A mixed-effects model approach to height–diameter relationships
DOI:
https://doi.org/10.4025/actasciagron.v48.i1.76526Palavras-chave:
Eucalyptus; mixed effects models; height-diameter relationship; sustainable forest practices.Resumo
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.
Downloads
Referências
Assmann, E. (1970). The principles of forest yield study: studies in the organic production, structure, increment and yield of forest stands. Pergamon Press.
Atkinson, A. C. (1985). Plots, transformations, and regression an introduction to graphical methods of diagnostic regression analysis. Oxford University Press.
Bani-Mustafa, A., Matawie, K. M., Finch, C. F., Al-Nasser, A., & Ciavolino, E. (2019). Recursive residuals for linear mixed models. Quality & Quantity, 53, 1263-1274. https://doi.org/10.1007/s11135-018-0814-6
Beckman, R. J., Nachtsheim, C. J., & Cook, R. D. (1987). Diagnostics for mixed-model analysis of variance. Technometrics, 29(4), 413-426. https://doi.org/10.2307/1269452
Butler, D. G., Cullis, B. R., Gilmour, A. R., & Gogel, B. J. (2009). Mixed models for S language environments: ASReml-R reference manual. ASReml estimates variance components under a general linear mixed model by residual maximum likelihood. Department of Primary Industries and Fisheries.
Cook, R. D. (1986). Assessment of local influence. Journal of the Royal Statistical Society: Series B Statistical Methodology, 48(2), 133-155. https://doi.org/10.1111/j.2517-6161.1986.tb01398.x
Cordeiro, G. M., Demétrio, C. G. B., & Moral, R. D. A. (2024). Modelos lineares generalizados e aplicações. Blucher.
Covarrubias-Pazaran, G. (2016). Genome-assisted prediction of quantitative traits using the R package sommer. PLoS ONE, 11(6), 1-15. https://doi.org/10.1371/journal.pone.0156744
Chenge, I. B. (2021). Height-diameter relationship of trees in Omo strict nature forest reserve, Nigeria. Trees, Forests and People, 3, 100051. https://doi.org/10.1016/j.tfp.2020.100051
da Silva, B. G., Demétrio, C. G. B., Sermarini, R. A., Molenberghs, G., Verbeke, G., Behling, A., Marques, E., Accioly Y., & Figura, M. A. (2024). Height-diameter models: a comprehensive review with new insights on relationships to generalized linear models and differential equations. International Forestry Review, 26(4), 398-419. https://doi.org/10.1505/146554824839334687
da Silva, B. G., Demétrio, C. G. B., Sermarini, R. A., Behling, A., Molenberghs, G., Verbeke, G., Marques, E., Accioly Y., & Figura, M. A. (2026). Modeling Pinus tree taper data using mixed-effects models. Journal of Forestry Research, 37(1), 75. https://doi.org/10.1007/s11676-026-01985-5
Dunn, P. K., & Smyth, G. K. (1996). Randomized quantile residuals. Journal of Computational and Graphical Statistics, 5(3), 236-244. https://doi.org/10.1080/10618600.1996.10474708
Hilden-Minton, J. A. (1995). Multilevel diagnostics for mixed and hierarchical linear models. University of California. https://hdl.handle.net/10568/81585
Hinde, J., & Demétrio, C. G. B. (1998). Overdispersion: models and estimation. Computational Statistics & Data Analysis, 27(2), 151-170. https://doi.org/10.1016/S0167-9473(98)00007-3
Hofiço, N. S. A., Costa, E. A., Fleig, F. D., Finger, C. A. G., & Hess, A. F. (2020). Height-diameter relationships for Eucalyptus grandis Hill ex. Maiden in Mozambique: using mixed-effects modeling approach. Cerne, 26(2), 183-192. https://doi.org/10.1590/01047760202026022677
Indústria Brasileira de Árvores. (2019). Relatório 2019 Indústria Brasileira de Árvores. IBA. https://iba.org/datafiles/publicacoes/relatorios/iba-relatorioanual2019
Indústria Brasileira de Árvores. (2022). Relatório 2022 Indústria Brasileira de Árvores. IBA. https://www.iba.org/datafiles/publicacoes/relatorios/relatorio-anual-iba2022
Kenward, M. G., & Roger, J. H. (1997). Small sample inference for fixed effects from restricted maximum likelihood. Biometrics, 53(3), 983-997.https://doi.org/10.2307/2533558
Lesaffre, E., & Verbeke, G. (1998). Local influence in linear mixed models. Biometrics, 54, 570-582. https://doi.org/10.2307/3109764
Lima, R. B., Görgens, E. B., Elias, F., Abreu, J. C., Baia, A. L., Oliveira, C. P., Silva, D. A. S., Batista, A. P. B., Lima, R. C., Sotta, E. D., Ferreira, R. L. C., Silva, J. A. A., & Carneiro Guedes, M. (2021). Height-diameter allometry for tropical forest in northern Amazonia. PLoS ONE, 16(12), 1-13. https://doi.org/10.1371/journal.pone.0255197
Loy, A. M. M. (2013). Diagnostics for mixed/hierarchical linear models. Iowa State University.
