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 -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

qDN

qDN lnlnln

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:

vuXfy ),( 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

),0( 2vN

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

),0( 2uN

),( 2uN

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:

)ln()ln( qDN

),0( iid 2 N

Models Fit the following Reineke model using SFR:

vuDN q )ln()ln(

),0( iid 2vNv

),( or ),0( 22uu NiidNiidu

Parameter estimation for SFR fits performed with Frontier Version 4.1 with ML (Coelli 1996). To obtain ML estimates:

by replaced are & 22uv

222uv

22

2

uv

u

Model Fits – Douglas Fir Least Squares

(Reineke’s) Half-Normal

(SFR) Truncated-Normal

(SFR)

Coefficient Estimate

Standard

Error t Ratio Estimate

Standard

Error T Ratio Estimate

Standard

Error t Ratio

10.307 0.1291 79.835 10.857 0.1443 75.244 10.895 0.1333 81.895

-0.956 0.0416 -22.994 -1.050 0.0452 -23.218 -1.077 0.0429 -25.130

2 0.0499 0.11849 0.0145 8.150 0.34678 0.3208 1.081

2u 0.10893 0.33361

2v 0.00956 0.01317

0.91926 0.0316 29.048 0.96202 0.0343 28.056

-1.1552 1.6297 -0.7088

Llog 22.375 37.348 39.017

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 = -0.956 (0.0416) SFR slope = -1.050 (0.0452)

Model Fits – Radiata Pine Least Squares

(Reineke’s) Half-Normal

(SFR) Truncated-normal

(SFR)

Coefficient Estimate

Standard

Error t Ratio Estimate

Standard

Error t Ratio Estimate

Standard

Error t Ratio

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

2u 0.1049 0.3175

2v 0.0103 0.0124

0.9177 0.0153 60.011 0.9625 0.0093 104.014

-1.127 0.4716 -2.3898

Llog 75.325 143.936 153.268

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