Advertisement

Annals of Forest Science

, Volume 70, Issue 7, pp 707–715 | Cite as

Selection of mixed-effects parameters in a variable–exponent taper equation for birch trees in northwestern Spain

  • Esteban Gómez-GarcíaEmail author
  • Felipe Crecente-Campo
  • Ulises Diéguez-Aranda
Original Paper

Abstract

Context

Taper equations predict the variation in diameter along the stem, therefore characterizing stem form. Several recent studies have tested mixed models for developing taper equations. Mixed-effects modeling allow the interindividual variation to be explained by considering both fixed-effects parameters (common to the population) and random-effects parameters (specific to each individual).

Aims

The objective of this study is to develop a mixed-effect variable exponent taper equation for birch trees in northwestern Spain by determining which fixed-effects parameters should be expanded with random-effects parameters.

Methods

All possible combinations of linear expansions with random effects in one and in two of the fixed-effects model parameters were tested. Upper stem diameter measurements were used to estimate random-effects parameters by the use of an approximate Bayesian estimator, which calibrated stem profile curves for individual trees.

Results

Parameter estimates for more than half of the mixed models investigated were nonsignificant. A first order autoregressive error structure was used to completely remove the autocorrelation between residuals, as mixed-effects modeling were not sufficient for this purpose.

Conclusion

The mixed model with the best fitting statistics did not provide the best calibration statistics for all upper stem diameter measurements. From a practical point of view, model calibration should be considered an essential criterion in mixed model selection.

Keywords

Betula pubescens Ehrh Taper equation Mixed-effects modeling Autocorrelation Calibration Galicia 

Notes

Funding

The present study was financially supported by the Spanish Ministry of Education and Science through the research project Modelos de evolución de bosques de frondosas caducifolias del noroeste peninsular (AGL2007-66739-C02-01/FOR), co-funded by the European Union through the European Regional Development Fund.

