On estimating the exponent of power-law frequency distributions.
ABSTRACT Power-law frequency distributions characterize a wide array of natural phenomena. In ecology, biology, and many physical and social sciences, the exponents of these power laws are estimated to draw inference about the processes underlying the phenomenon, to test theoretical models, and to scale up from local observations to global patterns. Therefore, it is essential that these exponents be estimated accurately. Unfortunately, the binning-based methods traditionally used in ecology and other disciplines perform quite poorly. Here we discuss more sophisticated methods for fitting these exponents based on cumulative distribution functions and maximum likelihood estimation. We illustrate their superior performance at estimating known exponents and provide details on how and when ecologists should use them. Our results confirm that maximum likelihood estimation outperforms other methods in both accuracy and precision. Because of the use of biased statistical methods for estimating the exponent, the conclusions of several recently published papers should be revisited.
- SourceAvailable from: Raúl Medina[Show abstract] [Hide abstract]
ABSTRACT: Long-term simulations (3000 yr) of an idealized basin using different tidal ranges (1, 2 and 3 m) and grain sizes (120, 480 and 960 μm) have been performed in order to cover a range of hydrodynamic and sedimentary conditions. Two different cell sizes (50 and 100 m) have been used to study the impact of cell size on tidal network development. The probability distributions of the drainage area and the drainage volume have been computed for every simulation (during an ebb and a flood phase). Power law distributions are observed in drainage area and drainage volume distribution. As an objective estimation of the exponent of a power law is an open issue, different approaches (linear binning, normalized logarithmic binning, cumulative distribution function and maximum likelihood) proposed by White et al. (2008) to estimate the exponent have been used to carry out a sensitivity analysis. Our findings indicate that although all methods results in high and significant correlation coefficients, more work is needed to develop a universal, objective estimation of the exponent.03/2014;
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ABSTRACT: Aim An understanding of the relationship between forest biomass and climate is needed to predict the impacts of climate change on carbon stores. Biomass patterns have been characterized at geographically or climatically restricted scales, making it unclear if biomass is limited by climate in any general way at continental to global scales. Using a dataset spanning multiple climatic regions we evaluate the generality of published biomass–climate correlations. We also combine metabolic theory and hydraulic limits to plant growth to first derive and then test predictions for how forest biomass should vary with maximum individual tree biomass and the ecosystem water deficit.Location Temperate forests and dry, moist and wet tropical forests across North, Central and South America.Methods A forest biomass model was derived from allometric functions and power‐law size distributions. Biomass and climate were correlated using extensive forest plot (276 0.1‐ha plots), wood density and climate datasets. Climate variables included mean annual temperature, annual precipitation, their ratio, precipitation of the driest quarter, potential and actual evapotranspiration, and the ecosystem water deficit. The water deficit uniquely summarizes water balance by integrating water inputs from precipitation with water losses due to solar energy.Results Climate generally explained little variation in forest biomass, and mixed support was found for published biomass–climate relationships. Our theory indicated that maximum individual biomass governs forest biomass and is constrained by water deficit. Indeed, forest biomass was tightly coupled to maximum individual biomass and the upper bound of maximum individual biomass declined steeply with water deficit. Water deficit similarly constrained the upper bound of forest biomass, with most forests below the constraint.Main conclusions The results suggest that: (1) biomass–climate models developed at restricted geographic/climatic scales may not hold at broader scales; (2) maximum individual biomass is strongly related to forest biomass, suggesting that process‐based models should focus on maximum individual biomass; (3) the ecosystem water deficit constrains biomass, but realized biomass often falls below the constraint; such that (4) biomass is not strongly limited by climate in most forests so that forest biomass may not predictably respond to changes in mean climate.Global Ecology and Biogeography 01/2011; 20(5). · 7.22 Impact Factor
Article: Scaling properties of tidal networks[Show abstract] [Hide abstract]
