I
PARAMETER E TIMATIO F R A HORTL F
PINE (Pinus echinolo Mill.) SA L ARE
GROWTH MODEL
By
CHARLES EDWARD ROSE Jr.
Bachelor of Science
Northern Arizona University
Flagstaff, Arizona
1995
Submitted to the Faculty of the
Graduate College of the
Oklahoma State University
in partial fulfillment of
the requirements for
the Degree of
MA TER OF SCIE CE
May, 1998
PARAMETER ESTIMATIO FOR A SHORTLEAF
PI E (Pinus echinarQ Mill.) BASAL AREA
GROWTH MODEL
Thesis Approved:
to;), g £Z.t! ~ean of the Graduate College
11
PREFACE
The purpose of this study was to model basal area growth using a system of
equations that accounts for tree interdependency within a plot by using seemingly
unrelated regression (SUR) to estimate the parameters. A major regression assumption is
that the error tenus are independent. For Forestry applications, trees within a plot are not
independent. If the independent observation assumption is violated the parameter
estimates standard errors may be underestimated and the mean square error may be
overestimated. Using seemingly unrelated regression to estimate the parameters of a
system of equations accounts for the correlation between error terms and tree
interdependency within a plot.
I wish to express my sincere gratitude to my major adviser, Dr. Thomas Lynch for
his support, understanding, and patience in guiding me through the research and
development process. I wish to express my gratitude to my other committee members,
Drs. Lawrence Gering and Mark Payton for their support and encouragement. I wish to
thank Michael Huebschmann for his guidance and patience with me in using SAS. I
would also like to express my gratitude to the Oklahoma State University Department of
Statistics for their guidance and support. Finally I would like to express my sincere
gratitude to the Oklahoma State University Department of Forestry and the USDA Forest
Service for this research opportunity and their financial assistance.
111
Chapter
TABLE OF CONTENTS
Page
1. INTRODUCTION 1
Objectives 2
II. LITERATURE REVIEW .4
Data Classification 4
Biological Considerations 6
Model Types 9
Stand Models ] 0
Densityfree models 10
Variabledensity models 12
Diameter Class Models 15
Individual Tree Models 17
Distancedependent 17
Distanceindependent 20
System of Equations 32
III. DATA 37
IV. METHODS .43
Model Considerations .43
Revised Data Set .45
Development of a System of Equations .47
Models 48
Calibration and Validation 54
Model Evaluation Criteria 57
V. RESULTS 61
Calibration 6]
Validation 65
lV
Chapter Page
VI. DISCUSSION 77
Complete Data Set 79
Conclusion 87
LITERATURE CITED 88
APPENDICES 95
APPENDIX A  SAS SUBROUTINE PROGRAM THAT RANKS A D
CREATES FOUR CLASSES WITHIN EACH PLOT
CORRESPONDING TO THE INDIVIDUAL TREE
DBHWlTHIN A PLOT 96
APPENDIX B  SAS SUBROUTINE PROGRAM THAT COMPUTES
THE MEANS FOR A NONLINEAR FUNCTION BY
DBH RANK CLASS WITHIN EACH PLOT BETWEEN
EACH ITERATION FOR FITTING A NONLINEAR
SYSTEM OF EQUATIONS 98
v
Table
LIST OF TABLES
Page
1. Attributes and class ranges for the USDA Forest ServiceOklahoma State
University cooperative research plots for the natural evenaged shortleaf
pine growth and yield study 38
2. Summary statistics for the complete data set of the Ouachita Highlands natural
evenaged shortleaf pine study for developing a basal area growth model
(N = 8928) 41
3. Example of ranking and placing trees in correct DBHrank class by plot
for use in a system of equations 46
4. Summary statistics for the calibration data set of the Ouachita Highlands natural
evenaged shortleaf pine study for developing a basal area growth model
(N = 6099) 55
5. Summary statistics for the validation data set of the Ouachita Highlands natural
evenaged shortleaf pine study for developing a basal area growth model
(N = 2829) 56
6. Parameter estimates, standard errors, and descriptions for Models I and 2
when fitted to the calibration data set 61
7. Parameter estimates, standard errors, and descriptions for Model 3
when fitted to the calibration data set 62
8. Parameter estimates, standard errors, and descriptions for Model 4
when fitted to the calibration data set 62
9. Fit index and mean square error for all models
using the calibration data set 64
10. Summary statistics for all models
using the validation data set 66
11. Average deviation for all models by DBH class
using the validation data set 66
Vi
Table Page
12. Average deviation by site index, basal area per acre, and age classes
for all models using the validation data set 67
13. Mean square error for all models by DBH class
using the validation data set. 68
14. Mean square error for all models by site index, basal area per acre. and
age classes using the validation data set 69
15. Mean absolute deviation for all models by DBHclass
using the validation data set 69
16. Mean absolute deviation for all models by site index, basal area per acre,
and age classes using the validation data set 70
17. Average absolute error as a percentage of mean average annual basal area
growth by DBH class for all models using the validation data set 71
18. Average absolute error as a percentage of mean AABAG by site index,
basal area per acre, and age classes for all models using the
validation data set 72
19. Parameter estimates, standard errors, and descriptions for
Models 1 and 2 when fitted to complete data set 80
20. Parameter estimates, standard errors, and descriptions for
Model 3 when fitted to complete data set RO
21. Models 1, 2, and 3 fit index, mean square error, and error sum of squares
when fitted to the entire data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Vll
Figure
LIST OF FIGURES
Page
1. Number of trees by diameter class for the
complete data set (N = 8928) 42
2. Number of trees by DBH class for the calibration (N=6099) and
validation (N=2829) data sets 57
3. Box plots of residuals by DBH class for all models using the
validation data set 73
4. Box plots of residuals by site index class for all models using the
validation data set 74
5. Box plots of the residuals by plot basal area class for all models
using the validation data set 75
6. Box plots of residuals by plot age class for all models using the
validation data set 76
7. Average deviation by DBH class and mean average annual basal area growth
for Models 1, 2, and 3 when fi tted to the complete data set 83
8. Average deviation by site index class and mean average annual basal
area growth for Models 1,2, and 3 when fitted to the complete data set R4
9. Average deviation by plot basal area class and mean average annual basal
area growth for Models 1, 2, and 3 when fitted to the complete data set 85
1O. Average deviation by age class and mean average annual basal area growth
for Models 1, 2, and 3 when fitted to the complete data set 86
Vlll
CHAPTER I
INTRODUCTION
The shortleaf pine (Pinus echinata Mill.) forest type is classified by the USDA
Forest Service as forests in which pine occupy at least 50% of the stocking of all live
trees with shortleafbeing the main pine species (USDA Forest Service 1972). Shortleaf
pine has a range of over 440,000 square miles in twentytwo states and is the most widely
distributed of the southern yellow pines (Willet 1986). Shortleaf pine distribution ranges
from Texas to New York and is second to loblolly pine (Pinus taeda L.) in terms of total
softwood volume for the southern pines. In recent years there has been concern about
loblolly pine being planted outside its natural geographic range which has renewed an
interest in the shortleaf pine resource (Willet 1986).
Oklahoma has approximately 765,000 acn:s of shortleaf pine located in the
Ouachita Highlands in the southeastern area of the state. Shortleaf pine has a slower
juvenile growth rate and regeneration is difficult when compared with the other southern
pines, but stern and crown form are generally better (Guldin 1986), and although shade
intolerant, shortleaf pine has the ability to tolerate drier upland sites (MeWilliams et al.
1986). More than 94% of shortleaf pine stands originate from natural regeneration and
these stands outperfonn other southern pines in areas where there are cold temperatures,
ice, and drought conditions (Williston and Balmer 1980).
2
The primary timber region for shortleafpine is found in the Ouachita Highlands
of eastern Arkansas and western Oklahoma and south into Louisiana and Texas (Braun
1950). Shortleafpine volume currently accounts for approximately 22% of the southern
pine compared with 57% for loblolly pine for the southern pines (McWillams et a1.
1986), but in the Ouachita Highlands shortleaf pine is the dominant pine species and
represents more than 50% of the softwood volume (van Hees 1980). Shortleafpine has
been steadily declining since the 1960's, mainly due to the replacement of mature stands
with other southern pines, predominantly loblolly pine (McWilliams et al. 1986).
Because shortleaf pine in the Ouachita Highlands has been characterized as slow growing
with below average volume per acre when compared with the other southern pines of the
region (Smith 1986), national corporations such as Weyerhaeuser have steadily converted
harvested shortleaf pine stands to loblolly plantations. However, nonindustrial private
individuals own the majority of the timberland throughout the Ouachita Highlands.
Because of the high cost of planting, shortleaf pine should continue to be a valuable
resource. Loblolly pine has been the dominant southern pine for commercial use in the
South and has been extensively managed and planted, but there is renewed interest in the
management of shortleaf pine.
Objectives
The purpose of this study is to develop a basal area growth model for natural
evenaged shortleaf pine in the Ouachita Highlands of southeastern Oklahoma and eastern
Arkansas. Currently the forest projection system for southeastern Oklahoma and eastern
Arkansas uses a basal area growth model developed by Hitch (1994). This study seeks to
3
improved upon Hitch' s basal area growth model used in the shortleaf pine stand simulator
(SLPSS) (Huebschmann et al. 1998) by:
(l) developing a system of equations based on tree diameter rank classes and
using seemingly unrelated regression (SUR) to estimate the common
parameters;
(2) compromising between a tree and standlevel model and accounting for
error correlation between tree diameter classes;
(3) using variables that are biologically reasonable while minimizing the use of
highly correlated variables; and
(4) recommending the suitability of the model through validation and evaluation.
CHAPTER II
LITERATURE REVIEW
This literature review focuses on growth and yield models for shonleaf pine and
models of other species that penain to developing a basal area growth model for natural
evenaged shortleaf pine of the Ouachita Highlands. Extensive research has been
conducted for growth and yield models of the southern pines with most studies
concentrating on loblolly and slash pine. Growth and yield models typically use linear
and nonlinear regression techniques to estimate the parameters. The classifications for
system of equations and parameter estimation for seemingly unrelated regression (SUR)
will be discussed. Murphy (1986) compiled a summary of growth and yield studies for
shortleaf pi.ne.
Data Classification
Growth and yield studies begin with data collection. Moser and Hall (1969)
suggested that forest growth can be considered a time series and that data collection over
time approximates a record of forest growth. They suggested three data classifications:
"real growth series", "abstract growth series", and "approximated real growth series."
The ideal data source for developing growth and yield equations would be a
complete chronological record of several stands from establislunent to harvest known as a
4
5
"real growth series.' Because of the time and expense involved in collecting data from
establishment to harvest, the "real growth series" is not practical or efficient for data
collection. While the 'real growth series' has an advantage of following a stand through
its entire life span, it is difficult to maintain and record data from stands representing a
wide variety of stand conditions for a species study.
One data collection method that has often been employed in yield studies is
known as an "abstract growth series." The "abstract groVvth series" consists of data
collected from numerous temporary plots covering a wide range of sites and ages to
accurately reflect stand conditions. The "abstract growth series" is desirable for efficient
data collection but individual trees can't be monitored for growth over time.
A common method for growth and yield research data collection is a compromise
between the two preceding methods and is known as an "approximated real growth
series" which consists of permanent plots that are remeasured at fixed intervals to
approximate the rate of growth within a geographical location. While an "approximated
real growth series" lacks the complete chronological history of the stand, since the
remeasurement of permanent plots may approximate the ideal data source for a particular
geographical location in a relatively short time it has been widely used by researchers.
The "real growth series" would be the ideal method for data collection for growth
and yield equations. However, the most economical and practical method for developing
growth models is based on a few repeated measurements of stands representing a variety
of ages, site indices, and densities. The "approximated real growth series" gives a good
approximation of actual forest growth through remeasurement of numerous plots that
incorporate a variety of stand ages and site qualities.
6
Biological Considerations
Mathematical models for describing growth should be biologically reasonable in
light of what is known concerning natural laws and biological processes. Empirical
equations may be developed for data sets that accurately describe the data but have no
biological basis and therefore may not accurately predict future growth (Vandal' 1994).
Among the biological considerations for models of tree growth are that there is an upper
asymptote relating to the maximum tree size for a given species, the growth rate after the
juvenile stage is inversely related to age, and that growth rate is inversely related to the
amount of competition. There are numerous biological processes restricting and limiting
the growth and size of a tree that should be considered when developing a model to
describe and predict growth. A common teclmique is to predict the maximum potential
growth for a species and then modify the potential growth based on the competition for
resources.
Zeide (1989) stated that growth results from cell division and is an inherently
exponential process. However, except in the earliest stages of tree development, a simple
exponential function may not accurately describe growth because of catabolic processes
that restrict the growth of a tree. Some growth equations have two components to
account for the anabolic and catabolic interaction of tree growth (e.g. Bertalanffy 1951).
Growth equations may differ in structure, but all growth models should conform to
reasonable biological behavior.
Bertalanffy (1951) hypothesized that growth of an organism could be thought of
as the difference between the anabolic and catabolic rates. The anabolic and catabolic

