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 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 = β 0 S β1 N β2  where B is stand biomass per unit area, N is stand density, S is relative site index, and β i ’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) = β 10 - β 11 ln(QMD) + ε 11  QMD is quadratic mean diameter and TPA is trees per acre Compared to fits obtained using quantile regression Maximum stand density index (SDI max ) 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

Data SpeciesData SourceTotal AgeDensity (# acre) Site index (ft) Douglas-firSMC, SNCC (base age 50) Red alderHSC (base age 30) Lodgepole pine BC Ministry of Forests – 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 Q it = exp(ß 0 + ß 1 ln(x it )) * exp(v it ) * exp(-u it ) Deterministic component Random 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 density SpeciesMeanStd. Dev.MinMax Douglas-fir Red alder Lodgepole pine Plot-specific SDI max showed no relationship with any other covariates

Results: Self-thinning boundary line SpeciesSFAQuantile regression InterceptSlopeInterceptSlope Douglas-fir (0.2246) (0.0708) (0.3604) (0.1256) Red alder (0.3017) (0.1171) (0.1849) (0.0666) Lodgepole pine (1.6751) (0.1591) (1.5949) (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 SDI max 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