ABSTRACT: A new methodology is developed to extract tidal network from hydrodynamic conditions, and use data derived from numerical modelling or field observations to test the hypothesis that tidal networks are characterized by scale-invariant properties. Different tidal network configurations have been obtained from long-term numerical simulations in an idealized basin. These simulations show the influence of hydrodynamic conditions (tidal range, TR) and sediment (grain size sediment, D50) on the final configuration of the network. One of the signatures of scale- invariant behaviour is related to the presence of a power law relationship in the probability distribution of geometrical characteristics. For each model configuration and field site the probability distribution of drainage area and the drainage volume has been calculated, and in both cases tidal networks show scale invariant characteristics. After assessing the sensivity of the results, an energy expenditure analysis shows that tidal basins evolve towards a state with less morphodynamic activity, with a lower energy expenditure compare with the initial state.Water Resources Research 05/2014; 50(6):4585-4602. · 3.15 Impact Factor
Ecology, 89(4), 2008, pp. 905–912
? 2008 by the Ecological Society of America
ON ESTIMATING THE EXPONENT OF POWER-LAW
ETHAN P. WHITE,1,2,4BRIAN J. ENQUIST,2AND JESSICA L. GREEN3
1Department of Biology and the Ecology Center, Utah State University, Logan, Utah 84322 USA
2Department of Ecology and Evolutionary Biology, University of Arizona, Tucson, Arizona 85721 USA
3Center for Ecology and Evolutionary Biology, University of Oregon, Eugene, Oregon 97403 USA
phenomena. In ecology, biology, and many physical and social sciences, the exponents of
these power laws are estimated to draw inference about the processes underlying the
phenomenon, to test theoretical models, and to scale up from local observations to global
patterns. Therefore, it is essential that these exponents be estimated accurately. Unfortunately,
the binning-based methods traditionally used in ecology and other disciplines perform quite
poorly. Here we discuss more sophisticated methods for fitting these exponents based on
cumulative distribution functions and maximum likelihood estimation. We illustrate their
superior performance at estimating known exponents and provide details on how and when
ecologists should use them. Our results confirm that maximum likelihood estimation
outperforms other methods in both accuracy and precision. Because of the use of biased
statistical methods for estimating the exponent, the conclusions of several recently published
papers should be revisited.
Power-law frequency distributions characterize a wide array of natural
Key words:binning; distribution; exponent; maximum likelihood estimation; power laws.
Power laws have a long history in ecology and other
disciplines (Bak 1996, Brown et al. 2002, Newman 2005).
Power-law relationships appear in a wide variety of
physical, social, and biological systems and are often cited
as evidence for fundamental processes that underlie the
dynamics structuring these systems (Bak 1996, Brown et
al. 2002, Newman 2005). There are two major classes of
power laws commonly reported in the ecological litera-
ture. The first are bivariate relationships between two
variables. Examples of this type of relationship include
the species–area relationship and body-size allometries.
Standard approaches to analyzing this type of data are
generally reasonable and discussions of statistical issues
related to this kind of data are presented elsewhere (e.g.,
Warton et al. 2006). The second type of power law, and
the focus of this paper, is the frequency distribution,
where the frequency of some event (e.g., the number of
individuals) is related to the size, or magnitude, of that
event (e.g., the size of the individual).
Frequency distributions of a wide variety of ecological
phenomena tend to be, at least approximately, power-
law distributed. These phenomena include distributions
of species body sizes (Morse et al. 1985), individual body
sizes (Enquist and Niklas 2001), colony sizes (Jovani and
Tella 2007), abundance among species (Pueyo 2006),
trends in abundance of species through time (Keitt and
Stanley 1998), step lengths in animal search patterns
(i.e., Levy flights; Reynolds et al. 2007), fire magnitude
(Turcotte et al. 2002), island size (White and Brown
2005), lake size (Wetzel 1991), flood magnitude (Mala-
mud and Turcotte 2006), landslide magnitude (Guzzetti
et al. 2002), vegetation patch size (Kefi et al. 2007), and
fluctuations in metabolic rate (Labra et al. 2007).
Frequency distributions are usually displayed as simple
histograms of the quantity of interest. If a distribution is
well-characterized by a power law then the frequency of
an event (e.g., the number of individuals with mass
between 10 and 20 g), f, is related to the size of that
event, x, by a function of the following form:
fðxÞ ¼ cxk
where c and k are constants, and k is called the exponent
and is typically negative (i.e., k , 0). Because f(x) is a
probability density function (PDF) the value of c is a
simple function of k and the minimum and maximum
values of x (Table 1). The specific form of the PDF
depends on whether the data are continuous or discrete,
on the presence of minimum and maximum values, and
on whether k is ,?1 or .?1. The different forms are
often given distinct names for clarity (see Table 1).
