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## Modeling regional variation in the self-thinning boundary line

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**Modeling regional variation in the self-thinning boundary**line Aaron Weiskittel Sean Garber Hailemariam 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 Pinusradiata • 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’sareparameters**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**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 component Inefficiency Random noise**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 density**• Plot-specific SDImax showed no relationship with any other covariates**Results: Self-thinning boundary line**• 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 • SDImaxvalues 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