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

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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.

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References

  • 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 255:2189–2203

    Article  Google Scholar 

  • Avery TE, Burkhart HE (2002) Forest measurements, 5th edn. McGraw-Hill, New York

    Google Scholar 

  • Bi HQ (2000) Trigonometric variable-form taper equations for Australian eucalyptus. For Sci 46:397–409

    Google Scholar 

  • Bruce D, Curtis RO, Vancoevering C (1968) Development of a system of taper and volume tables for red alder. For Sci 14:339–350

    Google Scholar 

  • Burkhart HE (1977) Cubic-foot volume of loblolly pine to any merchantable top limit. South J Appl For 1:7–9

    Google Scholar 

  • Burkhart HE, Walton SB (1985) Incorporating crown ratio into taper equations for loblolly pine trees. For Sci 31:478–484

    Google Scholar 

  • Cao QV, Burkhart HE (1980) Cubic foot volume of loblolly pine to any height limit. South J Appl For 4:166–168

    Google Scholar 

  • Cao QV, Burkhart HE, Max TA (1980) Evaluation of two methods for cubic–volume prediction of loblolly pine to any merchantable limit. For Sci 26:71–80

    Google Scholar 

  • 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), Madrid

  • Clutter JL, Fortson JC, Pienaar LV, Brister GH, Bailey RL (1983) Timber management: a quantitative approach. Krieger Publishing Company, New York

    Google Scholar 

  • Cochran WG (1963) Sampling techniques, 2nd edn. Wiley, New York

    Google Scholar 

  • 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 238:118–129

    Article  Google Scholar 

  • Davidian M, Giltinan DM (1993) Some general estimation methods for nonlinear mixed-effects models. J Biopharm Stat 3:23–55

    Article  PubMed  CAS  Google Scholar 

  • Davidian M, Giltinan DM (1995) Nonlinear models for repeated measurement data. Chapman & Hall, New York

    Google Scholar 

  • DGCONA (2002) Tercer Inventario Forestal Nacional, 1997–2006: Galicia. Dirección General de Conservación de la Naturaleza. Ministerio de Medio Ambiente, Madrid

    Google Scholar 

  • 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 36:1190–1205

    Article  Google Scholar 

  • Fang Z, Bailey RL (2001) Nonlinear mixed-effects modeling for slash pine dominant height growth following intensive silvicultural treatments. For Sci 47:287–300

    Google Scholar 

  • 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 46:1–12

    Google Scholar 

  • 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 84:49–60

    Article  Google Scholar 

  • 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 179:507–522

    Article  Google Scholar 

  • Goulding CJ, Murray JC (1976) Polynomial taper equations that are compatible with tree volume equations. N Z J For Sci 5:313–322

    Google Scholar 

  • Gregoire TG, Schabenberger O, Kong F (2000) Prediction from an integrated regression equation: a forestry application. Biometrics 56:414–419

    Article  PubMed  CAS  Google Scholar 

  • Hann DW, Walters DK, Scrivani JA (1987) Incorporating crown ratio into prediction equations for Douglas-fir stem volume. Can J For Res 17:17–22

    Article  Google Scholar 

  • Hartford A, Davidian M (2000) Consequences of misspecifying assumptions in nonlinear mixed effects models. Comput Stat Data Anal 34:139–164

    Article  Google Scholar 

  • ICONA (1993) Segundo Inventario Forestal Nacional. Ministerio de Agricultura, Pesca y Alimentación, Madrid

  • Jones RH (1990) Serial correlation or random subject effects? Commun Stat Simul Comput 19:1105–1123

    Article  Google Scholar 

  • Kozak A (1988) A variable–exponent taper equation. Can J For Res 18:1363–1368

    Article  Google Scholar 

  • Kozak A (2004) My last words on taper functions. For Chron 80:507–515

    Google Scholar 

  • Kozak A, Munro DD, Smith JHG (1969) Taper functions and their application in forest inventory. For Chron 45:278–283

    Google Scholar 

  • Laird NM, Ware JH (1982) Random-effects models for longitudinal data. Biometrics 38:963–974

    Article  PubMed  CAS  Google Scholar 

  • Leites LP, Robinson AP (2004) Improving taper equations of loblolly pine with crown dimensions in a mixed-effects modeling framework. For Sci 50:204–212

    Google Scholar 

  • Lindstrom MJ, Bates DM (1990) Nonlinear mixed-effects models for repeated measures data. Biometrics 46:673–687

