Application of Stochastic Frontier Regression (SFR) in the Investigation of the Size-Density Relationship

1
Application of Stochastic Frontier Regression
(SFR) in the Investigation of the Size-Density
Relationship

• Bruce E. Borders and Dehai Zhao

2
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
• Reinekes 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 -1.605 (Stand Density Index SDI)

3
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

4
Limiting Relationship Functional Form of Reineke
5
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)

6
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

7
Other Fitting Methods

• Partitioned Regression, Logistic Slicing (Thomson
  et al. 1996- 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)

8
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

9
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)

10
Stochastic Frontier Regression (SFR)

• SFR Model Form

y production (output) X k x 1 vector of input
quantities b 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
11
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

12
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

13
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

14
Data New Zealand

• Radiata Pine (Pinus radiata) Carter Holt
  Harveys 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

15
Models

• Fit the following Reineke model using OLS

16
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
17
Model Fits Douglas Fir
18
Model Fits Douglas Fir

• The g 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.

19
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 -0.956 (0.0416)
• SFR slope -1.050 (0.0452)

20
Model Fits Radiata Pine
21
Model Fits Radiata Pine

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

22
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 -1.208 (0.0279)
• SFR slope -1.253 (0.0252)

23
Limiting Density Lines Douglas Fir
24
Limiting Density Lines Radiata Pine
25
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

26
Summary/Conclusions

• Our fits indicate the slope of the self thinning
  line for Douglas Fir and Radiata Pine grown in
  New Zealand are -1.05 and 1.253, respectively
• This supports Wellers (1985) conclusion that
  this slope is not always near the idealized value
  of -3/2