components are opposing forces that are usually described by subtraction or division in a
differential equation describing gro\\.th. The ChapmanRichards (Chapman 196 J "
Richards 1959) function is a generalization of Bertalanffy 's (1951) growth model, which
although empirical has been used extensively in growth and yield equations and has the
following fonn:
7
(1)
where
ay r  =py qy at
y = tree size,
t = tree age, and
p, q, and r = constants (p, q > 0, 0 < r < 1).
The ChapmanRichards fimction's positive or anabolic term describes the cell
division while the negative or catabolic components describes environmental and selfregulatory
forces that oppose growth (Pienaar and Turnbull 1973). The anabolic and
catabolic components conceptually describe the biological processes of tree growth. The
ChapmanRichards function is widely used in forest growth and yield studies but is more
empirical than theoretical because it is based on a the generalization of Bertalanffy"s
growth model (Yang et al. 1978). Equations in the differential fonn usually describe
growth as either a linear function or a power function. The ChapmanRichards function
is typical of the family of growth equations with the catabolic term (qy) becoming more
prominent and restrictive as a tree ages.
Zeide (1989) stated that growth equations are usually combinations of power and
exponential functions but the relative growth rate of tree diameter is a power rather than
8
an exponential function of age. Therefore, he proposed the following growth equation
form:
(2)
where
y'
y
y = tree size,
t = tree age, and
a, b = constants (a, b > 0).
Zeide (1989) referred to equation 2 as the power decline since the incremental
increase at any given age is proportional to tree size. Because of the correlation between
diameter and crown size, the diameter indicates the amount of resources available. The
"a" parameter is interpreted as the initial relative growth rate and parameter "b"
represents the rate of aging. The power decline equation becomes a Schumacher type
equation (Schumacher 1939) when the "h" parameter equals two. Typically the power
decline predicts larger growth ratio than exponential equations because power functions
decrease more slowly.
There is an extensive body of work relating mathematical models to the biological
processes for describing growth processes. Zeide's (1989) study compared conventional
equation forms such as the ChapmanRichards function with the power decline. The
study results represented by the data set used show that the power decline equation
describes growth better than the conventional equations for different species, site
qualities, locations, and growth rates. A concern with the power decline was that all
predicted diameters were less than actual diameters for all site indices.
9
Martin and Ek (1984) conducted a study which concluded that empirical
equations may be more accurate than theoretical equations for a variety of data. but that
theoretical equations are usually more accurate for extrapolating predictions beyond the
range of the data. Yang et a1. (1978) found that a modified Weibull function was flexible
enough to describe most biological growth processes and possesses some desirable
theoretical characteristics for modeling growth. Regardless of whether an equation is
empirical or theoretical, careful consideration of explanatory variables is needed to
provide realistic and robust predictions by a model (Vanclay 1994).
Model Types
In order to meet the need for growth and yield information to effectively manage
the shortleaf pine resource, several types of shortleaf pine growth models have been
developed. Munro (1974) suggested the following three classifications for growth and
yield models: (1 ) standlevel models, (2) distanceindependent treelevel models, and (3)
distancedependent treelevel models. These three classifications can be divided into
subclasses (Davis and Johnson 1987).
According to Murphy (1986), regardless of the growth and yield model
classification, models are typically either inferential or predictive. Inferential studies are
designed statistically to answer specific questions about stand or tree structure.
Predictive studies are designed to produce mathematical models that are used to predict
growth and yield given certain stand characteristics.
The Davis and Johnson (1987) classificatiop. system will be used throughout this
paper for growth and yield model classification. Relatively little growth and yield

]0
research has been conducted for shortleaf pine in comparison to the other southern pines,
particularly loblolly pine.
Stand Models
Standlevel models use stand statistics such as basal area per acre, site index, trees
per acre, and volume and are classified by Davis and Johnson (1987) as either densityfree
or variabledensity models. Standlevel models require relatively little information
to predict stand growth and yield but only general information is obtained about future
stand conditions.
Densityfree models
Densityfree models use the concept of fully stocked stands to develop "normal"
yield tables or average stand density empirical yield tables. The term "normal" refers to
ideal fully stocked stands and is based upon the density of a stand that produces the
maximum cubictoot volume. Because "normal" fully stocked stands are subjective, few
stands in reality approach the yield of "nonnal" yield tables. "Normal" yield tables are
developed from temporary plots located in the fully stocked portion of a stand
representing various ages and site indices. The plot observations are sorted by volume
per unit area and site index classes, and volume is then plotted over age to obtained
"normal" yield curves. The earliest "normal" yield tables for natural evenaged shortleaf
pine are in the USDA Forest Service Miscellaneous Publication 50 (USDA Forest
Service 1929) which also provide site index curves. The Miscellaneous Publication 50
data were obtained from 188 temporary plots located throughout the southern United

11
States and results are presented in tabular and graphical form. ince plots wer elected
based on "normal" stocking, an adjustment for yield prediction should be made when
applying Miscellaneous Publication 50 to stands that are not normally stocked.
Sylvester (1938) constructed yield tables using data collected from 240 repeated
measurement plots of loblolly pine in Louisiana and Arkansas and compared anamorphic
guide curves and statistical methods for constructing yield tables with the yield tables of
Miscellaneous Publication 50, He inferred that the yield tables of Miscellaneous
Publication 50 were erroneous. In the comments appearing with Sylvester's (1938)
paper, F. X. Schumacher questioned the results because of a lack of information provided
in the analysis. Schumacher did note that the difference between the results and
Miscellaneous Publication 50 was probably due the subjective nature of what a fully
stocked or "normal" stand is and illustrated concerns associated with "normal" yield
tables. "Normal" yield tables have been used to predict the growth of a stand by
computing the periodic annual increment but are unreliable because the subject stand
usually has less density and therefore less growth and stocking than a "normal" stand.
Empirical yield tables are developed from plots having average stand density. A
volume versus age relationship is developed for these plots and thus the problem of
defining "normal" is eliminated, but the average stocking is still subjective.
The next major growth and yield study for natural evenaged shortleaf pine was
by Schumacher and Coile (1960) consisting of densityfree growth and yield equations
that have been extensively used but have some limitations concerning stocking because
density is not a variable in the model. The results are presented in equation form from
data collected from 74 temporary plots located in the North Carolina Piedmont district.
12
Variabledensity models
Variabledensity models use stand density as an explicit independent variable,
usually expressed as basal area per acre or number of trees per acre. Multiple regression
techniques are usually used to estimate parameters. Standlevel variabledensity models
were improvements over densityfree models. Schumacher (1939) developed the
prototype for the foHowing variabledensity model (Clutter et. a1. 1983):
where
v = an expression of per acre yield.
A = stand age,
f(S) = function of site index,
g (Ds ) = function of stand density, and
Pi = parameters.
Buckman (1962) published the first study that directly predicted growth from
current stand variables in a way that was compatible with yield using data from Red pine
in Minnesota. Compatible growth and yield models are defined as yield models that are
derived by the mathematical integration of growth models (Davis and Johnson 1987).
Clutter (1963) at about the same time as Buckman (1962) developed a compatible growth
and yield model for loblolly pine. The general steps involved in Clutter's (1963)
compatible equations are: (1) obtain models for current cubic foot volume and basal area
per acre as functions of current age, site index, and basal area. (2) derivatives of the
volume and basal area models are taken with respect to age to obtain growth models, (3)
13
coefficients for the growth equations are estimated using linear regression, and (4) growth
equations are integrated to obtain the volume and basal area projection models. The
major contribution by Buckman (1962) and Clutter (1963) is the compatibility of growth
and yield equations.
Murphy and Beltz (1981), using permanent plot data in Arkansas, Louisiana.
eastern Oklahoma, and eastern Texas, developed the first variabledensity models for
natural evenaged shortleaf pine. The volume growth prediction is obtained by using the
basal area growth projection equation in conjunction with the stand volume equation.
The future basal area per acre is projected as a function of stand density and age and then
used to predict volume as a function of site index, age, and stand density (Murphy and
Beltz 1981). The following year Murphy (1982) used the same data and basal area
projection equation to predict sawtimber volumes for natural evenaged shortleaf pine.
Stand volume variabledensity equations for natural evenaged shortleaf pine of
eastern Oklahoma and western Arkansas (Lynch et a1. 1991) were developed using a
stand volume equation that is related to the "Schumacher type yield model" through a
logarithmic transfonnation. The data are from 191 permanent plots that were established
in 19851987 in a cooperative effort between the USDA Forest Service and Oklahoma
State University. These equations may be used to estimate per acre merchantable cubicfoot,
sawtimber cubicfoot, boardfoot volumes of natural evenaged shortleaf pine. The
volume equations have the following general fonn:
14
wh re
v= volume per unit area,
B = basal area per unit area,
H =average total height of dominants and codominants, and
fJi = parameters.
Future volume is predicted by using the projected basal area and predicted height
from the site index equation. The basal area is projected using an equation that was
developed by Murphy and Beltz (1981) for shortleaf pine. The projected height for the
dominant and codominant shortleafpines for the Ouachita region is obtained using the
following equation (Graney and Burkhart 1973):
where
H = average height of dominants and codominants,
Sf = site index,
AGE =stand age, and
Qj = regression coefficients.
Equation 5 is used to predict the average height of the dominant and codominant
trees given stand age and site index. An alternative for predicting the average dominant
and codominant heights is to use the site index curves in Miscellaneous Publication 50
(USDA Forest Service 1929). Once the height and basal area are projected, equation 4
can be used to predict volume. The basal area projection equation developed by Murphy

15
and eltz (1981) has a tendency to underestimate future basal area because the model
development did not account for ingrowth. A basal area projection equation is presently
being developed for natural evenaged shOltleaf pine of the Ouachita Highlands which
should improve the volume predictions.
Diameter Class Models
Diameter class models provide more information than stand level models since
they provide volumes by diameter classes. Tree diameters are placed in diameter classes
and volume is computed for each diameter class. The stand volume is calculated by
aggregating the diameter class volumes. Diameter class models typically use a
probability distribution function such as the Weibull distribution function to allocate trees
to diameter classes. Other probability distribution functions have been used with varying
success such as the exponential and beta distributions. Bailey and Dell (1973) found the
Weibull to be a flexible funchon that has the capability to assume the full range of
unimodal continuous shapes of diameter distributions. The Weibull probability
distribution function has the following form:
(6)
( )
c Xa CI [( Xa )C]
f(X)=[; b exp  .b· a~X<C()
where
I(X) =0 otherwise
x = random variable,
a;::: 0, and
b, c >0.
16
The Weibull probability function has three parameters commonly denoted as a, b.
and c. The "a" parameter is the location parameter that indicates the lower end of the
diameter distribution and must be greater than or equal to zero for forest stands (Clutter et
al. 1983). The "b" parameter is referred to as the spread parameter and indicates the
width of the function and thus indicates the width of the diameter classes. The "c"
parameter describes the shape of the function (Avery and Burkhart 1994). When the "a"
and "c" parameters are zero and one respectively, the Weibull distribution function is
reduced to the exponential distribution function with the inverse ".T' shape that is
characteristic of unevenaged forests (Clutter et a1. 1983). When the "c" parameter is
approximately 3.6, the distribution is approximately normal (Johnson and Kotz 1970).
Once the parameters of the Weibull probability function have been estimated the
probabilities associated with each diameter class can be calculated and multiplied by the
number of trees in a stand to derive the number of trees for each diameter class.
Smalley and Bailey (1974) used a Weibull distribution function as part of a yield
prediction system for shortleafpine plantations in the Highlands of Tennessee, Alabama,
and Georgia. Tree diameter, height, and age were recorded on 104 plots of shortleaf pine
plantations. The tree height, survival, and age data were used to estimate the Weibull
distribution parameters. Mortality was estimated by the presence of dead or dying trees.
The maximumlikelihood estimates of the Weibull function parameters on each plot were
related to plot age, density, and site by regression analysis. The Weibull function was
then used to estimate the number oftrees surviving in each diameter class and a stand
table was constructed. The volumes by diameter classes were aggregated to obtain
volume per acre.

