ELECTRONIC SUPPLEMENTARY MATERIAL
Appendix C. Relationship between significance and sample size
To find out if the significance of our results was dependent on the number of individuals of
the species pairs or on the univariate spatial structure of the component patterns (measured by
the univariate pair correlation function) we followed the approach taken in Wiegand et al.
[22]. We calculated for all species pairs the Spearman rank correlation between the rank u0 of
the goodness-of-fit test and the number n1 of individuals of species 1, the number n2 of
individuals of species 2, and the value of the univariate pair correlation functions g11(r) and
g22(r) at various distances r. Because the focus of analysis 1 was on given neighbourhoods r
we conducted the goodness-of-fit test for single neighbourhoods r = 6m, 10m, and 30m.
However, in analysis 2 the focus was rather on the behaviour of the bivariate pair correlation
function over the distance interval 2-30m. We therefore used here the rank of the goodnessof-fit test calculated for this distance interval.
This analysis serves two purposes. First, we may suspect that ignoring species with
low abundance would severely bias our results. However, if the correlation between number
of individuals and the rank of the goodness-of-fit tests would be weak no severe bias would
occur. This was the case in the analysis of the Sinharaja data [22]. Second, if stochastic effects
in species richer communities would dilute species associations we would expect that the
correlation between number of individuals and the rank of the goodness-of-fit test should be
in general weaker in species richer forests but stronger in species poorer forests.
For analysis 1 we tested the correlation between the ranks of the goodness-of-fit tests
of the M axis (which is related to the total number of neighbours of species 2 within distance r
around the individuals of the focal species) at different neighbourhoods r and the number n1
and n2 of individuals of species 1 and 2, respectively. Table A2 shows that the rank of the
goodness-of-fit test was positively correlated with n1 and n2, but the correlation coefficient
remained for all three forests relatively low (rSp < 0.2 and rSp < 0.3, respectively). Thus, if one
of the species of a pair was more abundant, the association of this species pair was somewhat
more likely to be significant.
For analysis 2, the rank u0 of the goodness-of-fit test (conducted for the bivariate pair
correlation function over the distance interval 0-30m with respect to the heterogeneous
Poisson process) correlated weakly and positively with the number of individuals of species 2
(BCI: rSp = 0.13; p < 0.01; Sinharaja: rSp = 0.15; CBS: 0.34, p < 0.01), and with the number of
individuals of species 1 (BCI: rSp = 0.06, p < 0.01; Sinharaja: rSp = 0.03; CBS: 0.4, p < 0.01).
Thus, the significant interspecific associations detected did not primarily depend on the
sample sizes although as expected, significant effects tended to be more frequent for larger
sample sizes and the correlation was stronger for the species poorer CBS forest. The latter is
compatible with the dilution hypothesis.
We also found negative correlations of the rank of u0 of the goodness-of-fit test
(conducted for the bivariate pair correlation function over the distance interval 0-30m with
respect to the heterogeneous Poisson process) with the univariate pair correlation function of
focal species 1 and species 2 (figure A5a). The correlations were weak for BCI and Sinharaja
but strong for CBS, and there was little dependence on scale (figure A5a). This result
indicated that it was less likely to find significant small-scale associations with more
aggregated species, and the correlation was strong for the species poorer CBS forest. The
latter is compatible with the dilution hypothesis. We found similar results for the pair
correlation function of the second species (figure A5b), but here the weak correlation were
positive for BCI and Sinharaja. The quantitative agreement of the correlation coefficients
found for BCI with those obtained in the study of the Sinharaja forest is surprising.