There is substantial interest in using the parameters of
these power-law distributions to make inferences about
the processes underlying the distributions, to test
mechanistic models, and to estimate and predict patterns
and processes operating beyond the scope of the
observed data. For example, power-law species abun-
dance distributions with k ’ ?1 are considered to
represent evidence for the primary role of stochastic
Manuscript received 6 August 2007; revised 9 November
2007; accepted 20 November 2007. Corresponding Editor:
A. M. de Roos.
birth–death processes, combined with species input, in
community assembly (Pueyo 2006, Zillio and Condit
2007); quantitative models of tree size distributions
make specific predictions (e.g., k ¼ ?2; Enquist and
Niklas 2001) that can be used to test these models
(Coomes et al. 2003, Muller-Landau et al. 2006); and
power-law frequency distributions of individual size
have been used to scale up from individual observations
to estimate ecosystem level processes (Enquist et al.
2003, Kerkhoff and Enquist 2006).
One concern when interpreting the exponents of these
distributions is that there are a wide variety of different
approaches currently being used to estimate the
exponents (Sims et al. 2007, White et al. 2007). These
include techniques based on: (1) binning (e.g., Enquist
and Niklas 2001, Meehan 2006, Kefi et al. 2007); (2) the
cumulative distribution function (e.g., Rinaldo et al.
2002); and (3) maximum likelihood estimation (e.g.,
Muller-Landau et al. 2006, Edwards et al. 2007, Zillio
and Condit 2007). There has been little discussion in the
ecological literature of how the choice of methodology
influences the parameter estimates, and methods other
than binning are rarely used. If different methods
produce different results this could have consequences
for the conclusions drawn about the ecology of the
system (Edwards et al. 2007, Sims et al. 2007).
Here, we: (1) describe the different approaches used to
quantify the exponents of power-law frequency distri-
butions; (2) show that some of these approaches give
biased estimates; (3) illustrate the superior performance
of some approaches using Monte Carlo methods; (4)
make recommendations for best estimating parameters
of power-law distributed data; and (5) show that some
of the conclusions of recent studies are effected by the
use of biased statistical techniques.
METHODS FOR ESTIMATING THE EXPONENT
Perhaps the most intuitive way to quantify an
empirical frequency distribution is to bin the observed
data using bins of constant linear width. This generates
the familiar histogram. Specifically, linear binning
entails choosing a bin i of constant width (w ¼ xiþ1?
xi), counting the number of observations in each bin
(i.e., with values of x between xiand xiþw), and plotting
this count against the value of x at the center of the bin
(xi/2þxiþ1/2). If the counts are divided by the sum of all
the counts, this plot is an estimate of the probability
density function, f(x). The traditional approach to
estimating the power-law exponent is to fit a linear
regression to log-transformed values of f(x) and x, with
the slope of the line giving an estimate of the exponent,
k. Bins with zero observations are excluded, because
log(0) is undefined, and sometimes bins with low counts
are also excluded (e.g., Enquist and Niklas 2001). While
in practice the choice of bin width is normally arbitrary,
this choice represents a trade-off between the number of
bins analyzed (i.e., the resolution of the frequency
distribution) and the accuracy with which each value of
f(x) is estimated (fewer observations per bin provide a
poorer density estimate; Pickering et al. 1995).
and parameter values over which it applies, its probability density function (PDF; or probability mass function) f(x), its
cumulative distribution function F(x), and the maximum likelihood estimate (MLE) for k based on the PDF.
Descriptions of different power-law frequency distributions, including the name of the distribution, the range of data
f(x)F(x)MLE for k
Range a ? x , ‘
Parameters k , ?1, a . 0
2) Truncated Pareto?
Range a ? x ? b
Parameters k 6¼ ?1, a ? 0, b ? 0
?(k þ 1)a?(kþ1)xk
1 ? a?(kþ1)xkþ1
^k ¼ ?1 ?
