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Introduction Size-density relations have been quantified for a variety of species and it has been suggested that: ▫A universal slope exists (-3/2) ▫Intercept varies by species, but is not influenced by other factors Previous analyses have relied on ordinary least squares (OLS) or principal components analysis (PCA) to examine trends ▫Assumptions are violated and tests of parameter significance are invalid
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Modeling regional variation in the self-thinning boundary line Aaron WeiskittelSean GarberHailemariam Temesgen
Introduction•Although self-thinning constraints may
not be needed for individual tree growth models (Monserud et al. 2005; For. Sci. 50: 848), they are still important for:▫Stand-level projections
▫Developing stand management diagrams
▫Understanding basic stand dynamics
Introduction• Size-density relations have been quantified for
a variety of species and it has been suggested that:▫A universal slope exists (-3/2)▫Intercept varies by species, but is not influenced
by other factors
• Previous analyses have relied on ordinary least squares (OLS) or principal components analysis (PCA) to examine trends▫Assumptions are violated and tests of parameter
significance are invalid
Introduction• Zhang et al. (2005; CJFR 35: 1507) compared
several different methods for estimating the self-thinning boundary line▫OLS and PCA performed the poorest
sensitive to the data subjectively selected for fitting may produce lines with the inappropriate slope
▫Statistical inference is difficult with quantile regression and deterministic frontier functions
▫Stochastic frontier functions (SFF) performed the best
Introduction• Bi (2001; For. Sci. 47, 361) used SFF to examine
the self-thinning surface in Pinus radiata▫SFF successfully separated the effects of density-
dependent and density-independent mortality
▫SFF allows statistical inferences on the model coefficients
▫Generalized model form proposed: B = β0Sβ1Nβ2 where B is stand biomass per unit area, N is stand
density, S is relative site index, and βi’s are parameters
Objectives•Utilize SFF to examine maximum size-
density relations in coastal Douglas-fir, red alder, and lodgepole pine▫Test the generality of Bi’s (2001) model
▫Examine the influence of other covariates
▫Compare the results to a more traditional approach
Analysis• Used Frontier v4.1 (Coelli 1996) and R library
micEcon to fit the SFF▫ ln(TPA) = β10 - β11ln(QMD) + ε11
QMD is quadratic mean diameter and TPA is trees per acre
• Compared to fits obtained using quantile regression
• Maximum stand density index (SDImax) was estimated for each plot and regressed on other covariates similar to Hann et al. (2003)
• Significance of covariates evaluated using log-likelihood ratio tests
DataSpecies Data
SourceTotal Age Density (#
acre)Site index (ft)
Douglas-fir SMC, SNCC 5-65 92-1208 85.8-164(base age 50)
Red alder HSC 1-17 56-1524 75.4-114.8(base age 30)
Lodgepole pine
BC Ministry of Forests
16-146 136-3638 47.9 – 86.3(base age 50)
Stochastic frontier analysis•Used in econometrics to study firm
efficiency and cost & profit frontiers
•Model error has two components▫Random symmetrical statistical noise▫Systematic deviations from the frontier
•Qit = exp(ß0 + ß1 ln(xit)) * exp(vit) * exp(-uit)
Deterministic componentRandom noise Inefficiency
Stochastic frontier analysis•Fit using maximum likelihood
•u and v are assumed to be distributed independently of each other and the regressors
•u represents the difference in stand density at any given point and the estimated maximum density
▫Eliminates the subjectively of choosing stands that other techniques rely on
Results: Maximum stand densitySpecies Mean Std. Dev. Min Max
Douglas-fir 511 215 213 989
Red alder 484 226 122 1005
Lodgepole pine
725 406 136 1997
• Plot-specific SDImax showed no relationship with any other covariates
Results: Self-thinning boundary line
Species SFA Quantile regressionIntercept Slope Intercept Slope
Douglas-fir 9.9571(0.2246)
-0.9467(0.0708)
11.2289(0.3604)
-1.3309(0.1256)
Red alder 10.3891(0.3017)
-1.0359(0.1171)
10.6492(0.1849)
-1.1379(0.0666)
Lodgepole pine
10.0975(1.6751)
-0.8564(0.1591)
7.5188(1.5949)
-0.4664(0.5729)
•Stochastic frontier analysis and quantile regression produce significantly different results
Results: Self-thinning boundary line•Likelihood ratio tests indicated that the
inclusion of site index improved the model for Douglas-fir and red alder, but not for lodgepole pine
•The effect of fertilization in Douglas-fir was insignificant
•Red alder was also influenced by slope and aspect as well as soil water holding capacity
Conclusion• Stochastic frontier functions proved very useful
for this type of analysis and provided insights that other statistical techniques obscure
• SDImax values higher in this analysis slightly different than previously published values▫Lower for Douglas-fir, but higher for red alder
and lodgepole pine
• Douglas-fir and red alder support Bi’s general model, but lodgepole does not▫Site index only capture some of the variation for
red alder
Next Steps•Compare plantation to natural stands
•Use a more extensive red alder database
•Western Hemlock
Acknowledgements•Thanks to SMC, SNCC, HSC, BC Ministry
of Forests and their supporting members for access to the data