    Article  PubMed  CAS  Google Scholar 

  • Littell RC, Milliken GA, Stroup WW, Wolfinger RD, Schabenberger O (2006) SAS for mixed models, 2nd edn. SAS Institute Inc., Cary, NC

    Google Scholar 

  • Max TA, Burkhart HE (1976) Segmented polynomial regression applied to taper equations. For Sci 22:283–289

    Google Scholar 

  • Muhairwe CK (1999) Taper equations for Eucalyptus pilularis and Eucalyptus grandis for the north coast in New South Wales, Australia. For Ecol Manage 113:251–269

    Article  Google Scholar 

  • 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 24:252–259

    Article  Google Scholar 

  • Newnham RM (1988) A variable-form taper function. Information Report PI-X-83. Petawawa National Forest Institute. Canadian Forest Service, Petawawa, Ontario, Canada

    Google Scholar 

  • Newnham RM (1992) Variable-form taper functions for four Alberta tree species. Can J For Res 22:210–223

    Article  Google Scholar 

  • Pérez D, Burkhart HE, Stiff C (1990) A variable-form taper function for Pinus oocarpa Schiede. in Central Honduras. For Sci 36:186–191

    Google Scholar 

  • Pinheiro JC, Bates DM (1995) Approximations to the log-likelihood function in the nonlinear mixed effects model. J Comput Graph Stat 4:12–35

    Google Scholar 

  • Pinheiro JC, Bates DM (2000) Mixed-effects models in S and S-PLUS. Springer, New York

    Book  Google Scholar 

  • SAS support (2011) Sample 25032: %NLINMIX macro to fit nonlinear mixed models. http://support.sas.com/kb/25/032.html. Accessed 7 July 2011

  • Tasissa G, Burkhart HE (1998) An application of mixed effects analysis to modeling thinning effects on stem profile of loblolly pine. For Ecol Manage 103:87–101

    Article  Google Scholar 

  • Trincado G, Burkhart HE (2006) A generalized approach for modeling and localizing stem profile curves. For Sci 52:670–682

    Google Scholar 

  • Valentine HT, Gregoire TG (2001) A switching model of bole taper. Can J For Res 31:1400–1409

    Article  Google Scholar 

  • VanderSchaaf CL, Burkhart HE (2007) Comparison of methods to estimate Reineke's maximum size–density relationship species boundary line slope. For Sci 53:435–442

    Google Scholar 

  • Vonesh EF (1996) A note on the use of Laplace's approximation for nonlinear mixed effects models. Biometrika 83:447–452

    Article  Google Scholar 

  • Vonesh EF, Chinchilli VM (1997) Linear and nonlinear models for the analysis of repeated measurements. Marcel Dekker Inc, New York

    Google Scholar 

  • West PW, Ratkowsky DA, Davis AW (1984) Problems of hypothesis testing of regressions with multiple measurements from individual sampling units. For Ecol Manage 7:207–224

    Article  Google Scholar 

  • Wolfinger RD, Lin X (1997) Two Taylor-series approximation methods for nonlinear mixed models. Comput Stat Data Anal 25:465–490

    Article  Google Scholar 

  • 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)

  • 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 128:415–429

    Article  Google Scholar 

Download references

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.

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Authors and Affiliations

  1. Departamento de Ingeniería Agroforestal, Universidad de Santiago de Compostela. Escuela Politécnica Superior, R/Benigno Ledo, Campus Universitario, 27002, Lugo, Spain

    Esteban Gómez-García, Felipe Crecente-Campo & Ulises Diéguez-Aranda

Authors
  1. Esteban Gómez-García
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  2. Felipe Crecente-Campo
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  3. Ulises Diéguez-Aranda
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Corresponding author

Correspondence to Esteban Gómez-García.

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Handling Editor: Barry Alan Gardiner

Contribution of the co-authors

Esteban Gómez-García and Ulises Diéguez-Aranda analyzed the data and wrote the manuscript. Felipe Crecente-Campo provided technical assistance in model fitting and revised the text.

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Gómez-García, E., Crecente-Campo, F. & Diéguez-Aranda, U. Selection of mixed-effects parameters in a variable–exponent taper equation for birch trees in northwestern Spain. Annals of Forest Science 70, 707–715 (2013). https://doi.org/10.1007/s13595-013-0313-9

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  • DOI: https://doi.org/10.1007/s13595-013-0313-9

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