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 Title:
 Modeling single slash pine tree growth and spacing relationships
 Creator:
 Reich, Robin Michael, 1954
 Publication Date:
 1984
 Language:
 English
 Physical Description:
 xii, 139 leaves : ill. ; 28 cm.
Subjects
 Subjects / Keywords:
 Diameters ( jstor )
Error rates ( jstor ) Mathematical procedures ( jstor ) Mathematical variables ( jstor ) Pine trees ( jstor ) Stand age ( jstor ) Statistical discrepancies ( jstor ) Surface areas ( jstor ) Tree growth ( jstor ) Trees ( jstor ) Dissertations, Academic  Forest Resources and Conservation  UF Forest Resources and Conservation thesis Ph. D
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 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph. D.)University of Florida, 1984.
 Bibliography:
 Includes bibliographical references (leaves 131138).
 Additional Physical Form:
 Also available online.
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by Robin Michael Reich.
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 University of Florida
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 University of Florida
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 The University of Florida George A. Smathers Libraries respect the intellectual property rights of others and do not claim any copyright interest in this item. This item may be protected by copyright but is made available here under a claim of fair use (17 U.S.C. Â§107) for nonprofit research and educational purposes. Users of this work have responsibility for determining copyright status prior to reusing, publishing or reproducing this item for purposes other than what is allowed by fair use or other copyright exemptions. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder. The Smathers Libraries would like to learn more about this item and invite individuals or organizations to contact the RDS coordinator (ufdissertations@uflib.ufl.edu) with any additional information they can provide.
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MODELING SI NGLE SLASH PINE TREE GROWHTE
AND SPACING RELATIONSHIPS
EY
ROBIN MICHAEL REICH
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA 1984
This research is dedicated to my parents, Rudolph and Edna Reich, who consistently lent their support and encouragement in this endeavor. I would also like to dedicate this work to Nadine B. Mack, a very special friend, whom I could never repay for the companionship and understanding that she has given during the past year.
ACKNOWLEDGEMENTS
The author wishes to express appreciation to those who guided, assisted, and contributed to the completion of this study. Special appreciation is extended to Dr. Loukas G. Arvanitis, chairman of his supervisory committee, for his exceptional patience, guidance, and encouragement during the course of this research.
Appreciation is also extended to Mr. Lewis R. Grosenbaugh, Dr. Tony E. Squillace, Dr. James E. Keesling, and Dr. Mark C. Yang, for their contributions as members of the supervisory committee and for their advice and assistance during preparation of this dissertation. Grateful acknowledgement is also given to Dr. Ronald H. Randles who graciously substituted for Dr. Yang while he was on sabbatical.
Special appreciation is also due to Drs. Arnett C. Mace, Jr., and Robert Schmidt, Director and former Assistant Director of the School of Forest Resources and Conservation, for providing financial support in the form of a Graduate Assistantship and computer time from McIntireStennis Project No. 1874, and to Container Corporation of America for suppling field data from the McArthur spacing study.
Finally, the author is grateful to Mrs. Ruth R. Byrne for the excellent job of typing this dissertation.
iii
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS .......... . .. iii
LIST OF TABLES . . . . . vii
LIST OF FIGURES .................. ix
ABSTRACT . . ...................... xi
INTRODUCTION . . . . . 1
Objectives . . . . . 3
LITERATURE REVIEW . ............. 4
Total Tree Height ............... 6
Tree Diameter . . . . 8
Cubic Foot Volumes ............... 9
Tree Form . . . . . 10
Stand Mortality . . . . 12
Competition . . . . 13
Distance Sampling ..... .......... 15
System of Equations ..... ......... 17
STUDY PROCEDURES ................ 19
Population . . . . . 19
Spatial Analysis ................ 21
Microsite Variation .............. 22
Effects of Spacing on Individual Tree Growth 23 Volume and Surface Area Estimation .. ...... 24 Tree Form . . . . . 25
iv
Page
Data Interpolation ........ .... ... 25
Growth Function ................ 27
Model Developement ............... 29
Field Data Screening ............. 31
Description of the System of Equations . .. 35 ThreeStage LeastSquares. .......... 37
Identification of the System of Equations. . 39 Contemporaneous Correlations. ......... 41
CrossSectional TimeSeries Adjustments. ... 43 Model Validation. ..... ........... 51
RESULTS AND DISCUSSION ............. 53
Spatial Analysis ................ 53
Effects of Microsite Differences on Individual Tree
Growth ........ . . . 58
Effects of Spacing on Individual Tree Growth 71 Volume and Surface Area. .. .......... 71
Tree Form Equations .............. 78
Parameter Estimation .............. 81
First Stage Estimates .......... 81
Second Stage Estimates. .......... 83
Third Stage Estimates ...... ..... 84
Reduced Form Equations ............. 92
Final Form Equations .............. 92
V
Page
Model Validation ............ .... 103
Evaluation of the Coefficients. ........ 103 GoodnessofFit Statistics. ........ 105
Cumulative Distribution of Errors ...... 113
Correlation Among Equations. ........ .. 118
Efficiency of the OLS and 3SLS Procedures. ... 118 SUMMARY AND CONCLUSIONS. .............. 123
FUTURE RESEARCH .................. 128
LITERATURE CITED .................. 131
BIBLIOGRAPHY .................. .. 137
BIBLIOGRAPHICAL SKETCH ............... 139
vi
LIST OF TABLES
Table Page
1 Variables used to identify the "best" combination of explanatory variables ........ 34
2 Spatial analysis for healthy trees and dead
trees at four years of age. ......... 55
3 Analysis of variance of the spatial distribution
of healthy trees in the McArthur spacing study
at four years of age. ............. 56
4 Analysis of variance of the spatial distribution
of dead trees in the McArthur spacing study at
four years of age .............. 57
5 Average height of major understory species for
the McArthur study area at 21 years of age. 61
6 Total cubic foot volume outside bark, for slash pine
of the McArthur study area. ......... 79
7 Total bole surface area outside bark, for slash pine
of the McArthur study area. ......... 80
8 Comparison of tree form inside (FI) and outside
(F ) bark by diameter and total height . 82
9 Crosssectional timeseries adjustments for
diameter breast height ............ 85
10 Crosssectional timeseries adjustments for total tree height. ............... 86
11 Crosssectional timeseries adjustments for cubic foot volume. .............. 87
12 Crosssectional timeseries adjustments for bole surface area ................. 88
13 Concentrated maximum likelihood values for DBH
(D), total height (H), cubic volume (V) and surface
area (S). ................... 89
vii
Table Page
14 Error sum of squares for DBH (D), total height (H), cubic volume (V) and surface area (S). ..... 90
15 Comparison of the estimated coefficients for the DBH equation at three levels of 6. ...... 91
16 Estimates of the model parameters and standard errors for the DBH equation by OLS, 2SLS, and 3SLS. 93
17 Estimates of the model parameters and standard errors for the total height equation by OLS, 2SLS,
and 3SLS ................. 94
18 Estimates of the model parameters and standard errors for the cubic volume equation by OLS, 2SLS,
and 3SLS .................. .. 95
19 Estimates of the model parameters and standard errors for the surface area equation by OLS, 2SLS,
and 3SLS . . . . . 96
20 Reduced form coefficients for the DBH (D), total height (H), cubic volume (V) and surface area (S)
equations. . . . . 97
21 Final form coefficients for the DBH (D), total height (H), cubic volume (V) and surface area (S)
equations. .................. 99
22 Correlation matrix for the DBH equation. . 106
23 Correlation matrix for the total height equation. 107
24 Correlation matrix for the cubic volume equation. 108
25 Correlation matrix for the surface area equation. 109
26 Variance inflation factors for the 3SLS growth model. .................. .. 110
27 Evaluation of prediction equations for DBH, total height, cubic volume and surface area by OLS and
3SLS . . . . .. 112
28 Correlation across equations ......... 119
29 Efficiency of the OLS and 3SLS estimation procedure for DBH (D), total height (H), cubic volume (V)
and surface area (S) equations ....... 122
viii
LIST OF FIGURES
Figure Page
1 Slash pine spacing study. McArthur Tract, near
Callahan, Florida, installed in February, 1962. 20
2 Diameter breast height corrected for spacing and
replication by stand age and plot location 62
3 Total tree height corrected for spacing and
replication by stand age and plot location 63
4 Cubic volume corrected for spacing and replication by stand age and plot location. .... 64
5 Surface area corrected for spacing and replication by stand age and plot location. .... 65
6 Survival corrected for spacing and replication
by stand age and plot location ........ 66
7 Diameter breast height by stand age and plot
location . . . . 67
8 Total tree height by stand age and plot location 68 9 Cubic volume by stand age and plot location. 69 10 Surface area by stand age and plot location. 70 11 Effects of spacing on diameter breast height corrected for microsite and replication by stand age 72 12 Effects of spacing on total tree height corrected for microsite and replication by stand age 73 13 Effects of spacing on cubic volume corrected for microsite and replication by stand age ..... 74 14 Effects of spacing on surface area corrected for microsite and replication by stand age ..... 75 15 Effects of spacing on survival corrected for microsite and replication by stand age ..... 76
ix
Figure Page
16 Relative change in the interim multipliers associated with the exogenous variables 1/Age**2
and (IAlag/Age)*PAR ............ 100
17 Relative change in the interim multipliers associated with the exogenous variables PAR and
Comp . . . . . 101
18 Relative change in the interim multipliers associated with the exogenous variables Alag/Age
and Age. ...... ............ 102
19 Cumulative distribution of errors associated with the diameter breast height equation ..... 114
20 Cumulative distribution of errors associated with the total tree height equation ........ 115
21 Cumulative distribution of errors associated with the cubic volume equation ........ ... 116
22 Cumulative distribution of errors associated with the surface area equation. ........ ... 117
x
Abstract of Dissertation Presented to the Graduate Council of
the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
MODELING SINGLE SLASH PINE TREE GROWTH
AND SPACING RELATIONSHIPS
By
Robin Michael Reich
August, 1984
Chairman: Loukas G. Arvanitis Major Department: Forest Resources and Conservation
This study deals with modeling the effects of competition and available growing space on the growth and development of individual slash pine trees over time. A model to predict the cumulative growth of diameter at breast height, total height, cubic volume, and bole surface area has been developed using a modified form of the Verhulst growth function.
In the past, singleequation estimation procedures, such as Ordinary Least Squares (OLS), have been employed to estimate the structural coefficients associated with a system of equations. This study advocates ThreeStage Least Squares (3SLS), which allows estimation of the structural coefficients simultaneously using a systems approach. OLS and 3SLS are examined under two assumptions concerning the correlation of residual errors: contemporaneous covariance and serial correlation.
xi
The validity of the models was scrutinized following datasplitting (crossvalidation) and comparison of the models against data fit and prediction accuracy. In addition, the signs of the estimated coefficients were compared with theoretical expectations. Parameter estimates for each procedure were obtained from data accumulated in a spacing study which was established in 1962 and remeasured approximately every two years for the past 21 years.
Results of this study indicate that the suggested growth model
depicts the actual process quite well by using either OLS or 3SLS. The structural equations had a weighted k**2 value of approximately 93 percent. However, the 3SLS procedure was more efficient in estimating the coefficients of the growth model than OLS. The OLS procedure does not take into consideration the correlation of errors among equations or the presence of endogenous variables as explanatory ones. This results in a specification error which affects the variancecovariance matrix used in the estimation procedure.
Both OLS and 3SLS yield unbiased estimates of the model parameters. However, only the latter procedure assures asymptotically efficient parameter estimates with respect to the identification of the variancecovariance matrix. These two desirable properties of 3SLS lend validity to the growth model, allow for hypothesis testing, and construction of confidence intervals.
xii
INTRODUCTION
Recently, a renewed interest has emerged in modeling the effects of growing space and competition among individual trees within a forest plantation. Two basic approaches have been used: stand descriptive models and single tree models. In stand models, the variation associated with tree competition is evaluated indirectly in terms of stand averages and stocking densities. While such characteristics are useful under certain conditions, they tend to obscure the cause and effect relationship among neighboring trees.
If one considers single trees as the major component of the system, stand development may be modeled as the interaction between growth and competition of individual trees (Daniels et al. 1979). This feature allows the use of both tree and stand parameters to describe the interactions among competing neighborhood trees.
Models incorporating the spatial relationship of individual trees in a given plantation offer a greater opportunity to study the development of a stand of trees over time (Pielou 1977, Newnham and Smith 1964, Daniels et al. 1979, Wensel 1975). One aspect of tree growth and stand development that has not been adequately addressed is the effect of size and spatial distribution of vacancies, created by mortality and/or thinning on the growth and development of the remaining trees in a forest stand over time.
1
2
One of the major disadvantages of modeling the effects of vacancies on individual tree growth has been the use of singleequation estimation procedures employed to estimate the structural coefficients of a system of equations (Theil 1971, Sullivan and Clutter 1972). In such models it is assumed that the variables simultaneously satisfy certain assumptions pertaining to the standard linear model which assumes homoscedasticity (equal variances among regressors) and that the explanatory variables (i.e. endogenous and exogenous variables) are independent of the residual errors. The use of a singleequation estimation procedure in fitting individual equations to a set of data does not take into account the interrelationship between regressors as well as the interaction among equations (Hock 1958). Likewise, the procedure does not consider the joint distribution of the endogenous variables among equations or the properties associated with the disturbances. It also disregards any restrictions imposea by other equations on certain variables or coefficients in the system of equations (Haavelmo 1943).
