Annals of Forest Science

, Volume 68, Issue 2, pp 295–309 | Cite as

Nonlinear fixed and random generalized height–diameter models for Portuguese cork oak stands

  • Joana Amaral PauloEmail author
  • José Tomé
  • Margarida Tomé
Original Paper



The objective of the research was to develop a generalized height–diameter model for Quercus suber L. in Portugal, which can be applied both to undebarked and debarked trees, with diameter at breast height over cork larger than 2.5 cm.


A nonlinear fixed effects model (NLFEM) and a nonlinear mixed effects model (NLMEM) approaches were used. Parameters estimates were obtained using the SAS macro NLINMIX, which uses a linear approximation to the marginal likelihood function by expanding it with a first-order Taylor series on the random effects. The option of expanding on the random effects at their current empirical best linear unbiased predictors (EBLUP) was used. The fitted models were evaluated using an independent data set, together with an existing model specific for undebarked trees. To obtain subject specific predictions with the NLMEM, a conventional and an improved calibration procedures were applied, considering four different tree sub-sampling designs. Both proposed models included dominant height and stand density as covariates to explain plot variability.


Validation indicated that, even in the situations where the NLMEM calibration is not possible, this model should be preferred. The differences between the validated models, which were more evident for young stands, were considered. No large differences in predictive accuracy were found between the calibrated NLMEM using the conventional or the improved calibration procedures, for all the considered sub-sampling designs.


Height–diameter relationship Quercus suber L. Nonlinear mixed effects model Nonlinear fixed effects model Model calibration 



Financial support was provided by project CarbWoodCork (POCI/AGR/57279/2004 and PPCDT/AGR/57279/2004) financed by the Fundação para a Ciência e Tecnologia (Portugal) and by project Motive (Grant Agreement 226544) financed by the European Commission under the Seventh Framework Program for Research and Technological Development. This paper is part of the PhD project of the first author, which is funded by a scholarship (SFRH/BD/23855/2005) granted by Fundação para a Ciência e Tecnologia (Portugal).


