Maximum likelihood outperforms binning methods for detecting differences in abundance size spectra across environmental gradients

Abstract

Individual body size distributions (ISD) within communities are remarkably consistent across habitats and spatiotemporal scales and can be represented by size spectra, which are described by a power law. The focus of size spectra analysis is to estimate the exponent ( $\lambda$ ) of the power law. A common application of size spectra studies is to detect anthropogenic pressures. Many methods have been proposed for estimating $\lambda$ most of which involve binning the data, counting the abundance within bins, and then fitting an ordinary least squares regression in log-log space. However, recent work has shown that binning procedures return biased estimates of $\lambda$ compared to procedures that directly estimate $\lambda$ using maximum likelihood estimation (MLE). While it is clear that MLE produces less biased estimates of site-specific λ's, it is less clear how this bias affects the ability to test for changes in λ across space and time, a common question in the ecological literature. Here, we used simulation to compare the ability of two normalised binning methods (equal logarithmic and log₂ bins) and MLE to (1) recapture known values of $\lambda$ , and (2) recapture parameters in a linear regression measuring the change in $\lambda$ across a hypothetical environmental gradient. We also compared the methods using two previously published body size datasets across a natural temperature gradient and an anthropogenic pollution gradient. Maximum likelihood methods always performed better than common binning methods, which demonstrated consistent bias depending on the simulated values of $\lambda$ . This bias carried over to the regressions, which were more accurate when $\lambda$ was estimated using MLE compared to the binning procedures. Additionally, the variance in estimates using MLE methods is markedly reduced when compared to binning methods. The error induced by binning methods can be of similar magnitudes as the variation previously published in experimental and observational studies, bringing into question the effect sizes of previously published results. However, while the methods produced different regression slope estimates, they were in qualitative agreement on the sign of those slopes (i.e. all negative or all positive). Our results provide further support for the direct estimation of $\lambda$ and its relative variation across environmental gradients using MLE over the more common methods of binning.

Keywords: body size-abundance relationships; community biomass distributions; individual size distributions; logarithmic binning methods; maximum likelihood; size spectra; statistical methods.