Thus, studies similar to those aimed at evaluating the growth and development of stands over time should not be confined to individual equations but rather be extended to a system of equations (Zellner and Theil 1962). The aim of such a system is to describe a set of endogenous variables in terms of a set of exogenous ones and disturbances. The exogenous variables which occur in a system are determined outside of the system in question. On the other hand, the endogenous variables are determined jointly and simultaneously by the exogenous variables and the disturbances generated by the structural equations in the systems (Theil 1971).
3
This approach to estimating the parameters of a system of equations has two main advantages over the singleequation estimation procedure: it allows restrictions to be placed on certain variables, and permits the interaction of variables between equations to be considered in the estimation procedure.
The use of simultaneous equations could lend itself readily to the development of simulation and forecasting models designed to quantify the effects of changes in the degree of competition and growing space on the growth of individual trees in a given forest stand.
Objectives
The objectives of this study were (a) to model the growth and development of individual slash pine trees over time and (b) to quantify the effects of competition and available growing space on the growth of individual trees.
LITERATURE REVIEW
As the demand for wood products increases and the land base suitable for growing wood biomass continues to decline, forest management practices have intensified. Silvicultural practices aimed at increasing the productivity per unit land area include use of genetically improved seedlings, fertilization, thinning, bedding, prescribed burning, and drainage. The assessment of silvicultural practices in terms of stand management is complicated by uncertainties in the growth and development of individual stands due to differences in mortality, site, climatic factors, etc. (Lembersky and Johnson 1975). Recent advances in computer technology and its application to forestry have led to the development of realistic growth and yield models capable of assisting forest managers in assessing the effects of various silvicultural treatments and stand conditions on the growth and development of individual stands over time (Nance et al. 1982, Allen and Duzan 1980, Harms 1982). Knowledge gained from these models has helped foresters in developing optimal management strategies to effectively achieve specific objectives (Allen and Duzan 1980).
To fully exploit the potential of a given site for timber
production, there is a need to better understand how competition and available growing space affects tree growth (Harms and Langdon 1977). Two of the more important factors are stand density and site quality.
4
5
Site is defined by the Society of American Foresters (FordRobertson 1971, p. 242) as "an area considered in terms of its environment, particularly as this determines the type and quality of the vegetation the area can carry". Sites may be classified either qualitatively or quantitatively.
Stand density is defined as "a measure of the degree of crowding of trees within a given area". It is commonly measured as a ratio of crown length to tree height, crown diameter to DBH, or crown diameter to tree height (FordRobertson 1971, p. 255). In a given plantation, the initial spacing exerts a strong influence over the size and quality of individual trees making up the stand. Density is one of the most important characteristics a forest manager has available for controlling the development of a stand over time (Shepard 1974). Formerly, planting densities were selected on the basis of prior experience in stands grown on similar sites. Recently, stocking levels have been determined on the basis of the cost of establishment, interest rates, rotation length, product utilization and other economic factors affecting the production of wood.
A reliable tree growth model must be capable of predicting the growth response of individual trees over a wide array of planting densities for a given site. However, the biological potential of the various sites for timber production varies considerably throughout the range of the slash pine ecosystem (Boyce et al. 1975). Even within relatively small areas site productivity may vary due to microsite differences (Swindel and Squillace 1980).
Extensive research has shown that as stand density increases there is a progressive decrease in the average size and value of an individual
6
tree as well as a change in its morphological and physiological characteristics (Davis 1966, Jorgenson 1967, Harms and Langdon 1977). This phenomenon is attributed to the increased competition for available growing space, light, nutrients and water. As stand density increases the ability of some trees to compete with their neighboring trees diminishes. The effects of stand density on tree growth is more complex than a simple reduction in overall growth due to competition, but is a result of a combination of factors such as the initial planting density, genetic traits, disease and insect infestation, spatial distribution of mortality and climatic conditions (Davis 1966).
Competition between neighboring trees results in a natural
selection of those trees having the best possible complement of genetic traits for a given site and stand condition (Wright 1976, Jorgenson 1967). Inferior trees are gradually suppressed and shaded out. The number and distribution of suppressed trees depends upon the initial spacing of the stand, early mortality, the age at which the tree canopy closes, and the quality of the site. In the past, forest managers tended to overplant in order to ensure an adequate stocking level at the end of the rotation due to uncertainties in the survival rate. The effects of spacing on the size of individual tree variables are discussed below.
Total Tree Height
It has long been assumed that height growth is independent of stand density (Bennett 1963, Lloyd and Jones 1982, Husch et al. 1982). Usually, the mean height of a stand increases with increased spacing
(Shepard 1974, Jorgensen 1967, Bennett 1971). The closer the spacing the greater will be the intensity of competition among neighboring trees. In such cases, the proportion of suppressed trees will increase.
Bennett (1963) found that slash pine planted on oldfield sites
(site index 65, base age 25) grew approximately three feet annually for the first 15 years. Height growth increased with site quality and decreased with stand age. In a later study, Bennett (1971) observed that height growth declines after age ten years for all sites and densities.
Bennett (1963) also showed that during a given year height growth increases at a decreasing rate. By the end of March approximately 25 percent of the height growth is completed. The remaining percent of the height growth is completed by the end of June.
Lloyd and Jones (1982) pointed out that a substantial bias can
result in growth predictions based on densityheight independent models developed from data which include an effect of density on tree heights. The authors suggest that stand density should be used in modeling individual tree heights in order to obtain a more realistic height growth function. If a significant heightdensity relationship exists in the data, the omission of this information may result in a specification error which in turn affects the estimation of the coefficients in the model. The magnitude and direction of this bias depend solely upon the heightdensity relationship present in the development of the site index curves. Standard regression techniques may be employed to test the significance of individual variables in a given model.
8
Lloyd and Jones (1982) concluded that any attempt to modify site index to reflect stand density would be inappropriate since site index, by definition, is independent of density.
Tree Diameter
In general, the increase in the mean stand diameter is inversely proportional to the planting density. The configuration of the spacing (square versus rectangular) is not correlated with diameter growth (Bennett 1963). Campbell and Mann (1974) demonstrated that the initial planting density has a stronger influence on diameters than site index. As spacing and site index increase, the mean diameter of the stand shifts to the right resulting in a distribution skewed to the left. The interrelationships between stand density, individual tree size, competition and other factors control the rate of tree movement to higher diameter classes over time and, thus, the shape of the diameter distribution.
Stand density does not have a significant effect on average stand diameter until the fifth growing season (at standard stocking levels); it intensifies thereafter (Bennett 1963). Several researchers have demonstrated that most of the differences in diameters occur during the first 12 to 20 years (Shepard 1974, Jorgensen 1967). At closer spacings, growth is reduced due to increase in competition between trees. The mean periodic annual diameter growth is directly related to stand density and decreases with stand age (Lohrey 1961, Shepard 1974). When the differences in diameter growth between spacings stabilize, competition is severe for all spacings. On poor sites, it takes longer for differences in diameter growth to stabilize over time (Lohrey 1961).
Cubic Foot Volumes
For a given species, volumes of individual trees and stands are
directly related to age, site index, and planting density of which only spacing can be controlled by the forester. Maximum yields are realized in higher planting densities and sites. The increase in volume is proportional to stand density (Bennett 1963, Lohrey 1961, Campbell and Mann 1974). On sites below 50, spacing has little or no effect on yields because of the inefficiency of the trees in the stand to fully utilize the site (Bennett 1963). The percentage increase in yields per unit area due to an increase in site is greater for lower sites than for higher sites, but the actual volume increase is larger for the higher sites. Also, yields tend to be higher on oldfield sites than on natural forest sites due to a reduction in competing understory vegetation and residual effects from past fertilization (Bennett 1963). On an individual tree basis, volume growth increases with an increase in spacing. The economic advantage of the wider spacings are due mainly to the increase in wood quality which yields higher monetary returns. However, total wood productivity of the wider spaced stand may decrease.
Bennett (1971) indicates that current annual volume increment of slash pine plantations culminates at age 12 years in stands with approximately 200 trees per acre over all sites greater than 50. Culmination occurs later for higher densities but seldom takes place beyond 25 years after planting. Knowledge of the change in the current annual increment is important because it provides the forester with a better understanding of stand growth over time, and a basis for
10
determining optimal rotation lengths. Culmination of volume growth does not coincide with peak periods of height and diameter growth because of differences in growth rates. For a given stand, the combination of height and diameter growth results in maximum volume growth per unit land area for that site and stand density.
Tree Form
The effect of competition and available growing space not only
affect the size of individual trees but also the form of the main stem (Lohrey 1961, Horn 1961, Meng 1980). In a study with red pine (Pinus resinosa Ait.) plantations Horn (1961) observed that stem form improved with competition and age while thinning impaired tree form of dominant and codominant trees. Meng (19b0) found that there was a statistically significant change in stem form following heavy thinning (50 percent of the basal area removed) and fertilization (448 kg/ha) of black spruce (Picea mariana (Mill.) B.S.P.) stands. The change in the shape of the tree bole is in response to a decrease in competition and an increase in the available growing space (Horn 1961, Husch et al. 1982).
The diminution in diameters along the main stem is known as taper and is the fundamental reason for the variation in cubic volumes of trees with the same diameter at breast height and total tree height (Husch et al. 1982, Grosenbaugh 1966). Stem form also varies with species, diameter, age, and the quality of the site. Because of the large variation in stem forms among trees, there rarely exists any resemblance to one of the most common geometric solids (Grosenbaugh 1966, Husch et al. 1982).
11
Forest mensurationists have sought to discover a simple twovariable function involving only a few parameters which could be used to specify the entire tree profile (Grosenbaugh 1966). For example, Demaerschalk (1972) developed a system in which taper equations can be derived from existing volume equations; both equations yield identical results for total cubic volume if some taper data are available. In a similar study, Bailey et al. (1982) developed taper equations for siteprepaired slash pine plantations.
Grosenbaugh (1966) demonstrated that total height (THT), bole surface area (SUR), and volume (VOL) of a given tree are highly correlated with other variables that are affected by the size and form of the tree stem, such as quality or quantity of lumber, plywood, kraft pulp yields as well as specific gravity. Not only are heights, surface areas, and volumes useful in estimating various product yields, but comparisons of ratios involving the three variables THT, VOL, and SUR provide additional information on the form of an individual tree. Grosenbaugh (1966) introduced three measures of tree form based on the ratios of height, volume, and surface areas, namely:
Quadratic mean dia. (in.), Q = 13.5406 (VOL/THT)1/2 Arithmetic mean dia. (in.), A = 3.8197 SUR/THT, and
(Quad. Mean)2/Arith. mean (in.), Q2/A = 48.0000 VOL/SUR.
The three means coincide if a tree is a cylinder (no taper) and become increasingly different in the presence of taper. The ratio
(Arith. Mean)2/ (Quad. Mean)2, F = 0.0796 SUR2/THT*VOL
which will always be less than one, measures the inverse of the relative
12
variation in diameter along the main stem. Zero variation is indicated by a ratio of unity or when the tree is cylindrical. As relative variation increases, the ratio becomes increasingly smaller than unity.
Stand Mortality
One aspect of forest growth that has not been adequately addressed in the past pertains to the size, spatial distribution, and abundance of vacancies created by mortality in forest stands as well as the anticipated effect on individual tree growth (Bormann and Likens 1981).
If vacancies in a given stand are randomly or uniformly distributed in space, adjacent trees may utilize the additional growing space to accelerate their growth. The net result may be an increase in stand productivity because of the reduced competition among neighboring trees. If vacancies are aggregated, the added growing space may not be fully utilized by adjacent trees. In such cases a loss in stand yield may be realized which is proportional to the area occupied by the dead trees (Newnham and Smith 1964, Stauffer 1978). Jones (1977) demonstrated that the spatial distribution of residual trees within a given stand significantly contributed to differences in individual tree volumes as well as total stand volumes.
Tree mortality is extremely variable and difficult to predict (Jones 1977). There are two basic types of mortality that occur in forest stands: regular and irregular (Lee 1971). Regular mortality include trees that die due to competition for limited resources such as light, moisture, growing space and nutrients. Mortality caused by insect and disease infestations, as well as windfall and fire, is termed
13
irregular. Predictions for both types of mortality are cumbersome and highly variable no matter how much mortality data are available (Davis 1966, Mitchell 1969, Hamilton 1974).
Competition
During the development of a stand, the growth of some trees is
accelerated, while in others growth is suppressed or nonexistent. When openings in the stand are created by some kind of nonpersistent disturbances, adjacent trees tend to utilize the vacancies to increase their dominance within the stand (Mitchell 1969). The rate of growth depends upon the vigor of the adjacent trees to successively compete for the additional growing space (Bormann and Likens 1981).
A problem arises when analyzing the growth of individual trees due to the difficulty of quantifying the level of competition among neighboring trees (Daniels 1976, Opie 1968, Zedaker 1982). Initially, various measures of stand density in the vicinity of a given tree were used as a measure of competition. Such measures include basal area, angle count, point density, distances to neighboring trees, crown diameter, and the number of primary and secondary competing trees (Bella 1971). The concept of "zone of influence" as a measure of crown competition among neighboring trees has also been advocated. It is defined as the area of crown overlap between neighboring trees (Bella 1971, Venzina 1963).
