Download Application of Stochastic Frontier Regression (SFR) in the

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 evenaged 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)
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

N  Dq
ln N  ln    ln Dq
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. 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)
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  f (X ,  )  u  v
y = production (output)
X = k x 1 vector of input quantities
 = vector of unknown parameters
2
v = two-sided random variable assumed to be iid N (0,  v )
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 N  (0,  u2 ) the model
is referred to as the normal-half normal model
If u is assumed N  ( ,  u2 ) 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 remeasured (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 treenumber 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:
ln( N )     ln( Dq )  
 iid N (0,   )
2
Models

Fit the following Reineke model using SFR:
ln( N )     ln( Dq )  u  v
v iid N (0,  )
2
v
u iid N (0,  ) or iid N (  ,  )

2
u

2
u
Parameter estimation for SFR fits performed with Frontier Version 4.1 with ML
(Coelli 1996). To obtain ML estimates:
 &  are replaced by
2
v
2
u
 2   v2   u2
   
2
u
2
v
2
u
Model Fits – Douglas Fir
Coefficient
Least Squares
(Reineke’s)
Half-Normal
(SFR)
Truncated-Normal
(SFR)
Standard
Standard
Standard
Estimate
Error
t Ratio
Estimate
Error
T Ratio
Estimate

10.307
0.1291
79.835
10.857
0.1443
75.244
10.895
0.1333
81.895

-0.956
0.0416
-22.994
-1.077
0.0429
-25.130
2
0.0499
0.34678
0.3208
1.081
0.96202
0.0343
28.056
-1.1552
1.6297
-0.7088
-1.050
0.11849
0.0452
0.0145
-23.218
8.150
 u2
0.10893
0.33361
 v2
0.00956
0.01317

0.91926

log L
22.375
37.348
0.0316
29.048
39.017
Error
t Ratio
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 halfnormal and truncated-normal models – thus
we will use the half-normal model.
Model Fits – Douglas Fir

Comparison of slopes for OLS and halfnormal 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)
Model Fits – Radiata Pine
Coefficient
Least Squares
(Reineke’s)
Half-Normal
(SFR)
Truncated-normal
(SFR)
Standard
Standard
Standard
Estimate
Error
t Ratio
Estimate
Error
t Ratio
Estimate

10.696
0.0868
123.2387
11.084
0.0837
132.410
11.042
0.0793
139.175

-1.208
0.0279
-43.2406
-1.251
0.0264
-47.306
-1.253
0.0252
-49.799
2
0.0498
0.1143
0.0072
15.789
0.3299
0.0921
3.582
0.9625
0.0093
104.014
-1.127
0.4716
-2.3898
 u2
0.1049
0.3175
 v2
0.0103
0.0124

0.9177

log L
75.325
143.936
0.0153
60.011
153.268
Error
t Ratio
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 = -1.208 (0.0279)
SFR slope = -1.253 (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 -1.05 and 1.253,
respectively
This supports Weller’s (1985) conclusion that
this slope is not always near the idealized
value of -3/2