(k þ 1)(bkþ1? akþ1)?1xk
ln x ¼
ð^k þ 1Þþb^kþ1ln b ? a^kþ1ln a
3) Discrete Pareto?,§
Range x ¼ a, a þ 1, a þ 2, ... ‘
Parameters k , ?1, a ? 1
4) Power Functionjj
Range 0 ? x ? b
Parameters k . ?1, b . 0
ln x ¼?f0ð?^k;aÞ
(k þ 1)b?(kþ1)xk
^k ¼ logðbÞ ?1
Notes: The minimum value of x for which a distribution is valid is given by a, which is defined to be greater than zero. The
maximum value of x for which a distribution is valid is given by b, which is defined to be less than infinity.
? Sources are as follows: Pareto (Johnson et al. 1994); Truncated Pareto (Page 1968); Discrete Pareto (Clauset et al. 2007); Power
Function (Evans et al. 2000). There is an error in the MLE solution given by Evans et al. (2000) that has been corrected. Note that
MLEs are only guaranteed to be minimum variance unbiased estimators in the limit of large n. If n is small, corrections to the MLE
are available (Johnson et al. 1994, Clark et al. 1999, Clauset et al. 2007). All solutions assume that a and b are known.
? The MLE equations for these distributions cannot be solved analytically forˆk, so they must be solved using numerical methods.
§ f(?k, a) ¼ R‘
jj The Power Function distribution is often ignored in discussions of power-law distributions because it rarely occurs in natural
systems (Newman 2005). We include it here for completeness and because it has been suggested that in some groups individual size
distributions based on mass may be approximately power-law distributed with k . ?1 (e.g., Enquist and Niklas 2001).
k¼0(k þ a)kis the generalized zeta function and f0(?k, a) is its derivative with respect to ?k.
ETHAN P. WHITE ET AL.906Ecology, Vol. 89, No. 4
Simple logarithmic binning.—This approach is similar
to linear binning, except that instead of the bins having
constant linear width, they have constant logarithmic
width, b ¼ log(xiþ1) ? log(xi). The estimate of k is
obtained by log-transforming the values of x and
following the procedure described in the previous
section. Since the x data are transformed to begin with,
it is not necessary to transform the bin centers again
prior to fitting the regression. For power-law-like
distributions, an advantage of logarithmic binning is
the reduction of the number of zero and low-count bins
at larger values of x because the linear width of a bin
increases linearly with x; i.e., wi¼ xi(eb? 1). However,
this means that the number of observations within each
bin is determined not only by x, but also by the linear
width of the bin. Therefore, the slope of the regression
will give an estimate of k þ 1, not k (Appendix A; Han
and Straskraba 1998, Bonnet et al. 2001, Sims et al.
Normalized logarithmic binning.—The problem of
increasing linear width of logarithmic bins can be dealt
with by normalizing the number of observations in each
bin by the linear width of the bin, w. This converts the
counts into densities (number of observations per unit of
x; Bonnet et al. 2001, Christensen and Moloney 2005).
The linear width of a logarithmic bin can be calculated
as xi(eb? 1) (Appendix A). This normalization approach
is typically used in the characterization of aquatic size-
spectra and power-law distributions in physics (Kerr
and Dickie 2001, Christensen and Moloney 2005). It
removes the artifact from traditional logarithmic bin-
ning while maintaining the advantage of using larger
bins where there are fewer values of x. An alternative
approach is to use simple logarithmic binning and
subtract one from the estimated exponent (Han and
Straskraba 1998, Bonnet et al. 2001).
Fitting the cumulative distribution function
An alternative to binning methods is to work with the
cumulative distribution function (CDF). The CDF
describes the probability that a random variable, X,
drawn from f(x) is ?x. The CDF is straightforward to
construct for a set of observed data, and no binning is
required. To construct the CDF, first rank the n
observed values (xi) from smallest to largest (i ¼
1 ... n). The probability that an observation is less than
or equal to xi(the CDF) is then estimated as i/n (this is
the Kaplan-Meier estimate; Evans et al. 2000). Analyz-
ing the CDF avoids the subjective influence of the choice
of bin width and the problem of empty bins. Having
determined the CDF for a power-law distribution, the
exponent, k, of the probability density function (PDF)
can be estimated using regression. The traditional
approach is to transform the equation for the CDF
such that the slope of a linear equation is a function of k.