The basic assumptions of various competition indices are
a. The ability of a tree to compete for limited resources such as
growing space and light can be represented by a circle or zone
14
surrounding the tree, with the radius being a function of tree size such as volume, bole surface area, crown radius, DBH and total tree
height, and
b. The competition stress experienced by a tree is a function of the
degree to which its influence zone is overlapped by those of its
neighbors.
Several modifications have been made in defining the influencezone, including the use of weighted variables to denote the relative importance of the influence area overlap of a given tree to compete with neighboring trees (Daniels 1976, Venzina 1963). To estimate the growth potential of individual trees, Ek and Monserud (1974) used a weighted competition index based on the ratio of the total height of the subject tree to the average height of its competing neighbors multiplied by the crown radius of the subject tree. Other approaches advocated as a measure of competition are based on the ratio of diameters between the subject tree and its competing neighbors. This approach assigns weights proportional to the size of trees while the effects between trees of similar size remain essentially unweighted (Daniels 1976). Basal area has also been used as a measure of competition among neighboring trees. The basic assumption is that competition among individuals increases proportionally with the basal area of the stand (Bella 1971). Minor (1951) showed a slightly curvilinear relationship between diameter at breast height (DBH) and crown diameter. Minor also demonstrated that the ratio of DBH to crown diameter remains relatively constant once the basal area of a stand is stabilized.
15
Distance Sampling
Competition among individual trees within a stand is also directly related to the spatial distribution and size of the vacancies created by mortality. Therefore, it is important to approximate as closely as possible the spatial distribution of a given tree population in order to make valid inferences. Most sampling techniques and analyses were developed under the assumption of randomly distributed plant populations (Clark and Evans 1954). This assumption may not be valid for many plant communities (Ashby 1935). If the distribution of a population departs significantly from the underlying assumption of randomness, the resulting statistics and the reliability of the inferences may be in serious error (Cole 1946).
Distance sampling is one of the methods employed to study spatial patterns of plant populations over a continuous surface. Basically, this method involves distance measurements either from randomly selected points and/or from randomly selected plants to their nearest neighbor. The empirical frequency distribution of such distances and the associated statistics such as means and variances may be used in specific cases to test the degree of departure from randomness. Since random spatial patterns of plants are not very common in the real world, distance sampling methods have been applied to study the degree of aggregation or clustering of plants in a given area.
Recent findings (Reich 1980) indicate that Hopkins' modified coefficient of aggregation is an accurate and efficient computer sampling technique capable of analyzing the spatial patterns of natural and planted pine stands.
16
Hopkins' coefficient of aggregation, A, is based on the ratio of the squared distances between randomly selected points and the nearest tree over the squared distances between randomly selected trees and their nearest neighbor:
1
A =
E W2
where
A = Hopkins' coefficient of aggregation,
ECl = Sum of squared pointtotree distances, Em2 = Sum of squared treetotree distances.
Pielou (1977) introduced a modified form of Hopkins' index that is more sensitive in detecting nonrandom spatial patterns. Hopkins' modified coefficient of aggregation is defined as
A* =
I +A
which has a betatype distribution with mean 0.5 and variance 1/4(2n+1), where n is the number of pointtotree and treetotree distances in the sample. With a large sample size (n>50) Hopkins' index approaches a normal distribution. In such cases Z = 2 (A* 0.5)*(2n +1)1/2 can be viewed as a standard normal variate. The null hypothesis of a random spatial distribution would be rejected when
IZI> Za/2
where Za/2 is the critical Zvalue for the standard normal distribution. The population is said to be aggregate if A* is
17
significantly greater than 0.5 and uniformly distributed if A* is significantly less than 0.5.
System of Equations
Most growth simulation models include equations to predict DBH,
total tree height, crown width, and volume. They also incorporate some measure of competition to quantify the degree of interaction between neighboring trees. The predictive equations are not independent. When they are combined together they form a system of equations with correlated errors. This interdependence among regressors violates the assumption of a single equation estimation procedure which assumes that no correlation exists among the explanatory variables.
The above discussion suggests that studies of single tree growth
should not be confined to individual equations but rather be extended to a system of L equations commonly referred to as simultaneous equations.
In the development of any tree growth model the most important
consideration is the form of the relationship describing the interaction of individual tree parameters with growth. It is sometimes argued that the theory related to forest growth should provide insight into the relationship that the individual equations take on (Fisher 1970). The problem thus is to decide which set of basic characteristics adequately relates individual tree growth to cubic volume, surface area, tree diameter, total tree height, competition and available growing space.
Over the years numerous techniques have been developed to estimate the parameters associated with a system of equations. Several of the more common estimation procedures include LimitedInformation aximum
18
Likelihood (LIML) procedure, FullInformation MaximumLikelihood (FIML) procedure, Zellner's seemingly unrelated least squares procedure, TwoStage Least Squares (2SLS), and ThreeStage Least Squares (3SLS) to name but a few. The appropriate procedure to use in estimating the parameters of a system of simultaneous equations will depend upon the structure and characteristics of the equations making up the system.
For example, LIML takes into account the absence of certain variables from the Ith structural equation but does not assume the absence of any variables from the remaining L1 equations in the system when estimating the parameters of the Ith equation. On the other hand, FIML handles restrictions on the coefficients of all L equations simultaneously (i.e. zero coefficients in the Ith equation). The 2SLS procedure is similar to LIML while the 3SLS procedure is similar to FIML. The FIML and LIML procedures are rarely used because the estimation procedure involves solving a very complicated set of estimation equations (Theil 1971).
A recent contribution to the estimation of a set of coefficients for a system of simultaneous linear equations is the 3SLS procedure developed by Zellner and Theil (1962). This procedure yields estimates which are asymptotically at least as efficient as the 2SLS procedure (Madangly 1964). The 2SLS procedure is applied to each of the L structural equations one at a time but computes a joint asymptotic standard error for the system as a whole, while the 3SLS procedure (an extension of the 2SLS procedure) handles the unknown coefficients of all L structural equations at the same time.
STUDY PROCEDURES
Population
The data used in this study were provided by Container Corporation of America. The study area is located near Callahan, Florida, in Nassau County.
Slash pine seedlings were hand planted in February 1962 in twelve oneacre plots measuring 2.5x4 chains at six different spacings: 5x10, 7x10, 9x10, 11x10, 13x10, and 15x10 feet. Each plot was replicated twice and consisted of 14 evenly spaced rows. Each plot was surrounded by a buffer strip. Two spacing plots were replanted in February 1963 due to low planting survival (Figure 1).
Prior to planting, the study area was part of an improved pasture. The pasture was burned prior to planting in 1962. The site is somewhat poorly drained with a spodic horizon 1.2 to 2.0 meters in depth.
All trees were assessed for survival at two years of age and then approximately every two years thereafter, starting at six years of age. Tree measurements were collected on a subsample of trees from each plot. Sample data included the condition of the tree (healthy or dead), the relative location within the study area, diameter at breast height, total tree height, merchantable tree height to a four inch top, tree vigor (position in the canopy) and the number of primary and secondary competing trees.
19
20
cn o
04
0oo
e4
o
LJ
m 4
rr)
80
co
a P
OL U LU2
a
~4Q C )
)
oU
H ~4 p 
21
Prior to the fifth set of measurements, sample trees were selected from two diagonal transects established in each plot. On each transect 14 trees, one from each row, were sampled (28 trees per plot). Commencing from the fifth set of measurements on, four rows were systematically selected from each plot and all trees within the selected rows were measured. The number of sample trees per row varied by spacing and degree of survival.
Spatial Analysis
Hopkins' modified coefficient of aggregation was used to analyze
the spatial pattern of healthy and dead trees in the study area when the trees were four years old. Age four was selected for analysis because almost all mortality in the study area had already occurred by that time. In each plot, ten independent estimates of Hopkins' index A* were obtained based on 150 pointtotree and 150 treetotree distances. A zstatistic was used to determine the degree of departure of the observed spatial distribution from the theoretical (random) one. Analysis of variance was employed to evaluate the magnitude of differences in the spatial pattern of healthy and dead trees among the six spacings. Mean separation was determined using Duncan's multiple range test (a = 0.05).
22
Microsite Variation
Swindel and Squillace (1980) demonstrated the existence of
microsite variation on the growth of individual trees. To examine this extraneous variation on tree growth over time, the effects of spacing and replications were removed from the survival, DBH, total tree height, cubic volume, and surface area data for each time period. Spacing effects (Si) were estimated as the difference between the spacing means (SPi) and the grand mean (GM); replication effects (Rj) were estimated as the difference between the replication means (REPj) and the grand mean. Microsite was estimated by subtracting the effects of spacing and replications from the individual plot means: Y =Yij S R Si = SPi GM
Rj = REPj GM
where
YT. = corrected plot mean for spacing i and replication j, Yij = uncorrected plot mean for spacing i and replication j,
1 2
SP = E Yi spacing mean,
J=1
1 6
REP = z Y..ij replication mean,
i=1
1 6 2.
GM = E Y.. grand mean,
12 ij
i=1 j=1
Si spacing effect, R = replication effect.
23
The effect of microsite on survival and individual tree size was analyzed by (a) comparing the corrected plot means at each age, and
(b) comparing corrected plot means over time for a given plot.
Effects of Spacing on Individual Tree Growth
To estimate the effects of spacing on survival, DBH, total tree
height, cubic volume and surface area, individual plot means (Yij) were corrected for microsite and replication: = .. M.. R.
l ij ij 3
where
MiJ = Yj GM,
Y* = plot mean corrected for microsite and replication,
Mij = microsite effect.
Plot means for survival, DBH, total height, cubic volume, and surface area corrected for microsite and replication effects were regressed on spacing by age using the standard linear model:
Y? = B + 8 (spacing) + E.. 1 o 1 1
The significance of spacing on survival and individual tree size was tested using regression analysis (a = 0.05).
24
Volume and Surface Area Estimation
A sample of 150 standing trees were selected at random for
dendrometry measurements in the Spring of 1983. Approximately 12 trees were selected from each spacing x replication plot. Heights and corresponding upper stem diameters of standing sample trees were measured with a magnifying Barr and Stroud optical dendrometer, Model FP15. Measurements were taken at stump height (0.5 feet above ground), breast height (4.5 feet above ground), and various other points along the tree stem. This procedure divides each standing tree into segments that are relatively uniform with respect to taper, quality, and defects. Diameters at breast height were also measured with a diameter tape; bark thickness at breast height was measured with a bark guage.
Cubic foot volumes, bole surface area and heights inside and
outside bark for each sample tree were computed and accumulated over all measured tree segments. Product yield estimates were computed with Grosenbaugh's (1974) STX computer program. Regression equations were developed to predict total cubic foot volumes (VOL) and bole surface areas (SUR), outside bark, of individual trees as a function of diameter at breast height (DBH) and total tree height (THT) using the functional forms:
VOL = a DBHB THTY
SUR = a'DBH'THTY'
where a(a'), B(8'), and y(y') are regression coefficients estimated from the sample dendrometry data obtained from each spacing. The two regression equations were fitted using a standard linear model after a logrithmic transformation.
25
The regression equations were used to estimate individual tree volumes and surface areas at 6, 8, 10, 12, 14, 16, 19, and 21 years after planting for all sample trees used in the analysis.
Tree Form
Using the volume and surface area equations described in the
previous section, it is possible to express Grosenbaugh's (1966) four ratios of stem form in terms of diameter at breast height and total tree height by substituting the volume and surface area equations into the appropriate ratio. The four ratios will have the same functional form as the volume and surface area equations. Since trees can assume an infinite number of shapes, the coefficients in the above equations would tend to vary from tree to tree. Thus, equations developed for a given population can only predict the average change in tree form, cubic volume and surface area with respect to changes in diameters and heights of the same population.
Data Interpolation
The proposed growth model assumes a constant time interval (i.e. two years) between current and lagged exogenous and endogenous variables. To satisfy this constraint, three interpolation procedures (linear, Lagrangian, and quadratic) were examined to select an appropriate one to estimate DBH and total tree height at ages 4, 18, and 20 years after planting. The three interpolation procedures are described below
26
Linear Interpolation
If DBH and total tree height (y) are functions of the age of a tree
(x), and yO and yl are known functional values corresponding to x0 and Xl, respectively, the functional value of y can be approximated in a linear manner by
(Y1 Yl) x + XlYoX0 Y1
y = where, x0< x < xl x x0
Lagrangian Interpolation
Given any three points on a plane (x0, y0), (xl, y1) and (x2, y2)' it is possible to evaluate the Lagrangian interpolation of a second degree polynomial P2(x) which passes through the three points. The functional value of x, denoted by P2(x), can be approximated by
2
P2(x) = E Li(x)yi i=0
where
2 (x x.)
Li(x) = H ( xx j=0 i j
itj
Quadratic Interpolation
The quadratic procedure assumes no functional relationship between x and y. Given two values yo and y1, the value of y for a given x can be approximated by y = 1 Yi]2 where, yo < y < y1. Evaluation of the three interpolation procedures indicated the linear method diameters and total heights during the juvinile growth phase and underestimates during the latter years. The quadratic procedure was not considered because it estimates a single geometric
27
mean regardless of whether X is much closer to X2 than X1. As a result, the Lagrangian procedure was used to estimate DBH and total tree height at ages 4, 18, and 20 years. The interpolated data was then used to estimate total cubic volume and surface area.
Growth Function
The cumulative growth of an individual tree over time resembles an S  or sigmoid  shaped curve. Although the exact form of the cumulative growth curve changes with the dimensions of the tree (diameter, height, volume and surface area) the cumulative growth curve has certain properties that hold for all dimensions of a tree (Husch et al. 1982).
