Analysis
We estimated the effect of turf species on canopy-forming species as the
log-transformed response ratio (LNRR; Hedges et al. 1999). LNRR was
chosen for its similarity to the calculation of non-trophic species
interaction strengths from experimental data (Armas et al. 2004). The
direction of effects from all studies can be interpreted as species
interactions. A positive LNRR results from the negative effect that the
understory has on the canopy, such that, for example, the removal of
turfs results in higher canopy abundance in the experimental plot than
in the unmanipulated control. Because several metrics resulted in a
negative data point (e.g., negative growth rates), we standardized data
by study by adding the minimum data value to each data point in the
study to make the minimum data value zero, and then a small amount
(0.001) was added to all zero means and standard deviations. We used a
version of LNRR corrected for small sample size (equations 10 and 11 in
Lajeunesse 2015).
We analyzed the data using mixed effects meta-regression, with random
effects to account for variation within and among studies. Separate
models were run for observational and experimental studies, as the
methods differed significantly among the two study types. The random
effect structure for both experimental and observational data allowed
intercepts to vary among studies. Experimental random effects included
an additional term allowing intercepts to vary among different response
types (density, proportion, canopy length, growth rate), as, unlike in
observational studies, experimental studies often reported the results
of multiple different experiments that measured responses in multiple
ways (Appendix S2).
In addition to the fundamental differences between experimental and
observational methods, experimental interactions were often estimated at
multiple time points. There are many ways to model repeated measures in
meta-analysis (Koricheva et al. 2013). Because we were interested in
whether the effect of the understory changed through time, the number of
days since the start of the experiment was included in each experimental
model. For certain nested models, likelihood ratio tests were used to
determine whether additive or multiplicative model structures should be
used (Appendix S2 Table S3). In all analyses, model coefficients are
weighted by study precision (inverse variance) to give more weight to
more highly replicated studies.
To quantify the overall effect of turfs on the canopy, across all
studies, we used a random effects model with no fixed effects, removing
studies that manipulated herbivory. To quantify the effects of our
explanatory variables (functional group, depth, latitude), we used a
mixed effects model with the same random effect structure as our overall
effects model. To quantify the effects of herbivores, we subsetted the
data to only studies that included an herbivory treatment. We also ran
each experimental model separately for canopies in the order
Laminariales (“kelps”) and canopies in the order Fucales (Appendix
S3). Other orders present in the dataset (Dictyotales and
Desmarestiales) were not independently tested as there were too few
samples (n Dictyotales = 2, n Desmarestiales = 5).
Pseudo-R2 was calculated for each mixed model as the
proportion of total variance explained relative to a random-effects only
model. Models were fit with restricted maximum-likelihood estimation and
the significance of model coefficients and of linear contrasts was
determined using Wald-Type Tests (Viechtbauer 2010). We conducted a
series of analyses (Appendix S2) demonstrating that LNRR was not related
to study year, plot size, or inclusion of artificial understory.
Finally, publication bias was tested for all data (Appendix S2).