Chris J. Cieszewski, Mike Strub


In this study, we compare the properties of self-referencing models, such as various site dependent growth and yield models for predictions of height, diameter, basal area, volume, and density, developed using Nonlinear-Fixed-Effects (NFE) versus Nonlinear-Mixed-Effects (NME) modeling approaches. The properties investigated include the following core traditional well-behaved model characteristics applicable to self-referencing functions: Base-Age-Invariance, Path-Invariance, Indifference Under Model Reparameterization, and Model Conditioning to have the predictions at the base-age equal to the reference point, as well as estimation and prediction statistics such as bias and variance of the fitted versus predicted residuals. The results of this investigation demonstrate that self-referencing models based on the NFE approach possess all the desirable properties associated with logical behavior of the model and estimation statistics, while the NME based self-referencing models lack the well-behaved model properties. We illustrate these properties using an example of fitting self-referencing models to panel data of loblolly pine age-height measurements on a range of sites from the South Africa Correlated Curve Trend Study.


Mixed-Effects; Fixed-Effects; Self-Referencing; Base Age Invariance; Path Invariance; Invariance under Reparameterization; Well-Behaved Models; Model Conditioning; Site Models.

Full Text:



Patricia, A, Del Río, M., & Isabel, C. 2008. A mixed nonlinear height-diameter model for pyrenean oak (Quercus pyrenaica Willd.). For. Ecol. Manag. 256(1-2): 88-98.

Adame, P.I., Canellas, S. Roig, & Del Río, M. 2006. Modelling dominant height growth and site index curves for rebollo oak (Quercus pyrenaicaWilld.). Ann. For. Sci. 63: 929-940.

Bailey, R.L., & Clutter, J.L. 1974. Base-age-invariant polymorphic site curves. For. Sci. 20: 155–159.

Biging, G.S. 1985. Improved estimates of site index curves using a varying-parameter model. For. Sci. 31: 248-259.

Burkhart, H.E., & Tomé, M., 2012. Modeling Forest Trees and Stands. Springer, Dordrecht, 457.

Calama R, & Montero, G. 2004. Interregional nonlinear height-diameter model with random coefficients for stone pine in Spain. CJFR 34: 150-163.

Calegario, N., Daniels, R.F., Maestri, R., & Neiva, R. 2005. Modeling dominant height growth based on nonlinear Mixed-Effects model: a clonal Eucalyptus plantation case study. For. Ecol. Manag. 204: 11-20.

Cieszewski, C.J. 2001. Three methods of deriving advanced dynamic site equations demonstrated on inland Douglas-fir site curves. Canadian Journal of Forest Research 31(1): 165-173.

Cieszewski, C.J. 2002. Comparing fixed-and variable-base-age site equations having single versus multiple asymptotes. Forest Science 48(1): 7-23.

Cieszewski, C.J. 2003. Developing a Well-Behaved Dynamic Site Equation Using a Modified Hossfeld IV Function Y 3=(axm)/(c+ x m–1), a Simplified Mixed-Model and Scant Subalpine Fir Data. Forest Science 49(4): 539-554.

Cieszewski, C.J. & Bailey, R.L. 2000. Generalized algebraic difference approach: theory based derivation of dynamic site equations with polymorphism and variable asymptotes. For. Sci. 46(1): 116-126.

Cieszewski, C.J. & Bella, I.E. 1989. Polymorphic height and site index curves for lodgepole pine in Alberta. Can. J. For. Res. 19: 1151–1160.

Cieszewski, C.J., W.M. Harrison and S.W. Martin. 2000. Examples of practical methods for unbiased parameter estimation in self-referencing functions. In C.J. Cieszewski (ed.). Proceedings of the First International Conference on Measurements and Quantitative Methods and Management and The 1999 Southern Mensurationists Meeting. D.B. Warnell School of Forest Resources. University of Georgia. Athens, GA. 207 p.

Hall, D.B. & Bailey, R.L. 2001. Modeling and Prediction of Forest Growth Variables Based on Multilevel Nonlinear Mixed Models. For. Sci. 47: 311–321.

Diéguez-Aranda, U., Burkhart, H.E., Amateis, R.L. 2006a. Dynamic Site Model for Loblolly Pine (Pinus taeda L.) Plantations in the United States. For. Sci., 52: 262-272.

Diéguez-Aranda, U., Grandas-Arias, J.A., Álvarez-González, J.G., v. Gadow, K. 2006b. Site quality curves for birch stands in north-western Spain. Silva Fennica 40: 631-644.

Dorado F.C., Dieguez-Aranda, U., Anta, M.B., Rodríguez, M.S., & v. Gadow, K. 2006. A generalized height-diameter model including random components for radiata pine plantations in northwestern Spain. For. Ecol. Manag., 229: 202-213.

DuPlat, P. & Tran-Ha, M. 1986. Modeles de Croissance en Hauteur Dominante pourle hêtre, le sapin pectiné et le pin sylvestre (dans le Massif de l'Aigoual). Document No. 86.1. Office National de Forêts. France. (In French).

Elfving, B., Kiviste, A. 1997. Construction of site index equations for Pinus sylvestris L. using permanent plot data in Sweden. For. Ecol. Manag. 98: 125-134.

Eriksson, H., Johanssen, U., & Kiviste, A. 1997. A site-index model for pure and mixed stands of betula pendula and betula pubescens in Sweden. Scand. J. For. Res. 12: 149-156.