Grosenbaugh (1965) investigated numerous sigmoid and nonlinear functions capable of generating a wide variety of sigmoid and other monotonically changing shapes consistent with the growth of individual trees over time. Several nonlinear growth functions were examined to select one that is intrinsically linear with respect to the parameters in the growth model and that are consistent with the growth behavior of individual trees subjected to various competitive stresses. A few of the relationships examined by Grosenbaugh include the JohnsonSchumacher, Gompertz, Gauss, von Bertalanffy, Verhulst and the PearlReed growth functions. Principal features and asymptotic properties of the above growth functions are discussed by Grosenbaugh (1965).
The growth function selected to model tree growth in this study was that of Verhulst which has the following form:
28
Y = A (1 + C exp ( BX))1 where
Y = dependent variable, X = independent variable, A = asymptotic maximum of Y, (YMAX), C,B = regression coefficients. The Verhulst growth function was modified to include more than one independent variable in the growth model and to express individual tree growth on a percentage basis. The latter allows for comparison of the effects of competition and relative growing space on diameters, heights, volumes and surface areas over time. The modified Verhulst function has the following form: Y* = (1 + exp(Y'))1 where,
Y* = Y/A or Y/YMAX, 0 < Y* < 1, and Y' = BO + B1X1 + B2X2 + ... + BkXk + E
Taking the first derivative of Y* with respect to Y' yields
dY*
S= Y* (1 Y*) < 0. dY"
The second derivative is given by
d2y*
d2 = (1 2Y*)(1 Y*) Y* > 0.
29
The asymptotic properties of the modified Verhulst function are
Lim Y* = 1
Y, + C
Lim Y* = 0.
To utilize the modified Verhulst growth function in modeling
individual tree growth over time, the value of YMAX must be defined in terms of the dependent variables being modeled (diameter, height, volume and surface area). Since YMAX by definition is the maximum size an individual tree can attain, YMAX was equated to the site productivity of the stand. Maximum values for DMAX, HMAX, VMAX and SMAX were defined as follows:
HMAX = site index base age 50,
DMAX = maximum tree diameter for a given site index (HMAX),
VMAX = maximum cubic foot volume; f(DMAX, HMAX), and
SMAX = maximum bole surface area; f(DMAX, HMAX).
Model Development
When one performs several statistical tests to identify the
structural form of a regression equation and uses the data to estimate the parameters of the model, certain problems arise which can affect the sampling distribution of the estimators (Theil 1971, Snee 1977). This problem is overcome when the identification and estimation procedures are based on independent data sets. It is usually recommended that the
30
data set be split into two groups; the first set is used to identify the structural form of the equations, while the second set provides estimates of the coefficients of the model.
The main advantage of data splitting is that one can obtain an independent estimate of the prediction accuracy of the model. The disadvantage of data splitting is that the variance of the estimated coefficients in the model increases. Snee (1977) presented a simple formula to approximate the percent increase in the variance when the data are split into two groups of equal size:
Percent Increase = L(2n ) I1/2 ] 100 (n 1)
where n is the size of the estimation and prediction data sets. With a
sample size n = 30 the increase in the variance is approximately 42.6 percent and rapidly approaches 41.4 percent as the sample size goes to
infinity. Because of this increase in the variance most researchers use the same data set to identify and estimate the structural form of the equations.
In this study the sample data were sorted by age, spacing, and
replication and numbered from 1 to 5199. The data were sorted in this manner to increase the correlation among the independent variables over the six planting densities and nine time periods. Subsequently, the data were divided into two groups of equal size (NI = 2600, N2 = 2599). All odd observations were included in data set one; even observations were included in the second data set. The first data set (Ni) was used to identify the structural form of the growth equations using the OLS procedure, while the second data set (N2) was used to estimate the coefficients of the system of equations using the 3SLS procedure.
31
Field Data Screening
REX (Grosenbaugh 1967), a FORTRANIV computer program, was used for combinatorial screening of variables used in the growth model for this study. This approach explores all possible linear combinations of variables to find the "best" combination of independent variables that best explains the variation of the dependent variables. REX uses the mean squared residual, MSE (or variance of the regression estimate), as an indicator of a "best fit" since the MSE does not change monotonically with the number of independent variables in the equation. As opposed to other procedures such as stepwise, the selection of the "best" fitting model by REX is independent of the steps used in the exploratory analysis and provides unbiased estimates of the coefficients, but tends to underestimate the variance of prediction.
Fourteen variables were screened through REX for the diameter and height equations; thirteen variables were screened for the volume and surface area equations (Table 1). The final equations for diameter, total height, cubic volume and surface area had the following functional form:
D1= 0+11D 1 + 12 H_1+ 13PAR +14PAR2 +B15R2 +616A +018/A2 + 11COMP
+ a112DMS + 1
H2= a20 21D_+ 22 i1 +23PAR + 24PAR2 +B25R2 +26A +27A2 + 29RIP
+ 8211COMP/SP + E2
V3= 31D + 32H +30 +33PAR + 34PAR2 +a35R2 +336A +638/A2+
839R1P + B311COMP/SP + E3
32
S4 = r41D + 42H +a40+ 43PAR + B44PAR2 +45R2 +846A +B47A2 +
849RIP +B411COMP/SP + Eq where
D1 = Ln ( 1)
D1 = DBH/DMAX
DMAX = maximum DBH at 50 years of age D_1 = D1 lagged one period (two years)
H2 = Ln (I 1)
H2 = THT/HMAX
HMAX = site index, base index 50 years
H_1 ~ H2 lagged one period
V3 = In ( 1)
V3 = VOL/VMAX
VMAX = maximum cubic volume at age 50 years for a given DMAX
and HMAX
S4 = Ln ( 1)
S4 = SUR/SNAX
SMAX = maximum bole surface area at 50 years of age for a given
DMAX and HMAX
SP
PAR = relative growing space = S
5(kL+1)
KL = number of nearest neighbors lagged one time period,
KL = 0,1,2
33
SP = spacing within rows (feet)
A = tree age (years)
A_I = tree age lagged one period
R2 = A_1/A
RIP = (1 R2) PAR
2
COMP = E DBHk/(2 DBH)
k=l
DBHk = DBH of neighboring trees within rows (inches)
DBH = DBH of subject tree (inches)
DMS = diameter microsite (inches)
k = error term for the Ith structural equation
Ln = natural logarithm
rij' Bij = parameters of the endogenous and exogenous variables,
respectively.
The equations for diameter, total height, cubic volume and surface area were fitted using REX to obtain OLS estimates of the coefficients in the model. The coefficients were estimated assuming that diameter and height were exogenous variables in the cubic volume and surface area equations and that the disturbances among equations were uncorrelated. The four equations were also fitted using SAS/SYSREG (Helwig and Council 1979) procedure to obtain 3SLS estimates. This procedure assumes a correlation of residuals among the four equations along with the presence of endogenous variables as explanatory ones. Following is a detailed description of the 3SLS procedure.
34
Table 1. Variables used to identify the "best" combination of
explanatory variables.
Independent Equation Variable DBH(D) THT(H) VOL(V) SUR(S) DBH(D) N.U. N.U. A A Total tree height (H) N.U. N.U. A A DBH Lagged (D_1) A 2_ A N.U. N.U. Total height lagged (H_) A A N.U. N.U. PAR G G G C PAR **2 G G G G COMP *2 G 3 G G G COMP S S S S COMP/SP S S S S Alag/Age G G G G
Age G G G G 1/Age S S S S Age **2 G G G G 1/Age **2 S S S S (1Alag/Age)*PAR G G G G DBH Microsite G N.U. N.U. N.U. THT Microsite N.U. G N.U. N.U.
I/ Not used in the screening process. 2/ Variable was automatically included in the model and was not subject
to the screening process.
3/ G = group; S = set; members of sets within a group are treated as
nonfixed, and are included in regressions only when called for by specific combinatorial rules. For example, the competition group contains three sets; only one set will be included in the model if
the group is selected in the screening process.
35
Description of the System of Equations
The system of equations described in the previous section contains L linear stochastic structural equations consisting of L jointly dependent variables and K predetermined variables. It is assumed that
(a) the system can be solved for the jointly dependent variables and
(b) the disturbances of the structural equations have zero mean and
(c) the disturbances are serially independent and homoscedastic. The latter implies that their variances and contemporaneous covariances are finite and constant over time. A test for crosssectional and serial correlation will be discussed in a later section as well as the appropriate estimation procedure.
If there are n observations then any structural equation, say the Ith equation, can be written in the following form for all n observations combined:
y = Yy + Xk X + EX
where y, is a nxl column vector of observations for one of the L jointly dependent variables in the system; Y is a nxL matrix of values taken on by the explanatory dependent variables in the th equation; yQ is the corresponding Lx1 coefficient vector; X is a nxK matrix of values taken on by the exogenous variables; 8 is the corresponding Kxl coefficient vector; and 6 is a nxl column vector of structural disturbances. Using matrix notation the system as a whole may be written as follows: YF +X8 = E
where Y is a nxL matrix of jointly dependent variables; r is a LxL
36
matrix of parameters; X is a nxK matrix of predetermined variables; a is a KxL matrix of parameters; and E is a nxL matrix of disturbances. Since there are as many equations as jointly dependent variables, the matrix r is square and nonsingular by definition. Thus, the system may be solved for Y by
1 1
Y = XBF + Er
This system of equations is referred to as the reduced form. It describes the endogenous variables in terms of the current and lagged independent variables, lagged dependent variables and disturbances. The reduced form coefficients can be estimated indirectly from the structural equations as shown above or directly by the reduced form: Y = Xr + U
where T = 5I and U = EF i However, if the coefficients of the reduced form equations are estimated directly, it is impossible to obtain estimates of the coefficients associated with the structural equations. The latter approach was used by Sullivan and Clutter (1972) to estimate the coefficients of a simultaneous growth and yield model for loblolly pine stands. Since the reduced form equations do not contain current endogenous variables, as explanatory variables, the L reduced form equations can be considered unrelated even though they still have correlated errors among equations. In such cases Zellner's seemingly unrelated least squares procedure would be the appropriate one to use in estimating the coefficients of the reduced form equations (Theil 1971).
37
Threestage LeastSquares Procedure
The 3SLS procedure uses the results of the 2SLS procedure to estimate the coefficients of the entire system simultaneously.
The matrix E(Y ) is a submatrix of the matrix E(Y) which
corresponds to the jointly dependent variables that occur on the right hand side of the equation. The th structural equation may then be expressed in the following form:
S= [E(Y) Y] + + [Y E(Y)]
The first stage of the 3SLS procedure yields an ordinary least squares (OLS) estimate of E(Y ) which is given by
yy = [Y U: X] + E + U 6 where
Uz = M Y
M = I X(X'X)1X'
1
E(Y ) = X(X'X) X'Y Y In the second stage OLS is used to provide an estimate of the coefficient vector 6 Zellner and Theil (1962) and Theil (1971) have shown that the 2SLS estimate of 6 is given by
d = [Z' X(X'X)1 X'Z ]' Z' X(X'X)Ix' Y
R R .
38
where Z. = [Yk Uk: XZ]. The same authors discuss in detail the estimation procedure in the two cited references. An estimate of the variancecovariance matrix S. is given by Sz [Z' x(x'x)x'zX]
from which asymptotic standard errors can be computed by taking the square roots of the diagonal elements. The scalar S., is an estimate of the variance a which is defined as the mean square of the associated residuals.
S = (Y Z d ) (Y Z d ) SU2 n 2 z To obtain the third stage estimates, the L structural equations, for all n observations, are stacked in the following manner:
Yl Z1 0 .. 61 61 Y2 0 Z2 O0 62 :2
YL 0 0 ZL L EL or simply
y = Z6 + E .
Premultiplying the above equation by the transpose of a LnxKL Kronecker product of the form:
39
X 0 0
0 X 0
0 0 X
where X is an Kxn matrix of explanatory variables yields
(I X') y = (I 0 X') Z6 + (I X')E where y is a Lnxl vector of endogenous variables.
Applying the Generalized Least Squares (GLS) procedure to the above system of equation leads to the following estimate of 6:
6= (Z'[t'eX(X'X) x' Z)1' z'[lx(x')X']y.
If the variancecovariance matrix t is replaced by S = (sj), the matrix of mean squares and products of the 2SLS residuals leads to:
6 = (Z'[Xl X(X'X) X']Z) Z'[S1X(XX)1 'y the threestage least squares (3SLS) estimator.
Identification of the System of Equations
To obtain an efficient estimate of the coefficients of the structural system of equations and an estimate of the variancecovariance matrix, the sytem of equations must satisfy an order condition of identifiability (Theil 1971). A necessary condition for
40
the existence of the 3SLS estimator is that K > N., where K is the total number of predetermined variables in the system and N. is the number of parameters to be estimated in the th structural equation. Three possible conditions exist K N. If K < N, the th equation is said to be under identified and the 2SLS estimator does not exist. This implies that at least one row or column of the variancecovariance matrix cannot be estimated. In other words the parameters of the Ath equation are not estimable because of a linear combination of the other equations that contains only variables which do not occur in the Zth equation. In such a situation, Zellner and Theil (1962) recommend the elimination of all under identifiable equations from the system and confinement of the 3SLS procedure only to identifiable equations.
If K = N the Zth equation is said to be just identified. If all the equations in the system are just identified or if the variancecovariance matrix f is diagonal, the 3SLS estimator provides no gain over the 2SLS procedure (Theil 1971, Zellner and Theil 1962).
