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value in the correct position within the whole size hierarchy of sampled
individuals. If the scaling follows a power law, the plot built using the
logarithm of P(M) in the y-axis and the untransformed size M in the x-axis
will form a straight line with slope
1. This method captures well the scaling
structure, providing a better visualization of the scaling patterns and, more
importantly, returning a much more accurate estimation of slope values (i.e.
scaling exponent) than binned methods.
The other unbinned method most commonly employed is based on an ML
estimation of the scaling exponent (
Clauset et al., 2009; Newman, 2005
). This
method is based on the equation
a
"
#
n
X
n
x
i
x
min
a ¼
1
þ
ln
¼
i
1
where
is the ML estimation of the scaling parameter, x
i
is the body size of
the ith individual, and x
min
is the smallest value comprising the range of sizes
for which the power law holds true (
Newman, 2005
). Frequency or cumula-
tive distributions, even when fitting a real power law for a wide range of body
sizes, usually deviate from this regime at smaller sizes (
Clauset et al., 2009;
Marquet et al., 1995
). Previous results based on several reviews of power-law
model fitting in which parameter estimation procedures were tested against
simulated data show that ML gives more accurate and less biased estimates
than other available methods for scaling exponent estimation (
Edwards,
2008; Clauset et al., 2009; Newman, 2005; White et al., 2008
).
An interesting extension of the above-mentioned method can be proposed
here: plotting the estimates of
a
as a function of x
min
would be useful to break
up the entire scaling pattern into main and secondary scaling regions, reveal-
ing the possible existence of regime shifts whenever the slope estimates
approach zero (
Figure 4
B).
a
B. Bivariate Relationships
Binning methods artificially treat data as bivariate by relating frequency to
size in the context of a regression analysis applied to data after histogram
construction. However, this analysis is fundamentally univariate, and as
such, the real functional dependence between density and size, as implied in
DMR, is only approximate. However, some DMR studies were originally
based on the analysis of the relationship between density and body size
measured independently. This typically involves a variable representing pop-
ulation or species mean body size and the associated mean or maximum
density. This is the case for the analysis of population abundances within a
single community (
Cyr et al., 1997a
), the combination of data from different