The linearized equation differs among distributions
(Appendix A). The slope of the regression will be equal
to k þ 1, making it necessary to subtract 1 to obtain k
(Bonnet et al. 2001, Rinaldo et al. 2002).
Maximum likelihood estimation
Maximum likelihood estimation (MLE) is one of the
preferred approaches for estimating frequency distribu-
tion parameters (e.g., Rice 1994). MLE determines the
parameter values that maximize the likelihood of the
model (in this case, a power law with an unknown
exponent) given the observed data. Specifically, MLE
finds the value of k that maximizes the product of the
probabilities of each observed value of x (i.e., the
product of f(x) evaluated at each data point; see Rice
 for a good introduction to maximum likelihood
methods). The specific solution for the maximum
likelihood estimate of k and whether the solution is
closed form or requires numerical methods to solve
depends on the minimum and maximum values of x and
on the value of k (Table 1). Alternatively, the likelihood
can be maximized directly using numerical methods
(Clauset et al. 2007, Zillio and Condit 2007). While
MLE does not provide an opportunity for visual
inspection of the distribution to determine if the
assumption of the power-law functional form is
reasonable, the validity of this assumption can be
assessed using simple goodness-of-fit tests such as the
Chi-square on binned data (Clark et al. 1999, Clauset et
al. 2007, Edwards et al. 2007), or by visually assessing
the linearity of binned data, or the CDF (Benhamou
2007), under the appropriate transformation.
COMPARING THE METHODS
While uncorrected simple logarithmic binning clearly
provides incorrect estimates of k, the alternative
approaches discussed above all seem reasonable and
intuitive. However, the different approaches do not
perform equally well, and some produce biased esti-
mates of the exponent (e.g., Pickering et al. 1995, Clark
et al. 1999, Sims et al. 2007). We applied Monte Carlo
methods to illustrate the advantages and disadvantages
of the various approaches and to explore cases relevant
to ecology that have not been previously addressed.
Monte Carlo methods generate data that are, by
definition, power-law distributed with known exponents,
making it possible to compare the performance of the
different techniques in estimating the value of k.
We generated power-law distributed random numbers
using the inverse transformation method for the Pareto
distribution (Ross 2006), and using the rejection method
for the discrete Pareto distribution (Devroye 1986).
Each analysis consisted of the following: (1) generating
10000 Monte Carlo data sets for each point in the
analysis (e.g., for each sample size), (2) estimating the
exponent for each data set using the methods described
previously, and (3) comparing the performance of the
methods based on bias (i.e., accuracy) and on the
variance in the estimate (i.e., precision). We report on
simulated distributions generated using k¼?2 and a¼1.
April 2008907ON ESTIMATING POWER-LAW EXPONENTS
The results for other combinations of parameters are
qualitatively similar. We also evaluated the influence of
sample size on the various estimation techniques, and
for binning-based approaches we evaluated the effect of
bin width on the analysis.
Uncorrected simple logarithmic binning gives the wrong
exponent.—Non-normalized logarithmic binning does
not estimate k; it estimates k þ 1 (Han and Straskraba
1998, Bonnet et al. 2001, Sims et al. 2007). Therefore if
simple logarithmic binning is used, and an estimate of k
is the desired result, then it is necessary to subtract 1
from the slope of the logarithmically binned data. Not
doing so will give the wrong value for the exponent.
Binning-based approaches perform poorly.—Linear
binning performs poorly by practically any measure.
In most cases it produces biased estimates of the
exponent and its estimates are highly variable (Figs. 1
and 2). In addition, the estimated exponent is highly
dependent on the choice of bin width, and this
dependency varies as a function of sample size (Fig. 3).
While normalized logarithmic binning performs better
than linear binning, its estimates are also dependent on
the choice of bin width and are more variable than
alternate approaches. Our results are based on recom-
mended practices in binning analyses (following Picker-
ing et al. 1995). Many alternative approaches to
constructing bins and performing regressions on binned
data are conceivable, and it is possible that some of these
may improve the performance of the estimates. Howev-
er, this highlights the fact that binning-based methods
are sensitive to a variety of decisions, and it appears that
no amount of tweaking will be able to produce a
consistent binning-based method for estimating the
exponent. In general, binning results in a loss of
information about the distributions of points within a
bin and is thus expected to perform poorly (Clauset et al.