The 3SLS procedure provides an asymptotic gain over 2SLS when the structural disturbances have nonzero contemporaneous covariances and at least some of the equations are overidentified, K > N (Zellner and Theil 1962). When the above conditions hold, the asymptotic covariance matrix of the 2SLS estimator dQ will exceed that of the corresponding subvector of 6 by a positive semidefinite matrix. The asymptotic superiority of the 3SLS procedure is due to the optimality of the GLS procedure which underlies the joint estimation procedure (Theil 1971, Madangly 1964, Zellner and Theil 1962).
41
Contemporaneous Correlations
If the disturbances of the growth model are correlated across equations, a systems approach is required to assure asymptotically efficient parameter estimates. If the assumptions of the standard linear model hold for all equations, the elements of the disturbance vector E have zero mean but with unequal variances for each equation in the system. In general, there are three types of disturbances associated with a system of simultaneous equations. The first E(E ,E )
a
and E(c ,,E .), for a # n concerns disturbances of different years but the same equation (serial correlation). In the standard linear model these covariances are assumed to be zero. The second type of covariance E( ,E ,n) for a # n pertains to the disturbances of both different years and different equations. It is also assumed that these covariances are equal to zero. The third type of covariance E(e ,E ,) deals with disturbances of different equations but of the same year. This type of covariance is referred to as the contemporaneous covariance.
The disturbances associated with the four structural equations,
diameter, total height, cubic volume and surface area are assumed to be correlated. There are several reasons for suggesting the contemporaneous nature of the disturbances. If two variables are correlated with a third common variable, they will also appear to be correlated with each other. Since both tree height and tree diameter are correlated with age, height also appears to be correlated with diameters (Husch et al. 1978). Since tree diameters and tree heights are commonly used to predict volumes and surface areas, it would be natural to assume that volumes and surface area are correlated with diameters and heights as well as with age.
42
The contemporaneous covariance matrix for a given observation has the following form:
011 012 IL 021 022 2L
E (E aE) =
aL1 GL2 LL
The diagonal elements aii are the variances associated with the structural equations and the offdiagonal elements aij are the contemporaneous covariances in the ath time period. The condition that (i.. = ji ) and the independence of the observations over time is an extension of the homoscedasticity constraint on the standard linear model. It also indicates that the disturbances are random variables from a multivariate population with zero mean and constant covariance matrix. The disturbance vector s implies that the covariance matrix is block diagonal with ndiagonal submatrices all equal to .
1 o O 0 0 0 f2 0
V(q:) =
.
0 0 .n TIn
43
The offdiagonal submatrices are all zero since their elements are of the first E( ,en ), E(a'.,Cn ,) and second type E(ca E ,) for a n. The inverse of V(es) is also block diagonal with I in the diagonal blocks. Estimates of the parameters of the ith equation in the system are obtained by the use of an Aitkens' estimator: = (X't X) Xi Zy
where BX is the coefficient vector for the Ith equation; X is a n x K matrix of explanatory variables; and y. is an n x 1 vector of dependent observations for the ith equation. The difficulty in applying the Aitken estimator is that the matrix t is usually unknown and must be estimated from the sample data. However, the variancecovariance matrix t under 3SLS will be misspecified if a cross sectional and/or time series correlation exists among the equations in the system.
Cross Sectional Time Series Analysis
Data obtained by repeated measurements on an individual will
contain correlated errors (Sullivan and Clutter 1972). Measurements taken closer together in time are more highly correlated than measurements taken further apart in time. Furthermore, unequal variances may be present in the data (McHugh and Wall 1962). In practice serial correlations and heterogeneous errors are frequently ignored and parameter estimates are obtained by ordinary least squares (OLS). The OLS estimates of the coefficients are unbiased but have inflated errors due to a specification error. Also, the estimate of the residual mean
44
squared error is biased downward (Sullivan and Clutter 1972). The presence of nonzero covariances and nonhomogeneous variances provide a basis for considering the inclusion of this additional information into the estimation procedure to obtain a more efficient estimate of the error components of the model (McHugh and Wall 1962).
Consider the following crosssectional time series model. Suppose N individual trees, n = 1,2,...,N have been sampled over T periods of time, t = 1,2,...,T. The variable to be explained is denoted Ynt* The dependent variable is explained by K truly exogenous variables, and L lagged values of the dependent variable. The model may be expressed in the following manner:
y = YO r + X8 + E where
Y11
Yl YlT
YN YN1
YNT
an NT x 1 vector;
X1 XK 11
XNT NT
45
an NT x K matrix of exogenous variables;
1 L
ye =
1 L y NT1 y NTe an NT x L matrix of lagged values of the dependent variable; 611
Eli
CNI 6Nl
N N1
NT
an NT x 1 vector of residuals. Also, a is a K x 1 vector of constant coefficients of the exogenous variables, and r is a L x 1 vector of coefficients for the lagged endogenous variables. The model can be simplified by letting Z = [X: Ye] an NT x K + 6 matrix, and 6 = [B:F] a K + 8 x 1 vector of coefficients:
y = Z6 + C.
Rao (1959) considered the estimation of the parameters of a linear model for data containing correlated errors and derived estimators requiring an independent estimate of the variancecovariance matrix. Elston and Grizzle (1962) compared Rao's GLS procedure with that of OLS and suggested that OLS might in some cases provide unbiased estimates
46
because of its simplicity. Sullivan and Clutter (1972) suggested that OLS should not be ignored in the analysis of growth and yield data with correlated errors because of its simplicity and when parameter estimates rather than confidence intervals are desired for the parameters and predictions.
A large proportion of the theory relating to the growth and yield of individual trees is aimed at explaining the effects of various stand factors (i.e. stand density, age, and competition) on tree growth over time. However, the majority of the growth models currently in use have frequently been developed from data covering a large crosssection of stand conditions within a given population and the specification of the model is based on the overall response observed in the aggregated sample data. For example, the growth of an individual tree is dependent upon the initial planting density, level of competition, the age of the tree, and productivity of the site. Thus, for a given site and initial planting density, a theoretical growth function can be developed to explain the growth as a function of stand age and the level of competition. However, the actual specification and subsequent estimation of the model parameters are based on data consisting of a large crosssection of individual trees from different sites and stand densities making up the population. It is not selfevident that an individual growth function could be developed to model the growth of individual trees over a wide range of stand conditions (Theil 1971, Behr 1981). For a given level of competition, site productivity, and age it would be reasonable to assume that the growth response would differ over a wide range of planting densities. It would be unrealistic to think that one variable such as planting density could be used to estimate the
47
growth response of an individual tree for various levels of competition over time. In such a case, all sample observations are grouped together and the estimation of the model parameters are performed on the combined sample of crosssectional and timeseries data (Balestra and Nerlov 1966).
Often in pooling data across a wide range of planting densities, it is difficult if not impossible to capture all of the important crosssectional effects. There may be specific factors associated with each crosssection effect that help explain the observed variation in the dependent variables but are unavoidably excluded from the model (Behr 1981, Balestra and Nerlove 1966).
Inferences on the estimated model parameters and any confidence
statements are based on the assumptions pertaining to the properties of the residual vector E. The specific ignorance due to the exclusion of specific crosssectional relationships results in overestimating the disturbance term. Each residual Ent for the nth observation during time period t, may be decomposed into two statistically independent parts:
Ent = Ynt + Pn
where ynt is the residual variation due to general ignorance (sampling error), and on is the residual variation due to specific crosssectional ignorance. It is assumed that ynt has zero mean and variance a2 and
nt Y
covariance matrix a21. It is further assumed that pn is a random variable with mean zero and variance a2. Thus the residual cnt has zero
Sat
mean and variance a2 = a2 + a2 and covariance p 02 of Ynt and y for any pair of observations (nt,n't') where nt#n't' is equal to 02. Hence, p = o2/(02 + a2). It is further assumed that the crosssectional
P' Y W'
48
disturbances pn are serially independent and that the disturbances are independent from one observation to the next. In other words: G2 n = n'
0 otherwise
2 n = n', t = t' E(y nt'Y't') =
0 otherwise
E(Ynt'in = 0 for all n,t.
The assumption that the crosssectional disturbances wn are
independent from one observation to the next needs mentioning. If the sample trees are selected at random and are independent of each other, then this assumption can readily be satisfied. However, if the trees are systematically selected as in this study, this assumption may not be well satisfied and must be taken into consideration when interpreting the results of the analysis.
From the above discussion the variancecovariance matrix can be written as follows: Al 0 0 0 A2 O E(,c'] = a2
0 0 AN
where
49
1 p p p p 1 p p E(,n E) = 02A = 02
n' n E E
p p p 1 is a T x T matrix, and 02 = a2 + a2
S y i
p = 2/2 .
In a single equation model with a crosssectional timeseries component, OLS estimates of the coefficient vector are inefficient. Consequently, GLS is required to obtain efficient parameter estimates: ^ x 1 1 1
= (X X) x0Y.
However, since 02 is usually not known, it must be estimated from the sample data. A technique often employed to solve for the parameter vector 8 is to search over all values of p and choose the corresponding parameter set which maximizes the likelihood function (Behr 1981).
If E is distributed according to a multivariate normal distribution with zero mean and covariance matrix ~, then the probability density function P(e) may be written as follows: P(S) = (2H)NT/21 /2 exp(1/2 e21 E) where E = y Xa. The logarithmic likelihood function for the parameters p, y and a2 may be written as follows:
L(p,y,2) = LTn(2n) 1/2Ln ll 1/2 'c .
L(P'y' 2 = 2
50
The maximum of the likelihood function L(p,y,o2) is obtained when the quadratic form
E' s = [y Xy] l[y Xy]
is minimized. This occurs when
y = (X'aX) 1 ly.
The value for a2 may be found by differentiating L(p,y,a2) with respect
E C
to aC and setting the results to zero.
Behr (1981) presented an alternative maximum likelihood estimator called the concentrated likelihood estimator which is given by N N N 1p + T.p
2L(p,y,2) = E T.Ln(2H) + E T.Ln(a2(1p)) + E Ln( )
E 1 1 E Ip
Using the GLS procedure, estimates of a2 are obtained for each value of p along with the corresponding estimates of y. The value of p that maximizes the concentrated maximum likelihood function is the maximum likelihood estimate of p which implies that the corresponding estimates of y and a2 are also maximum likelihood estimates (Maddala 1977, Behr 1981, Balestra and Nerlove 1966, Theil 1971). Since p varies over the closed interval, zero to one, L(p,y,a2) must reach a maximum within the interval or at one of the boundaries. If the maximum likelihood estimator of p is zero, this implies that the cross sectional error is equal to zero u2 = 0 and that the total variability of the model 02 = a2 is due to general ignorance and not to a specification error in the model.
In the presence of a crosssectional timeseries error component, the normal 2SLS procedure will be inefficient. The suggested procedure
51
is to apply the ML estimator to each of the four first stage structural equations (Maddala 1977, Behr 1981). For each equation, search over the values of p and choose the p value and corresponding parameter set that maximizes the concentrated likelihood function. Thus, for each equation a set of modified 2SLS estimates and corresponding p values are obtained. The third stage is then applied to the data using the variancecovariance matrix obtained in the second stage.
Model Validation
The efficiency of the OLS and 3SLS estimation procedure was evaluated by four goodnessoffit statistics (Murphy 1983):
n (Yi Yi)2
Fit Index, R2 = 1 E
(Y. )2
n .
Mean Difference, D = (Y. Y.
n 1 1
Root Mean Square Error, RMSE= [ Yi)2] /2
n Y. Y.
Percent Mean Difference, D=  ( T ) 100
1
Percent Root Mean Square Error, RMSE% = ( Y Y)2] I2 100
where
52
n = number of observations
Yi = observed value
Y = mean of observed values
Y. = predicted value.
Other checks on the validity of the growth model included
(a) An examination of the model coefficients for wrong signs and
associated variance inflation factors (VIF). The VIF is defined as the inverse of partial correlation coefficients associated with the
dependent variables in the model,
(b) Comparison of the correlations of errors across equations for the
3SLS procedure, and
(c) The efficiency of the 3SLS procedure relative to the OLS estimation
procedure. An Ftest was used to examine the equality of variances
of the two estimation procedures:
2 ^
F OLS(B)
Fij =
1 2
3SLS(B)
where Fij is the Fstatistic for the ith equation and jth parameter.
RESULTS AND DISCUSSION
Spatial Analysis
Hopkins' modified coefficient of aggregation was used to depict the spatial pattern of healthy and dead trees in the study area at four years of age. Hopkins' index yields a point estimate of the spatial distribution ranging from 0.2 to 0.87 (fairly uniform to highly aggregated). A value of 0.5 corresponds to a random spatial pattern. A population is said to be distributed at random if the relative location of an individual tree is independent of the location of any other tree within an infinitely large area. An aggregated population is one in which the trees occur in clumps of varying densities and sizes. A uniform spatial distribution is one in which the trees are evenly distributed over a given area on the vertices of a triangular, square or hexagonal lattice (Reich 1980).
Two transition levels, occurring at approximately 0.4 and 0.6, separate the three basic spatial patterns, uniform, random and aggregated. Values near the transition levels are not clearly discriminating. Following is a discussion of the spatial patterns of healthy and dead trees observed in the study.
