Data
preparation
All data manipulations and later analyses were conducted using the free
software R 3.6.2 (R Core Team 2019) and Rstudio 1.1.442 (RStudio Team
2016). To approximate normal distributions, all raw data
(EFraw) were boxcox-transformed
(EFboxcox) using two lambda-values (λ and
λ2) estimated with the package ”geoR” (Ribeiro Jr 2020):
\begin{equation}
\text{EF}_{\text{boxcox}}=\ boxcoxtransformed\ \left(\text{EF}\right)=\ \frac{\text{EF}_{\text{raw}}+\ \lambda_{2}^{\text{\ \ \ λ}}-1}{\lambda}\nonumber \\
\end{equation}To scale all EFs to a comparable range of 0 to 1, the
EFboxcox were minmax-transformed
(EFminmax):
\begin{equation}
\text{EF}_{\text{minmax}}=minmaxtransformed\ \left(\text{EF}\right)=\ \frac{\text{EF}_{\text{boxcox}}-\ min(\text{EF}_{\text{boxcox}})}{\left(\text{EF}_{\text{boxcox}}\right)\ -min(\text{EF}_{\text{boxcox}})}\nonumber \\
\end{equation}Variation in individual
EFs
The
variation of individual EFs was quantified as a standard deviation over
all data points (individual measures on plots). The individual EFs are
often measured at the same time (Supporting information A, Table S2).
Thus, variation of individual EFs is expected to be comparable and not
biased by the identity of years and seasons measurements were taken.
However, we tested whether the variation of individual EFs depended on
the number of repeated measures, meaning how often in time EFs were
measured (number of years * number of seasons). Therefore, a model with
the standard deviation per individual EF depending on the explanatory
variable “number of repeated measures” (number of years * number of
seasons an individual EFs was measured) was run.
The drivers of the variation in
individual EFs (EFminmax), were tested in a linear model
with the explanatory terms ”block” (factor with four levels), ”SR”
(initial number of species planted, log-transformed continuous
variable), ”plotID” (factor with 80 levels), ”season” (factor with 3
levels, as no measurements were done in winter), ”year” (continuous
variable), and their interactions. The plot identity (plotID) effect
mainly accounts for differences among the initially planted communities.
This set of terms is referred to as ”drivers” in the following. The same
model was conducted for EFs measured only once per year excluding
”season” and the respective interaction terms.
To analyse whether different
classes of EFs were affected differently by drivers, the variance in
individual EFs explained by individual drivers was calculated by
dividing the sum of squares explained by the driver by the total sum of
squares in the repective model of the individual EFs explained above. In
a subsequent model the explained variation per EF and per driver were
used as meta-data. The variation was tested against the classes of EFs
(with the different classes of EFs as levels) and the drivers (with the
levels ”block”, ”SR”, ”plotID”,
”season”, ”year”, and their interactions) as independent variables.
Relationships between pairs of
EFs
Relationships between EF pairs were statistically investigated using
covariances and correlations. In correlations, the relationship between
two EFs was standardised by the variation of the individual EFs (product
of their standard deviations), enabling us to compare relationships
between different EF pairs. To calculate correlation coefficients, we
used the R-package Hmisc 4.4-2 (Harrell Jr 2020). We used the
non-standardised relationships (covariances) to analyse the influence of
drivers on relationships among EFs.
Variation in EF
correlations
To quantify the general strength and variation of EF correlations, we
calculated the mean and the standard deviation of Fisher’s Z-transformed
correlation coefficients for each EF pair. Correlation coefficients were
calculated among measurements on all plots at a particular time point
and then averaged across time points. Hence we refer to this correlation
as the mean correlation. It includes the effects of species richness and
plot identity. In order to plot the EF relationships as correlation
coefficients on a scale of –1 (perfect negative) to 1 (perfect positive
correlation), the mean correlation coefficients were back-transformed
from Z-scale.
