Power-law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man-made phenomena. Unfortunately, the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution—the part of the distribution representing large but rare events—and by the difficulty of identifying the range over which power-law behavior holds. Commonly used methods for analyzing power-law data, such as least-squares fitting, can produce substantially inaccurate estimates of parameters for power-law distributions, and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all. Here we present a principled statistical framework for discerning and quantifying power-law behavior in empirical data. Our approach combines maximum-likelihood fitting methods with goodness-of-fit tests based on the Kolmogorov-Smirnov (KS) statistic and likelihood ratios. We evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches. We also apply the proposed methods to twenty-four real-world data sets from a range of different disciplines, each of which has been conjectured to follow a power-law distribution. In some cases we find these conjectures to be consistent with the data, while in others the power law is ruled out.

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Power-Law Distributions in Empirical Data

Until recently, large apex consumers were ubiquitous across the globe and had been for millions of years. The loss of these animals may be humankind's most pervasive influence on nature. Although such losses are widely viewed as an ethical and aesthetic problem, recent research reveals extensive cascading effects of their disappearance in marine, terrestrial, and freshwater ecosystems worldwide. This empirical work supports long-standing theory about the role of top-down forcing in ecosystems but also highlights the unanticipated impacts of trophic cascades on processes as diverse as the dynamics of disease, wildfire, carbon sequestration, invasive species, and biogeochemical cycles. These findings emphasize the urgent need for interdisciplinary research to forecast the effects of trophic downgrading on process, function, and resilience in global ecosystems.

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Trophic Downgrading of Planet Earth

We live amid a global wave of anthropogenically driven biodiversity loss: species and population extirpations and, critically, declines in local species abundance. Particularly, human impacts on animal biodiversity are an under-recognized form of global environmental change. Among terrestrial vertebrates, 322 species have become extinct since 1500, and populations of the remaining species show 25% average decline in abundance. Invertebrate patterns are equally dire: 67% of monitored populations show 45% mean abundance decline. Such animal declines will cascade onto ecosystem functioning and human well-being. Much remains unknown about this “Anthropocene defaunation”; these knowledge gaps hinder our capacity to predict and limit defaunation impacts. Clearly, however, defaunation is both a pervasive component of the planet’s sixth mass extinction and also a major driver of global ecological change.

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Defaunation in the Anthropocene

BACKGROUND: Comparing patterns of ter- restrial and marine defaunation helps to place human impacts on marine fauna in context and to navigate toward recovery. De- faunation began in ear- nest tens of thousands of years later in the oceans than it did on land. Al- though defaunation has been less severe in the oceans than on land, our effects on marine animals are increasing in pace and impact. Humans have caused few complete extinctions in the sea, but we are responsible for many ecological, commercial, and local extinctions. Despite our late start, humans have already powerfully changed virtually all major marine ecosystems. ADVANCES: Humans have profoundly de- creased the abundance of both large (e.g., whales) and small (e.g., anchovies) marine fauna. Such declines can generate waves of ecological change that travel both up and down marine food webs and can alter ocean ecosystem functioning. Human harvesters have also been a major force of evolutionary change in the oceans and have reshaped the genetic structure of marine animal popula- tions. Climate change threatens toaccelerate marine defaunation over the next century. The high mobility of many marine animals offers some increased, though limited, ca- pacity for marine species to respond to cli- mate stress, but it also exposes many species to increased risk from other stressors. Be- cause humans are intensely reliant on ocean ecosystems for food and other ecosystem ser- vices, we are deeply affected by all of these forecasted changes. Three lessons emerge when comparing the marine and terrestrial defaunation ex-

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Marine defaunation: Animal loss in the global ocean

Large wild herbivores are crucial to ecosystems and human societies. We highlight the 74 largest terrestrial herbivore species on Earth (body mass ≥100 kg), the threats they face, their important and often overlooked ecosystem effects, and the conservation efforts needed to save them and their predators from extinction. Large herbivores are generally facing dramatic population declines and range contractions, such that ~60% are threatened with extinction. Nearly all threatened species are in developing countries, where major threats include hunting, land-use change, and resource depression by livestock. Loss of large herbivores can have cascading effects on other species including large carnivores, scavengers, mesoherbivores, small mammals, and ecological processes involving vegetation, hydrology, nutrient cycling, and fire regimes. The rate of large herbivore decline suggests that ever-larger swaths of the world will soon lack many of the vital ecological services these animals provide, resulting in enormous ecological and social costs.

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Collapse of the world's largest herbivores.

The purpose of this data set was to compile body mass information for all mammals on Earth so that we could investigate the patterns of body mass seen across geographic and taxonomic space and evolutionary time. We were interested in the heritability of body size across taxonomic groups (How conserved is body mass within a genus, family, and order?), in the overall pattern of body mass across continents (Do the moments and other descriptive statistics remain the same across geographic space?), and over evolutionary time (How quickly did body mass patterns iterate on the patterns seen today? Were the Pleistocene extinctions size specific on each continent, and did these events coincide with the arrival of man?). These data are also part of a larger project that seeks to integrate body mass patterns across very diverse taxa (NCEAS Working Group on Body Size in Ecology and Paleoecology: linking pattern and process across space, time, and taxonomic scales). We began with the updated version of D. E. Wilson an...

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John P. Haskell

Papers

Body mass of late Quaternary mammals