Generalized graphical mixed models connect ecological theory with widely used statistical models

Abstract Ecological dynamics are often analysed across multiple sites, times, and variables. Ecologists typically represent interactions across space, time, and variables using generalized linear mixed models (GLMMs), generalized additive models (GAM), and structural equation models (SEM). Here, I introduce the family of generalized graphical mixed models (GGMMs) and show that it extends GLMMs, GAMs, and SEMs. GGMMs represent ecological systems using a mathematical graph, where each analytic unit (node) has a direct effect on other units via specified linear interactions contained in a joint path matrix (edges). The GGMM first requires defining the dimensions and resolution of analytical units, and I illustrate concepts using combinations of sites, times, ages, and taxa. Next, the path matrix is constructed by combining elementary ecological relationships like ecological interactions, evolutionary trade‐offs, time lags, and spatial diffusion. Finally, the GGMM is expressed as a simultaneous equation, efficiently estimated as a Gaussian Markov random field, and can be used for prediction, inference, and causal analysis. I demonstrate GGMMs using three contrasting case studies and a simulation experiment, using a range of simple or complex software including an R‐package ( tinyVAST ), base‐R code for simulating dynamics, and bespoke estimation code using RTMB. Analysing age‐structured population dynamics using tinyVAST shows that correlations by cohort can improve out‐of‐sample predictions. Simulating spatial dynamics in R shows that diffusive movement can be represented (or eliminated) in spatio‐temporal models by including correlations across space, time, and a space–time interaction. Estimating separate Ornstein‐Uhlenbeck parameters (OU) in a phylogenetic structural equation model using RTMB shows that geographic range has faster stabilizing selection than body size or specific metabolism across mammals. Finally, a simulation experiment confirms that the OU parameters can be better estimated while jointly interpolating missing data, which is not feasible using existing R‐packages ( phylosem or phylolm ). I conclude that GGMMs connect ecological theory with statistical models that are applied for inference, prediction, and causal analysis throughout ecology. In particular, GGMMs allow analysts to sequentially add (and test) mechanisms during bespoke development of simulation or estimation models while also reverting to common statistical models when appropriate.