Random‐effects ordination: describing and predicting multivariate correlations and co‐occurrences

Ecology is inherently multivariate, but high‐dimensional data are difficult to understand. Dimension reduction with ordination analysis helps with both data exploration and clarification of the meaning of inferences (e.g., randomization tests, variation partitioning) about a statistical population. Most such inferences are asymmetric, in that variables are classified as either response or explanatory (e.g., factors, predictors). But this asymmetric approach has limitations (e.g., abiotic variables may not entirely explain correlations between interacting species). We study symmetric population‐level inferences by modeling correlations and co‐occurrences, using these models for out‐of‐sample prediction. Such modeling requires a novel treatment of ordination axes as random effects, because fixed effects only allow within‐sample predictions. We advocate an iterative methodology for random‐effects ordination: (1) fit a set of candidate models differing in complexity (e.g., number of axes); (2) use information criteria to choose among models; (3) compare model predictions with data; (4) explore dimension‐reduced graphs (e.g., biplots); (5) repeat 1–4 if model performance is poor. We describe and illustrate random‐effects ordination models (with software) for two types of data: multivariate‐normal (e.g., log morphometric data) and presence–absence community data. A large simulation experiment with multivariate‐normal data demonstrates good performance of (1) a small‐sample‐corrected information criterion and (2) factor analysis relative to principal component analysis. Predictive comparisons of multiple alternative models is a powerful form of scientific reasoning: we have shown that unconstrained ordination can be based on such reasoning.