Hierarchical change-point regression models including random effects to estimate empirical critical loads for nitrogen using Bayesian Regression Models (brms) and JAGS

The concept of critical loads is used in the framework of the Convention on Long-range Transboundary Air Pollution (UNECE) to define thresholds below which no damaging effects on habitats occur based on the latest scientific knowledge. Change-point regression models applied in a Bayesian framework are useful statistical tools to estimate critical empirical loads. While hierarchical study designs are common in ecological research, previous methods to estimate critical loads using change-point regression did not allow to analyse data collected under such a design. This method update provides an implementation of hierarchical data structure by including random effects such as study sites or as in this example tree species within the Bayesian approach of change-point regression models using two different approaches. The example data set is an European wide gradient study of the impact of climate change and air pollution on forest tree health assessed by foliar nutrient status of nitrogen (N) to phosphorus (P) from 10 different conifer tree species originated from 88 forest sites and 9 countries covering 22 years (1995-2017). Both modelling approaches using JAGS and Bayesian Regression Models using ‘Stan’ (brms) resulted in reasonable and similar estimations of the critical empirical load for nitrogen (CL(emp)N) for temperate forests. These methodological examples of using different approaches of Bayesian change-point regression models dealing with random effects could prove useful to infer CL(emp)N for other ecosystems and long-term data sets. • Hierarchical change-point regression models are suitable for estimating critical empirical loads. • The Bayesian framework of these models provides the inclusion of the current critical load and various confounding or modifying variables. • Here we present two ways of implementing hierarchical data sets in Bayesian change-point regression models using JAGS and brms.