Location–scale models in ecology: heteroscedasticity in continuous, count and proportion data
Bibliographic record
Abstract
Ecological data seldom meet the assumption of constant variance. Yet patterns of heteroscedasticity often reflect biologically meaningful variation, such as differences in plasticity or variable responses to environmental stresses. However, most studies model only the mean, treating variance as statistical noise. Here, we describe location–scale regression modeling, which estimates mean (location) as well as variance (scale) coefficients. We introduce three increasingly flexible formulations: (1) fixed-effect location–scale models, (2) models with random effects on the mean, and (3) double-hierarchical models with random effects on both mean and variance. We extend location–scale models from Gaussian to non-Gaussian data, including over-dispersed counts, proportions, and zero-inflated outcomes, features common to ecological datasets. Beyond overdispersion, we address underdispersion in count data and one-inflation in continuous proportions, providing a flexible framework for complex variance structures. We show that location–scale models can uncover informative variance patterns with minimal additional code. To support implementation, we provide an online tutorial, model selection workflow, and diagnostic guidance. Finally, we refer to new frontiers including multivariate, meta‑analytic, phylogenetic, and location-scale shape models. By treating variance as a biological response, instead of a nuisance, location–scale models enrich our understanding of organism and ecosystem dynamics in a changing world.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.031 | 0.084 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.004 |
| Bibliometrics | 0.002 | 0.004 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.005 | 0.003 |
| Research integrity | 0.003 | 0.004 |
| Insufficient payload (model declined to judge) | 0.006 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".