References

  1. Adu-Bredu S, Bi AFT, Bouillet JP, Mé MK, Kyei SY, Saint-André L (2008) An explicit stem profile model for forked and unforked teak (Tectona grandis) trees in West Africa. For Ecol Manage 255:2189–2203CrossRefGoogle Scholar
  2. Avery TE, Burkhart HE (2002) Forest measurements, 5th edn. McGraw-Hill, New YorkGoogle Scholar
  3. Bi HQ (2000) Trigonometric variable-form taper equations for Australian eucalyptus. For Sci 46:397–409Google Scholar
  4. Bruce D, Curtis RO, Vancoevering C (1968) Development of a system of taper and volume tables for red alder. For Sci 14:339–350Google Scholar
  5. Burkhart HE (1977) Cubic-foot volume of loblolly pine to any merchantable top limit. South J Appl For 1:7–9Google Scholar
  6. Burkhart HE, Walton SB (1985) Incorporating crown ratio into taper equations for loblolly pine trees. For Sci 31:478–484Google Scholar
  7. Cao QV, Burkhart HE (1980) Cubic foot volume of loblolly pine to any height limit. South J Appl For 4:166–168Google Scholar
  8. Cao QV, Burkhart HE, Max TA (1980) Evaluation of two methods for cubic–volume prediction of loblolly pine to any merchantable limit. For Sci 26:71–80Google Scholar
  9. Castroviejo S, Laínz M, López González G, Monteserrat P, Muñoz Garmendia F, Paiva J, Villar L (1990) Flora ibérica: Plantas vasculares de la Península Ibérica e Islas Baleares. Volumen II: Platanaceae-Plumbaginaceae (partim.). RJB (CSIC), MadridGoogle Scholar
  10. Clutter JL, Fortson JC, Pienaar LV, Brister GH, Bailey RL (1983) Timber management: a quantitative approach. Krieger Publishing Company, New YorkGoogle Scholar
  11. Cochran WG (1963) Sampling techniques, 2nd edn. Wiley, New YorkGoogle Scholar
  12. Corral-Rivas JJ, Diéguez-Aranda U, Corral Rivas S, Castedo Dorado F (2007) A merchantable volume system for major pine species in El Salto, Durango (Mexico). For Ecol Manage 238:118–129CrossRefGoogle Scholar
  13. Davidian M, Giltinan DM (1993) Some general estimation methods for nonlinear mixed-effects models. J Biopharm Stat 3:23–55PubMedCrossRefGoogle Scholar
  14. Davidian M, Giltinan DM (1995) Nonlinear models for repeated measurement data. Chapman & Hall, New YorkGoogle Scholar
  15. DGCONA (2002) Tercer Inventario Forestal Nacional, 1997–2006: Galicia. Dirección General de Conservación de la Naturaleza. Ministerio de Medio Ambiente, MadridGoogle Scholar
  16. Diéguez-Aranda U, Castedo-Dorado F, Álvarez-González JG, Rojo A (2006) Compatible taper function for Scots pine plantations in northwestern Spain. Can J For Res 36:1190–1205CrossRefGoogle Scholar
  17. 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
  18. Fang Z, Borders BE, Bailey RL (2000) Compatible volume-taper models for loblolly and slash pine based on a system with segmented-stem form factors. For Sci 46:1–12Google Scholar
  19. Fonweban J, Gardiner B, Macdonald E, Auty D (2011) Taper functions for Scots pine (Pinus sylvestris L.) and Sitka spruce (Picea sitchensis (Bong.) Carr.) in Northern Britain. Forestry 84:49–60CrossRefGoogle Scholar
  20. Garber SM, Maguire DA (2003) Modeling stem taper of three central Oregon species using nonlinear mixed effects models and autoregressive error structures. For Ecol Manage 179:507–522CrossRefGoogle Scholar
  21. Goulding CJ, Murray JC (1976) Polynomial taper equations that are compatible with tree volume equations. N Z J For Sci 5:313–322Google Scholar
  22. Gregoire TG, Schabenberger O, Kong F (2000) Prediction from an integrated regression equation: a forestry application. Biometrics 56:414–419PubMedCrossRefGoogle Scholar
  23. Hann DW, Walters DK, Scrivani JA (1987) Incorporating crown ratio into prediction equations for Douglas-fir stem volume. Can J For Res 17:17–22CrossRefGoogle Scholar
  24. Hartford A, Davidian M (2000) Consequences of misspecifying assumptions in nonlinear mixed effects models. Comput Stat Data Anal 34:139–164CrossRefGoogle Scholar
  25. ICONA (1993) Segundo Inventario Forestal Nacional. Ministerio de Agricultura, Pesca y Alimentación, MadridGoogle Scholar
  26. Jones RH (1990) Serial correlation or random subject effects? Commun Stat Simul Comput 19:1105–1123CrossRefGoogle Scholar
  27. Kozak A (1988) A variable–exponent taper equation. Can J For Res 18:1363–1368CrossRefGoogle Scholar