At four years of age healthy trees were uniformly distributed
throughout the study area (Table 2). But significant differences were observed in the spatial patterns of healthy trees among spacings and
53
54
replications (Table 3). The significant interaction suggests a lack of consistency of spacing and replication effects. However, the effect was very small. All but one of the spacings had higher coefficients in replication A than in replication B. The significant replication suggests a location effect (Table 2). In general, the spatial patterns of healthy trees were significantly more uniformly distributed at wider spacings. It should be noted that Hopkins' index is independent of tree density and that the index is actually measuring the distribution and size of the vacancies created by mortality. Since the survival rate increased with spacing, the wider spacings retained more of the original planting configuration than the smaller spacings which had a lower survival rate.
The spatial patterns of dead trees were randomly distributed
throughout the study area, but varied significantly among spacings and replications (Table 2 and 4). No spacing (x) replication interaction was observed. In five of the 12 plots a transition was observed among spatial patterns for the dead trees. Four of the transitions were from random to uniform while one was from random to aggregate. The transitions between spatial patterns are indicated by a two letter code: U/R, R/U, A/R, and R/A. The sequence of letters is based on the frequency of the observed spatial pattern. For example R/U indicates that at least 50 percent (or five out of ten) of the observed spatial patterns were random while the remaining estimates indicated a uniform spatial pattern.
As mortality in the stand continues over time, the spatial
distribution of the healthy and dead trees will change with time. In general, healthy trees will tend to converge from uniform to more of a
55
Table 2. Spatial analysis for healthy and dead trees at
four years of age.
Healthy/ Spatial3/ Deadl/ Spatial Spacing Rep Trees Pattern Trees Pattern
5 A .3277a,2 2/ U .4899cd,l R
B .3352a,2 U .4335cde,1 R
7 A .2829b,2 U .5255abc,1 R
B .2387c,2 U .3966de,l R/U
9 A .2161c,2 U .5085cd,' R
B .1888e,2 U .5202ac,l R
11 A .2337c,2 U .4503cde,'l R/U
B .2008de,2 U .4011dc,1 U/R
13 A .1002f,2 U .6247a,1 R/A
B .0994f,2 U .6240ab,'l R
15 A .2359c,2 U .4442cde,'l R
B .2266c,2 U .3392e,'l U/R
1/ I/ Mean value for Hopkins' modified coefficient of aggregation, A,
based on ten independent samples of 150 pointtotree and 150 treetotree distances.
2/ Means with different letter superscripts indicate significant
differences among spacing and replications. Means with different
number superscript indicate significant difference between healthy
and dead trees (a = 0.05).
U = uniform spatial pattern; R = random spatial distribution
A = aggregated spatial distribution.
56
Table 3. Analysis of variance of the spatial distribution of healthy
trees in the McArthur spacing study at age four years. Source df SS MS F Spacing (S) 5 .541 .108 173.06** Replication (R) 1 .009 .009 14.29** S x R 5 .009 .002 2.92* Error 98 .061 .001
TOTAL 109 .620
** Significant at the a = 0.01 level.
* Significant at the a = 0.05 level.
57
Table 4. Analysis of variance of the spatial distribution of dead
trees in the McArthur spacing study at age four years. Source df SS MS F Spacing (S) 5 .614 .123 11.02** Replication (R) 1 .079 .079 7.07** S x R 5 .067 .013 1.21 Error 98 1.080 .011 Total 109 1.840
** Significant at the a = 0.01 level
58
random spatial distribution while the dead trees will converge toward a uniform or aggregated spatial pattern (Kent and Dress 1979, Kent and Dress 1980, Reich 1980). The convergence of spatial patterns over time are evident in certain stands infected with fusiform rust and pitch canker.
Effects of ticrosite Differences on Individual Tree Growth
Analysis of variance was used to study the effects of spacing on
survival, DBH, total height, cubic volume and bole surface area at 2, 6, 8, 10, 12, 14, 16, 19 and 21 years after planting. Results indicate that survival and tree size (DBR, total height, cubic volume, and surface area) are significantly influenced by spacing. No significant differences were observed between replications; however, there was a significant spacing (x) replication interaction. The significant interaction suggests a lack of consistency of spacing and replication effects and suggests that the location of the plot played an important role in the growth of the individual trees in this study. Even though the 12 acre study area appears to be homogeneous, in reality there seems to be an intricate pattern between "good" and "poor" microsites within the study area.
After correcting DBH, survival, tree heights, cubic volumes, and surface area data for spacing and replication effects and plotting the corrected values over stand age and plot location, a definite microsite pattern emerge which help explain the significant spacing X replication interaction. The effects of microsite on survival, DBH, total height, cubic volume, and surface area are depicted in Figures 2 through 6 by
59
plot location and stand age. Uncorrected plot means are shown in Figures 7 through 10.
In the center of the study area there is a clearing with very little understory vegetation. East and west of the clearing the understory vegetation increases both in density and height. The dominant understory species are wax myrtle (Myrica cerifera L.) which averaged 15.5 feet in height (Table 5).
At four years of age, survival, DBH, and tree heights were somewhat larger in the center of the study area and decreased in size and numbers along the eastwest direction. The observed microsite pattern corresponds approximately to the distribution of the understory vegetation of the study area. The microsite effect on tree heights decreases with age and diminishes by the age of 21 years (Figure 3).
Microsite noticeably influenced DBH, cubic volume, surface area and survival over the 21 year period (Figures 2,4,5, and 6). Over time, the microsite pattern changed; diameters, volumes, surface area and survival rates were higher in the eastwest portion of the study area and decreased toward the center. No silvicultural practices, such as burning, have been conducted on the study area since establishment in 1963. Most likely the observed microsite pattern is a biological response caused by microtopography and underground drainage variation.
A recent study by Permar and Fisher (1983) indicated that wax myrtle is an important species for fixing nitrogen in southern flatwoods. It has been shown that nitrogen fertilizer can result in substantial increases in volume growth of slash pine on spodosal soil sites in Southeastern U.S. (Fisher and Garbett 1980). These flatwood soils are generally low in available nitrogen and respond to added
60
nitrogen with increased growth. Stands fertilized with nitrogen have a gain of approximately 4.5 m3 /ha/year compared to unfertilized ones (Fisher and Garbett 1980). Also, the nitrogen content of both the litter and soil tend to be greater beneath wax myrtle than pines. Average annual accumulation of nitrogen under wax myrtle was 2.4 g N/m3 /year (Permar and Fisher 1983).
It may be assumed that the understory vegetation became established shortly after or during the planting of the stand. During the first four to six years, understory species (wax myrtle, in particular) competed with the young pine seedlings in the east and west portion of the stand. This competition resulted in a decrease in both diameter and height growth compared to trees located in the center of the study area where there was not competing understory vegetation. When the pine trees were able to compete with the understory vegetation for growing space, nutrients, light and water, they were able to efficiently utilize the additional nitrogen fixed by the wax myrtle. The net result is a higher survival rate and larger than average trees in the east and west portion of the stand compared to trees in the center where little or no additional nitrogen from the understory vegetation was available.
This hypothesis may explain the microsite patterns observed in survival and tree size, but it still needs to be tested with a study designed specifically to analyze the effects of microsite and/or nitrification of slash pine by wax myrtle on the growth of individual trees over time.
61
Table 5. Average height of major understory species for the McArthur
study area at 21 years of age.
Understory Average Species Height (FT) Range
Acer rubrum L. 1.4 11.316.0 Ilex glabra (L.) Gray 6.6 4.9 8.0 Myrica cerifera L. 15.5 5.025.0 Persea borbonia L. 10.9 6.815.5 Quercus incana Bartr. 15.0 15.0 Quercus laurifolia Michx. 10.2 4.818.0 Quercus nigra L. 13.8 6.825.5 Quercus virginiana Mill. 6.4 6.4 Sabal palmetto (Walt.) Lodd. ex. Sholt. 5.7 4.3 8.0 Serenoa repens (B.) Small 4.2 3.5 4.7 Vaccinium stamineum L. 11.2 11.2
62
z
Z
olO
C(5 C<
O
0 u 8

W 2L
78 158 98 13B 18 58 9A 'SA 7A 13A IA A PLOT LAYOUT Figure 2. Diameter breast height corrected for spacing and replication by stand age and plot location.
63
70
z
0 6O
__ 9
050
LLLLJ < 40
z 2
20
C0
LU
78 158 98 38 !He 58 9A 5 TA A A SA PLOT LAYOUT Figure 3. Total tree height corrected for spacing and replication
by stand age and plot location.
64
14
AGE
12 2
LL
010
U
L
 8 16
I
66
2
o 66 4
7B 15B 98 13B 11B 5B 9A 15A 7A 13A IIA 5A
PLOT LAYOUT
Figure 4. Cubic volume corrected for spacing and replication by
stand age and plot location.
65
O
L, 100 t
LJ AGE
c
Q 21
0 4
co 80
LL C
4 0 LU0
<' 40 12
10
O 0 20
7B 15B 98 138 118 5B 9A 15A 7A 13A IIA 5A PLOT LAYOUT
Figure 5. Surface area corrected for spacing and replication by stand
age and plot location.
66
110
100
AGE
O 8 LL 10
Z 12
o 0 908 0
Z 19 ...0 2I
S0
>z /
D CL
CO 0 607B 158 9B 136 118 58 9A 15A 7A 13A IIA 5A PLOT LAYOUT
Figure 6. Survival corrected for spacing and replication by stand age
and plot location.
67
10
z
(D 21 W 66
22
4
78 158 98 IBE lIB 58 9A 15A 7A !3A 11A 5A
PLOT LAYOUT
Figure 7. Diameter breast height by stand age and plot location.
68
70r
GOGE LL 501
PLOT LAYOUT
Fiure 8. Total tree height by stand ae and lot location.
69
LL 10 AGE
W 21
J O
6 16
14
4 12
10
2 8
7B 158 98 13B 118 58 9A 15A 7A 13A IIA 5A PLOT LAYOUT
Figure 9. Cubic volume by stand age and plot location.
70
100
L AGE
80
/8 21
UJ
C 60
16 L) 4
LL
X 40
12
6
O
7B 15B 9B 13B 118 5B 9A 15A 7A 13A IIA 5A PLOT LAYOUT
Figure 10. Surface area by stand age and plot location.
71
Effects of Spacing on Individual Tree Growth
Regression analysis indicates that spacing had no significant
effect on height growth after correcting for microsite and replications (Figure 12). However, spacing significantly affected survival rate and size of individual trees, excluding height (Figures 11 and 15). The effects of spacing on survival, DBH, cubic volume and surface area increased with age.
Several studies (Bennett 1974, Shepard 1974) have reported both positive and negative correlations between height growth and spacing. These discrepancies may be in part due to the exclusion of microsite effects from the analysis of the data. In this study conventional analysis of variance indicated that spacing had a significant effect on tree heights. After correcting for microsite and replication effects, regression analysis indicated that spacing had no significant effect on tree heights.
Volume and Surface Area
Regression analysis was used to develop volume and surface area
equations to predict cubic foot volumes and bole surface area in square feet for the sample trees used in the study. Regression equations were developed for both inside and outside bark. An exponential equation was used to estimate total cubic volume and surface area as a function of diameter at breast height outside bark (DBH) and total tree height (THT). The form of the equations was based on the residuals and resulting coefficient of multiple determination. The final form of the
72
10 AGE
0 21 89
0.80
LLU 6
LLJ 0
R2=0.1 8 Z 4
0
.OO
2
I
5 7 9 II 3 5
SPACING (FT)
Figure 11. Effects of spacing on diameter breast height corrected
for microsite and replication by stand age.
73
70
AGE
RZ= O.I 21 60 R2 =0.04 9
0
LL R2=O20 6 R 2=0.20
o R=O0.OI
O R2= .20 I 30= C !Z
I I
 WR2O.I
S20
R2=0.15
O 6
10 r
R2=O.O!
5 7 9 ii i 15 SPACING (FT')
Figure 12. Effects of spacing on total tree height corrected
for microsite and replication by stand age.
74
(Z
LUo
<
0 AGE
OL16 21
8r
22
SLLJ 00
5 7 9 1i 5
SPACING (FT)
Figure 13. Effects of spacing on cubic volume corrected for
microsite and replication by stand age.
75
AGE
0 100 ,6 21
Uo
(r 0"
o 80
0 7
O l 0. I6
L.
LL0 60
oz
S 40
r 10 LLJ
U 0 Rz= 0.00 () R2=0.04
6
5 7 9 II1 13 15 SPACING (FEET)
Figure 14. Effects of spacing on surface area corrected for microsite
and replication by stand age.
76
AGE
101 012
IO 60
;~ (D. 16
2
R2 0.10 2= 0.1
 8416
80
 I s ~ ~ '
50
SPACING (FT)
Figure 15. Effects of spacing on survival corrected for microsite
and replication by stand age.
77
equation used to estimate cubic volume and surface area is Y = aDBH THTY
where a,B and y are regression coefficients estimated from the sample dendrometry data using log transformation. Volume
VOB = exp[5.3036+2.1963Ln(DBH)+0.7492Ln(THT)] R2 = 0.970 SLn(VOB) = 0.098 Relative variation: 0.91 to 1.10 VIB = exp[5.9612+2.3004Ln(DBH)+0.7928Ln(THT)I R2 = 0.964 SLn(VIB) = 0.114 Relative variation: 0.89 to 1.12
Surface Area
SOB = exp[1.4180+1.1376Ln(DBH)+0.8358Ln(THT)] R2 = 0.969 SLn(SOB) = 0.060 Relative variation: 0.94 to 1.06 SIB = exp[1.7201+1.1859Ln(DBH)+0.8559Ln(THT)J R2 = 0.966 SLn(SIB) = 0.066 Relative variation: 0.94 to 1.07 where
78
VOB = cubic foot volume outside bark
VIB = cubic foot volume inside bark SOB = bole surface area outside bark
SIB = bole surface area inside bark
exp = exponential
Ln = natural logarithm.