17
Individual Tree Models
Individual tree models may have advantages over standlevel models for certain
applications since they can simulate the competitive environment of each tree
individually by simulating the growth of each individual tree in diameter, height, and
crown. These models usually contain equations for prediction of individual tree growth,
probability of survival, and volume. Results for all trees in the simulated stand are
aggregated for per acre attributes, volumes, and growth rates (Davis and Johnson 1987).
Mortality and growth of an individual tree is dependent upon its relative position and size
in comparison to neighboring trees. A major distinction between stand level and
individual tree models is that individual tree models aggregate the stand volume after
each individual tree's growth and volume is calculated whereas the stand level model
aggregates individual tree data into stand characteristics before development of model
equations. While individual tree models are data intensive and more time consuming to
develop than stand level models, they provide more information about stand and tree
dynamics.
There are two types of individual tree models, distancedependent and distanceindependent.
The primary difference between the two types of individual tree models is
competition accountability.
Distancedependent
Distancedependent individual tree models have been developed to more
accurately simulate competition measures between neighboring trees. According to
18
Clutter et al. (1983), distancedependent models may provide more detail concerning tree
growth and the relationships between tree biological and ecological interactions but
require a tree list with the spatial separation between trees as a major component of the
input data. If spatial locations are unknown the simulator must generate a reasonable
map of tree locations before beginning the simulation.
The main assumption of distancedependent models is that better predictions of
individual tree growth can be obtained if each neighboring tree size and location is
known, but this assumption has not been empirically validated (Clutter et a1. 1983). In
addition to basic tree measurements, each tree's location must be plotted on XY
coordinates which gives the location of each tree within a plot. Distancedependent
growth projections usually proceed through the following steps:
(1) competition index is computed for each tree,
(2) mortality probabilities are computed as functions of the competition indices,
(3) periodic growth rate of each tree is predicted over the projection period
(usually one year), and
(4) individual tree volumes are predicted from the final projection
and aggregated for stand level statistics.
PTAEDA is a distancedependent model developed for loblolly pine (Daniels and
Burkhart 1975) which grows trees individually and assumes that each tree has a
theoretical maximum growth potential. PTAEDA was developed for managed loblolly
pine plantations to estimate the influence of different sites, spatial patterns, and
silvicultural regimes on tree growth. The growth potential is based upon the maximum

19
growth of an open grown tree and tree vigor and competition factors modify the potential
growth. The initial stand is determined by the location of each tree in the planting spatial
pattern. The PTAEDA model works in two stages: first a Weibull probability distribution
function determines the diameter for each individual juvenile tree and the model grows
the initial juvenile stand until intertree competition begins. Then in the second stage the
model accounts for intertree competition. The PTAEDA model grows in the juvenile
stage until the CCF' (Crown Competition Factor) (Krajicek et a1. 1961) reaches 100
percent, at which time it is assumed the intertree competition begins (Davis and Johnson
1987). The distancedependent competition measure used by the PTAEDA model is a
distanceweighted size ratio (DR) index developed by Hegyi (1974). Distanceweighted
size ratio indices are defined as the sum of the ratios between the dimensions of each
competitor to the subject tree weighted by a function of intertree distance. This definition
ofcompetitors has been preferred in recent years and has an advantage of being easy to
compute and explaining variation in growth with precision similar to that of other
distancedependent competition indices (Tome and Burkhart 1989). The PTAEDA
model has proven to be effective for southern pine plantations but has shortcomings for
natural regenerated stands (Davis and Johnson 1987).
In addition to the distanceweighted size ratio competition index used in
PTAEDA, numerous competition indices have been developed in an attempt to increase
the precision of gro'Wth and yield predictions. Most distancedependent competition
indices begin with Staebler's (1951) concept of a circular influence zone around a subject
ICCF is a stand density measure that describes the available area for the average in relation to maximum
area of an open grown tree (Avery and Burkhart 1994).
20
tree in which the competing trees reduce the rate of growth of the subject tree. The
overlap from circular influence zones of competing trees measures the amount of
competition. Spurr (1962) developed a point density measure in which trees included in a
fixed point sampling angle gauge (prism) sweep are considered to be competitors. Brown
(1965) developed the area potentially available index as a measure of point density in
which the area available for each tree was calculated as the area of the smallest polygon
bisecting the intertree lines.
Studies show that distancedependent competition indices contribute little or no
improvement in growth prediction when compared to use of distanceindependent
measures of competition (Tome and Burkhart 1989).
Di stanceindependent
Distanceindependent models usually project tree growth as a function of diameter
and stand level variables, typically having three main components: diameter growth.
height gro\\'1h, and mortality (Avery and Burkhart 1994). Distanceindependent models
assume that the spatial separation and tree diameters are uniformly distributed throughout
the stand (Davis and Johnson 1987). Since the exact location of each tree within a stand is
unknown, competition is normally defined by a comparison of a tree's characteristics
with that of other trees within a stand. Since distanceindependent models do not use
spatial information to formulate competition indices, they are less data intensive than
distancedependent models. An assumption of distanceindependent models is that if a
tree is smaller than the average tree within the stand r~garding crown, diameter, and
height then the tree lacks competitive vigor in comparison with larger trees.
21
Two commonly used competition indices are DD (ratio ofthe quadratic mean
diameter to an individual tree DBH) and BAL (the cumulative basal area of trees larger
than the su~ject tree). As the DD index decreases the tree is considered to be increasingly
vigorous and will grow at rates closer to its maximum theoretical potential. The BAL
index indicates that as the cumulative basal area of the trees larger than the subject tree
decreases, the subject tree is more competitive. The BAL index is used in Stage's (1973)
PROGNOSIS forest projection system. Krumland (1982) developed a competition index
that is based upon a tree's crown and is defined as the percentage ofland covered by a
live tree crown (measured at a height of 66 percent up the Jive crown of a subject tree).
This competition index has been shown to be effective but is difficult to measure.
Basal area growth is often used as the dependent variable for increase in stem
thickness because studies show that the correlation between various competition indices
are higher with tree basal area increment than with diameter increment (Bella 1971;
Johnson 1973). Since competition among trees for resources is a major component of
individual tree models it seems intuitive that basal area growth might yield the better
model. A study by West (1979) compared the results of growth equations using both
basal area growth and diameter growth as the dependent variable and found that the
correlation between tree basal area growth and parameters detennining the growth were
higher than for tree diameter increments. There was no evidence that the precision of
estimates of predicted diameters made with either diameter or basal area growth as the
dependent variable differed significantly. West (1979) concluded that the higher
correlation of basal area growth equations was probably due to the partial dependence of
basal area growth on the initial tree diameter.
22
The two primary methods for modeling distanceindependent tree growth are
composite modeling of tree growth as a function of tree, site, and stand characteristics
and the potentialmodifier growth function. The potentialmodifier fW1ction models tree
growth as a theoretical maximum potential based on an individual's tree characteristic
which is multiplied by a modifier to account for stand and tree characteristics as well as
competition.
PROGNOSIS is a composite distanceindependent individual tree model
developed by the USDA Forest Service for the western United States (Stage 1973) that
directly predicts the growth of a tree. The PROGNOSIS forest projection system uses the
composite growth model approach because the difficulty in obtaining dominantage site
relationships for mixed species stands makes potential growth difficult to estimate. The
PROGNOSIS growth model has the following general form (Wykoff 1990):
l7) In (dds) = COMP + SITE + f3 l In (dbh) + f3 2 dbh 2
where
CONI? = function of competition measures,
SITE = function of site quality,
dbh = diameter at breast height,
dds = IOyear periodic change in squared diameter (inches) and
f3i = regression coefficients.
The PROG:.JOSIS model predicts the natural logarithm of new growth in square
inches (dds) as a function of site, tree, stand, and competition characteristics and is easily
converted to either basal area or diameter. Since PROGNOSIS was developed for mixed
23
conifers it uses habitat type, geographic location, slope, aspect, and elevation to express
site quality rather than site index. Crown ratio, crown competition factor (CCF), and the
cumulative basal area of trees larger than the subject tree (BAL) measure competition.
The composite model approach has performed adequately and is used extensively
throughout the western United States but is not often applied to southern pines. Wykoff
(1990) presents a detailed discussion of the model development and performance.
The potentialmodifier growth function used in many forest projection systems
consists of the potential component which estimates the theoretical maximum diameter or
basal area growth of a tree growing free of competition and a modifier component that
reduces the potential based on competition factors. Parameters of potentialmodifier
growth functions have been fitted in two ways: (1) parameters for a tree's potential
growth are fitted based on an individual tree characteristics, then the modifier parameters
which are a function of tree and stand characteristics are fitted while holding the potential
parameters constant, and (2) parameters for the potentialmodifier function are fitted
simultaneously.
The STEMS forest projection system, which was developed for the Lake States of
the United States, uses a potentialmodifier function to estimate individual tree growth
(Belcher et al. 1982). The STEMS potential diameter growth function that was
developed for 26 species by Hahn and Leary (1979) using data collected from the
dominant and codominant trees throughout the Lake States has the following form:
(8) Potential Growth = j3 J + j3 2 DfJ 3 + j3 4 (SI) (CR) (D jP 5
24
where
Potential Growth = potential annual dbh growth (inches/year),
D = initial tree dbh (inches).
Sf = plot site index (base age = 50),
CR = tree crown ratio, and
fii = regression coefficients.
The potential diameter grov.rth was derived for each tree species over a tenyear
projection period. It uses the initial diameter, crown ratio. and site index as the
independent variables. In developing the potential growth function, Hahn and Leary
(1979) grouped the independent variables by species, oneinch diameter classes, 10foot
site index classes and 1Opercent crown ratio classes. It was assumed that the diameter
growth was normally distributed around the mean diameter growth. The potential growth
was estimated as the mean growth plus 1.65 standard deviations corresponding to the
95th percentile of dominant and codominant tree diameter growth. The mathematical
fonn of the potential growth model was developed as a generalization of the Richards
function (Richards 1959) and has a speciesspecific intercept along with anabolic and
catabolic terms. The catabolic term is an allometric relation of tree diameter and the
anabolic term is a product of crown length, site index and an allometric relation of tree
diameter. Leary and Holdaway (1979) developed a modifier function for use with Hahn
and Leary's (1979) potential growth function which is a function of size, site, and
competitive status and has the following form:
(9) Competition Modifier = 1 e f(R)g(AD)[ (BAmax  BA)/ BA] 0.5
25
where
Competition Modifier =competition index (bounded between 0 and 1),
f(R) = function characterizing the individual tree's relative diameter effect on
the modifier,
g(AD) = function characterizing the average stand diameter effect.
BAmax = maximum basal area per acre, and
BA = current basal area per acre.
In estimating the parameters of the modifier function, the parameters of the
potential growth function are held constant and because the competition modifier adjusts
potential growth which is constrained to be between zero and one. As the stand basal
area and competition increase the competition modifier approaches zero, and conversely,
the competition modifier approaches one as stand density and competition decrease to
allow trees growing relatively open and free of competition to approach the theoretical
potential growth.
The first distanceindependent growth model for shortleafpine was developed as
part of the TWIGS and STEMS forest projection system for the central United States
(Shifley 1987). The growth model data came from Forest Inventory and Analysis (FIA)
plots in Missouri, Ohio and I.ndi.ana and the model uses a potentialmodifier growth
function. Unlike the Lake States potential growth function. which is the sum of the two
components plus an intercept, the Central States potential growth function is the product
of two components and was developed as a function of tree size, crown ratio, and site
index for each species. The Central States potential growth function has the following
form:

26
v. here
Potential Growth = potential tree basal area growth (ft2/year),
BA = tree basal area (ft2/ac),
Sf = plot site index (base age = 50),
CR = individual tree's crown ratio, and
Pi = regression coefficients.
The first bracketed component in the potential growth function above is the
growth form of the ChapmanRichards function which is used to model maximum
potential growth and the second bracketed component adjusts potential basal area growth
by the tree crown ratio and site index, increasing or decreasing maximum potential
growth. The potential growth function was constrained to achieve maximum tree
diameter within biologically reasonable limits by comparing results from the potential
growth model based on the fastest growing five percent of the trees by species and
diameter class to the National Register of Big Trees (American Forestry Association
1982).
Shifley and Brand (1984) used a variation of the ChapmanRichards function to
account for a maximum tree basal area since potential growth approaches zero as the
basal area approaches the biological upper limit. The constraint was implemented by
using the first bracketed component of equation ]0 and setting the potential growth to
zero. Solving for basal area yields the following equation:
27
I
(11) Max Basal Area = (;;)1 fJ2
where
Max Basal Area = species estimated maximum basal area (ft2) and
/3i = estimated parameters.
The maximum basal area constrains equation 10 to a prespecified upper
asymptote and was developed to ensure that the model behaves realistically when
predicting growth for large diameter trees. Solving equation 11 for /33 and substituting
into equation 10 yields the Central States TWIGS present potential growth equation form.
The Central States modifier component reduces the potential growth in response
to competition and expresses growth as a percentage of the estimated potential growth.
The modifier has the following form:
(12)
where
Mod~fier =/39
1
(!!l+P8 D2 )(I. BA )2
1
BAL BA e max
lvfodijier = competition modifier (bounded between 0 and 1),
BAL = basal area (ft2/ac) of all trees larger than subject tree,
D = current tree dbh (inches),
BAmax = maximum basal area per acre,
BA = stand basal area (ft2/ac), and
/3i = regression coefficients.