2007, Edwards et al. 2007). Therefore, while binning is
useful for visualizing the frequency distribution, and
normalized logarithmic binning performs well at this
task, binning-based approaches should be avoided for
parameter estimation (Clauset et al. 2007).
Maximum likelihood estimation performs best.—While
fitting the cumulative distribution function (CDF)
generally produces good results, estimates of k using
the CDF approach are often biased at small sample sizes
and are consistently more variable than those using
maximum likelihood estimation (MLE; Fig. 2; Clark et
al. 1999, Newman 2005). This probably results because
the logarithmic transformation used in fitting the CDF
weights a small number of points more heavily, and
because the points in the CDF are not independent thus
violating regression assumptions (see Clauset et al.
 for other issues with regression-based approach-
es). While alternative approaches to fitting the CDF
(e.g., nonlinear regression) could improve the perfor-
mance of this estimator, MLE has been shown
mathematically to be the single best approach for
estimating power-law exponents (i.e., it is the minimum
variance unbiased estimator; Johnson et al. 1994, Clark
et al. 1999, Newman 2005). In addition, MLE produces
valid confidence intervals for the estimated exponent
methods of fitting the power-law exponent. (a) A single Monte
Carlo sample from a Pareto distribution plotted as 1 minus the
cumulative distribution, F(x). Data are plotted as gray open
circles along with the fits to the data using the four different
methods: linear binning (red; linear), normalized logarithmic
binning (blue; Nlog), maximum likelihood estimation (green;
MLE), and cumulative distribution function fitting (black;
CDF). (b) Kernel density estimates of the distribution of
exponents from 10000 Monte Carlo runs. Line colors are the
same as for (a), and the value of k used to generate the data is
indicated by the dashed line. Parameter values: n¼500 random
points in each simulation; k¼?2; 1 ? x , ‘; linear bin width¼
3; logarithmic bin width ¼ 0.3. The binning analyses used the
minimum value of x and excluded the last bin and bins
containing ?1 individual. Exclusion of the last bin is not
necessary, but it improves the performance of binning-based
approaches and is thus conservative in the context of our
conclusions. The single sample for (a) was chosen to illustrate
the general results shown in (b). Binning methods generate
biased estimates of the exponent and result in more variable
estimates than approaches based on MLE and CDF.
Example of Monte Carlo results for the different
ETHAN P. WHITE ET AL. 908Ecology, Vol. 89, No. 4
(Appendix A), which the other methods do not (Clark et
al. 1999, Newman 2005, Clauset et al. 2007).
Minimum and maximum values.—Minimum and
maximum attainable values of ecological quantities can
result either from natural limits on the quantity being
measured (e.g., trees cannot grow above some maximum
size), or from methodological limits on the values that
can be observed (e.g., fires ,1 ha are not recorded). In
addition, the power-law form of the distribution may
not hold over the entire range of x, making it necessary
to select a restricted range of x on which to estimate the
exponent. While binning-based approaches do not
assume particular limits on x (but see Pickering et al.
1995), CDF and MLE approaches assume the minimum
and maximum attainable values of x given in Table 1. In
some cases these limits may be known, but if not, it may
be necessary to estimate them (e.g., Kijko 2004, Clauset
et al. 2007). Because maximum likelihood estimation for
the truncated Pareto requires numerical methods, it has
been suggested that in some cases with both a minimum
and maximum value that the error introduced by
assuming that there is no maximum is small enough
that it is reasonable to estimate the exponent using the
maximum likelihood estimate for the Pareto distribu-
tion. Clark et al. (1999) suggest this approximation in
cases where the maximum value is at least two orders of
(a) linear and (b) normalized logarithmic binning for three
different sample sizes: n ¼ 200 (solid black line), n ¼ 500 (solid
red line), and n ¼ 1000 (solid blue line); based on 1000 Monte
Carlo runs from the Pareto distribution per point. Parameter
values were k ¼?2 (dashed black line) and 1 ? x , ‘. Error
bars are 62 SE. Other binning methods are as in Fig. 1.