Cubic volume and bole surface area outside bark by DBH and total height are given in Tables 6 and 7.
Tree Form Equations
Substituting the cubic volume and surface area equations into
Grosenbaugh's (1966) four ratios involving solids of revolution yields the following relationship between tree form, diameter at breast height and total height for the sample trees used in this study:
Outside Bark
Q = 0.9549(DBH2196 THT0250) 1/2
Ao = 0.9251 DBH1.138 THT0.164
Qo/Ao = 0.9857 DBH1.059 THT0.087
Fo = 0.927 DBH0079 THT0.076
79
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81
Inside Bark
Q = 0.6873(DBH2.300 TH0.207) 1/2
AI = 0.6839 DBH186 THT0.144
QI/AI = 0.6908 DBH1.114 THT0.063
F = 0.9880 DBH0071 THT0081"
I
Table 8 compares average tree form, inside and outside bark, by diameter and total height.
Parameter Estimation
The growth model for diameter, total height, cubic volume and
surface area was fitted to a sample of 2600 crosssectional and timeseries observations using a 3SLS procedure. Although the thirdstage estimates are the parameters of interest, the results of the first and second stages will also be discussed.
First Stage Estimates
In the first stage, estimates are obtained for the endogenous
variables (diameter and total height) which appear as explanatory ones in the volume and surface area equations. This is accomplished by regressing all predetermined variables on diameter and height using OLS. The estimated equations represent the unrestricted reduced form parameter estimates for diameter and total height. The predicted values for diameter and total height are uncorrelated with the disturbances associated with the cubic volume and surface area equations.
82
Table 8. Comparison of tree form inside (FI) and outside (Fo)
bark by diameter and total height.
Total Height
Inside Bark (F1) Outside Bark (Fo) DBH 50 70 90 50 70 90 5 81 79 77 79 77 75 6 82 80 78 80 78 76 7 83 81 79 81 79 77 8 84 81 80 82 80 78 9 84 82 80 82 80 79 10 85 83 81 83 81 79 11 86 83 82 84 82 80 12 86 84 82 84 82 81 13 87 84 83 85 83 81 14 87 85 83 85 83 82 15 87 85 83 86 84 82
83
Second Stage Estimates
In the second stage, maximum likelihood estimates of the structural parameter vector and variancecovariance matrix are obtained based on the assumptions regarding the crosssectional and timeseries characteristics of the disturbances. The endogenous explanatory variables are replaced by the predicted values for diameter and total height, which are obtained from the first stage. The GLS procedure was used to estimate the structural parameters of the growth model for the second stage. The variancecovariance matrix was estimated iteratively.
Because of the large data set (N1=2600) (requiring excessive time and space to invert a 2600 square matrix) the total data set (NI+N2=5199) was divided into 52 groups with approximately 100 observations per group. A random sample of 15 groups was selected to obtain a maximum likelihood estimate of p. Searching over all values of p within the interval 0 to 1 in increments of 0.1, a unique set of structural parameters and model variances was obtained for each equation and sample group. The ML estimate of p, and the corresponding model variance and concentrated likelihood function are given in Tables 9 through 12.
The parameter p is the ratio of the crosssectional and timeseries error component to the total model variance. All ML estimates of p for the four equations and 15 samples were zero. A p value of zero implies that the model variance arises from random sampling error with no crosssectional or timeseries error component. Table 13 shows the absolute value of the concentrated likelihood function for the four equations over the range of p. Table 14 depicts the relationship between the error sum of squares and the ML estimate of p. The error sum of squares
84
increases exponentially as p approaches one, while the concentrated maximum likelihood function decreases in a somewhat linear fashion. The estimated structural parameters and model variance are very stable near the region of the ML value of p. The changes in the estimated parameters for the diameter equation are given in Table 15 for selected values of p.
Although the ML estimator has many desirable properties, there have been questions concerning its use in the variancecomponents models (Behr 1981). Monte Carlo simulations have shown that the estimated p values were zero when the true value was actually larger. It was argued that the serial nature of the exogenous variables interacted strongly with the autoregressive nature of the problem under study, causing problems in estimating p. This was not considered a problem in this study.
Third Stage Estimates
In the third stage an asymptotically efficient estimate of the
growth model is obtained by regressing the four equations together as a system. Since the ML estimate of p was zero, the standard 3SLS procedure was used to obtain the third stage estimates of the parameters. Since the disturbances from the four equations are contemporaneously correlated with variancecovariance matrix Q, and since Q is not known, it is estimated from the second stage using data set Ni. In the second stage each equation is transformed in accordance with the GLS procedure when Q is unknown.
85
Table 9. Crosssectional timeseries adjustments for diameter
at breast height
RUN OBS TI/ p CONLY2/ RSQ VAR ESS31 100 9 0 1.329 0.974 0.013 1.196 2 100 9 0 1.878 0.964 0.019 1.690 3 100 9 0 2.122 0.965 0.021 1.909 4 100 9 0 2.477 0.956 0.024 2.229 5 100 9 0 1.981 0.965 0.020 1.783 6 100 9 0 2.902 2.943 0.029 2.612 7 100 9 0 4.073 0.920 0.041 3.666 8 100 9 0 2.859 0.934 0.029 2.573 9 100 9 0 3.518 0.935 0.035 3.167 10 100 9 0 1.405 0.971 0.014 1.265 11 100 9 0 4.293 0.935 0.043 3.863 12 100 9 0 1.413 0.974 0.014 1.272 13 100 9 0 2.800 0.937 0.028 2.520 14 100 9 0 1.860 0.958 0.019 1.674 15 100 9 0 4.191 0.909 0.042 3.772
Number of time periods
 Concentrated likelihood value
 Error sum of squares
86
Table 10. Crosssectional timeseries adjustments for total tree
height.
RUN OBS T/ p CONLK2/ RSQ VAR ESS3/ 1 100 9 0 1.031 0.985 0.010 0.928 2 100 9 0 0.845 0.988 0.008 0.761 3 100 9 0 1.353 0.981 0.014 1.218 4 100 9 0 1.373 0.982 0.014 1.235 5 100 9 0 1.096 0.985 0.011 0.986 6 100 9 0 3.412 0.947 0.034 3.071 7 100 9 0 3.151 0.953 0.032 2.836 8 100 9 0 0.798 0.988 0.008 0.718 9 100 9 0 3.208 0.955 0.032 2.887 10 100 9 0 0.833 0.987 0.008 0.750 11 100 9 0 3.342 0.956 0.033 3.008 12 100 9 0 1.230 0.983 0.012 1.102 13 100 9 0 0.968 0.985 0.010 0.871 14 100 9 0 0.914 0.985 0.009 0.823 15 100 9 0 4.178 0.930 0.042 3.760
 Number of time periods
 Concentrated likelihood estimate
Error sum of squares
87
Table 11. Crosssectional timeseries adjustments for cubic foot
volume.
RUN OBS T p CONLK/ RSQ VAR ESS3/ 1 100 9 0 3.494 0.986 0.035 3.144 2 100 9 0 4.626 0.983 0.046 4.164 3 100 9 0 9.248 0.968 0.092 8.323 4 100 9 0 6.907 0.974 0.069 6.216 5 100 9 0 8.283 0.970 0.083 7.455 6 100 9 0 13.562 0.945 0.136 12.205 7 100 9 0 20.176 0.921 0.202 18.158 8 100 9 0 5.391 0.974 0.054 4.852 9 100 9 0 17.360 0.933 0.174 15.624 10 100 9 0 4.718 0.980 0.047 4.246 11 100 9 0 21.632 0.934 0.216 19.469 12 100 9 0 6.026 0.978 0.060 5.423 13 100 9 0 5.347 0.976 0.053 4.812 14 100 9 0 5.327 0.974 0.053 4.795 15 100 9 0 17.511 0.921 0.175 15.760
Number of time periods
2/ Concentrated likelihood estimate
Error sum of squares
88
Table 12. CrossSectional timeseries adjustments for bole surface
area.
RUN OBS T_/ p CONLK2/ RSQ VAR ESS3/ 1 100 9 0 1.459 0.989 0.015 1.313 2 100 9 0 1.829 0.988 0.017 1.556 3 100 9 0 3.747 0.975 0.037 3.372 4 100 9 0 3.031 0.979 0.030 2.728 5 100 9 0 3.302 0.978 0.033 2.972 6 100 9 0 6.672 0.949 0.067 6.005 7 100 9 0 9.098 0.933 0.091 8.188 8 100 9 0 2.604 0.978 0.026 2.344 9 100 9 0 8.880 0.937 0.089 7.992 10 100 9 0 1.825 0.985 0.018 1.643 11 100 9 0 9.181 0.946 0.092 8.263 12 100 9 0 2.346 0.984 0.023 2.111 13 100 9 0 2.134 0.982 0.021 1.920 14 100 9 0 2.328 0.980 0.023 2.096 15 100 9 0 8.659 0.927 0.087 7.793
Number of time periods
 Concentrated likelihood estimate
Error sum of squares

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MODELING SINGLE SLASH PINE TREE GRCWTK AMD SPACING RELATIONSHIPS BY ROBIN MICHAEL REICH A DISSERTATION FRESENTIED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNTVERSITY OF FLORIDA 1984
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This research is dedicated to my parents, Rudolph and Edna Reich, who consistently lent their support and encouragement in this endeavor. I would also like to dedicate this work to Nadine B. teck, a very special friend, whom I could never repay for the companionship and understanding that she has given during the past year.
