competitive response. The hierarchical trait distance model occurs when the traits conferring a high competitive effect also confer a high competition tolerance[^compreponse]. During the first day of the workshop we discussed the possibility of including a model with separate

links of traits with competitive effect and competitive response. This

model is connected to several papers by Goldberg *et al.*, where

competition is framed in term of effect and response and their links to

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@@ -111,7 +110,7 @@ the multiplicative effect-response model[^equmult].

[^compreponse]:Through out the document I will use competitive response as the inverse of competition tolerance.

[^equmult]:The equations of the multiplicative models are given in the Appendix.

[^equmult]:The equations of the multiplicative models are given in the [Appendix 1](#multi).

[^inter]:There was also a detailed discussion of more complex model that would include both effect and response and interactions among both.

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@@ -148,21 +147,21 @@ most complex model eq. \ref{response_effect_trait} this gives:

\begin{equation}

\sum_{n=1}^{N_p} \lambda_{n,f} \times B_n = \sum_{n=1}^{N_p} (a +b \times t_{f}

We (mainly Francis Hui, PhD student from UNSW) formatted all tree

data set to match common unit and names (see Appendix

[Variables description and units](#units)). We tested to check whether the range of

data set to match common unit and names (see [Appendix 2,

Variables description and units](#units)). We tested to check whether the range of

variables values (mean and quantile) were within sensible limits and

visually inspected plots of $G$ per $D$ to check for errors.

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@@ -210,8 +209,8 @@ validate the trait extraction (see Figure \ref{trait} for the range of traits va

## Data processing

Next we split each dataset by ecoregion, keeping only

ecoregions where at least three species contributed more than

5% of the average basal area to each plot (this sentence not very clear), thereby excluding quasi-monospecific stands.

ecoregions where in average at least three species contributed more than

5% of the average basal area of the plots, thereby excluding quasi-monospecific stands.

First we computed the local basal area ($cm^2/m^2$) of neighborhood

competitor per species for each individual tree. For NFI data the

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@@ -221,7 +220,7 @@ plot size depending on tree dbh and the New Zealand data the plots are

20x20 m). For the large tropical plots, the neighborhood was defined as

a 15 m radius around the tree.

The community weight mean $t_c$ of the neighboring trees and the absolute

The community weight mean of the neighboring trees and of the absolute

trait distance between the focal tree and neighborhood trees was calculated using the species level traits data, or filling

missing value with genus level data when it was possible, or

filling the remaining value with the community mean of the trait. All traits were

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@@ -251,7 +250,7 @@ the functions for the size and competition effect (mainly following the work of

@uriarte_trait_2010), and

2. using lmer was much faster than an estimation with JAGS or [Stan](http://mc-stan.org/).

When the analysis is more advanced I will verify whether choice of model-fitting technique influences results, byt running the same model using Stan.

When the analysis is more advanced I will test whether choice of linear or non-linear functions for functions $s$ and $g$ influences results, by running the same model using Stan.

## Fitted models

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@@ -264,7 +263,7 @@ The models fitted were based on Equation \label{G1}:

where:

- $\log {G}$ is the log basal area growth\color{red} (I think you should use G rather than BAG because that's what you used earlier, why introduce a new term?)\color{black}

- $\log {G}$ is the log basal area growth,

- $\log{G\textrm{max}}$ is the intercept representing log basal area growth with no competition [^Gmax] including a plot $p$ random effect to account for variation of abiotic conditions between plots in NFI or the quadrats in large tropical plots (assuming the same

variance for all species), a random focal species $f$ effect and a random individual $i$ effect when multiple

census are present,

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@@ -275,26 +274,28 @@ census are present,

We compared two alternative models for $\lambda_{n,f}$:

(i) $\lambda$ is a

(i) $\lambda$ is a function effect and response traits ($\lambda_{n,f} = a +b \times t_{f} +c

\times t_{n}$) and

(ii) $\lambda$ is a

function the absolute trait distance ($\lambda_{n,f} = a + b \times

$) and

(ii) $\lambda$ is a function effect and response traits ($\lambda_{n,f} = a +b \times t_{f} +c

\times t_{n}$).

$).

These two models can be expressed in terms of $t_c$ (community weighted mean trait value) as follows. For the absolute trait distance:

\begin{equation} \label{logGabs}

These two models can be expressed in terms of community weighted mean trait value as follows. For the trait effect-response model:

\begin{equation} \label{logG-ER}

\log{G}_{f,p,i,t} = \log{G\textrm{max}}_{f,p,i} + \alpha_f \times D_{i,t} + a \times

\log{B}_\textrm{tot} + b \times \log{B}_\textrm{tot} \times |t_{c} - t_{f}|.

\log{B}_\textrm{tot} + b \times \log{B}_\textrm{tot} \times t_f + c \times \log{B}_\textrm{tot} \times \overline{t_{n}}.