  28. Kozak A (2004) My last words on taper functions. For Chron 80:507–515Google Scholar
  29. Kozak A, Munro DD, Smith JHG (1969) Taper functions and their application in forest inventory. For Chron 45:278–283Google Scholar
  30. Laird NM, Ware JH (1982) Random-effects models for longitudinal data. Biometrics 38:963–974PubMedCrossRefGoogle Scholar
  31. Leites LP, Robinson AP (2004) Improving taper equations of loblolly pine with crown dimensions in a mixed-effects modeling framework. For Sci 50:204–212Google Scholar
  32. Lindstrom MJ, Bates DM (1990) Nonlinear mixed-effects models for repeated measures data. Biometrics 46:673–687PubMedCrossRefGoogle Scholar
  33. Littell RC, Milliken GA, Stroup WW, Wolfinger RD, Schabenberger O (2006) SAS for mixed models, 2nd edn. SAS Institute Inc., Cary, NCGoogle Scholar
  34. Max TA, Burkhart HE (1976) Segmented polynomial regression applied to taper equations. For Sci 22:283–289Google Scholar
  35. Muhairwe CK (1999) Taper equations for Eucalyptus pilularis and Eucalyptus grandis for the north coast in New South Wales, Australia. For Ecol Manage 113:251–269CrossRefGoogle Scholar
  36. Muhairwe CK, LeMay VM, Kozak A (1994) Effects of adding tree, stand, and site variables to Kozak's variable–exponent taper equation. Can J For Res 24:252–259CrossRefGoogle Scholar
  37. Newnham RM (1988) A variable-form taper function. Information Report PI-X-83. Petawawa National Forest Institute. Canadian Forest Service, Petawawa, Ontario, CanadaGoogle Scholar
  38. Newnham RM (1992) Variable-form taper functions for four Alberta tree species. Can J For Res 22:210–223CrossRefGoogle Scholar
  39. Pérez D, Burkhart HE, Stiff C (1990) A variable-form taper function for Pinus oocarpa Schiede. in Central Honduras. For Sci 36:186–191Google Scholar
  40. Pinheiro JC, Bates DM (1995) Approximations to the log-likelihood function in the nonlinear mixed effects model. J Comput Graph Stat 4:12–35Google Scholar
  41. Pinheiro JC, Bates DM (2000) Mixed-effects models in S and S-PLUS. Springer, New YorkCrossRefGoogle Scholar
  42. SAS support (2011) Sample 25032: %NLINMIX macro to fit nonlinear mixed models. http://support.sas.com/kb/25/032.html. Accessed 7 July 2011
  43. Tasissa G, Burkhart HE (1998) An application of mixed effects analysis to modeling thinning effects on stem profile of loblolly pine. For Ecol Manage 103:87–101CrossRefGoogle Scholar
  44. Trincado G, Burkhart HE (2006) A generalized approach for modeling and localizing stem profile curves. For Sci 52:670–682Google Scholar
  45. Valentine HT, Gregoire TG (2001) A switching model of bole taper. Can J For Res 31:1400–1409CrossRefGoogle Scholar
  46. VanderSchaaf CL, Burkhart HE (2007) Comparison of methods to estimate Reineke's maximum size–density relationship species boundary line slope. For Sci 53:435–442Google Scholar
  47. Vonesh EF (1996) A note on the use of Laplace's approximation for nonlinear mixed effects models. Biometrika 83:447–452CrossRefGoogle Scholar
  48. Vonesh EF, Chinchilli VM (1997) Linear and nonlinear models for the analysis of repeated measurements. Marcel Dekker Inc, New YorkGoogle Scholar
  49. West PW, Ratkowsky DA, Davis AW (1984) Problems of hypothesis testing of regressions with multiple measurements from individual sampling units. For Ecol Manage 7:207–224CrossRefGoogle Scholar
  50. Wolfinger RD, Lin X (1997) Two Taylor-series approximation methods for nonlinear mixed models. Comput Stat Data Anal 25:465–490CrossRefGoogle Scholar
  51. Xunta de G (2001) O monte galego en cifras. Dirección Xeral de Montes e Medio Ambiente Natural. Consellería de Medio Ambiente. Santiago de Compostela (Spain)Google Scholar
  52. Yang Y, Huang S, Trincado G, Meng SX (2009) Nonlinear mixed-effects modeling of variable–exponent taper equations for lodgepole pine in Alberta, Canada. Eur J For Res 128:415–429CrossRefGoogle Scholar

Copyright information

© INRA and Springer-Verlag France 2013

Authors and Affiliations

  • Esteban Gómez-García
    • 1
    Email author
  • Felipe Crecente-Campo
    • 1
  • Ulises Diéguez-Aranda
    • 1
  1. 1.Departamento de Ingeniería AgroforestalUniversidad de Santiago de Compostela. Escuela Politécnica Superior, R/Benigno Ledo, Campus UniversitarioLugoSpain

Personalised recommendations