Why Fourth-Corner Latent Variable Models Overstate Confidence in Trait–Environment Relationships and What to Use Instead

R script to the paper: Cajo J.F. ter Braak (2025) Why Fourth-Corner Latent Variable Models Overstate Confidence in Trait–Environment Relationships—and What to Use InsteadThe shareable link to the published version is https://rdcu.be/eXukf The abstract isTrait-based ecology seeks to explain species’ responses to environmental gradients through their functional traits, aiming for generalizable predictions of community assembly. The fourth-corner problem formalizes this challenge by integrating species traits, environmental variables, and species abundances. Recently, generalized latent variable models (GLLVMs) have been proposed as a comprehensive solution, claiming to capture both unobserved ecological gradients and trait–environment interactions.This paper shows—through theoretical analysis, empirical data, and simulations—that GLLVMs often overstate confidence in inferred relationships, inflating type I error rates. In contrast, double-constrained correspondence analysis (dc-CA) and a generalized linear mixed model with random species-specific environmental and random site-specific trait effects (GLMM3) offer more reliable inference. While GLMM3, potentially extended with latent variables, may provide the most powerful framework, its practical use is limited by computational demands. dc-CA emerges as a robust and accessible alternative, balancing statistical rigor with interpretability by treating both species and sites as units of analysis.I argue that trait–environment models should not only fit observed data but also generalize to new trait values and ecological contexts. Overconfident conclusions from any supervised learning model that ignores species- or site-level variation risk misleading ecological understanding and misinforming conservation and management decisions.