28
The Central states modifier decreases growth as the basal area of larger trees
increases and larger diameter trees receive a smaller proportion of potential growth.
Shifley (1987) defined a maximum stand density of 200 square feet of basal area per acre
(BAMAX ) to constrain the modifier within biological reasonable limits. The shortleaf pine
potential modifier function of TWIGS has perfonned adequately but was fitted to data
outside the Ouachita Highlands geographical region.
Hitch (1994) developed the first shortleaf pine distanceindependent individual
tree for the Ouachita HigWands, which is currently being used in the SLPSS forest
projection system for evenaged natural shortleaf pine. The data for this study came from
permanent research plots established cooperatively by the USDA Forest Service and the
Forestry Department at Oklahoma State University. The plots were established in 198587
with remeasurements occurring at four or five year intervals. All shortleafpine having
a diameter of greater than or equal to one inch were recorded and each tree permanently
numbered. The following is a summary of basal area growth models Hitch (1994) fitted
and evaluated.
A modified PROGNOSIS type model was fitted using a single intercept in place
of the location effects and site effects were estimated by site index. PROGNOSIS was
developed for unevenaged stands of mixed conifers in the western United States.
consequently Hitch (1994) modified the model to include an independent variable to
represent stand age. The model performed adequately with a fit index and MSE of 0.593
and 0.000046, respectively. However, the model predicted a maximum tree size of more
than 70 inches DBH. The large maximum tree diameter is a result of the fact that the
research plots do not include stands in which the basal area growth has culminated.
29
Potentialmodifier functions were fitted to the data and evaluated in an effort to
better describe groVv1h. Hitch (1994) used three methods to constrain the maximum
growth of the potential fW1ction. The first was a modified form of Hahn and Leary's
(1979) STEMS potential growth model with twoinch diameter classes in which the
potential growth was estimated as the mean growth plus 1.65 standard deviations
corresponding to the 95th percentile of basal area growth. The second method fitted the
fastest growing fivepercent of the trees using the dominant and codominant trees in each
one inch diameter class with a modi.fied ChapmanRichards function (Shifley and Brand
1984; Shifley 1987) with tree size constrained to a biological reasonable maximum size.
The maximum tree size was derived as 36 inches (basal area = 7.068 W) by averaging the
maximum diameter found from local records and the largest shortleaf pine recorded in the
National Records of Big Trees (American Forestry Association 1992). The third method
was suggested by Amateis et al. (1 989) who proposed using the data from open grown
trees to approximate the potential growth for trees growing without competition. The
data for the potential function came from a study of open growing shortleaf pine in the
West Gulf Region (Smith et al. 1992). The potentialfunction was a variation of the
ChapmanRichards function that included an intercept component. All three methods
appear to constrain the maximum growth within biologically reasonable bounds and were
coupled with a modifier to determine which would best fit the data.
Hitch (1994) considered two modifier functions. The first modifier was Shifley's
(1987) variation ofthe STEMS and TWIGS function for the Central States. The second
modifier was a modified logistic function developed by Murphy and Shelton (1993).
This modifier easily adapts to a variety of stand and tree conditions by including more

30
independent variables to explain more variation in tree growth and has the following
form:
(13 )
where
Mod(fier
1 + exp [,B I BAL + fJ 2 S1 + ,B 3 AGE + ,B 4 BA + ..... ]
Modifier = competition index (bounded between 0 and I),
BAL = basal area (ft2/ac) of all trees larger than subject tree,
S1 = plot site index (base age = 50),
AGE = plot age (years),
BA = stand basal area (ft2), and
/3i = regression coefficients.
The final model Hitch (1994) developed used the modified ChapmanRichards
potential growth function (Shifley and Brand 1984) coupled with the modifier developed
by Murphy and Shelton (1993). The model predicts grov,,1h adequately with a fit index of
0.609 and MSE of 0.000044. Although the model adequately predicts basal area growth.
there is concern about model bias because of underpredictions for stands with kss than
45 square feet of basal area per acre and a underrepresentation of young stands with a
site index greater than 65 (base age equals 50 years). The model was developed using
data from stands that have not reached culmination of basal area per acre and may over or
under predict for stands beyond the range of the data.
Bitoki et al. (1997) developed a distanceindependent shortleaf pine individual
basal area growth model for unevenaged stands in the Ouachita Highlands. The data
came from CFI plots established by the Deltic Farm and Timber Company Inc. in 1965
66 with remeasurement occurring at fiveyear intervals. Individual trees were measured

31
and recorded for all trees greater than or equal to fiveinch dbh. Plots used to develop the
model had no harvesting activities or silvicultural treatments during the fiveyear
measurement interval and shortleaf pine comprised at least 70% of the basal area of each
plot. The basal area growth model was developed using a potentialmodifier basal area
growth model for which the parameters were fitted in two steps. First the theoretical
potential growth was fitted separately using a variation of the ChapmanRichards
function (Shifley and Brand 1984) with one parameter eliminated by using Hitch's (1994)
estimate for maximum basal area of an individual shortleaf pine. The other two
parameters were fitted using nonlinear regression. The second step consisted of fitting
the modifier function parameters to the complete potential modifier model by using
nonlinear regression while holding the potential growth constant. This was a variation of
the model used in TWIGS and described by Shifley (1987). The model fit appears
adequate with a fit index of 0.44, but because the study data came from CFI plots. the
data do not represent all stand conditions equally. There is concern about diameter
growth predictions for large trees and underrepresented site index classes because there
are few observations for trees greater than 16 inches DBH and for site index classes 40
and 70 (base age equals 50 years).
Potentialmodifier models usually estimate the parameters by estimating the
maximum potential growth in isolation and then holding potential growth constant while
estimating the parameters for the modifier. The potential function is normally fitted
separately using either a subset of the data or open grown tree data and the modifier is
fitted to the complete data set while holding the potential function constant. Often a
series of iterations of the two steps are completed to stabilize the parameters. Murphy
32
and Shelton (1996) proposed fining the data to the potential modifier function
simultaneously. Previously Wensel et al. (1987) fitted a potential modifier function
simultaneously for northern California conifers but found that it confounded the potential
and modifier effects. Murphy and Shelton's (1996) data are from a study on the growth
and development of loblolly pine in Arkansas and Louisiana. They selected a growth
function that is biologically reasonable for achieving a maximum growth rate. The
potential growth function selected bounded the function to an upper asymptote. Then a
variation of the logistic function (equation 13) was selected as the modifier, which is
constrained between zero and one. The potential function was fitted to obtain an estimate
of the parameters and then the potentialmodifier function was fined simultaneously
using nonlinear OLS (Ordinary Least Squares) using the potential growth function
estimates as initial values. The results reveal the model achieved a good fit with a fit
index of 0.69 and root mean square error of 0.56 square centimeters. The model exhihits
logical results when compared with biological processes. The logistic function modifier
can be easily adapted to add more variables as needed. However because the components
are fined simultaneously, the potential component cannot be analyzed in isolation and
does not necessarily equal the maximum theoretical potential growth.
System of Equations
Growth and yield studies often use a system of equations to describe stand
development. Early applications of systems of equations in forestry fitted the parameters
of each equation independently using OLS (e.g. Moser 1972). Furnival and Wilson
(1971) suggested that fitting the parameters for each equation in a system independently
33
was not satisfactory because a variable may be dependent in one equation and
independent in another equation. Therefore, coefficients of one equation may be
functionally related to coefficients in another equation. and the residuals of each equation
may be correlated. Fumival and Wilson (1971) proposed that parameters for a system of
equations describing forest growth and yield could be fitted simultaneously using known
econometric techniques. Simultaneously fitting parameters of a system of equations
provides an increase in parameter estimation efficiency.
The general simultaneous paranleter estimation technique for a system of
equations may be applied to linear or nonlinear systems with small, large, or unequal
sample sizes and with or without imposing constraints between parameters of the system
(Reed 1987). The optimal parameter estimation technique is defined by a given system
and empirical studies suggest efficiency is gained because of a reduction in the SSE (error
sum of squares) (Reed 1987). Parameter estimation for equations within a system is
accomplished using either linear or nonlinear regression techniques to minimize the SSE.
There are three steps typically used for estimating parameters of a system of
equations. First the parameters for each equation within a system are independently
estimated by using linear or nonlinear OLS. Second, the variancecovariance matrix is
estimated using the error tenns of the independently estimated equations. Third, the
generalized least squares is used to estimate the parameters of the system using the
residual variancecovariance matrix (parameter constraints may be imposed) (Reed
1987).
Pindyck and Rubinfeld (1981) classified systems of equations as simultaneous
equations, recursive equations, or seemingly unrelated equations. Systems of
......
34
simultaneous equations have variables that are independent in one equation and
dependent in another equation and crossequation error correlation exists (equation 14).
Parameter restrictions may occur both within and across equations and independent
variables are either endogenous (determined previously in the system of equations) or
exogenous (determined independently of the system of equati.ons). Two stage and three
stage least squares may be used to estimate biased but consistent parameters (Borders
1989).
( 14)
where
Y 2 = ~20 + ~21YI + ~23Y3 + ~24XI + P25X2 + e2
Y J = ~30 + ~31YI + ~32 Y2 + ~34XI + 035X2 + e3
Yj = endogenous variables,
Xi = exogenous variables, and
ei =error term.
Recursive systems of equations have sequential relationships between endogenous
variables and OLS can be used for parameter estimation if no cross equation correlation
between the err components exists. A recursive system of equations by definition has
no correlation between righthand side endogenous variables and the error components of
the lefthand side endogenous variables. The OLS estimation for recursive system of
equations requires that error components for the system of equations be pairwise
uncorre1ated (Borders 1989). Because recursive systems of equations are not cross
correlated there is no gain in efficiency for simultaneous parameter estimation unless
constraints are imposed between equation parameters (Reed 1987).


35
Seemingly unrelated equations hav 110 analytical relationship between equations
but are linked because the error terms across equations are correlated (Pindyck and
Rubinfeld 198]). If the cov (ei, ej) = 0 for all combinations of i and j then SUR is
inappropriate, but if the cov (ei, ~j) ;;t:. 0 then a correlation between the errors of the
equations exists and SUR may provide a gain in parameter estimation efficiency. The
three step procedure for estimating parameters for a system of equations discussed
previously is appropriate and parameter estimation efficiency is improved by accounting
for the cross equation correlation. Zellner (1962) suggl:sts that efficiency in parameter
estimation may be gained if the system is viewed as a single equation and is
accomplished by using the generalized least squares estimation.
Lynch and Murphy (1995) used SUR for parameter estimation in developing a
compatible height prediction and projection system for natural evenaged shortleaf pine
of the Ouachita Highlands. The study data are from 208 pennanent growth and yield
research plots established from 198589 and the data include individual tree heights,
diameters, plot densities, ages, and site indices. The two compatible height equations
developed are: (1) height prediction for time one and (2) height prediction/projection for
time two given the height at time one. The SUR parameter estimation technique was
used because error correlation was expected from the common parameters and because
heights measured at different times are correlated. SUR was used for efficient parameter
estimates for both equations by providing consistency across equations with parameter
restrictions placed across equations. OLS estimates for each individual equation were
presented for comparison to SUR estimates but the OLS individually estimated
.....


36
parameters differed by as much as 20% from the SUR parameter estimates. Since the
equations are compatible and interrelated using the first equation with OLS parameters to
project height in the second equation may be inaccurate. Studies have indicated that
parameter variances estimated from large samples using SUR may be less than parameter
variances obtained using OLS (Judge et al. 1988). The advantage of SUR in the
compatible height prediction and projection system is that the parameters may be used by
either equation depending upon available information for height prediction. SUR
provided a good fit for both equations while accounting for error correlation between the
two equations.
Hasenauer et al. (1998) recently fitted a system of three equations separately using
OLS, and simultaneously using two and threestage least squares. The three equations
are a basal area growth increment model, height increment model. and crown ratio model.
The data are from the Austrian National Forest Inventory which consists of over 7.500
Norway Spruce (Picea abies 1. Karst). Since the results indicated high crossequation
correlation, the threestage least squares was the most efficient technique.

CHAPTER III
DATA
The data are primarily from the cooperative study being conducted by the USDA
Forest Service and Oklahoma State University Department of Forestry to develop growth
and yield models for natural evenaged shortleaf pine stands of the Ouachita and Ozark
National Forests of southeastern Oklahoma and eastern Arkansas. These data constitute
an approximated real growth series. Original plot installation was during the dormant
season of 19851987 when basic forest measurements were recorded, subsequently
remeasurements were recorded on a four or five year interval for each plot. For a detailed
discussion of plot reconnaissance, installation, and location see the USDA Forest Service
establishment and progress report (Murphy 1988a).
A total of 191 plots located in the Ouachita and Ozark National Forests were
installed which covered a wide spectrum of site, age, and density classes. There are four
site index, age, and density classes for a total of 64 combinations (Table 1).
Each combination of the siteagedensity classes originally was to have 3
replicates for a total of 192 plots. However only two plots were located for the age 20
years, site index> 75 feet, and residual basal area 30 square feet combination. Therefore,
only 191 plots were actually installed (Murphy 1988a).
37

38
Table 1. Attributes and class ranges for the USDA Forest ServiceOklahoma State
University cooperative research plots for the natural evenaged shortleaf pine
gro~rth and yield study.
Attribute Class Range Class Midpoint
Basal area per acre (sq.ft.)
Site index (base age = 50 years)
Age (years)
16~5 30
4675 60
76105 90
106135 120
< 56
5665 60
6675 70
> 75
1130 20
3150 40
5170 60
7190 80
The original plots were selected based upon agesitedensity classes and the
following stand criteria:
(1) naturally regenerated stands containing at least 70 percent shortleaf pine in terms
of basal area for trees 0.6 inches DBH and larger;
(2) maximum age range of dominant and codominant trees was 10 years or less;
(3) less than 10foot variation for site index within a stand;
(4) evenaged forest distribution with no obvious holes or clumping and no more than
two age classes per plot; and
(5) no significant insect, disease, or fire damage and no harvesting during the
previous five years.
For plot reconnaissance, the shortleaf pine and other species plot basal area were
tallied separately using a 10factor prism. Five dominant or codominant shortleaf pines
39
were selected at each plot for height and age measurement. The initial stand information
was used for plot siteagedensity classifications and to assign silviculture prescriptions.
The study design called for 0.2 acre circular plots, surrounded by a 33 foot
isolation butfer for each siteagedensity combination. Silviculture prescriptions were
required to control any existing hardwoods greater than or equal to oneinch diameter at
ground level and shortleaf pine were thinned when necessary to achieve the desired basal
area for both the plot and buffer. The residual shortleaf pines on the 0.2 acre plot were
numbered, measured, located from plot center, and tallied for all trees greater than or
equal to one inch DBH. In addition, the crown class of each tree was recorded.
At each plot representative shortleaf pines for each diameter class were selected
for measuring total height and the height to live crown. The age for the representative
dominant and codominant sample trees was determined using an increment borer. The
annual rings of each increment core were counted and five years added to derive tree age.
The plot site index was calculated using a site index equation developed by Graney and
Burkhart (1973) for shortleaf pine of the Ouachita HigWands based on the average total
height and age of the representative dominant and codominant trees. The DBI! was
measured to the nearest tenth of an inch for all shortleaf pine greater than or equal to one
inch. Stand basal area was derived by summing the basal area of individual trees on a
plot and expanding to a per acre basis. The crown ratio was computed as the ratio of the
crown length to the total height for trees that were measured for height. For trees not
measured for height, the crown ratio was predicted using the model form developed by
Dyer and Burkhart (1987) with the parameters fitted using nonlinear regression in SAS
(SAS Institute Inc. 1989). The following distanceindependent individual tree