Changing bin width changes the estimated exponent for all
Effect of bin width on the estimated exponents for
exponent and (b) the variance of that exponent, for the four
estimation methods: linear binning (red), normalized logarith-
mic binning (blue), maximum likelihood estimation (green), and
cumulative distribution function fitting (black). Values for each
sample size were generated using 10000 Monte Carlo runs from
the Pareto distribution with parameter values of k¼?2 (dashed
line), and 1 ? x , ‘; linear bin width¼7.5, and logarithmic bin
width ¼ 0.75. Other binning methods are as in Fig. 1. Linear
binning fails to converge to the correct estimate. While the
other methods all appear to converge at large sample sizes,
maximum likelihood estimation always yields the lowest
variance in the estimated exponent.
(a) Effect of sample size on the mean estimated
April 2008 909ON ESTIMATING POWER-LAW EXPONENTS
magnitude greater than the minimum; that is, max (x) .
100 3 min (x).
Deviations from the power law.—Empirical data are
rarely perfectly power-law distributed over the entire
range of x (Brown et al. 2002, Newman 2005). MLE and
CDF approaches respond to deviations differently
because the traditional MLE analysis implicitly weights
data on a linear scale while the traditional CDF
approach weights it on a logarithmic scale (McGill
2003). The CDF approach will therefore respond more
strongly to deviations from the power law at large values
of x (such as those observed in individual size
distributions; e.g., Coomes et al. 2003) than the MLE
approach, whereas MLE will respond more strongly to
deviations at small values of x (commonly observed in
many power-law distributions; e.g., Newman 2005). It is
common to truncate data in the tails that exhibit
deviations from the power law before fitting the
exponent (e.g., Newman 2005). However, these devia-
tions also should not be ignored, as they may help
identify important biological processes (e.g., Coomes et
al. 2003). In some cases deviations may suggest that the
power law is in fact not the appropriate model for the
data. This can be evaluated using goodness-of-fit tests
on binned data (Clark et al. 1999, Clauset et al. 2007,
Edwards et al. 2007) or by using model selection
techniques to compare the power-law to alternative
distributions (Muller-Landau et al. 2006, Clauset et al.
2007, Edwards et al. 2007).
Discrete data.—Most of the MLE and CDF methods
presented here assume that the data are continuously
distributed, as is often the case (e.g., body size).
However, some ecological patterns (e.g., species-abun-
dance distributions) are comprised of discrete observa-
tions (e.g., it is impossible to census 4.3 individuals). It is
therefore necessary to use analogous discrete distribu-
tions. In the case of the Pareto distribution a discrete
analog exists in the form of the aptly named discrete
Pareto distribution (Johnson et al. 2005, Newman 2005;
Table 1; also called the Zipf or Riemann-zeta distribu-
tion). In some cases continuous distributions can
reasonably approximate discrete data; but in the case
of the Pareto, using the continuous maximum likelihood
estimate instead of that derived from the discrete
distribution produces strongly biased results and should
be avoided (Appendix C; Clauset et al. 2007).
IMPLICATIONS FOR PUBLISHED RESULTS
One of the most important implications for published
results is that studies that have estimated exponents
using uncorrected simple logarithmic binning (e.g.,
Morse et al. 1988, Meehan 2006) have reported the
wrong exponent. This is particularly important in cases
where the exponent is used to test quantitative
predictions. For example, an analysis in Meehan
(2006) evaluates whether observed individual size
distribution exponents were consistent with those
predicted, using simple logarithmic binning. Meehan
concluded that the empirical data matched the predic-
tions (Fig. 4a). However, since the reported exponents
are equal to k þ 1, the analysis suggests that the size
distribution is substantially steeper than expected, thus
refuting, rather than supporting, the hypothesized
mechanism (Fig. 4a; Appendix B).
