A mixed-effects model approach to height-diameter relationships

Abstract

Height-diameter models are widely used to estimate tree height from diameter at breast height (DBH) and play a crucial role in forest inventories by reducing fieldwork effort. However, statistical challenges such as nonlinearity, heteroscedasticity, nonnormality, and outliers can compromise model accuracy. To address these issues, this study proposes a generalization of Scolforo's height-diameter model (Scolforo, 1998), which incorporates random effects to improve flexibility and fit. With observational data from Eucalyptus urograndis plantations, we present a step-by-step framework for model fitting, inference, and validation. Our approach considers hierarchical structures and variability across stands to improve predictive performance. To rigorously assess model adequacy, we conducted a simulation study under various scenarios, evaluating goodness-of-fit with deviance, randomized quantile residuals, and least-confounded residuals. These diagnostics identify misspecification and enable robust parameter estimation. Additionally, we provide a detailed tutorial (Appendix B) for implementing the model in R that encompasses (i) inference for fixed and random effects, (ii) local influence analysis to detect sensitive observations, and (iii) residual-based diagnostics adapted to mixed models. Our results reveal the adaptability of the model to complex data structures while maintaining interpretability. The proposed framework provides forest researchers a reliable tool for height prediction that combines theoretical rigor and practical applicability. The accompanying R tutorial increases reproducibility and facilitates the integration of the framework into forest inventory workflows.