Meng, S. X., Huang, S., Lieffers, V. J., Nunifu, T., & Yang, Y. (2008). Wind speed and crown class influence the height–diameter relationship of lodgepole pine: nonlinear mixed effects modeling. Forest Ecology and Management, 256(4), 570-577.
Moral, R. A., Hinde, J., & Demétrio, C. G. B. (2017). Half-normal plots and overdispersed models in R: the hnp package. Journal of Statistical Software, 81(10), 1-23. https://doi.org/10.18637/jss.v081.i10
Nobre, J. S., & Singer, J. M. (2007). Residual analysis for linear mixed models. Biometrical Journal: Journal of Mathematical Methods in Biosciences, 49(6), 863-875. https://doi.org/10.1002/bimj.200610341
Nobre, J. S. (2026). Subrotinas computacionais. https://sites.google.com/dema.ufc.br/juvencio/subrotinas-computacionais
Pinheiro, J. (2011). nlme: Linear and nonlinear mixed effects models. R package version, 3, 1.
R Core Team (2021). R: A language and environment for statistical computing. R Foundation for Statistical Computing.
Raptis, D. I., Kazana, V., Kazaklis, A., & Stamatiou, C. (2021). Mixed-effects height–diameter models for black pine (Pinus nigra Arn.) forest management. Trees, 35, 1167-1183. https://doi.org/10.1007/s00468-021-02106-x
Säfken, B., Rügamer, D., Kneib, T., & Greven, S. (2021). Conditional model selection in mixed-effects models with cAIC4. Journal of Statistical Software, 99(8), 1-30. https://doi.org/10.18637/jss.v099.i08
Schützenmeister, A., & Piepho, H.-P. (2012). Residual analysis of linear mixed models using a simulation approach. Computational Statistics & Data Analysis, 56(6), 1405-1416. https://doi.org/10.1016/j.csda.2011.11.006
Scolforo, J. R. S. (1998). Biometria florestal: modelagem do crescimento e da produção de florestas plantadas e nativas. UFLA/FAEPE.
Singer, J. M., Rocha, F. M., & Nobre, J. S. (2017). Graphical tools for detecting departures from linear mixed model assumptions and some remedial measures. International Statistical Review, 85(2), 290-324. https://doi.org/10.1111/insr.12178
Verbeke, G., & Molenberghs, G. (2000). Estimation of the marginal model. In Linear mixed models for longitudinal data (pp. 41-54). Springer. https://doi.org/10.1007/978-1-4419-0300-6
Waternaux, C., Laird, N. M., & Ware, J. H. (1989). Methods for analysis of longitudinal data: blood-lead concentrations and cognitive development. Journal of the American Statistical Association, 84(405), 33-41. https://doi.org/10.2307/2289844
West, B. T., Welch, K. B., & Galecki, A. T. (2022). Linear mixed models: A practical guide using statistical software. Chapman and Hall/CRC. https://doi.org/10.1201/9781003181064
Downloads
Publicado
Edição
Seção
Licença
Copyright (c) 2026 Breno Gabriel da Silva, Clarice Garcia Borges Demétrio, Renata Alcarde Sermarini, Alexandre Behling, Geert Molenberghs, Geert Verbeke, Eduardo Resende Girardi Marques, Yuri Accioly, Marco Aurélio Figura (Autor)
Este trabalho está licenciado sob uma licença Creative Commons Attribution 4.0 International License.
DECLARAÇÃO DE ORIGINALIDADE E DIREITOS AUTORAIS
Declaro que o presente artigo é original, não tendo sido submetido à publicação em qualquer outro periódico nacional ou internacional, quer seja em parte ou em sua totalidade.
Os direitos autorais pertencem exclusivamente aos autores. Os direitos de licenciamento utilizados pelo periódico é a licença Creative Commons Attribution 4.0 (CC BY 4.0): são permitidos o compartilhamento (cópia e distribuição do material em qualquer meio ou formato) e adaptação (remix, transformação e criação de material a partir do conteúdo assim licenciado para quaisquer fins, inclusive comerciais.
Recomenda-se a leitura desse link para maiores informações sobre o tema: fornecimento de créditos e referências de forma correta, entre outros detalhes cruciais para uso adequado do material licenciado.
Como Citar
