Appendix 2 Equations for models of body size evolution
Consider a single island-mainland pair, sharing a common ancestor at a time t in the
past, which had body size Sa. The mainland taxon evolves by geometric Brownian
motion (i.e., a random walk on a logarithmic scale), and the island taxon evolves by
the Ornstein-Uhlenbeck process (i.e., a random walk with a tendency to converge
towards the value  at a rate ). Their body sizes can be written as
ln Sm  ln Sa  mW (t),
(1a)
ln Si  ln Sa e
(1b)
1 e 2t 
 1 e   iW 
,
 2 
t
t

where W(t) is a Weiner process describing the cumulative effects of a random walk,
and m and i denote the rates of evolution on the mainland and island.

Now consider a data set of such pairs, and define k as the variance in evolutionary
change undergone by each mainland taxon divided by the variance in the ancestral
states across the data set:
k
(2)
 2m t pair
Varln Sa 
.
The value k is an important indicator of biases in tests of the Island Rule. For
example, under null B, size evolution is unaffected by insular habitation and so i =

m and  = 0. In this case, given a large data set, the correlation coefficient between
mainland body size and the body-size ratio is expected to be
rln Sm ,ln R   k /2
(3)
which will always be negative, yielding false support for the Island Rule. Equation (3)
assumes that R is log transformed, but a similar result holds if this is not so. The same

result can also be shown to hold approximately for independent contrasts.
If the ancestral body sizes of each pair also evolved from a single common ancestor
by Brownian motion, then k can also be written as
k
(4)

t pair
t pair

 1 (n 1) 
1 troot
1
troot


n
where n is the number of pairs, and (1-) is the mean proportion of the time since the
root that two distinct mainland species evolved as separate lineages ( = 0.3 for the
topology of primate phylogeny in Figure 1). (Equation (4) is increased by a factor of
2 if non-insular evolution also has a centralising tendency.) Equations (2)-(4) show
the importance of the parameter tpair/troot for determining the outcome of simulations,
and were used to set the rate of evolution, m, such that the standard deviation of
simulated mainland body sizes resembled that of the real data.
Appendix 3 Regression diagnostics for head-body length data
This appendix presents diagnostics for the SMA regression of Si on Sm, when the body
sizes are head-body lengths (mm) from primates. This test was found to yield
significant support for the Island Rule by Bromham and Cardillo (2007). Diagnostic
plots explained by Warton et al. (2006) and implemented in R (R Development Core
Team 2006) are shown in Figure A1. Part (a) shows standardised residuals, defined as
lnSi - lnSm where  is the best-fit SMA slope, plotted against “fitted axis scores”
defined as lnSi + lnSm; Part (b) shows a qq plot of standardised residuals against
normal
scores.
These
plots
show
clear
evidence
of
non-normality and
heteroscedasticity (Warton et al. 2006), suggesting that the p-value of this regression
is doubtful under a standard t-test. Log transforming the measurements improves
model fit, as expected if larger absolute changes in body size are more probable in
larger animals (Felsenstein 1988), and renders the test non-significant. But model fit
remains poor (not shown), making preferable the non-parametric permutation
approach used in the main text.
References
Bromham, L. & Cardillo, M. 2007 Primates follow the ‘island rule’: implications for
interpreting Homo floresiensis. Biol. Lett. 3, 398-400.
Felsenstein, J. 1988 Phylogenies and quantitative characters. Ann. Rev. Ecol. Syst. 19,
445-471.
R Development Core Team 2006 R: A language and environment for statistical
computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3900051-07-0, URL http://www.R-project.org
Warton, D. I., Wright, I. J., Falster, D. S. & Westoby, M. 2006 Bivariate line-fitting
methods for allometry. Biol. Rev. 81, 259-291.