\end{equation}

For the trait effect-response model:

\begin{equation} \label{logG-ER}

We also fitted version of the model that only included the effect part (not including $b \times \log{B}_\textrm{tot} \times t_f$) or only the response part (not including $c \times \log{B}_\textrm{tot} \times \overline{t_{n}}$).

For the absolute trait distance:

\begin{equation} \label{logGabs}

\log{G}_{f,p,i,t} = \log{G\textrm{max}}_{f,p,i} + \alpha_f \times D_{i,t} + a \times

\log{B}_\textrm{tot} + b \times \log{B}_\textrm{tot} \times T_f + c \times \log{B}_\textrm{tot} \times t_{c}.

\log{B}_\textrm{tot} + b \times \log{B}_\textrm{tot} \times \overline{|t_{n} - t_{f}|}.

\end{equation}

We also fitted version of the model that only included the effect part (not including $b \times \log{B}_\textrm{tot} \times t_f$) or only the response part (not including $c \times \log{B}_\textrm{tot} \times t_{c}$).

where $\overline{|t_{n} - t_{f}|} = \sum_{n=1}^{N_p} P_n \times |t_n -t_f|$.

We then compared these two models to a null model where competition is

constant and independent of focal and neighborhood species trait.

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@@ -313,11 +314,11 @@ trait-based model as the difference in $R^2$ to the constant competition model e

For several of the NFI data sets (Spain, France, US) the absolute trait distance model was selected as the best model in more ecoregions than any version of the effect-response model (number of best model over all trait and ecoregions: absolute distance=102, Effect=48,Response=5, Effect-response= 26, no competition effect=28, simple competition (no trait)=18). (see Tables \ref{table-aic-SLA} \ref{table-aic-Leaf.N} \ref{table-aic-Wood.density} and \ref{table-aic-Max.height} for full details on model selection by AIC).

The effect size of the models shows a different picture on the figure \ref{boxplot-effectsize} and the figure \ref{boxplot-effectsize-MAP}. The effect size of the effect-response model was often much higher value than the absolute distance models. This was not the case for all ecoregions, with a large proportion still showing low effect sizes[^EffectSize]. Only for maximum height the absolute distance models resulted in effect size similar to the effect-response models.

The effect size of the models shows a different picture on the figure \ref{boxplot-effectsize} and the figure \ref{boxplot-effectsize-MAP}. The effect size of the effect-response model had often much higher value than the absolute distance models. This was not the case for all ecoregions, with a large proportion still showing low effect sizes[^EffectSize]. Only for maximum height the absolute distance models resulted in effect size similar to the effect-response models.

[^EffectSize]:The effect size represents the increase in $R^2$ of a particular model over the basic diameter growth model (diameter growth variance). It would be better expressed as a percentage of competition explained (species and diameter effect explain more variation than competition so the effect size will always be low) but I need to work more on that point (try to fit a model with a random effect per focal species x neighborhood species in $\lambda$?).

Overall the effect-response models give is strongest at low mean annual precipitation (

Overall the effect-response models is strongest at low mean annual precipitation (

Figure \ref{boxplot-effectsize-MAP}). This was the case for all traits. This pattern is also visible on the plots of the parameters in function of the MAP of the ecoregion where the maximum value of the parameters is reached for low MAP (see figure \ref{param-trait}). From this figure it is also clear that the model of hierarchical trait distance I used in @kunstler_competitive_2012 is not able to represents adequately the link between traits and competition. In the effect-response models the effect and response parameters are not generally of opposite sign and not of the same magnitude ($b \ne -c$). This means that the competitive effect and response are not necessarily correlated and not related in the same way to the traits. The fact that competitive effect and response are not always correlated was already stressed out by @goldberg_competitive_1996.

Most of the effect-response models fitted show a competitive effect (negative value of the parameters on the figure \ref{param-BATOT}). And overall the average competitive effect of one unit of neighborhood basal area is higher (parameters more negative on figure \ref{param-BATOT}) in ecoregions with lower MAP.

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@@ -351,6 +352,8 @@ Most of the effect-response models fitted show a competitive effect (negative va

\pagebreak

\newpage

![**Correlation pairs over all data sets 9in log scale).** Each data set is drawn with a different symbols and colors. Traits SLA ($mm^2/mg$), Leaf N per mass ($mg/g$), wood density

($mg/mm^3$), maximum height ($m$). \label{trait}](traits-XY.pdf)

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@@ -686,7 +689,7 @@ As for the additive model it is then possible to develop the multiplicative mode

\end{equation}

## Comparison of the multiplicative and additive effect and response model

## Appendix 1. Comparison of the multiplicative and additive effect and response model {#multi}

Developing the multiplicative model gives

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@@ -713,7 +716,7 @@ The multiplicative model is more constraining than the additive model plus inter

\newpage

# Appendix. Variables description and units {#units}

# Appendix 2. Variables description and units {#units}