Determinants of Density–Body Size Scaling Within Food Webs and Tools for Their Detection - Advances in Ecological Research - page 29

In amphibians, this could be an expected pattern considering the occurrence
of several and separated mating events and larvae cohorts along the repro-
ductive period. If this bimodal pattern is ignored, the estimated DMR with
ML is 1.3. However, the fit of segmented regressions indicates the existence
of three breaks and four different slopes ( Figure 7 B). The slopes obtained
with ML for the two modes are 2.68 and 2.25, representing larger values than
those estimated without considering the occurrence of changes in the DMR.
E. Evaluation of Methodological Performance
Evaluation of the performance of statistics with ideal data following a power-
law distribution indicates much more stable and consistent estimations of
scaling parameters with ML approach in comparison to binned methods (see
Section IV and Clauset et al., 2009; White et al.,2008 ). These evaluations were
based on the simulation of random samples from a power-law distribution.
However, real observations rarely follow a perfect power-law relationship. The
comparative evaluation of statistics performance with real data accompanies
previous approaches and is needed in order to properly assess available meth-
odologies. To this aim, we performed a bootstrap analysis in a gradient of
sample sizes was performed to evaluate alternative methods. For this analysis,
we focused on the fish community, a database composed of 680 individuals.
To test the performance of the used methods for different sample sizes,
estimations were run using samples from 100 to 600 individuals randomly
taken from the fish community dataset. A 1000 samples were made for each
resample size, registering the confidence interval at the 95% for each one of
the methods considered. The minimum sample size for which estimations
could be obtained using segmented linear functions was 300 individuals for
histogram methods. Below this sample size the segmented linear functions do
not fit properly, that is, for 200 individuals more than 20% of these samples
resulted in models for which parameters could not be estimated. Notably,
cumulative distribution and ML methods could be fitted to all resample sizes.
As expected, all methods are more accurate and less biased as the resample
size increases ( Figure 9 ). However, histogram methods show biases, system-
atically underestimating the exponent when sample size is reduced. Cumula-
tive distribution estimations present a small bias towards steeper slopes,
which is gradually reduced for bigger samples ( Figure 9 C). While the lower
confidence band in this method is close to the real value the upper band
presents large deviations even at large sample sizes. In contrast, ML estima-
tions present a symmetric confident band and practically no biases even at
small sample sizes ( Figure 9 D). This result strongly supports the use of ML
methods in the estimation of scaling parameters.