2.4 | Statistical Analysis
Height data were recorded at regular intervals over the course of the
experiment to aid in determining when plant-soil feedbacks occurred and
to assess their strength and direction. The rate of plant growth is
variable over time, which means non-linear models will generally perform
better than linear models at capturing how height changes over time. We
chose to use generalized additive models (GAMs) to evaluate grass growth
over time. GAMs are similar to generalized linear models except that
they replace linear covariates with local smoothing functions that
enable modeling of non-linear processes (Hastie and Tibshirani 1986). To
help us understand the overall effect and timing of plant-soil feedbacks
on the four phytometers, we built GAMs of the height data of each
treatment group over time using the mgcv package (v1.8-34; Wood,
2011) in R. The following is a simplification of the generalized
additive model (GAM) formula that was used for each group of phytometers
(Yee and Mitchell 1991).
\begin{equation}
\mathbb{E}\log\mathbf{y}=x_{1}:x_{2}+(\sum{\mathcal{f(}\mathbf{t}_{\mathbf{i}}\mathbf{)+\ }\mathrm{\ }\mathcal{f(}\mathbf{t}_{\mathbf{i}}\mathbf{)}x_{1}:x_{2})}+(1|x_{3})\nonumber \\
\end{equation}The formula relates the expected value (\(\mathbb{E}\))
log10-transformed height (\(\log\mathbf{y}\)) as a
function of the interaction between the factors conditioning species
(\(x_{1}\)) and sterilization status (\(x_{2}\)), the sum (∑) of
smoothing (\(\mathcal{f}\)) variables time (\(\mathbf{t}_{\mathbf{i}}\))
and time given each level of the interaction of the two factors
(\(\mathcal{f(}\mathbf{t}_{\mathbf{i}}\mathbf{)}x_{1}:x_{2}\)), and a
random intercept (\(1|x_{3}))\) using the unique ID for each pot in the
phytometer phase of the experiment. The random intercept was selected to
account for repeated measures on each phytometer (Pedersen et al.,
2019). The models used the Gaussian family and identity link function.
Model selection was done by comparing the AIC for candidate models. We
found this model formulation to explain the most variance while
retaining only the variables that contribute to explanatory power of the
model. We plotted the output of these generalized additive models (GAMs)
using the tidymv R package to visualize and facilitate comparison
of plant height over time under different treatments (Coretta, 2022).Post hoc comparisons were done using the emmeans package
(v1.7.1-1; Russell, 2021). For each phytometer species, the mean
estimated height was contrasted between each treatment group.
Significance was determined using a Tukey post hoc comparison
adjustment for a family of ten estimates.
Table 1. The model type and R2 value for each
biomass type (Shoot, Root, or Total) and phytometer Andropogon
gerardi (ANGE), Schizachyrium scoparium (SCSC), Bromus
inermis (BRIN), and Pascopyrum smithii (PASM). Model types are
mixed effects (M) or linear (L) and either contain an interaction term
(I) between conditioning soil type and sterilization status or do not
include the interaction term (no I). Asterisks (*) denote models that
have significant main effects. Adjusted R2 (adj)
quantifies the explained variance of fixed effects in linear models.
Conditional R2 (cond) quantifies the variance
described by fixed and random effects in mixed models. SeeMethods section for detailed model description.