Appendix S4
Full description of meta-analysis
The effect sizes for primary and secondary consumers were estimated using mixed-effects models
(Borenstein et al. 2009; Viechtbauer 2010). Here, we performed separate mixed-effects models (n=6) with
trophic position (i.e. primary vs. secondary consumers) as categorical moderators to compare differences
in effect size for a given food web, using the restricted maximum likelihood estimator (REML)
(Mengersen & Schmid 2013). In mixed-effects models, a fixed effect is used to model variability among
groups (i.e. trophic level), while a random effect is used to model within-group variability (Viechtbauer
2010). The QM test was employed to test the difference in mean effects size between each trophic level,
i.e. to evaluate the probability that effect size differed between the levels of moderators (Rosenberg
2013). In addition, we report I2 values as a separate measure of heterogeneity (Higgins & Thompson
2002). I2 values typically range from 0 to 100% and is interpreted as the percentage of variability in
effect-size estimates that is due to between-study heterogeneity rather than sampling error (Rosenberg
2013). We report 1-I2 values to represent the actual proportion of variance in d explained by differences
between trophic levels.
We also estimated the influence of publication bias, which is the probability that statistically
significant (P<0.05) results are more likely to be published than non-statistically significant results, using
the trim-and-fill method (Duval & Tweedie 2000). This is a sensitivity analysis that corrects for funnelplot asymmetry by adding values for “missing” studies to generate a symmetric funnel plot from which a
new mean effect size can be estimated (Jennions et al. 2013).
REFERENCES
1.
Borenstein, M., Hedges, L.V., Higgins, J.P.T. & Rothstein, H.R. (2009). Introduction to Meta-analysis.
John Wiley & Sons, Chichester, UK.
2.
Duval, S. & Tweedie, R. (2000). A nonparametric “trim and fill” method of accounting for publication
bias in meta-analysis. J. Am. Stat. Assoc., 95, 89-98.
3.
Higgins, J.P.T. & Thompson, S.G. (2002). Quantifying heterogeneity in a meta-analysis. Stat. Med., 21,
1539-1558.
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Appendix S4
4.
Jennions, M., Lortie, C., Rosenberg, M. & Rothstein, H. (2013). Publication and Related Biases. In:
Handbook of Meta-analysis in Ecology and Evolution (eds. Koricheva, J, Gurevitch, J &
Mengersen, K). Princeton University Press Princeton, NJ, pp. 207-236.
5.
Mengersen, K. & Schmid, C. (2013). Maximum Likelihood Approaches to Meta-analysis. In: Handbook
of Meta-analysis in Ecology and Evolution (eds. Koricheva, J, Gurevitch, J & Mengersen, K).
Princeton University Press Princeton, NJ, pp. 125-144.
6.
Rosenberg, M. (2013). Moment and Least-Squares Based Approaches to Meta-analytic Inference. In:
Handbook of Meta-analysis in Ecology and Evolution (eds. Koricheva, J, Gurevitch, J &
Mengersen, K). Princeton University Press Princeton, NJ, pp. 108-124.
7.
Viechtbauer, W. (2010). Conducting meta-analyses in R with the metafor package. Journal of Statistical
Software, 36, 1-48.
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Appendix S4
Table A1 The results from the mixed-effect models with trophic level as a categorical moderator and the mean effect size calculated for each
trophic level individually. This analysis represents data without outliers. The P-value for the QM test is given for each model.
Trophic
Ecosystem Type
P-value of mixed-effects model 1-I2
level
Mean effect size
95% CI
n
Wetlands
Green
0.027
Brown
0.209
Woodlands
Green
0.552
Brown
0.080
Grasslands
Green
0.942
Brown
0.925
35% Primary
Secondary
12% Primary
Secondary
-0.65
0.56
0.32
-0.49
-0.97/-0.34
0.06/1.2
-0.14/0.79
-1.3/0.27
8
5
11
7
33% Primary
Secondary
11% Primary
Secondary
-0.57
-0.27
-0.58
0.83
-1.1/-0.02
-1.8/0.63
-1.2/0.04
-0.11/1.9
7
5
11
6
5% Primary
Secondary
12% Primary
Secondary
0.01
0.05
-0.23
0.04
-0.86/0.88
-1.3/1.4
-0.64/0.19
-0.68/0.76
9
6
15
8
Total
98
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Appendix S4
Table A2 This analysis represents the same information as Table A1 except that the data was calculated with outliers (in bold and underlined).
The outliers only occurred in the green food web of grassland ecosystems.
Trophic
Ecosystem Type
P-value of mixed-effects model 1-I2
level
Mean effect size
95% CI
n
Wetlands
Green
0.027
Brown
0.209
Woodlands
Green
0.552
Brown
0.080
Grasslands
Green*
0.581
Brown
0.925
35% Primary
Secondary
12% Primary
Secondary
-0.65
0.56
0.32
-0.49
-0.97/-0.34
0.06/1.2
-0.14/0.79
-1.3/0.27
8
5
11
7
33% Primary
Secondary
10% Primary
Secondary
-0.57
-0.27
-0.58
0.83
-1.1/-0.02
-1.8/0.63
-1.2/0.04
-0.11/1.9
7
5
11
6
0.5% Primary
Secondary
12% Primary
Secondary
1.35
-1.18
-0.23
0.04
-1.4/4.14
-5.4/3.0
-0.64/0.19
-0.68/0.76
10
8
15
8
Total
101
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Appendix S4
Trim-and-Fill Funnel Plots: P-values greater than 0.05 indicates no sensitivity to publication
bias. Open circles are estimated number of missing studies, whereas dark circles are observed
estimates from the meta-analysis. Data is shown without outliers.
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