Fang, Z., & Bailey, R.L. 2001. Nonlinear Mixed-Effects modeling for slash pine dominant height growth following intensive silvicultural treatments. For. Sci. 47: 287-300.

Friesen, M.C., Demers, P.A., Davies, H.W., Teschke, K., & Marion, S. 2002. Determinants of dust exposure in sawmills: A comparison of Fixed-Effects and Mixed-Effects predictive statistical models. Epidemiology. 13: S231-S231.

Kershaw, J.A., Benjamin, J.G., & Weiskittel, A.R. 2009. Approaches for Modeling Vertical Distribution of Maximum Knot Size in Black Spruce: A Comparison of Fixed- and Mixed-Effects Nonlinear Models. For. Sci. 55: 230-237.

Krumland, B., Eng, H. 2005. Site index systems for major young-growth forest and woodland species in northern California. California Forestry Report No. 4. California Department of Forestry and Fire Protection. Sacramento, CA. 219 p., found at: http: //

Laird, N.M., & Ware, J.H. 1982. Random-Effects Models For Longitudinal Data. Biometrics. 38: 963-974.

Lappi, J., & Bailey, R.L. 1988. A height prediction model with random stand and tree parameters: An alternative to traditional site index methods. For. Sci. 34: 907-927.

Lappi, J., & Malinen, J. 1994. Random-parameter height/age models when stand parameters and stand age are correlated. For. Sci. 40: 715-731.

Leites, L.P., & Robinson, A.P. 2004. Improving taper equations of loblolly pine with crown dimensions in a Mixed-Effects modeling framework. For. Sci. 50: 204-212.

Lindstrom, M., & Bates, D. 1990. Nonlinear Mixed-Effects Models For Repeated Measures Data. Biometrics. 46: 673-687.

Meng, S.X., & Huang, S.M. 2009. Improved Calibration of Nonlinear Mixed-Effects Models Demonstrated on a Height Growth Function. For. Sci. 55: 238-248.

Nord-Larsen, T. 2006. Developing Dynamic Site Index Curves for European Beech (Fagus sylvatica L.) in Denmark. For. Sci. 52: 173–181.

Overton, R.C. 1998. A comparison of Fixed-Effects and mixed (random-effects) models for meta-analysis tests of moderator variable effects. Psychological Methods. 3: 354-379.

Amaral, P.J., Tome, J., & Tome, M. 2011. Nonlinear fixed and random generalized height-diameter models for Portuguese cork oak stands. Annals For. Sci. 68: 295-309.

Ratkowsky, D.A. 1990. Handbook of nonlinear regression models. Marcel Dekker, Inc., New York, NY. 241.

Raulier, F., Lambert, M.C., Pothier, D., & Ung, C.H., 2003. Impact of dominant tree dynamics on site index curves. Forest Ecology and Management 184, 65-78.

Rivas, J.J.C., González, J.G.Á., González, A.D.R., & v. Gadow, K. 2004. Compatible height and site index models for five pine species in El Salto, Durango (Mexico). For. Ecol. Manag. 201: 145–160.

Rose, C.E., Cieszewski, C.J., & Carmean, W.H. 2003. Three methods for avoiding the impacts of incompatible site index and height prediction models demonstrated on jack pine curves for Ontario. For. Chron. 79: 928-935.

Saunders, M.R., & Wagner, R.G. 2008. Height-diameter models with random coefficients and site variables for tree species of Central Maine. Annals of For. Sci. 65: 203.

Schumacher, F.X. 1939. A new growth curve and its application to timber-yield studies. J. For. 37: 819-820.

Segura-Correa, J.C., Armendariz, I, & Santos, R. 2008. Comparison of fixed and mixed models for the analysis of random block designs with split plot fit. Cuban Journal of Agricultural Science. 42: 13-17.

Sharma, M., & Parton J. 2007. Height-diameter equations for boreal tree species in Ontario using a Mixed-Effects modeling approach. For. Ecol. Manag. 249: 187-198.

Lee, S.Y., & Xu, L. 2004. Influence analyses of nonlinear Mixed-Effects models. Computational Statistics & Data Analysis. 45: 321–341.

Strub, M. R., & Bredenkamp, B.V. 1985. Carrying capacity and thinning response of Pinus taeda in the CCT experiments. South African Forestry Journal, 133: 6-11.

Temesgen H., Monleon, V.J., & Hann, D.W. 2008. Analysis and comparison of nonlinear tree height prediction strategies for Douglas-fir forests. CJFR. 38: 553-565.

Trincado, G.V., Kiviste, A., v. Gadow, K. 2003: Preliminary site index models for native Roble (Nothofagus obliqua) and Rauli (N. alpina) in Chile. New Zealand J. For. Sci., 32: 322-333.

Wang, M.L., Borders, B.E., & Zhao, D. 2008. An empirical comparison of two subject-specific approaches to dominant heights modeling: The dummy variable method and the mixed model method. For. Ecol. Manag. 255: 2659-2669.

Wang, Y., Lemay, V.M, & Baker, T.G. 2007. Modelling and prediction of dominant height and site index of Eucalyptus globulus plantations using a nonlinear Mixed-Effects model approach. CJFR. 37: 1390-1403.

Weiskittel, A.R., Hann, D.W., Hibbs, D.E., Lam, T.Y., & Bluhm, A.A. 2009. Modeling top height growth of red alder plantations. For. Ecol. Manag. 258: 323–331.


  • There are currently no refbacks.


© 2008 Mathematical and Computational Forestry & Natural-Resource Sciences