The standard deviation of the individual correlations at the different
time points quantifies the temporal variation (among seasons and years)
of correlations among EFs. However, using all time points to calculate
the temporal variation, might be influenced by the number of time points
and the identity of time points (deviating years or seasons). Therefore,
first, we checked whether this temporal variation, based on all time
points, depended on the number of time points. We analysed the temporal
variation of the correlations per EF pair as a function of the number of
timepoints that EF pair was measured (number of years* number of
seasons). The number of repeated measures for pairwise EFs, meaning the
number of times two EFs were measured at the same time (same year and
same season), ranged from 0 to 36 times
(Supporting information B, Table
S2 contains an overview of the individual EFs and at what time (years
and seasons) they were measured). Second,
we checked whether the variation
of correlations per EF pair depended on the identity of the time point
that EF pair was measured. Therefore, for each EF-pair we randomly chose
four time points to calculate a standard deviation of the respective
correlation coeffcients. For each EF-pair this was done 20 times. The
range of these 20 standard deviations per EF pair was used to check
whether the standard deviation for that EF pair was stable (small range
indicating no identity effect of years and seasons) or not (large range
indicating strong identity effects of years or seasons).
Drivers of the covariance between EF
pairs
To analyse whether years, seasons, species richness, and plot identity
affect EF relationships by driving individual EFs in similar or opposing
ways, we partitioned overall covariances into contributions of the
different explanatory terms. Here, plot identity was further decomposed
in the effects of functional group richness, and the presence of the
functional groups legumes, herbs (tall and short herbs combined), or
grasses. This decomposition of covariances was based on an additive
partitioning of sums of products (SPs) in the same way as additive
partitioning of sum of squares (SS) is used in a decomposition of
variances in an analysis of variance (ANOVA). This type of covariance
analysis has previously been used to investigate, for example, the
influence of explanatory terms on trait-trait relationships (He, Wang et
al. 2009) and is frequently used in quantitative genetic and
phylogenetic approaches (Kempthorne 1957, Bell 1989).
The sums of products, which are equivalent to covariances, were obtained
per EF pair using the following formula:
\begin{equation}
SP(X,Y)=\frac{SS(X+Y)\ -\ SS(X)\ -\ SS(Y)}{2}\nonumber \\
\end{equation}where X and Y are the EFs of interest, and X+Y is the sum of the two
EFs. The SS were obtained from general linear models (implemented with
the lm() function in R (Mangiafico 2015)) with the explanatory terms
”block”, ”log(SR)”, ”plotID”, ”season”, ”year”, and the interactions
”season:year”, ”log(SR):(season + year + season:year)”, ” plotID:(season
+ year + season:year)” (note that here, following conventions of R, we
use the colon instead of a multiplication sign as interaction operator).
For each EF pair, three linear models were run: one for each of the
individual EFs (X and Y) and one for the sum of the two EFs (X + Y),
based on the measurements from different time points of
EFminmax. Like in ANOVA, SPs are divided by their
degrees of freedom to obtain mean SPs (MSPs), which are divided by
residual MSP to calculate F-ratios and significances. Because there are
nested effects, not all terms could be tested against ”Residuals”.
”Block” and ”log(SR)” had to be tested at the level of variation between
plots with different species compositions (plotID). Similarly, the
interaction terms ”log(SR):(season + year + season:year)” had to be
tested for the same reason against “plotID:(season + year +
season:year)”. All other terms were tested against ”Residuals”
(Supporting Information B, Table S3). It has been shown that for
balanced experimental designs such as the Jena Experiment this method is
comparable to linear mixed-model analysis using restricted maximum
likelihood methods (Schmid, Baruffol et al. 2017).
Because SPs are additive, we can express the influence of each driver on
EF covariation (i.e., the relationship between the EFs) by calculating
the percentage of the (absolute) total sum of products explained,
similar to a percentage variance explained (He, Wang et al. 2009).
However, unlike variances, covariances are either positive, indicating a
positive relationship between two variables, or negative, indicating an
inverse (i.e. trade-off) relationship between two variables. The sign of
SPs for each explanatory term informs us about whether covariances are
positive or negative. This means that we could deduce whether the
individual drivers affected the EFs in a pair in an trade-off (negative
covariance) or synergistic (positive covariance) way. Therefore, we show
”signed percentages” of covariance in the results by multiplying the
absolute percentages with the sign of the respective covariance.
RESULTS
Variation in individual EFs
First, we compared the variation of
individual EFs and EF classes. The average standard deviation,
calculated by averaging all standard deviations of all EFs, was 0.17.
While some EFs varied strongly in time among replicated measures, other
EFs showed a low variation (Table 2; minimum standard deviation was 0.07
for plant carbon, maximum standard deviation was 0.38 for plant sodium).
The variation of individual EFs did not depend on the number of times
(number of years * number of seasons) they were measured
(F1,29= 0.753, p= 0.393)(Supporting information C, Fig.
S3). Classes of EFs did not differ significantly in their variation
(F7,23=0.76, p=0.63; Table 2).