40
competition measures were computed for each plot. The basal area of all trees as large or
larger than the subject tree (BAL) was calculated and expanded to a per acre basis. The
CCF was computed using a technique by Rogers and Sander (1984) for shortleaf pine and
the ratio of the quadratic mean diameter to individual tree dbh (DD) was calculated. For
individual growth models discussed previously additional attributes were computed when
appropriate. The midpoint of an attribute was calculated when appropriate by using the
observed values attained by an attribute at time one and two and averaging the two
observations. If mortality occurred for a given tree between measurement periods then
halfofthe initial measurement was used, a technique described by Bolton and Meldahl
(1990).
In addition, data from 25 plots comprising an "approximated real growth" series
were available from a study initiated by Frank Freese in thinned stands of evenaged
natural shortleaf pine in the Ouachita Highlands. The study was initiated in 196364 and
25 of the 35 O.2acre plots that were installed in the Ouacita National For(:st still exist.
The initial 35 plots were installed with the foHowing stand residual basal area per acre:
45,65, 85, 105, and 125 square feet. The remaining 25 plots were assigned to these
residual basal area levels in 1988. Remeasurements of these plots during 1988 were
consistent with the methods discussed previously. In addition, the other 10 plots were
available for developing the growth model by utilizing the historical data on these plots.
One of these 10 plots was removed from consideration because of fire damage. This left
a total of 34 plots for developing the growth model. Murphy (1988b) provides further
details for this study.
41
Eight plots from the cooperative growth and yield study were decommissioned
because of stand damage or failure to follow silvicultural prescriptions. The remaining
183 plots from this study, together with the 34 plots from Freese's study, constitute a
total of 217 plots that are available for developing growth and yield equations for natural
evenaged shortleaf pine of the Ouachita Highlands. The data set consists of 8928
individual tree observations for which the summary statistics (Table 2) were computed.
Table 2. Summary statistics for the complete data set of the Ouachita Highlands natural
evenaged shortleaf pine study for developing a basal area growth model (N =
8928).
Attribute Minimum Maximum Mean Standard Error
Plot Age (years) 21.0 96.0 45.1 18.71
Plot Site Index 38.9 87.1 57.3 9.70
(base age=50)
Stand Basal Area 22.53 177,12 106.80 30.93 ...
(sq. ft.)
i';
ce,
DBH (inches) l.2 24.9 7.93 3.74
Crown Ratio o1310 0.7636 0.3799 0.063
CCF 13.72 255.44 132.32 48.86
Avg. Annual 0.010 0.07l8 0,0124 0.0102
Individual Tree
Basal Area
Growth
Individual Tree 0.00723 3.383 0.4196 0.3795
Basal Area
Quadratic Mean 0.4424 4.241 1.0875 0.3205
Diameter to
Individual Tree
DBH(DD)
Basal Area of all 0 171.37 66.00 37.80
Trees as Large or
Larger than the
Subject Tree
(BAL)


42
The number of trees by DBH class was calculated for the complete data set
(Figure 1). As illustrated by the graph, there are fewer observations in the 2, 14. 16. and
18inch DBH classes in comparison with the other DBH classes. The number of trees in
the 14, 16, and 18 DBH classes should increase over time as trees grow into these classes.
Since the study entails thinning from below. the future number of trees in the 2inch
DBH class may not increase and could decrease.
200J
1800
1600
1400
~
1200
'0.... 100J
~§
z 800
600
400
200
0


n~
~

,;

 ~

 
 r n I I I I I I I I I

2 4 6 8 10 12 14 16 18
DBHClass
Figure 1. Number of trees by diameter class for the complete data set (N = 8928).


CHAPTER IV
METHODS
Model Considerations
Among the major linear regression assumptions is that the errors are independent
and identically normally distributed. Nonlinear and linear regression models differ in that
the nonlinear model least squares estimators of their parameters are not unbiased,
normally distributed, or minimum variance estimators. Ratkowsky (1990) stated that the
regression assumptions for nonlinear regression need only be correct approximately
because the least squares criterion tends to be robust in minor departures from the
assumptions. Major departures from the regression assumptions such as a dependency
between the error terms can lead to significant estimate errors. Except for the
independence assumption, nonlinear regression models tend to conform to the linear
regression assumptions asymptotically as the sample size approaches infinity. Trees
within a plot have some interdependency because of the competition for resources on the
plot. Also trees within a plot share a similar microenvironment, which may be above or
below average for tree growth. When the assumption of independent errors is violated.
the MSE and standard error of the parameter estimates may seriously underestimate the
variance of the error terms and the standard error of the parameter estimates when
calculated according to the OLS procedures (Neter et a1. 1996).. A system of equations
43
44
was developed to account for tree interdependency within a plot and SUR was used for
parameter estimation. The SUR parameter estimation is an appropriate technique because
the error terms from trees within a plot are correlated. To determine the feasibility of
using a system of equations to model evenaged natural stands of shortleaf pine some of
the model forms discussed previously were examined.
Since the basal area growth model may be applied outside the data geographical
range, the model should behave within known biological limitations when predicting the
maximum basal area or diameter for shortleafpine. The model should follow a realistic
growth pattern and behave logically with respect to the independent variables. As growth
approaches zero the individual tree basal area should approach an upper asymptote that
corresponds to a biologically reasonable diameter.
Individual tree growth may be predicted as basal area increment or diameter
increment or as a function of either basal area or diameter. As discussed previously, West
(1979) concluded that for the species in his growth study, there was no significant
difference between the predictions obtained using either basal area or diameter increment
to model growth. Individual tree diameter growth culminates before basal area growth
and thus may have an advantage over basal area growth for predicting growth in young
stands. The fit index is nonnally higher for basal area growth than for diameter growth
using the same data because the range of diameter growth is smaller in comparison to the
range of basal area growth (Shiftey 1987). Consequently, the denominator of the fit
index (corrected total sum of squares) is smaller for diameter growth than for basal area
growth. Individual tree volumes are nonnally proportional to the product of tree height
and basal area and consequentiy it is logical to predict basal area growth directly. Also,
45
because the initial diameter is knoVvTI for this study, diameter growth can be computed
from basal area growth. Whether to model growth using basal area or diameter is
normally detennined by study objectives. Since the growth model developed may be
used in the shortleaf pine simulator for the Ouachita HigWands, it is logical to use a basal
area growth model. The average annual basal area growth (AABAG) was used to model
growth since remeasurements occurred over different intervals. The AABAG facilitates
using the model over any projection period.
The dependent variables in model development focused on variables that are
currently obtained during the inventory and variables that can be derived from basic
forest measurements. Some measurements typically taken during an inventory include
DBH, site index, stand age, and basal area per acre. Competition measures such as CCF
can be computed from the inventory measurements. The model will be distanceindependent
and attributes discussed earlier for distanceindependent models will be
considered. Independent variables used in the model development will be examined for
significance using an alpha level of 0.05. Independent variables deemed insignificant
will be removed from the model.
Revised Data Set
The data set was revised for modeling basal area growth using a system of
equations. The current data set contains one record for each individual tree. This data set
was revised to create four classes corresponding to the individual tree diameters within
each plot. The revised data set for modeling basal area growth using a system of four
....
r.;

46
equations was developed through the following steps. Individual tree diameters within
each plot were ranked in ascending order. The four DBH rank classes were computed by
dividing the ranked DBH tree list by four. If the ranked DBH tree list was evenly
divisible by four then each DBH rank class had an equal number of trees. The DBH rank
class one corresponds to the DBH of the smallest tree(s). A subroutine program was
written in SAS to place trees in the correct DBH rank classes if the number of trees on a
plot was not evenly divisible by four (Appendix A). For example (Table 3), if a plot has
six trees the first and second class would have two trees and the remaining two classes
would have one tree.
Table 3. Example ofranking and placing trees in correct DBH rank class by plot
for use in a system of equations.
Plot Number DBH (inches) Rank DBH Rank Class
134 6.3 I 1
134 7.1 2 1
134 8.2 3 2
134 10.3 4 2
134 10.9 5 3
134 13.2 6 4
A visual inspection of the revised data was conducted to insure that trees were
placed in the proper class. The revised data set has a total of 217 plot records. Plot 261
was removed when fitting the system of equations models to the complete data set
because it has only two trees and could not be used for parameter estimation in the system
of equations.
.......

47
Development of a System of Equations
The revised data set was used for modeling the system of four equations. The
system of four equations for each model has the following general form:
1 nj
AABAGI = IfCXj)+ej
nl )=1
1 n2
AABAG2 =IfCX2)+e2
n, . 1
 J=
1 n3
AABAG3 = IfCX3 )+e3
n3 )=1
1 n4
AABAG4 =IfCX4)+e4
n4 )=1
where
AABAGij = average annual basal area growth for tree) of class i within a plot,
f = a function of stand and individual tree characteristics,
Xi = a vector of stand and class i individual tree characteristics on a plot.
ni = number of trees in class i, and
ei = error component associated with each class within a plot.
Seemingly unrelated regression is an appropriate technique for estimating
parameters in this system of equations because a correlation is expected between the error
components within each plot for the four classes. The correlation is expected because
trees within a plot are ecologically interdependent and competing for finite resources.
(

48
The SUR parameter estimation technique has been proven to provide a gain in efficiency
when the error components are correlated and the structure of each equation differs
(Zellner 1962). When the generalized system of equations uses linear functions (f),
independent variables are the means of each attribute by class within each plot. Class
means could not be used for nonlinear functions (f) because a function evaluated at the
mean does not equal the mean of the function for nonlinear equations. Therefore a
program in SAS PROC MODEL (SAS Institute Inc. 1989) was written to estimate the
parameters using an iterative process. The program used 217 records corresponding to
the plots. Each individual tree within a plot and its respective attributes were arrayed
along one record. The program evaluated the function for each plot between iterations
and computed the mean of the function for each class within a plot (Appendix B).
Models
Three basal area growth models were developed for trial use in a system of
equations. Modell (Hitch 1994) is the current model being used in the shortl af pine
simulator and was used as a basis for comparison. The numerator of Model 1 is the
potential function developed by Shifley (1987) and is constrained for a biologically
reasonable tree size (M). A biologically reasonable tree size for shortleaf pine of this
geographical location was derived as having a 36inch DBH and the equivalent basal area
is 7.068384 square feet. The potential function initial parameters were estimated by
fitting the potential function separately using the fivepercent fastest growing trees by
oneinch diameter class. If a diameter class had less than 21 observations then aU trees
for that diameter class were used to estimate the initial potential function parameters.

49
The denominator of Model 1 is the competition modifier suggested by Murphy and
Shelton (1996) in which the parameters were estimated while holding the potential
function parameters constant. To reduce bias exhibited by Model I with respect to some
DBH classes, one iteration was completed to reestimate the parameters of both the
potential and competition functions as suggested by Wensel et al. (1987). The Modell
basal area growth model has the following form:
/3 (PI BA J PIBA 2 (1/32)
AABAG= M
1 (/33+/34SBA+/3sAGE+/36DD+/37BA) +e
where
AABAG =average annual basal area growth,
BA = individual tree basal area,
M = maximum basal area (M = 7.068384),
SBA = stand basal area,
AGE = stand age,
DD = diameter of subject tree divided by quadratic mean diameter, and
Pi = parameters.
..
0/
50
Model 2 uses an individual tree equation that is mathematically identical to Model
1 but to estimate parameters using SUR. a system of four equations was fonned
corresponding to the four diameter rank classes within each plot. Model 2 has the
following form:
P BA1h( PIBA} J 1 n3 1 J M (1 fJ 2)
AABAG3 = n3 I(fJ3+P4SBA~fJ;sA'C. E+P6D 
D'i+fJ7B 
Aj)
J=! 1+ e .
where
j =individual tree observation(s) within a class (class = 1,2,3,4) on a plot,
AABAG, BA, M.. SBA, AGE, 13i, and DD defined previously, and
ni = observations in class i on a plot.
...,
e....~
......

