Application of Stochastic Frontier Regression (SFR) in the Investigation of the Size-Density Relationship Bruce E. Borders and Dehai Zhao

Self-Thinning Relationship -3/2 Power Law (Yoda et al. 1963) – in log scale relationship between average plant mass and number of plants per unit area is a straight line relationship with slope = -1.5 Reineke’s Equation - for fully stocked even- aged stands of trees the relationship between quadratic mean DBH (Dq) and trees per acre (N) has a straight line relationship in log space with a slope of (Stand Density Index – SDI)

Self-Thinning Relationship Over the years there has been an on-going debate about theoretical/empirical problems with this concept One point of contention has been methods used to fit the self-thinning line as well as which data to use in the fitting method

Limiting Relationship Functional Form of Reineke

Self-Thinning Relationship Three general methods have been used to attempt “fitting” the self-thinning line  Yoda et al. (1963) arbitrarily hand fit a line above an upper boundary of data  White and Harper (1970) suggested fitting a Simple Linear Regression line through data near the upper boundary using OLS (least squares with Data Reduction)

Self-Thinning Relationship  Use subjectively selected data points in a principal components analysis (major axis analysis) (Mohler et al. 1978, Weller 1987) or in a Reduced Major Axis Analysis (Leduc 1987, Zeide 1991) NOTE – in this method it is necessary to differentiate between density-dependent and density-independent mortality  Clearly, the first approach is very subjective and the other two approaches result in an estimated “average maximum” as opposed to an “absolute maximum” size-density relationship

Other Fitting Methods Partitioned Regression, Logistic Slicing (Thomson et al limiting relationship between glacier lily (Erythronium grandiflorum) seedling numbers and flower numbers (rather subjective methods) Data trimming method (Maller 1983) Quantile Regression (Koenker and Bassett 1978) Stochastic Frontier Regression (Aigner et al. 1997)

Stochastic Frontier Regression (SFR) Econometrics fitting technique used to study production efficiency, cost and profit frontiers, economic efficiency – originally developed by Aigner et al. (1977) Nepal et al. (1996) used SFR to fit tree crown shape for loblolly pine

Stochastic Frontier Regression (SFR) SFR models error in two components:  Random symmetric statistical noise  Systematic deviations from a frontier – one-side inefficiency (i.e. error terms associated with the frontier must be skewed and have non-zero means)

Stochastic Frontier Regression (SFR) SFR Model Form: y = production (output) X = k x 1 vector of input quantities  = vector of unknown parameters v = two-sided random variable assumed to be iid u = non-negative random variable assumed to account for technical inefficiency in production

Stochastic Frontier Regression (SFR) SFR Model Form:  If u is assumed non-negative half normal the model is referred to as the normal-half normal model  If u is assumed then the model is referred to as the normal-truncated normal model  u can also be assumed to follow other distributions (exponential, gamma, etc.)  u and v are assumed to be distributed independently of each other and the regressors  Maximum likelihood techniques are used to estimate the frontier and the inefficiency parameter

Stochastic Frontier Regression (SFR) The inefficiency term, u, is of much interest in econometric work – if data are in log space u is a measure of the percentage by which a particular observation fails to achieve the estimated frontier For modeling the self-thinning relationship we are not interested in u per se – simply the fitted frontier (however – it may be useful in identifying when stands begin to experience large density related mortality) In our application u represents the difference in stand density at any given point and the estimated maximum density – this fact eliminates the need to subjectively build databases that are near the frontier

Data – New Zealand Douglas Fir (Pseudotsuga menziessi) – Golden Downs Forest – New Zealand Forestry Corp  100 Fixed area plots with measurement areas of 0.1 to 0.25 ha  Various planting densities and measurement ages  Initial stand ages varied from 8 to 17 years – plots were re- measured (most at 4 year intervals)  If tree-number densities for adjacent measurements did not change we kept only the last data point – final data base contained 269 data points

Data – New Zealand Radiata Pine (Pinus radiata) – Carter Holt Harvey’s Tokoroa forests in the central North Island  Fixed area plots with measurement areas of 0.2 to 0.25 ha  Various planting densities and measurement ages  Initial stand ages varied from 3 to 15 years (most 5 to 8 years) – plots were re-measured at 1 to 2 year intervals  We eliminated data from ages less than 9 years and if tree- number densities for adjacent measurements did not change we kept only the last data point – final data base contained 920 data points

Models Fit the following Reineke model using OLS:

Models Fit the following Reineke model using SFR: Parameter estimation for SFR fits performed with Frontier Version 4.1 with ML (Coelli 1996). To obtain ML estimates:

Model Fits – Douglas Fir

The  parameter is shown to be statistically significant for the SFR fits. Thus, we conclude that u should be in the model. Testing u and the log likelihood values show there is no difference between the half- normal and truncated-normal models – thus we will use the half-normal model.

Model Fits – Douglas Fir Comparison of slopes for OLS and half- normal model show they are very close and can not be considered to be different from one another  OLS slope = (0.0416)  SFR slope = (0.0452)

Model Fits – Radiata Pine

The  parameter is shown to be statistically significant for the SFR fits. Thus, we conclude that u should be in the model. For radiata  is different from zero and the log likelihood values shows that the SFR truncated-normal model is preferred

Model Fits – Radiata Pine Comparison of slopes for OLS and truncated half-normal model show they are very close and can not be considered to be different from one another  OLS slope = (0.0279)  SFR slope = (0.0252)

Limiting Density Lines – Douglas Fir

Limiting Density Lines – Radiata Pine

Summary/Conclusions SFR can help avoid subjective data editing in the fitting of self-thinning lines In our application of SFR we eliminated data points in stands that did not show mortality during a given re-measurement interval The inefficiency term, u, of SFR characterizes density-independent mortality and the difference between observed density and maximum density Thus – SFR produces more reasonable estimates of slope and intercept for the self-thinning line

Summary/Conclusions Our fits indicate the slope of the self thinning line for Douglas Fir and Radiata Pine grown in New Zealand are and 1.253, respectively This supports Weller’s (1985) conclusion that this slope is not always near the idealized value of -3/2