1
Appendix S5 – Calculation of effect sizes
2
3
We used Fisher’s z-transformation to calculate an effect size for each individual study: z = 0.5 × log
4
[(1 + r)/(1 – r)], with r the coefficient of correlation between parasitism rate or parasitoid species
5
richness and elevation. The asymptotic variance of z (s²z) was calculated as s²z = 1/(n–3), where n
6
corresponds to the number of point data used in regressions, plotted on figures or reported by the
7
authors (Borenstein et al. 2009). As the conditional variance is inversely related to n, studies with
8
larger sample sizes had a greater weight in the calculation of the mean effect size (Gurevitch &
9
Hedges 1999), which represents the weighted average of relationships between parasitism rate or
10
parasitoid richness and elevation. Effects were considered statistically significant if the 95% bias-
11
corrected bootstrap confidence interval (CI) calculated with 9999 iterations did not include zero.
12
Analyses were performed on Fisher’z statistics, but, for the sake of convenience, we back-
13
transformed z values to obtain correlation coefficients (r) in the ‘results’ section.
14
The magnitude of the effect of elevation on parasitism rate or parasitoid richness was
15
estimated by the slopes of linear regressions. Standardized regression slopes were generated by the
16
weighted least squares approach (Bini et al. 2001, Becker & Wu 2007): we calculated a weighted
17
mean of slopes (b), by combining slopes bi of each i study ranging from 1 to k as:
k
wb
i i
18
b
i 1
k
w
eqn(1)
i
i 1
19
where wi is the reciprocal of the variance of the slope of the ith study: wi = 1/s²(bi).
20
21
22
The variance of mean slope b was calculated as:
sb2 
1
w
i 1
23
eqn(2)
k
i
24
We used a random effect model to test the effect of elevation range, as a continuous variable, on
25
effect size (Gurevitch & Hedges 1999). We used a mixed-effect model to assess between-class
26
heterogeneity (for each categorical covariate, i.e. host and parasitoid types) and to evaluate the
27
significance of the class effect (Gurevitch & Hedges 1999), assuming a fixed effect across classes and
28
a random effect within classes (Borenstein et al. 2009). The weighted mean effect size and a bias-
29
corrected bootstrap confidence interval were then calculated for each class of covariate. We
30
calculated the variation in effect size explained by the categorical model (QBetween or QB). This
31
between-class heterogeneity was tested against a ² distribution, to evaluate the significance of the
32
class effect.
33
We checked the dataset for publication bias with the weighted fail-safe number, as modified
34
by Rosenberg (Rosenberg 2005). The fail-safe number is the number of non significant, unpublished
35
or missing studies that would have to be added to a meta-analysis to convert the result of the meta-
36
analysis from significant to non significant. This number was compared with the conservative 5k+10
37
(Rosenberg et al. 2000), where k is the total number of individual comparisons. We also drew Normal
38
quantile plots, which we assessed by eye to identify potential publication bias and abnormalities in
39
data structure (Rosenberg et al. 2000). All meta-analyses were conducted with Metawin 2.0 software
40
(Rosenberg et al. 2000).
41
42
References
43
Becker, B. J. & Wu, M. J. 2007 The synthesis of regression slopes in meta-analysis. Stat. Sci. 22, 414-
44
429.
45
Bini, L. M., Coelho, A. S. G. & Dini-Filho, J. A. F. 2001 Is the relationship between population density
46
and body size consistent across independent studies? A meta-analytical approach. Revista
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Brasileira de Biologia 61, 1-6.
48
49
Borenstein, M., Hedges, L. V., Higgins, J. P. T. & Rothstein, H. R. 2009 Introduction to meta-analysis.
Chichester: Wiley edition. John Wiley & Sons.
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51
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Gurevitch, J., and L. V. Hedges. 1999 Statistical issues in ecological meta-analyses. Ecology 80, 11421149.
Rosenberg, M. S. 2005 The file-drawer problem revisited: a general weighted method for calculating
fail-safe numbers in meta-analysis. Evolution 59, 464-468.
Rosenberg, M. S., Adams, D. C. & Gurevitch, J. 2000 MetaWin: Statistical software for Meta-analysis.
Version 2.0. Sunderland, MA. Sinauer Associates, Publishers.