The parameters for Model 2 were stimated using the technique described for
Model I with the following modifications. The parameters were estimated using the
SUR option in the SAS PROC MODEL procedure with parameter restrictions placed
across the four equations.
Model 3 uses a modified Weibull probability function as the potential and the
same competition modifier used for Models 1 and 2.
51
~
[P2J[BAjrr'l
fAj t i~
1 nl /3 0 PI /3 I e /31
~
AABAGI = I 04
1+ //33+ /3 4SBAt /3 SAGET /36 DD) +/3 7BA))
.
nl ) =1
.. ..
..
.'".
(P2 J( BA JrPr1
) t Jt .... .
oJ
/3 0 PJ /3 1
e PI ..
1 n2 .".,'
AABAG2 = I ~
1+ e(/3 3+ fJ 4 SBA +PSAGE+ fJ6DDj + /37 BAj) wi n2 )=J el
~
(P2 J( BAjrr1l
( HAjt ,
1 n) Po P
J
/3
1
e PI
AABAG3 = I (/3)+/3 4SBA+fJ SAGE+ /36 DD i +PTBA i) n3 ) = I l+e ..
where
AABAG, BA, SBA, AGE, DD, Pi, ni, and} are as defined previously.
52
The Weibul1 probability density function was modified by adding a parameter (/30)
that scaled the function. The modified Weibull probability density function was selected
as the potential function because of its flexibility and to detennine the viability of using
the function to model basal area gro\\tth. The potential function was fitted to the fastest
growing five percent of the trees using the method described previousJy. Then the
competition modifier was fitted while holding the potential function constant. One
iteration of refitting the potential and modifier functions was completed as descri bed for
Models I and 2 to reestimate the parameters and remove some bias with respect to
certain DBH classes.
The composite function (Mode14) selected is a variation of the PROGNOSIS
model (equation 7). Model 4 predicts the natural log of basal area growth as a linear
function of stand and tree characteristics. The system of four equations has the following
[onn:
·.. .0.{ ,..
·J ..
'.".' •.... ;..
~
:>

53
where
AABAG(i = average annual basal area growth oftreej in class i (square feet).
DBH = tree diameter (inches).
BAL = cumulative basal area of all trees larger than subjective tree,
CR = crown ratio,
CCF = crown competition factor,
Sf = site index (base age = 50),
AGE = stand age (years). and
Pi = estimated parameters.
The dependent variable in this model requires computing the natural log which is
a problem because of the negative or zero growth observations. The data set consisted of
approximately fivepercent of the average annual basal area growth measurements having
zero or negative growth. These observations may be attributed to logical reasons such as
peeling bark, which results in a smaller diameter being recorded during the second
measurement than at the first measurement. Because Model 4 required computing the
natural log of basal area growth, the observations that were less than or equal to zero were
constrained to be number that was less than the smallest observable basal area growth.
The smallest observable growth during a measurement period corresponds to a tree
growing from 1.0 to 1.1 inches with a basal area growth of 0.00 145 square feet. All
AABAG with less than or equal to zero growth were constrained to be 0.00125 square
feet.
54
Calibration and Validation
To calibrate and evaluate the performance of the growth models, the complete
data set was divided into a calibration and validation data set. The plots were strati tied
by the site indices, stand ages, and basal area per acre combinations. Approximately 113rd
of the plots within each combination of age, site index, and basal area classes were
selected randomly to fonn a validation data set. The calibration and validation data sets
contain 149 and 68 plots, respectively. The summary statistics for the calibration and
validation data sets are similar with no substantial differences. The summary statistics
for the calibration and validation data sets are presented in Tables 4 and 5, respectively.
The calibration data set was used to estimate parameters for the models. The
calibration of the models was focused on the statistical fitting process using either linear
or nonlinear regression techniques to assure that the models perform satisfactorily with
respect to the fit index and mean square error and to examine the residuals for
heterogeneity. The computer program was verified to ensure that the trees were placed in
the correct classes and that the program was operating correctly.
The validation process was used to detennine model performance using the
estimated parameters from the calibration data set to detennine the fit and performance of
the models on an independent data set. Validation tests were both subjective and
objective. The subjective tests were used to observe the predicted basal area growth and
determine if the projections were realistic with respect to current knowledge of shortlcaf
pine growth. Model predictions should follow known principles of tree and stand
development and should have biological integrity at the stand extremes. Models were
....
55
examined under extreme conditions. such as for young and small diameter trees to
evaluate model perfonnance. The objective tests are outlined under model evaluation
criteria.
Table 4. Summary statistics for the calibration data set of the Ouachita Highlands natural
evenaged shortleaf pine study for developing a basal area growth model (N =
6099).
Attribute Minimum Maximum Mean Standard
Error
Plot Age 21.0 94.0 45.1 18.53
(years)
~
4
Plot Site Index 38.9 85.9 57.6 9.50 1•
(base age=50) ~
Stand Basal 29.57 177.12 108.22 31.66
Area (sq. ft.)
DBH (inches) 1.5 24.9 8.04 3.69
Crown Ratio 0.1310 0.7636 0.3785 0.062
CCF 26.46 255.44 133.05 50.90
Avg. Annual 0.008 0.0718 0.0127 0.0102
Individual Tree
Basal Area
Growth
Individual Tree 0.0123 3.383 0.4268 0.3757
Basal Area
Quadratic 0.4424 4.2410 1.0875 0.3191
Mean
Diameter to
Individual Tree
DBH(DD)
Basal Area of 0 171.37 66.74 38.47
all Trees as
Large or
Larger than the
Subject Tree
(BAL)
"""""
56
Table 5. Summary statistics for the validation data set of the Ouachita Highlands natural
evenaged shortleaf pine study for developing a basal area growth model (N =
2829).
Attribute Minimum Maximum Mean Standard
Error
Plot Age 22.0 96.0 43.4 18.98
(years)
Plot Site Index 40.0 87.1 56.8 10.1
(base age=50)
Stand Basal 22.53 140.37 103.74 29.06
Area (sq. ft.)
DBH(inches) 1.2 23.2 7.70 3.86
Crown Ratio 0.1557 0.6619 0.3824 0.058
CCF 13.72 203.69 130.74 44.11
Avg. Annual 0.010 0.0700 0.0119 0.0100
Individual Tree
Basal Area
Growth
Individual Tree 0.00723 2.924 0.4041 0.3871
Basal Area
Quadratic 0.4520 3.725 1.0934 0.3235
Mean
Diameter to
Individual Tree
DBH(DD)
Basal Area of a 139.24 64.42 36.28
all Trees as
Large or
Larger than the
Subject Tree
(BAL)
.......
1400
1200
'c"; 1000
~
i 800 '
0...
.'0" 600
E
z'" 400
200
0
Calibration Data Set

 
rrr

r
  
 nn I I I I I I I
246 81012141618
DBHClass
800
700
600
500
400
300
200
100
o
Validation Data Set


r
 
rrrnn
2 4 6 8 1012141618
DBHCla~s
57
Figure 2. Number of trees by DBH class for the calibration (N = 6099) and validation (N
= 2829) data sets.
The number of trees by DBH class graphs for the calibration and validation data
sets exhibit the same general pattern. The 4inch DBH class has the largest deviation
from the complete data set number of trees by DBH class for both the calibration and
validation data sets. The calibration and validation data sets are under and overrepresented
for the 4inch DBH class, respectively.
Model Evaluation Criteria
Model evaluation was conducted by computing the fit index and MSE and
examining model perfonnance across the data range with respect to DBH, site index,
basal area per acre, and age classes. The goal was to determine model perfonnance and
to examine the models developed by using a system of equations to evaluate the validity
of this method.
58
The fit index for each model was computed to detennine how well the model fit~
the data. The fit index has the following form:
11 Ie;
FIT INDEX = I _',_'=..:1
11 I(y; r[
1=1
where
n = number of observations in validation, calibration, or complete data set
ei = residual of the ith observation, and
Yi = average annual basal area gro~1:h for tree i,
In addition to the fit index, several other statistics were computed to evaluate the
performance of the models for the calibration and validation data sets. The mean square
error (MSE) was computed as the sum of the squared difference between the predicted
basal area growth and the actual basal area growth divided by the degrees of freedom and
is a measure of the dispersion.
MSE
where
p = number of parameters, and
ei and n are defined previously.
II
IeT
;=1
np
...
59
The MSE was computed for the complete calibration and validation data sets. In
addition, the MSE was calculated using the validation data set for the DBH, age, site
index, and basal area classes. The average deviation has the following form.
where
Average Deviation
ei and n are defined previously.
n Iel
1=1
11
The average deviation by DBH, age, and basal area class was computed as the
sum of the errors divided by the total number of observations and is used to detect bias.
The mean absolute deviation was calculated as the sum of the absolute difference
between the predicted average annual basal area growth and the actual average annual
basal area growth divided by the number of observations. The mean absolute deviation
indicates the average absolute deviation from the mean AABAG and has the following
form:
17 Llel!
Average Absolute Deviation = ,1=,1__
n
where
ei and n are defined previously.
~..
)

60
The average absolute error as a percentage of the mean AABAG by attribute was
computed as the sum of the absolute deviations divided by the sum of the actual basal
area growth.
/I
Ileil
Average Percent Absolute Deviation = i=l (100)
n
LYi
i=l
where
ei, Yi, and n are defined previously.
The above model evaluation criteria are presented in tabular form. Box plots of
the residuals were constructed by DBH, site index, age, and basal area classes to illustrate
model biases and departures from normality.
The model(s) that performed best were then fitted to the complete data set 0 f 217
plots. Further analysis was then conducted by computing the fit index, MSE, and average
deviations by attribute classes for the complete data set. Bar charts of the average
deviation by attribute classes and AABAG were constructed using the complete data sel to
detect biases and compare model performance.
)
CHAPTER V
RESULTS
Calibration
All four models used the same calibration data set, which consists of 149 plots
and 6099 individual tree observations. Model 1 parameters were fitted using nonlinear
regression using the 6099 individual tree records. Models 2 and 3 used a system of four
nonlinear equations to fit the parameters using the plot mean of the function for each
DBH rank class. Model 4 used the plot means of each DBH rank class to fit the
parameters to a system of four linear equations. Models 2, 3, and 4 used the DBH rank
classes by plot with 149 plot observations used for fitting the parameters. The parameter
estimates, standard errors, and descriptions for Models I and 2, Model 3, and Model 4 are
presented in Tables 6, 7, and 8, respectively.
Table 6. Parameter estimates, standard errors, and descriptions for Models I and 2 when
fitted to the calibration data set.
Parameter
Modell
Estimate Standard Error
0.081555 0.001626
0.573986 0.010216
3.453924 0.095805
0.015943 0.00040 I
0.029879 0.001049
1.191026 0.075979
1.065994 0.049297
61
Model 2
Estimate Standard Error
0.081248 0.005159
0.572551 0.037990
2.816660 0.323730
0.015596 0.001332
0.023381 0.003361
0.895971 0.209410
0.982284 0.144010
Description
SA
BA Power
Intercept
SBA
AGE
DD
BA
62
Table 7. Parameter estimates, standard errors, and descriptions for Model 3 when fitted
to the calibration data set.
Description
Weibull Multiplier
Weibull Spread Parameter
Weibull Shape Parameter
Logistic Intercept
SBA
AGE
DD
BA
0.237509 0.029240
3.432234 0.383970
1.482893 0.039310
2.662908 0.312800
0.014920 0.001283
0.024019 0.003245
0.837411 0.202140
0.977300 0.138470
Parameter Estimate Standard Error
Models 1, 2, and 3 used the same competition modifier and consequently the signs
of parameters /33/37 are logically identical. The logical properties of the signs for the
estimated parameters were examined while holding the other estimated parameters
constant. Although illogical coefficient signs are possible in a valid predictive model due
to multicolinearity, logical signs are desirable. For Models 1, 2, and 3, the signs for the
estimated parameters /3~ (stand basal area). /35 (stand age), and /36 (quadratic mean
diameter to individual tree DBH) are logically positive. This implies that as these
attributes increase, such as stand age, the rate of growth decreases, and conversely,
parameter Pc (individual tree basal area) is negative and indicates that as the individual
tree basal area increases its growth rate increases.
Table 8. Parameter estimates, standard errors, and descriptions for Model 4 when fitted
to the calibration data set.
Parameter Estimate Standard Error Description
6.567868 0.232430
1.457768 0.088260
0.003002 0.000727
1.388672 0.344310
0.016536 0.001858
0.004905 0.000577
Intercept
Natural Log DBH
BAL
CR
CCF
AGE
po
63
Model 4 is a variation of the PROG OSIS model, with crown ratio squared, DBH
squared, and site index excluded from the final model because they were insignificant
(alpha level of 0.05). The signs of Model 4 attributes are logical and consistent with
known forest growth patterns. Model 4 estimated parameters /3J (basal area of all trees as
large or larger than the subject tree), 135 (plot age), and /36 (CCF) have negative signs
which are consistent with growth patterns. For instance, as CCF increases, tree
competition increases and hence the rate of growth decreases. The signs for the estimated
parameters /32 (DEli) and /34 (crown ratio) are logically positive.
Models 1 and 2 use the same potential function, which was initially fitted
separately from the competition modifier. The potential function has a fit index of
0.7593 and MSE of 0.000052. Model 3 uses a modified Weibull probability density
function as the potential function which has a fit index of 0.7749 and a MSE of 0.000048
when fitted to the fastest growing five percent of the trees by oneinch diameter class.
The individual tree growth function used in Models I and 2 are identical mathematically
but the degrees of freedom for the standard errors differ. Model I estimated parameters
are based upon 6099 individual tree observations whereas Model 2 uses a system of
equations that are based upon 149 plot observations.
The fit index and MSE for all four models are presented in Table 9. Modell had
the highest fit index (0.6270) and Model 4 the lowest (0.5740). Models 2 and 3 fit
indices and MSE's are similar to those of Modell. All models provide a reasonable fit
with a small variance of the error terms.
64
Table 9. Fit index and mean square error for aJI models using the calibration data set.
Model Fit Index MSE
1 0.6270 0.0000391
2 0.6206 0.0000397
3 0.6180 0.0000400
4 0.5740 0.0000450
All model residuals were plotted against their respective model attributes and
DBH to detect any trends. Except for DBH, there was no evidence of trends. The plots of
residuals versus the DBH classes for models I, 2, and 3 revealed some bias and slight
heterogeneity of variance. As discllssed in Methods (Chapter 4), one iteration for the
potential and modifier functions was performed to reestimate the parameters. The
iteration removed some of the bias with respect to DBH and provided a better fit.
To determine whether SUR was an appropriate technique, the correlation among
residuals was examined for the system of equations models. The residual correlation
matrix is presented below for Model 2 with Models 3 and 4 illustrating similar results.
The DBHrank classes are 14 (left to right, top to bottom). As illustrated by the matrix,
there is a moderate amount of correlation between the plot DBH rank classes.
Consequently SUR may provide a gain in parameter estimation efficiency.
fl.O 0.5684 0.6341 0.5099
l 1.0 0.6532 0.5165
1.0 0.6440
1.0
The pairwise correlation for DBH rank classes might be positive due to the fact
that the study entails thinning from below and consequently the suppressed trees are
F'"
65
removed. Since the microenvironment on a plot is similar for all DBH rank classes, if
growth in the first rank class is above the mean, growth in the second rank class tends to
be above the mean. Therefore, the residuals tend to be positively correlated.
The parameter correlation matrices were examined to detect multicollinearity
between independent attributes. The correlation matrix for Modell is presented below
with the other models showing similar results. There is a strong inverse correlation
between the intercept (/33) and both stand basal area (/34) and quadratic mean diameter to
individual tree DBH (/36)' There is also a significant inverse correlation between stand
age (P~) and individual tree basal area (/37)' The asymptotic correlation of parameters
matrix for Modell P.1/37 (left to right, top to bottom) is
1.0  0.64510426
1.0
0.04746676
 0.11238271
1.0
 0.81875629
0.29022445
 0.40950827
1.0
 0.33982952
0.18386998
 0.80374116
0.51353762
1.0
Multicollinearity may cause the parameter estimates to differ substantially when using
different techniques for parameter estimation or when the data are updated.
Validation
The validation data set consists of 68 plots and 2829 individual tree observations.
All models were evaluated using the individual tree observations in the validation data set
to determine model performance. The validation summary statistics for all models are
presented Table 10.

