Feedback and Modularization in a Bayesian Meta-analysis of Tree Traits Affecting Forest Dynamics

We describe a unique application of modularization in the context of a Bayesian meta-analysis of quantitative information obtained from the literature. Incomplete reporting, resulting in large amounts of missing data, is common in many meta-analyses, and, in this study, it led to poor mixing and identifiability problems in a fully Bayesian meta-analysis model. As an alternative to the full Bayesian approach, we modularized model components (e.g., modules of covariates, sample sizes, and standard errors) to prevent missing covariate data in these modules from allowing feedback that would otherwise affect parameters in the covariate module (direct feedback control) or affect covariate effects parameters in the mean model for the response (indirect feedback control). The combination of direct and indirect feedback control greatly improves mixing and facilitates convergence of Markov chain Monte Carlo (MCMC), yielding realistic pseudo-posteriors. Thus, our modularization approach allowed us to address important limitations of existing meta-analytic methods by accommodating incomplete reporting and by considering all model quantities as stochastic, including the response variable of interest (e.g., a sample mean) and sample sizes, standard errors, and all covariates, reported or not. We illustrate our approach using data summaries extracted from literature on specific leaf area (SLA) of trees, an important functional trait linked to tree growth and forest dynamics and a key parameter in models of forest responses to climate change. A hierarchical model based on taxonomic relationships allows borrowing of strength to infer SLA for 305 tree species in the United States based on information for 158 of those species. In the context of the SLA meta-analysis, we discuss problems that arise from feedback among model components and provide ecological arguments for modularization-for cutting feedback. We anticipate that our approach may be applied to meta-analyses of other important tree traits and to similar meta-analytical studies in general.