Appendix
Here we briefly present the equations used in our analysis (for a more detailed
explanation see Rosenberg et al. 2000; Hartung et al. 2008). First of all, we estimated
the Hedges’ difference (di,j, Eq.1) and its associated variance (vi,j, Eq.2), for each
morphological comparison as the difference between fat body size group mean (i.e.,
maximum minus minimum), expressed in units of pooled standard deviation and
corrected for small sample bias:
d i, j 
(X E  X C ) * J
S
[Eq.1]
vi , j 
NC  NE
d2

NCNE
2( N C  N E )
[Eq.2]
where XC and XE are the means of the minimum and maximum group, respectively; NC
and NE are the sample size of the minimum and maximum group, respectively; J is a
sample size correction factor (Eq.3), and S is the pooled standard deviation (Eq.4);
specifically, J and S were estimated as:
J  1
S
3
4( N  N E  2)  1
C
( N E  1) * ( s E ) 2  ( N C  1) * ( s C ) 2
( N E  N C  2)
[Eq. 3]
[Eq.4]
where sC and sE are the standard deviation of the minimum and maximum group,
respectively.
After that, we used random-effect meta-analytical models with continuous
predictor variables to evaluate the correlation between latitude, climatic variables and
some potential confounding variables (i.e., body size and sampling effort) with fat body
size flexibility. We selected random-effect models because they do not depend on the
assumption that one true effect size is shared by all the studies. Note that in these
models total heterogeneity (QT) is decomposed into the heterogeneity explained by the
model (QM) and error heterogeneity (QE), in a similar fashion as in a regression analysis
(see below).
To evaluate differences in the amount of flexibility (d) between sexes (levels:
females and males) and between the kind of dependent variable assessed in each study
(levels: percentage of body mass, absolute weights, adjusted weights and volume), we
estimated the cumulative effect size for each level of a given factor (Ei, Eq.5) and its
associated variance (s2(Ei), Eq.6), and then the cumulative effect size for the factor (E,
Eq. 7) and its associated variance (s2(E), Eq.8):
ki
Ei 
w
i, j
j 1
* d i, j
ki
 wi, j
[Eq.5]
j 1
s 2 ( Ei ) 
1
ki
 wi, j
j 1
[Eq.6]
ki
m
E
 w
i 1 j 1
m ki
i, j
di, j
[Eq.7]
 wi, j
i 1 j 1
s2 (E) 
1
m
[Eq.8]
ki
 w
i 1 j 1
i, j
where m is the number of levels of a factor, ki is the number of studies in the jth factor,
wi,j and dij are the weight and the effect size for the ith study in the jth factor; note that
individual weight (wi,j) is the reciprocal of the individual variance (wi,j = 1/vi,j). Finally,
we compare cumulative effect size among different levels of a factor using
heterogeneity tests. In these tests, total variation (QT) is partitioned, as in a conventional
analysis of variance, in two components, the variance explained by the model (QM) and
the residual error variance (QE) (Eqs.9 to 11):
QT  QM  QE
m
ki
[Eq. 9]
QT   wi , j d i2, j
 m ki

  wi , j d i , j 


i 1 j 1



m
ki
 w
i 1 j 1
i 1 j 1
ki
Q E   wi , j d i2, j
j 1
 ki

  wi , j d i , j 


j 1


ki
w
j 1
2
[Eq. 10]
i, j
2
[Eq. 11]
i, j
Both QM and QE can be tested against a X2-distribution with m – 1 degrees of
freedom for QM and n – m degrees of freedom for QE (where n is the number of
comparisons and m the number of levels of a factor). Finally, in the comparisons among
factors we used a mixed-effect model, because it not depends on the assumption that
one true effect size is shared by all of the studies in the same factor. This implies that a
fixed-effect model analysis is used to determine the summary statistic, and from these
statistics a between-study variance (σ2pooled, Eq. 12) is calculated:
2
 pooled

QE  (n  m)
kj


 kj
wij2 

m


i 1

  wij  k j

j 1 i 1

wij 

i 1


[Eq. 12]
where all the parameters are defined as above. From this between study variance new
individual weights [wi,j, Eq. 13] are estimated as:
wi , j 
1
2
v i , j   pooled
[Eq. 13]
and used in the mixed-effect model, which now is calculated using the equations
described above.
References
Rosenberg, M.S. et al. 2000. METAWIN, version 2.1. Statistical Software for MetaAnalysis. Sinauer Associates Inc.
Hartung, J. et al. 2008. Statistical meta-analysis with applications. John Wiley & Sons,
Inc.