66
Table 10. Summary statistics for all models using the validation data set.
Model Fit Index MSE Average Error Error Percentage Mean Absolute Error
I 0.6074 0.0000393 0.0007998 37.11 0.0044087
2 0.6036 0.0000397 0.0005774 37.47 0.0044513
3 0.6016 0.0000399 0.0005439 37.62 0.0044693
4 0.5821 0.0000418 0.00103900 38.84 0.0046148
Note: Error Percentage = average absolute error as a percentage ofmeanAABAG
The fit index and MSE results for the validation data set when compared to the
calibration data set for Models], 2, and 3 are inferior, while Model 4 improved. Model 3
illustrated the smallest absolute value of average error (0.0005439) while Model 4 has the
largest absolute value of average error (0.00 I039). All models except Model 4 exhibit an
overprediction for average error. The variance among the average absolute error as a
percentage of the mean AABAG for Models 1,1, and 3 is relatively small while Model 4
has the highest error percentage. The mean absolute error among the four models is
similar with Model 4 having the highest absolute error.
The validation data set was used to detect any trends by DBH, site index, basal
area, and age classes for average deviation, M.,,)E, average absolute error, and average
absolute error as a percentage of the mean AABAG by attribute class. The results for the
average deviation by DBH class are presented in Table 11.
Table 11. Average deviation for all models by DBH class using the validation data set.
DBH Class N Modell Model 2 Model 3 Model 4
2 215 0.0009295 0.0010123 0.0011487 0.0000192
4 683 0.0009268 0.0005659 0.0005602 0.0017151
6 496 0.0002946 0.0008933 0.0009947 0.0028879
8 433 0.0006813 0.0003399 0.0002422 0.0011852
10 391 0.0009385 0.0008093 0.0007911 0.0002121
12 321 0.0020258 0.0021786 0.0022184 0.0010329
14 ]73 0.0025150 0.0027801 0.0028366 0.0011660
16 67 0.0008632 0.0007556 0.0007553 0.0014970
18 50 0.0022730 0.0024204 0.0032815 0.0033731

67
Models 1,2, and 3 have identical signs by DBH class for average deviation.
Model 4 has the smallest average deviation for DBH class 2. Model 2 underpredicts for
the 2inch DBH class while the other models overpredict. Model 4 performs better for
DBH classes 10, 12, and 14. Model 3 performs better for the 4 and 8inch DBH classes.
Model 1 is superior for the 6inch DBH class and for the 16 and 18inch DBH classes all
models underpredict. The average deviation by site index. basal area, and age classes
were computed and are presented in Table 12.
Table 12. Average deviations by site index, basal area per acre, and age classes for all
models using the validation data set.
N Modell Model 2 Model 3 Model 4
Site Index
<56 1485 0.0004838 0.0001059 0.0000623 0.0018454
60 622 0.0006002 0.0005599 0.0005223 0.0005920
70 568 0.0019515 0.0019576 0.0019641 0.0005699
>75 154 0.0004048 0.0001030 0.0000358 0.0010047
Balac
30 229 0.0020835 0.0025523 0.0026437 0.0038334
60 239 0.0027902 0.0030303 0.0028908 0.0009413
90 608 0.0007943 0.0005074 0.0004648 0.0017232
120 1753 0.0009070 0.0006760 0.0006677 0.0007069
Age
20 1024 0.0002492 0.0008470 0.0008633 0.0029609
40 797 0.0017839 0.0012287 0.0012309 0.0000471
60 722 0.0022831 0.0024280 0.0024225 0.0014040
80 286 0.0019318 0.0008095 0.0010751 0.0030910
Models 1, 2, and 3 have similar trends for site index, basal area, and age classes.
For site index classes, Models 1,2, and 3 all overpredict for average deviation. Model 4
has underpredictions for site indices <56, 60, and >75 and performs best for the site index
class 70. Models 1,2, and 3 results are similar for all four basal area classes. They
..
68
underpredict for basal area of 30 square feet per acre and overpredict for the other three
basal area classes. Model 4 has more variation within attribute classes when compared
with the other models. Model 4 performs the best for basal area class 60 and is inferior
for the basal area class 90. Models 1, 2. and 3 underpredict for age classes 20 and 80, and
overpredict for the other two age classes. Model 4 performs best for age class 40. Model
1 perfonns best for the age class 20, and Model 2 is superior for the older age class (80).
The MSE for all models was computed by DBH class and presented in Table 13.
Table 13. Mean square error for all models by DBH class using the validation data set.
Class N Model 1 Model 2 Model 3 Model 4
2 215 0.0000023 0.0000024 0.0000027 0.0000019
4 683 0.0000077 0.0000073 0.0000073 0.0000113
6 496 0.0000266 0.0000295 0.0000297 0.0000385
8 433 0.0000382 0.0000412 0.0000409 0.0000401
10 391 0.0000522 0.0000519 0.0000517 0.0000463
12 321 0.0000626 0.0000593 0.0000600 0.0000581
14 173 0.0000921 0.0000913 0.0000917 0.0000882
16 67 0.0001854 0.0001835 0.0001826 0.0002026
18 50 0.0002300 0.0002309 0.0002380 0.0002652
With the exception of 4inch DBH class where Model 4 performs best, all models
perform similarly by DBH class for mean square error. The mean square error hy site
index, basal area per acre, and age classes are presented in Table 14.
69
Table 14. Mean square error for all models by site index basal area per acre, and age
classes using the validation data set.
N ModeJ 1 Model 2 Model 3 Model 4
Site Index
<56 1485 0.0000275 0.0000278 0.0000279 0.0000310
60 622 0.0000402 0.0000408 0.0000408 0.0000432
70 568 0.0000572 0.0000579 0.0000587 0.0000560
>75 154 0.0000920 0.0000914 0.0000926 0.0000974
Balac
30 229 0.0000710 0.0000766 0.0000766 0.0000759
60 239 0.0000967 0.0000937 0.0000933 0.0000982
90 608 0.0000355 0.0000348 0.0000350 0.0000382
120 1753 0.0000295 0.0000300 0.0000303 0.0000318
Age
20 1024 0.0000147 0.0000169 0.0000173 0.0000251
40 797 0.0000438 0.0000431 0.0000430 0.0000415
60 722 0.0000510 0.0000513 0.0000519 0.0000466
80 286 0.0000895 0.0000864 0.0000866 0.0000951
The mean square error for all models reveals little variation among models by site
index. basal area per acre. or age class. The validation data set results for mean absolute
deviations by DBH, site index, basal area per acre, and age classes for all models are
presented in Tables 15 and 16.
Table 15. Mean absolute deviation for all models by DBH class using the validation data
set.
DBHClass N Modell Model 2 Model 3 Model 4
2 215 0.001 J973 0.0012668 0.0013658 0.0010710
4 683 0.0022436 0.0021662 0.0021877 0.0026637
6 496 0.0039337 0.0041680 0.0041841 0.0049294
8 433 0.0046312 0.0048731 0.0048551 0.0048393
10 391 0.0054917 0.0054545 0.0054410 0.0051348
12 321 0.0062618 0.0061451 0.0061720 0.0058953
14 173 0.0077675 0.0077006 0.0077255 0.0074574
16 67 0.0101988 0.0102113 0.0101765 0.0100904
18 50 0.0108322 0.0108360 0.0110230 0.0119798

70
The mean absolute deviation by DBHclass reveals that all models behave
similarly and there is not a substantial amount of variation among models for the DBH
classes. All models exhibit the trend of mean absolute deviation increasing as the DBH
class increases with the 16 and 18inch DBH classes having the highest mean absolute
deviation for all models.
Table 16. Mean absolute deviation for all models hy site index, basal area per acre, and
age classes using the validation data set.
N Modell Model 2 Model 3 Model 4
Site Index
<56 1485 0.0034833 0.0035020 0.0035275 0.0038789
60 622 0.0048433 0.0049299 0.0049] 77 0.0049857
70 568 0.0057531 0.0058382 0.0058661 0.0055378
>75 154 0.0066185 0.0065563 0.0065871 0.0068090
BaJac
30 229 0.0060362 0.0062881 0.0063330 0.0067268
60 239 0.0077023 0.0076989 0.0076447 0.0074454
90 608 0.0040877 0.0040347 0.0040476 0.0045127
120 1753 0.0038584 0.0039130 0.0039391 0.0039884
Age
20 1024 0.0027089 0.0028279 0.0028731 0.003576]
40 797 0.0048471 0.0048167 0.0048185 0.0047709
60 722 0.0053553 0.0054140 0.0054262 0.0049689
80 286 0.0069086 0.0068150 0.0067949 0.0070047
All models behave similarly for mean absolute deviation by site index, basal area
per acre, and age classes. For both site index and age, all models exhibit higher mean
absolute error as the site index and age class increases. All models exhibit the same trend
for basal area per acre mean absolute deviation. Their smallest respective absolute
deviations occur in the basal area classes 90 and 120, and their largest respective absolute
deviations occurring in the basal area per acre 30 and 60 classes.
....
71
The average absolute error as a percentage of the mean AABAG by DBH, site
index, basal area per acre, and age classes for all models was computed and are presented
in Tables 17 and 18.
Table 17. Average absolute error as a percentage of mean average annual basal area
growth by DBH class for all models using the validation data set.
DBH Class N Modell Model 2 Model 3 Model 4
2 215 62.53 66.16 71.32 55.93
4 683 34.88 33.68 34.01 41.41
6 496 36.48 38.65 38.80 45.71
8 433 36.39 38.29 38.15 38.03
10 391 37.54 37.28 37.19 35.10
12 321 38.94 38.21 38.38 36.66
14 173 39.11 38.77 38.90 37.55
16 67 37.10 37.14 37.02 36.74
18 50 30.31 30.32 30.84 33.52
All models exhibit their respective highest average absolute error as a percentage
of average AABAG by DBH class for DBH class 2. Mode14 performs the best (55.93%)
and Model 3 performs the worst (71.32%) for the 2inch DBH class. The average
absolute error as a percentage of average AABAG by DBH class for the other DBH classes
reveals that all models perform similarly with no substantial difference between the
respective models. The average absolute error as a percentage of the average AABAG by
site index, basal area per acre, and age classes are presented in Table 18.
72
Table 18. Average absolute error as a percentage of mean AABAG by site index, basal
area per acre.. and age classes for all models using the validation data set.
N Model 1 Model 2 Model 3 Model 4
Site Index
<56 1485 35.93 36.12 36.38 40.01
60 622 37.29 37.96 37.86 38.39
70 568 42.79 43.42 43.63 41.19
>75 154 29.14 28.87 29.00 29.98
BaJac
30 229 26.59 27.69 27.90 29.63
60 239 39.23 39.21 38.94 37.92
90 608 31.10 30.69 30.79 34.33
120 1753 43.0 I 43.61 43.90 44.45
Age
20 1024 29.04 30.31 30.80 38.33
40 797 39.16 38.91 38.92 38.54
60 722 43.27 43.82 43.92 40.22
80 286 37.48 36.97 36.86 38.00
Models 1, 2, and 3 behave similarly for average absolute error as a percentage of
mean AABAG for site index, basal area per acre, and age classes. Model 4 behaves
similarly to the other models with a few exceptions. Model 4 behaves worse for site
index class <56 and slightly better for site index class 70. Model 4 average absolute error
as a percentage of the mean AABAG is higher with respect to basal area per acre classes
30,90, and 120 and lower for basal area class 60. All models have their respective
highest average absolute error as a percentage of the mean AABAG in the following
attribute classes: site index 70, basal area per acre 120, and age 60. Models 1, 2, and 3
have their respective lowest average absolute error as a percentage of the mean AABAG in
the following attribute classes: site index >75, basal area per acre 30, and age 20.
pa
73
Figure 3 illustrates the residuals box plots by DBHclasses. Models l, 2, and 3
illustrate similar trends with respect to bias by DBH class with Model I and 2 performing
about equally and Model 3 illustrating slightly higher bias for the lower and higher DBH
classes. Model 4 exhibits the worst bias for the largest DBH class and least bias for the
smallest DBH class. All models demonstrate an underprediction bias for the largest DBH
class (18+). All four models have their highest skewness in the 18inch DBH class.
Model 1 Model 2
0.025 ,",0., 0.025,, o
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oeH (inC1las)
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Figure 3. Box plots of residuals by DBH class for all models using the validation
data set.
..

