Abstract. Power-law relationships are among the most well-studied functional relationships in biology. Recently the common practice of fitting power laws using linear regression (LR) on log-transformed data has been criticized, calling into question the conclusions of hundreds of studies. It has been suggested that nonlinear regression (NLR) is preferable, but no rigorous comparison of these two methods has been conducted. Using Monte Carlo simulations, we demonstrate that the error distribution determines which method performs better, with NLR better characterizing data with additive, homoscedastic, normal error and LR better characterizing data with multiplicative, heteroscedastic, lognormal error. Analysis of 471 biological power laws shows that both forms of error occur in nature. While previous analyses based on log-transformation appear to be generally valid, future analyses should choose methods based on a combination of biological plausibility and analysis of the error distribution. We provide detailed guidelines and associated computer code for doing so, including a model averaging approach for cases where the error structure is uncertain.

Abstract. The species abundance distribution (SAD) is one of the most studied patterns in ecology due to its potential insights into commonness and rarity, community assembly, and patterns of biodiversity. It is well established that communities are composed of a few common and many rare species, and numerous theoretical models have been proposed to explain this pattern. However, no attempt has been made to determine how well these theoretical characterizations capture observed taxonomic and global-scale spatial variation in the general form of the distribution. Here, using data of a scope unprecedented in community ecology, we show that a simple maximum entropy model produces a truncated log-series distribution that can predict between 83% and 93% of the observed variation in the rank abundance of species across 15 848 globally distributed communities including birds, mammals, plants, and butterflies. This model requires knowledge of only the species richness and total abundance of the community to predict the full abundance distribution, which suggests that these factors are sufficient to understand the distribution for most purposes. Since geographic patterns in richness and abundance can often be successfully modeled, this approach should allow the distribution of commonness and rarity to be characterized, even in locations where empirical data are unavailable.

The maximum entropy theory of ecology (METE) is a unified theory of biodiversity that predicts a large number of macroecological patterns using information on only species richness, total abundance, and total metabolic rate of the community. We evaluated four major predictions of METE simultaneously at an unprecedented scale using data from 60 globally distributed forest communities including more than 300,000 individuals and nearly 2,000 species.METE successfully captured 96% and 89% of the variation in the rank distribution of species abundance and individual size but performed poorly when characterizing the size-density relationship and intraspecific distribution of individual size. Specifically, METE predicted a negative correlation between size and species abundance, which is weak in natural communities. By evaluating multiple predictions with large quantities of data, our study not only identifies a mismatch between abundance and body size in METE but also demonstrates the importance of conducting strong tests of ecological theories.

Taylor's law (TL) describes the scaling relationship between the mean and variance of populations as a power law. TL is widely observed in ecological systems across space and time, with exponents varying largely between 1 and 2. Many ecological explanations have been proposed for TL, but it is also commonly observed outside ecology. We propose that TL arises from the constraining influence of two primary variables: the number of individuals and the number of censuses or sites. We show that most possible configurations of individuals among censuses or sites produce the power-law form of TL, with exponents between 1 and 2. This "feasible set" approach suggests that TL is a statistical pattern driven by two constraints, providing an a priori explanation for this ubiquitous pattern. However, the exact form of any specific mean-variance relationship cannot be predicted in this way, that is, this approach does a poor job of predicting variation in the exponent, suggesting that TL may still contain ecological information.

The Maximum Entropy Theory of Ecology (METE) predicts a universal species–area relationship (SAR) that can be fully characterized using only the total abundance (N) and species richness (S) at a single spatial scale. This theory has shown promise for characterizing scale dependence in the SAR. However, there are currently four different approaches to applying METE to predict the SAR and it is unclear which approach should be used due to a lack of empirical comparison. Specifically, METE can be applied recursively or non-recursively and can use either a theoretical or observed species-abundance distribution (SAD). We compared the four different combinations of approaches using empirical data from 16 datasets containing over 1000 species and 300,000 individual trees and herbs. In general, METE accurately downscaled the SAR (R2 > 0.94), but the recursive approach consistently under-predicted richness. METE’s accuracy did not depend strongly on using the observed or predicted SAD. This suggests that the best approach to scaling diversity using METE is to use a combination of non-recursive scaling and the theoretical abundance distribution, which allows predictions to be made across a broad range of spatial scales with only knowledge of the species richness and total abundance at a single scale.