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