74
Figure 4 illustrates the box plots for residuals by site index class. Models I 2.
and 3 perfonn similarly with the difference between Models 1 and 2 negligible. Models 2
and 3 illustrates slightly less bias in the plot site index class >75 and slightly more in site
index class 70. Model 4 behavior is substantially different from the other models with
overprediction bias only in site index class 70. Model 4 does exhibit the least bias with
respect to site index classes 60 and 70.
Model 1 Model 2
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0020 I
00'5 a 0015 a ~ 0 a
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Model 3
0020,.. ,
Model 4
0020 I
0015
a a
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0.015
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Figure 4. Box plots of residuals by site index class for all models using the
validation data set.
..
75
Figure 5 illustrates the box plots of residuals by plot basal area class. Models 1 2
and 3 behave similarly with Models 2 and 3 performing slightly better for plot basal area
90 and Model 1 better for plot basal area 30. Model 4 has the least bias for basal area
class 120 and 60, but has substantially more underprediction bias for basal area class "'0.
Modell Model2
0.020  0020.
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0.01S. l , 0.0'5.
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Plot aas.. Area (squire Peell.Cte) Plot ea5.l1 ~. (SQuare feeVlae)
Model 3 Mod.14
C.020.
I
0015 :
0020 I
0015 I
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PkJt easal Are. (SQuire tee1JIUIIl) P)o\ Bual Atea (SQuire 'eev.ae)
Figure 5. Box plots of the residuals by plot basal area class for all models using the
validation data set.
....
76
The box plots of the residuals by age class are presented in Figure 6. Models 1, ,
and 3 behave similarly for all age classes. Models 1 and 2 perfoml the best for the age
classes 20 and 80. Model 4 has strong departures in biases with respect to the other
models. Model 4 exhibits the least bias for age class 40 but has significantly more
underprediction bias in age class 20 with respect to the other models.
Modell Model 2
0020, 0020 , I
0015 l
I 00'51 (,
0.010 ~ 0.010 I .. ~ 0.005· i OOOS
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Model J
Mod.14
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Figure 6, Box plots of residuals by plot age class for all models using the validation data
set.
......

CHAPTER VI
DISCUSSION
The calibration and validation results reveal that Models L 2, and 3 perform well
with no substantial differences among the models. Model 4 performs adequately but its
biological integrity is compromised because it predicts a diameter larger than 70 inches.
There could be some concern about using Model 4 to simulate the future forest structure
for old stands. Model 4 performs best for the 2inch diameter class for average error,
NISE, average absolute error, and average absolute error as a percentage of the mean
AABAG. Although Model 4 performs best for the 2inch DBH class, it still performs
poorly for this diameter class. Model 4 was excluded from further consideration because
for the model calibration and validation results it has the least favorable fit index, MSE,
average error, average absolute error as a percentage of the mean AABAG, and mean
absolute error.
Models 1, 2, and 3 perform similarly for the calibration and validation data sets.
Although the Model 1 fit index, MSE, average absolute error as a percentage of the mean
AABAG, and mean absolute error are better, the differences among the three models is
negligible. For example, the respective fit indexes using the validation data set for
Models 1,2, and 3 are 0.6074,0.6036, and 0.6016, respectively. Model 3 has the
77

78
smallest average error of 0.0005439, but the other models perform well with Models]
and 2 having average errors of 0.0007998 and 0.0005774, respectively.
Model 1 does perform best using the validation data set for the average deviation
criterion for DBH classes 2, 6, 12, 14 and] 8, but with the exception ofDBH class 6 the
difference between models is negligible. The difference between Models l, 2, and 3 for
the 6inch DBH class is 0.0002946, 0.0008933, and 0.0009947, respectively. Model 2
performs best overall for the average deviation by site index, basal area per acre and age
class. Although Model 1 performs best for some attribute classes, the difference between
Model 1 and Models 2 and 3 predictions is smaller than when Model 2 performs best.
There is little distinguishable difference in MSE and mean absolute deviation for
each of the attributes. Model 3 does perform worst overall for mean absolute deviation
but the difference is negligible among the models.
A concern with the shortleaf pine simulator has been the poor performance of the
2inch diameter class. The 2inch diameter class has the highest average absolute error as
a percentage of the mean AABAG. For the 2inch diameter class, Models 1,2, and 3 have
average absolute error as a percentage of the mean AABAG of 62.53,66.] 6, and 71.32,
respectively. The 2inch diameter class is more sensitive than the other diameter classes
because if all diameter classes have the same amount of measurable diameter growth the
2inch diameter class will have the highest relative amount of basal area growth. For
example, a tree growing from 2.0 to 2.linches has approximately 10% increase in basal
area growth, whereas a tree growing from] 0.0 to 1O.linches has approximately 2%
increase in basal area growth. The average absolute error as a percentage of the mean
•
79
AABAG for Models 1.2, and 3 by site index, age, and basal area per acre clas es provides
little basis for differentiation among respective models.
The box plots of Models 1, 2, and 3 reveal no substantial differences among the
three models with most attribute classes having a fairly symmetric distribution. Models
I, 2, and 3 have their highest DBH class skewness (negative) for the 16 and 18inch
diameter classes. This skewness was expected because there are few observations in the
16 and 18inch DBH classes. All three models exhibit their highest plot basal area
skewness (negative) for the plot basal area class 30. This was expected because there are
relatively few observations in this plot basal area class. The lOyear age class for all
three models has the greatest skewness (negative skewed) among age classes. Although
the 20year age class has the most observations among age classes there has been a
tendency for the young age class to perfonn poorly. The box plots illustrate no
substantial differences for the overall bias pattern for Models 1. 2, and 3.
Complete Data Set
Models I, 2, and 3 were fitted to the complete data set for further evaluation. Plot
261 was removed from the data set for fitting the parameters for Models 2 and 3. There
were only two trees on plot 261 and it could not be used for estimating the parameters for
a system of equations. Models I and 2 and Model 3 parameter estimates, standard errors,
and descriptions are presented in Tables 19 and 20, respectively.
po
Table 19. Parameter estimates, standard errors, and descriptions for Models 1 and 2
when fitted to complete data sel.
80
Parameter
Madej 1
Estimate Standard Error
0,0815392 0.0013261
0.5727113 0.0083359
3.4613420 0.0787408
0.0160711 0.0003341
0.0295606 0.0008705
1.2477951 0.0629028
1.0603063 0.0401967
Mode12
Estimate Standard Error
0.083670 0.004831
0.582591 0.033420
2.768013 0.274390
0.015550 0.00 to77
0.023695 0.002699
0.916957 0,186520
0.994703 0.120200
Description
BA
BA Power
Intercept
SBA
AGE
DD
BA
Table 20. Parameter estimates, standard errors, and descriptions for Model 3
when fitted to complete data set.
Parameter Estimate Standard Error Description
0.279316 0.03920
3.930101 0.49676
1.459578 0.03460
2.667933 0.26603
0.015163 0.00105
0.023272 0.00261
0.907143 0.18089
0.959767 0.11546
Weibull Multiplier
Weibull Spread Parameter
Weibull Shape Parameter
Logistic Intercept
SBA
AGE
DD
BA
SUR provides a gain in efficiency for large data sets but a comparison of the
standard errors for the estimated parameters for Model I and Models 2 and 3 could not bc
conducted. Model 1 estimated parameters are based upon 8928 individual tree
observations whereas Models 2 and 3 parameter estimates are based upon 216 plot
observations. Whil,e Model 1 appears to have a lower standard error for the estimates,
when the independence assumption of regression is violated there may be a significant
underestimation of the lvfSE and standard errors of the estimated parameters. Models 2
and 3 were fitted using OLS with no parameter restrictions placed across the equations

81
for comparison with the SUR fit. There was a large gain in efficiency when Models 2
and 3 were fitted using SUR with parameter restrictions placed across equations.
However, it cannot be concluded that Models 2 and 3 provide a gain in parameter
estimation efficiency when compared with Model I. The fit index, MSE, and SSE for
Models 1, 2, and 3 when fitted to the complete data set are presented in Table 21.
Table 21. Models 1,2, and 3 fit index, mean square error, and error sum of squares when
fitted to the entire data set.
Model
I
2
3
Fit Index MSE
0.6224 0.0000390
0.6154 0.0000397
0.6144 0.0000398
SSE
0.3482911
0.3547762
0.3556626
Modell has the highest fit index (0.6224) and lowest MSE (0.000039) but as
expressed previously, when the independence assumption is violated the MSE may be
significantly underestimated. If the MSE is underestimated then the SSE would be
underestimated and the fit index would be overestimated. There is not a substantial
difference between the three models, with Models 2 and 3 having fit indices of 0.6154
and 0.6144, and MSE's of 0.0000397 and 0.0000398, respectively. Although MSE and fit
indices for Models 2 and 3 are less favorable than Model l, these models account for the
interdependency among trees within a plot.
Horizontal bar charts of the average deviation by class attributes and mean
AABAG are presented in Figures 710. Figure 7 illustrates the average deviation by DBH
class and meanAABAG for Models 1,2, and 3. This illustrates that the 2inch DBH class
has the largest bias relative to basal area growth. Models 1 and 2 are relatively equal for
the 2inch diameter class with Model 3 having a larger bias than growth. All three
82
models overpredict for the 2inch diameter class. Model 2 has virtually no bias for the 4inch
DBH class with Models I and 3 illustrating a small overprediction bias. Modell has
the least bias for the 6, 8, and 1Oinch DBH classes with all three models underpredicting
on average. Model I has the least bias for the 16inch DBH class. Model 2 is superior
for the I8inch DBH class with all three models having their largest bias for the I8inch
DBH class. The 18inch DBH class has the largest absolute bias (underprediction) for all
models which was expected because this DBH class contains few observations.
The average deviation by site index class and mean AABAG is presented in Figure
8. All models exhibit their respective smallest biases in the extreme site index classes
«56 and >75). For site index class <56, Model I has a slight overprediction while
Models 2 and 3 have underpredictions. All three models perform about equally for all
site index classes.
The plot basal area average deviation and mean AABAG graph is presented in
Figure 9. All models have their respective smallest and largest bias for plot basal area
class 90 and 30, respectively. All models underpredict for plot basal area on average
except for plot basal area class 90. Model I performs best for plot basal area class 30 but
there is little substantial difference among the models.
The graph of average deviation by age class and mean AABAG is presented in
Figure 9. All models have underprediction bias for age classes 20, 40, and 80 with Model
I performing best for age class 20 and worst for age class 80. There is little
differentiation between Models 2 and 3 for all age classes. The AABAG decreased from
age 40 to 60 before increasing substantially from age 60 to 80 and may be the result of
age class 60 having substantially more basal area per acre than age classes 40 and 80.
83
0.040
c::::::J Model 1
0.035 ~ Model 2
~ Model 3
lIB AABAG
0.030
0.025
(V
(V LL 0.020
(V
~ro
c::::::Jr 0.015 en
0.010
0.005
0.000
0.005
2 4 6 8 10 12 14 16 18
DBHClass
Figure 7. Average deviation by DBH class and mean average annual basal area growth
for Models 1,2, and 3 when fitted to the complete data set.

0.025
CJ Model 1
0.020
fWAJ Model 2
~ Model 3
~ AABAG
0.015
.....
Q)
Q)
LL
Q) 0.0