  1. Adame P, Río M, Cañellas I (2008) A mixed nonlinear height-diameter model for Pyrenean oak (Quercus pyrenaica Willd.). For Ecol Manage 256:88–98CrossRefGoogle Scholar
  2. Calama R, Montero G (2004) Interregional nonlinear height-diameter model with random coefficients for stone pine in Spain. Can J For Res 34:150–163CrossRefGoogle Scholar
  3. Cañellas I, Sánchez-González M, Bogino SM, Adame P, Herrero C, Roig S, Tomé M, Paulo JA, Bravo F (2008) Silviculture and carbon sequestration in mediterranean oak forests. In: F. Bravo et al. (Ed.), Managing forest ecosystems: the challenge of climate change. Springer Science, Berlin, 338 ppGoogle Scholar
  4. Davidian M, Giltinan DM (1995) Nonlinear models for repeated measurement data. Chapman and Hall, New York, p 359Google Scholar
  5. Dorado FC, Diéguez-Aranda U, Anta MB, Rodríguez MS, Gadow KV (2006) A generalized height-diameter model including random components for radiate pine plantations in northwestern Spain. For Ecol Manage 229:202–213CrossRefGoogle Scholar
  6. Fang Z, Bailey RL (1998) Height-diameter models for tropical forest on Hainan Island in southern China. For Ecol Manage 110:315–327CrossRefGoogle Scholar
  7. Fang Z, Bailey RL (2001) Nonlinear mixed effects modeling for slash pine dominant height growth following intensive silvicultural treatments. For Sci 47:287–300Google Scholar
  8. Huang S, Price D, Titus S (2000) Development of ecoregion-based height–diameter models for white spruce in boreal forests. For Ecol Manage 129:125–141CrossRefGoogle Scholar
  9. Huang S, Titus SJ, Wiens DP (1992) Comparison of nonlinear height–diameter functions for major Alberta tree species. Can J For Res 22:1297–1304CrossRefGoogle Scholar
  10. Lappi J (1997) Longitudinal analysis of height/diameter curves. For Sci 43:555–570Google Scholar
  11. Lei Y, Parresol BR (2001) Remarks on height–diameter modelling. USDA For. Serv. Research Note SRS-10, Southern Research Station, Asheville, NC, 5 ppGoogle Scholar
  12. Lindstrom MJ, Bates DM (1990) Nonlinear mixed effects models for repeated measures data. Biometrics 46:673–687PubMedCrossRefGoogle Scholar
  13. Meng SX, Huang S (2009) Improved calibration of nonlinear mixed-effects models demonstrated on a height growth function. For Sci 55(3):238–247Google Scholar
  14. Natividade JV (1950) Subericultura. Direcção Geral dos Serviços Florestais e Aquicolas, Lisbon, p 387Google Scholar
  15. Patrone G (1963) Lezioni di Dendrometria. Firenze, Italy, p 392Google Scholar
  16. Paulo JA, Tomé M (2009) An individual tree growth model for juvenile cork oak stands in Southern Portugal. Silva Lus 17(1):27–38Google Scholar
  17. Pinheiro JC, Bates DM (2000) Mixed-effects models in S and S-Plus. Stat. And comput. series. Springer, New York, p 528CrossRefGoogle Scholar
  18. Prodan M (1968) Forest biometrics, English edn. Pergamon Press, Oxford, p 447Google Scholar
  19. Sánchez-González M, Cañellas I, González GM (2007) Generalized height-diameter and crown diameter prediction models for cork oak forests in Spain. Invest Agrar: Sist Recur For 16(1):76–88Google Scholar
  20. SAS Institute Inc. (2004) SAS/STAT® 9.1 user's guide. SAS Institute Inc., Cary NC, 5,136 ppGoogle Scholar
  21. Saunders M, Wagner RG (2008) Height-diameter models with random coefficients and site variables for tree species of central Maine. Ann For Sci 65(203):10. doi: 10.1051/forest:2007086 Google Scholar
  22. Schabenberger O, Pierce F (2002) Contemporary statistical models for the plant and soil sciences. CRC Press LLC, Boca Raton, 738 ppGoogle Scholar
  23. Schnute J (1981) A versatile growth model with statistically sTab. parameters. Can J Fish Aquat Sci 38:1128–1140CrossRefGoogle Scholar
  24. Sharma M, Parton J (2007) Height–diameter equations for boreal tree species in Ontario using a mixed-effects modeling approach. For Ecol Manage 249:187–198CrossRefGoogle Scholar
  25. Temesgen H, Monleon VJ, Hann DW (2008) Analysis and comparison of nonlinear tree height prediction strategies for Douglas-fir forests. Can J For Res 38:553–565CrossRefGoogle Scholar
  26. Temesgen H, von Gadow K (2004) Generalized height–diameter models for major tree species in complex stands of interior British Columbia. Eur J For Res 123:45–51Google Scholar
  27. Tomé M (2004) Modelo de crescimento e produção para a gestão do montado de sobro em Portugal. Relatório final do projecto POCTI/AGR/35172/99. Publicações GIMREF, RFP1/2005, Universidade Técnica de Lisboa, Instituto Superior de Agronomia, Centro de Estudos Florestais, Lisbon, 85 ppGoogle Scholar
  28. Toumi L, Lumaret R (1998) Allozyme variation in cork oak (Q. suber L.): the role of phylogeography and genetic introgression by other Mediterranean oak species and human activities. Theor Appl Genet 98:647–656CrossRefGoogle Scholar
  29. Trincado G, Burkhart H (2006) A generalized approach for modeling and localizing stem profile curves. For Sci 52(6):670–682Google Scholar
  30. Vanclay JK (2009) Tree diameter, height and stocking in even-aged forest. Ann For Sci 66(702):7. doi: 10.1051/forest/2009063 Google Scholar
  31. Vanclay JK, Skovsgaard JP, Hansen CP (1995) Assessing the quality of permanent sample plot databases for growth and yield modelling in forest plantations. For Ecol Manage 71:177–186CrossRefGoogle Scholar
  32. Vonesh EF, Chinchilli VM (1997) Linear and nonlinear models for the analysis of repeated measurements. Marcel Dekker, New York, p 560Google Scholar

Copyright information

© INRA and Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Joana Amaral Paulo
    • 1
    Email author
  • José Tomé
    • 1
  • Margarida Tomé
    • 1
  1. 1.Instituto Superior de AgronomiaLisbonPortugal

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