Analyses based on linearly binned data should also be
revisited due to the potential for biased estimates and
the strong influence of bin width on the estimated
exponent. In particular, studies that have used linear
binning to test the predictions of theoretical models or
(a) Meehan (2006) and (b) Enquist and Niklas (2001), using less
biased methods. Plots are probability densities of the estimated
exponents using the original methodology from these studies
(red line; simple logarithmic binning in Meehan, linear binning
in Enquist and Niklas), and using less biased methods (blue
line; normalized logarithmic binning for Meehan, maximum
likelihood estimation for Enquist and Niklas). Both studies
purported to support a theoretically derived exponent (dotted
line). However, when the exponents are estimated using less
biased methods it becomes clear that the observed data deviate
significantly from the theoretical prediction.
Reanalysis of individual size distribution data from
ETHAN P. WHITE ET AL. 910 Ecology, Vol. 89, No. 4
compare exponents from different data sets (e.g.,
Enquist and Niklas 2001, Coomes et al. 2003, Niklas
et al. 2003, Kefi et al. 2007) may have reached incorrect
conclusions. We reanalyzed the original data from
Enquist and Niklas (2001) and found that while the
original linear binning analyses suggested that observed
diameter distribution exponents were near the theoret-
ical prediction of ?2, MLE suggests that the observed
exponents are actually closer, on average, to ?2.5 (Fig.
4b; Appendix B). Our reanalysis indicates that the size–
frequency distributions in Gentry’s plots are not, in
general, adequately represented by a power law with an
exponent of ?2, as originally claimed by Enquist and
Niklas (2001; see Appendix B for an important caveat).
While normalized logarithmic binning performs better
than linear binning, it can still introduce biases of ;10%
depending on the bin width. While many analyses based
on normalized logarithmic binning are probably reason-
able, the recent suggestion that normalized logarithmic
binning is the best approach for fitting exponents (Sims
et al. 2007) is unwarranted, and MLE should be used
whenever possible (Clark et al. 1999, Clauset et al. 2007).
Compared to binning-based approaches, results from
fitting the CDF are probably reasonable. In cases with
low sample sizes, where small errors in the estimated
exponent could influence the conclusions of the study, or
where minimum or maximum attainable values of x
have been ignored (see Pickering et al. 1995), it may be
worth checking the results using MLE. Regardless,
MLE is the single best method for estimating exponents
and should be used in future studies.
The vast majority of ecological studies that estimate
exponents for power-law-like distributions use ap-
proaches based on binning the empirical data (e.g.,
Morse et al. 1988, Enquist and Niklas 2001, Coomes et
al. 2003, Niklas et al. 2003, Meehan 2006, Jovani and
Tella 2007, Kefi et al. 2007, Reynolds et al. 2007, Sims et
al. 2007). These binning-based methods tend to produce
results that are biased, have high variance, and are
contingent on the choice of bin width. Instead of
binning, maximum likelihood estimation should be used
when fitting power-law exponents to empirical data
(Clark et al. 1999, Newman 2005, Edwards et al. 2007).
We have focused on power laws because they, at least
approximately, characterize a number of distributions of
interest to ecologists. The issues raised here, and the
conclusions discussed, should apply broadly to frequen-
cy distributions in general, and in particular to other
distributions with heavy tails. Paying careful attention
to fitting methodologies and consultation of statistical
references (e.g., Johnson et al. 1994) should help
improve the estimation of distributional parameters.
We particularly thank Tim Meehan for generously sharing
his data with us and for comments on the manuscript. For the
Gentry data we thank Alwyn Gentry, the Missouri Botanical
Garden, and collectors who assisted Gentry or contributed data
for specific sites. We also thank David Coomes, Susan Durham,
Jim Haefner, Brian McGill, and Tommaso Zillio for comments
on the manuscript. This work was funded by the National
Science Foundation through a Postdoctoral Fellowship in
Biological Informatics to E. P. White (DBI-0532847). We used
the first–last author emphasis approach (sensu Tscharntke et al.
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Details of logarithmic binning, cumulative distribution function fitting, and maximum likelihood estimation (Ecological Archives
Details of reanalysis of results published in Meehan (2006) and Enquist and Niklas (2001) (Ecological Archives E089-052-A2).
Example of bias resulting from fitting discrete data with continuous maximum likelihood solutions (Ecological Archives E089-
Matlab files for fitting power-law exponents using different methods (Ecological Archives E089-052-S1).
ETHAN P. WHITE ET AL.912Ecology, Vol. 89, No. 4