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Abstract A fundamental pattern in ecology is that smaller organisms are more abundant than larger organisms. This pattern is known as the individual size distribution (ISD), which is the frequency distribution of all individual body sizes in an ecosystem.The ISD is described by a power law and a major goal of size spectra analyses is to estimate the exponent of the power law,λ. However, while numerous methods have been developed to do this, they have focused almost exclusively on estimatingλfrom single samples.Here, we develop an extension of the truncated Pareto distribution within the probabilistic modelling language Stan. We use it to estimate multipleλs simultaneously in a hierarchical modelling approach.The most important result is the ability to examine hypotheses related to size spectra, including the assessment of fixed and random effects, within a single Bayesian generalized mixed model. While the example here uses size spectra, the technique can also be generalized to any data that follow a power law distribution. 

The study explores the individual size distribution (ISD) pattern in ecological communities, characterized by a negative correlation between individual body size and abundance (N ∼ M^λ). The parameter λ denotes the rate of decline in relative abundance from small to large individuals. Despite known influences of temperature and resource availability on body size, their effects on λ remain diverse. Leveraging data from 2.4 million individual body sizes in continental freshwater streams, the research the hypothesis that λ varies as a function of temperature and resource supply. Surprisingly, despite varied environmental conditions and complete species turnover, minimal variation in λ (mean = −1.2, sd = 0.04) was observed, with no discernible impact from temperature or resource supply. The unexpected λ value of −1.2 suggests a higher-than-expected relative abundance of large individuals, challenging assumptions of metabolic scaling at 0.75 and implying large subsidy inputs to large predators. Simulation and mesocosm experiments support a metabolic scaling coefficient of ∼0.4 for freshwater macroinvertebrates. The findings underscore remarkable consistency of individual size distributions in freshwater streams, likely driven by shallow metabolic scaling and large subsidies to large consumers.