Choosing the Response Matrix: Generalised Linear Latent Variable Models for Multivariate Ecology and Evolution

Shinichi Nakagawa , Ayumi Mizuno , Russell Dinnage, Marija Purgar, Christine Sosiak, Sergio Poo Hernandez, Iwo Gross, Erick Lundgren, Malgorzata Lagisz , Eduardo S. A. Santos , Santiao Ortega

Multivariate responses are central to ecology and evolutionary biology, but their covariance is often difficult to model and interpret. Generalised linear latent variable models (GLLVMs) provide a parsimonious way to represent covariance among many responses using a smaller number of latent variables. They are widely used for Site X Species data in joint species distribution modelling and model-based ordination. Here, we argue that their broader value lies in treating the response matrix as an explicit modelling choice. Changing the response matrix changes the biological question: Site X Species, Site X Trait, Individual X Trait, and Species X Trait formulations target different kinds of covariance, even when they use the same basic idea of summarising shared variation with latent variables. We describe the basic structure of GLLVMs, the interpretation of latent variables, loadings, residual covariance or correlation, and communality, which measures how much of a response's variation is shared with other responses. We then use "the fourth-corner problem'' to show why the choice of response matrix matters, before developing Unit X Trait as an organising principle for applications in functional biogeography, behavioural syndromes, and phylogenetic trait integration. We conclude that GLLVMs are best viewed not as a method for one data type, but as a general modelling language for multivariate biological covariance, provided that the response matrix, level of inference, and limits of mechanistic interpretation are made explicit.