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).