## Introduction

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 literature. 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 (Malamud 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:

where *c* and λ are constants, and λ is called the exponent and is typically negative (i.e., λ < 0). Because *f*(*x*) is a probability density function (PDF) the value of *c* is a simple function of λ 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 λ is <−1 or >−1. The different forms are often given distinct names for clarity (see Table 1).

*f*(

*x*), its cumulative distribution function

*F*(

*x*), and the maximum likelihood estimate (MLE) for λ based on the PDF.

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 abundance distributions with λ ≈ −1 are considered to represent evidence for the primary role of stochastic 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., λ = −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 distributions; (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.