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ACKNOWLEDGEMENTS The author wishes to express appreciation to those who guided, assisted, and contributed to the completion of this study. Special appreciation is extended to Dr. Loukas G. Arvanitis, chairman of his supervisory committee, for his exceptional patience, guidance, and encouragement during the course of this research. Appreciation is also extended to Mr. Lewis R. Grosenbaugh, Dr. Tony E. Squillace, Dr. James E. Keesling, and Dr. Mark C. Yang, for their contributions as members of the supervisory committee and for their advice and assistance during preparation of this dissertation. Grateful acknowledgement is also given to Dr. Ronald H. Randies who graciously substituted for Dr. Yang while he was on sabbatical. Special appreciation is also due to Drs Arnett C. Mace, Jr., and Robert Schmidt, Director and former Assistant Director of the School of Forest Resources and Conservation, for providing financial support in the form of a Graduate Assistantship and computer time from MclntireStennis Project No. 1874, and to Container Corporation of America for suppling field data from the McArthur spacing study. Finally, the author is grateful to Mrs. Ruth R. Byrne for the excellent job of typing this dissertation. iii
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TABLE OF CONTENTS Page ACKNOWLEDGEMENTS iii LIST OF TABLES LIST OF FIGURES ABSTRACT INTRODUCTION 1 Objectives 3 LITERATURE REVIEW 4 Total Tree Height g Tree Diameter g Cubic Foot Volumes 9 Tree Form Stand Mortality 12 Competition Distance Sampling System of Equations u STUDY PROCEDURES Population j^g Spatial Analysis 21 Microsite Variation 22 Effects of Spacing on Individual Tree Growth ... 23 Volume and Surface Area Estimation 24 Tree Form 25 Iv
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Page Data Interpolation 25 Growth Function 27 Model Developement 29 Field Data Screening 3^ Description of the System of Equations 35 ThreeStage LeastSquares 37 Identification of the System of Equations 39 Contemporaneous Correlations 41 CrossSectional TimeSeries Adjustments 43 Model Validation 5]^ RESULTS AND DISCUSSION 53 Spatial Analysis 53 Effects of Microsite Differences on Individual Tree Growth 5g Effects of Spacing on Individual Tree Growth ... 71 Volume and Surface Area ji Tree Form Equations 73 Parameter Estimation qi First Stage Estimates 81 Second Stage Estimates 83 Third Stage Estimates 84 Reduced Form Equations 92 Final Form Equations 92 V
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Page Model Validation jq3 Evaluation of the Coefficients 103 GoodnessofFit Statistics 105 Cumulative Distribution of Errors 113 Correlation Among Equations 113 Efficiency of the OLS and 3SLS Procedures us SUMMARY AND CONCLUSIONS 123 FUTURE RESEARCH j28 LITERATURE CITED ^^1 BIBLIOGRAPHY BIBLIOGRAPHICAL SKETCH ion vi
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LIST OF TABLES Table 10 12 Page Variables used to identify the "best" combination of explanatory variables 34 Spatial analysis for healthy trees and dead trees at four years of age 55 Analysis of variance of the spatial distribution of healthy trees in the McArthur spacing study at four years of age 55 Analysis of variance of the spatial distribution of dead trees in the McArthur spacing study at four years of age 5y Average height of major understory species for the McArthur study area at 21 years of age. 61 Total cubic foot volume outside bark, for slash pine of the McArthur study area 79 Total bole surface area outside bark, for slash pine of the McArthur study area 80 Comparison of tree form inside (Fj^) and outside (Fq) bark by diameter and total height .... 11 Crosssectional timeseries adjustments for cubic foot volume Crosssectional timeseries adjustments for bole surface area 13 Concentrated maximum likelihood values for DBH (D), total height (H), cubic volume (V) and surface area (S) 82 Crosssectional timeseries adjustments for diameter breast height 85 Crosssectional timeseries adjustments for total tree height 86 87 88 89 VI 1
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Table p^ge 14 Error sum of squares for DBH (D), total height (K), cubic volume (V) and surface area (S) 90 15 Comparison of the estimated coefficients for the DBH equation at three levels of 6 91 16 Estimates of the model parameters and standard errors for the DBH equation by OLS 2SLS, and 3SLS 93 17 Estimates of the model parameters and standard errors for the total height equation by OLS, 2SLS, and 3SLS 94 18 Estimates of the model parameters and standard errors for the cubic volume equation by OLS, 2SLS, and 3SLS 95 19 Estimates of the model parameters and standard errors for the surface area equation by OLS, 2SLS, and 3SLS 95 20 Reduced form coefficients for the DBH (D), total height (H), cubic volume (V) and surface area (S) equations 97 21 Final form coefficients for the DBH (D), total height (H), cubic volume (V) and surface area (S) equations 99 22 Correlation matrix for the DBH equation 106 23 Correlation matrix for the total height equation. 107 24 Correlation matrix for the cubic volume equation. 108 25 Correlation matrix for the surface area equation. 109 26 Variance inflation factors for the 3SLS growth model j^jQ 27 Evaluation of prediction equations for DBH, total height, cubic volume and surface area by OLS and 3SLS Â•Â•Â•Â•Â•.....,,.,.,,,, 112 28 Correlation across equations 119 29 Efficiency of the OLS and 3SLS estimation procedure for DBH (D), total height (H), cubic volume (V) and surface area (S) equations 122 viii
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LIST OF FIGURES Figure Page 1 Slash pine spacing study. McArthur Tract, near Callahan, Florida, installed in February, 1962. 20 2 Diameter breast height corrected for spacing and replication by stand age and plot location ... 62 3 Total tree height corrected for spacing and replication by stand age and plot location ... 63 4 Cubic volume corrected for spacing and replication by stand age and plot location 64 5 Surface area corrected for spacing and replication by stand age and plot location 65 6 Survival corrected for spacing and replication by stand age and plot location 66 7 Diameter breast height by stand age and plot location 57 8 Total tree height by stand age and plot location 68 9 Cubic volume by stand age and plot location. 69 10 Surface area by stand age and plot location. 70 11 Effects of spacing on diameter breast height corrected for microsite and replication by stand age 72 12 Effects of spacing on total tree height corrected for microsite and replication by stand age ... 73 13 Effects of spacing on cubic volume corrected for microsite and replication by stand age 74 14 Effects of spacing on surface area corrected for microsite and replication by stand age 75 15 Effects of spacing on survival corrected for microsite and replication by stand age 76 ix
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Figure Page 16 Relative change in the interim multipliers associated with the exogenous variables 1/Age**2 and ( lAlag/Age)*PA.R 100 17 Relative change in the interim multipliers associated with the exogenous variables PAR and Comp 18 Relative change in the interim multipliers associated with the exogenous variables Alag/Age and Age 102 19 Cumulative distribution of errors associated with the diameter breast height equation 114 20 Cumulative distribution of errors associated with the total tree height equation 115 21 Cumulative distribution of errors associated with the cubic volume equation 116 22 Cumulative distribution of errors associated with the surface area equation 117 X
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Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MODELING SINGLE SLASH PINE TREE GROWTH AND SPACING RELATIONSHIPS By Robin Michael Reich August, 1984 Chairman: Loukas G. Arvanitis Major Department: Forest Resources and Conservation This study deals with modeling the effects of competition and available growing space on the growth and development of individual slash pine trees over time. A model to predict the cumulative growth of diameter at breast height, total height, cubic volume, and bole surface area has been developed using a modified form of the Verhulst growth function. In the past, singleequation estimation procedures, such as Ordinary Least Squares (OLS), have been employed to estimate the structural coefficients associated with a system of equations. This study advocates ThreeStage Least Squares (3SLS), which allows estimation of the structural coefficients simultaneously using a systems approach. OLS and 3SLS are examined under two assumptions concerning the correlation of residual errors: contemporaneous covariance and serial correlation. xi
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The validity of the models was scrutinized following datasplitting (crossvalidation) and comparison of the models against data fit and prediction accuracy. In addition, the signs of the estimated coefficients were compared with theoretical expectations. Parameter estimates for each procedure were obtained from data accumulated in a spacing study which was established in 1962 and remeasured approximately every two years for the past 21 years. Results of this study indicate that the suggested growth model depicts the actual process quite well by using either OLS or 3SLS. The structural equations had a weighted k**2 value of approximately 93 percent. However, the 3SLS procedure was more efficient in estimating the coefficients of the growth model than OLS. The OLS procedure does not take into consideration the correlation of errors among equations or the presence of endogenous variables as explanatory ones. This results in a specification error which affects the variancecovariance matrix used in the estimation procedure. Both OLS and 3SLS yield unbiased estimates of the model parameters. However, only the latter procedure assures asymptotically efficient parameter estimates with respect to the identification of the variancecovariance matrix. These two desirable properties of 3SLS lend validity to the growth model, allow for hypothesis testing, and construction of confidence intervals. xii
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INTRODUCTION Recently, a renewed interest has emerged in modeling the effects of growing space and competition among individual trees within a forest plantation. Two basic approaches have been used: stand descriptive models and single tree models. In stand models, the variation associated with tree competition is evaluated indirectly in terms of stand averages and stocking densities. While such characteristics are useful under certain conditions, they tend to obscure the cause and effect relationship among neighboring trees. If one considers single trees as the major component of the system, stand development may be modeled as the interaction between growth and competition of individual trees (Daniels et al. 1979). This feature allows the use of both tree and stand parameters to describe the interactions among competing neighborhood trees. Models incorporating the spatial relationship of individual trees in a given plantation offer a greater opportunity to study the development of a stand of trees over time (Pielou 1977, Newnham and Smith 1964, Daniels et al. 1979, Wensel 1975). One aspect of tree growth and stand development that has not been adequately addressed is the effect of size and spatial distribution of vacancies, created by mortality and/or thinning on the growth and development of the remaining trees in a forest stand over time. 1
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2 One of the major disadvantages of modeling the effects of vacancies on individual tree growth has been the use of singleequation estimation procedures employed to estimate the structural coefficients of a system of equations (Theil 1971, Sullivan and Clutter 1972). In such models it is assumed that the variables simultaneously satisfy certain assumptions pertaining to the standard linear model which assumes homoscedasticity (equal variances among regressors) and that the explanatory variables (i.e. endogenous and exogenous variables) are independent of the residual errors. The use of a singleequation estimation procedure in fitting individual equations to a set of data does not take into account the interrelationship between regressors as well as the interaction among equations (Hock 1958). Likewise, the procedure does not consider the joint distribution of the endogenous variables among equations or the properties associated with the disturbances. It also disregards any restrictions imposea by other equations on certain variables or coefficients in the system of equations (Haavelmo 1943). Thus, studies similar to those aimed at evaluating the growth and development of stands over time should not be confined to individual equations but rather be extended to a system of equations (Zellner and Theil 1962). The aim of such a system is to describe a set of endogenous variables in terms of a set of exogenous ones and disturbances. The exogenous variables which occur in a system are determined outside of the system in question. On the other hand, the endogenous variables are determined jointly and simultaneously by the exogenous variables and the disturbances generated by the structural equations in the systems (Theil 1971).
PAGE 15
3 This approach to estimating the parameters of a system of equations has two main advantages over the singleequation estimation procedure: it allows restrictions to be placed on certain variables, and permits the interaction of variables between equations to be considered in the estimation procedure. The use of simultaneous equations could lend itself readily to the development of simulation and forecasting models designed to quantify the effects of changes in the degree of competition and growing space on the growth of individual trees in a given forest stand. Objectives The objectives of this study were (a) to model the growth and development of individual slash pine trees over time and (b) to quantify the effects of competition and available growing space on the growth of individual trees.
PAGE 16
LITERATURE REVIEW As the demand for wood products increases and the land base suitable for growing wood bioraass continues to decline, forest management practices have intensified. Silvicultural practices aimed at increasing the productivity per unit land area include use of genetically improved seedlings, fertilization, thinning, bedding, prescribed burning, and drainage. The assessment of silvicultural practices in terms of stand management is complicated by uncertainties in the growth and development of individual stands due to differences in mortality, site, climatic factors, etc. (Lembersky and Johnson 1975). Recent advances in computer technology and its application to forestry have led to the development of realistic growth and yield models capable of assisting forest managers in assessing the effects of various silvicultural treatments and stand conditions on the growth and development of individual stands over time (Nance et al. 1982, Allen and Duzan 1980, Harms 1982). Knowledge gained from these models has helped foresters in developing optimal management strategies to effectively achieve specific objectives (Allen and Duzan 1980). To fully exploit the potential of a given site for timber production, there is a need to better understand how competition and available growing space affects tree growth (Harms and Langdon 1977). Two of the more important factors are stand density and site quality. 4
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5 Site is defined by the Society of American Foresters (FordRobertson 1971, p. 242) as "an area considered in terms of its environment, particularly as this determines the type and quality of the vegetation the area can carry". Sites may be classified either qualitatively or quantitatively. Stand density is defined as "a measure of the degree of crowding of trees within a given area". It is commonly measured as a ratio of crown length to tree height, crown diameter to DBH, or crown diameter to tree height (FordRobertson 1971, p. 255). In a given plantation, the initial spacing exerts a strong influence over the size and quality of individual trees making up the stand. Density is one of the most important characteristics a forest manager has available for controlling the development of a stand over time (Shepard 1974). Formerly, planting densities were selected on the basis of prior experience in stands grown on similar sites. Recently, stocking levels have been determined on the basis of the cost of establishment, interest rates, rotation length, product utilization and other economic factors affecting the production of wood A reliable tree growth model must be capable of predicting the growth response of individual trees over a wide array of planting densities for a given site. However, the biological potential of the various sites for timber production varies considerably throughout the range of the slash pine ecosystem (Boyce et al. 1975). Even within relatively small areas site productivity may vary due to microsite differences (Swindel and Squillace 1980). Extensive research has shown that as stand density increases there is a progressive decrease in the average size and value of an individual
PAGE 18
6 tree as well as a change in its morphological and physiological characteristics (Davis 1966, Jorgenson 1967, Harms and Langdon 1977). This phenomenon is attributed to the increased competition for available growing space, light, nutrients and water. As stand density increases the ability of some trees to compete with their neighboring trees diminishes. The effects of stand density on tree growth is more complex than a simple reduction in overall growth due to competition, but is a result of a combination of factors such as the initial planting density, genetic traits, disease and insect infestation, spatial distribution of mortality and climatic conditions (Davis 1966). Competition between neighboring trees results in a natural selection of those trees having the best possible complement of genetic traits for a given site and stand condition (Wright 1976, Jorgenson 1967). Inferior trees are gradually suppressed and shaded out. The number and distribution of suppressed trees depends upon the initial spacing of the stand, early mortality, the age at which the tree canopy closes, and the quality of the site. In the past, forest managers tended to overplant in order to ensure an adequate stocking level at the end of the rotation due to uncertainties in the survival rate. The effects of spacing on the size of individual tree variables are discussed below. Total Tree Height It has long been assumed that height growth is independent of stand density (Bennett 1963, Lloyd and Jones 1982, Husch et al. 1982). Usually, the mean height of a stand increases with increased spacing
PAGE 19
7 (Shepard 1974, Jorgensen 1967, Bennett 1971). The closer the spacing the greater will be the intensity of competition among neighboring trees. In such cases, the proportion of suppressed trees will increase. Bennett (1963) found that slash pine planted on oldfield sites (site index 65, base age 25) grew approximately three feet annually for the first 15 years. Height growth increased with site quality and decreased with stand age. In a later study, Bennett (1971) observed that height growth declines after age ten years for all sites and densities. Bennett (1963) also showed that during a given year height growth increases at a decreasing rate. By the end of March approximately 25 percent of the height growth is completed. The remaining percent of the height growth is completed by the end of June. Lloyd and Jones (1982) pointed out that a substantial bias can result in growth predictions based on densityheight independent models developed from data which include an effect of density on tree heights. The authors suggest that stand density should be used in modeling individual tree heights in order to obtain a more realistic height growth function. If a significant heightdensity relationship exists in the data, the omission of this information may result in a specification error which in turn affects the estimation of the coefficients in the model. The magnitude and direction of this bias depend solely upon the heightdensity relationship present in the development of the site index curves. Standard regression techniques may be employed to test the significance of individual variables in a given model.
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8 Lloyd and Jones (1982) concluded that any attempt to modify site index to reflect stand density would be inappropriate since site index, by definition, is independent of density. Tree Diameter In general, the increase in the mean stand diameter is inversely proportional to the planting density. The configuration of the spacing (square versus rectangular) is not correlated with diameter growth (B ennett 1963). Campbell and Mann (1974) demonstrated that the initial planting density has a stronger Influence on diameters than site index. As spacing and site index increase, the mean diameter of the stand shifts to the right resulting in a distribution skewed to the left. The interrelationships between stand density, individual tree size, competition and other factors control the rate of tree movement to higher diameter classes over time and, thus, the shape of the diameter distribution. Stand density does not have a significant effect on average stand diameter until the fifth growing season (at standard stocking levels); it intensifies thereafter (Bennett 1963). Several researchers have demonstrated that most of the differences in diameters occur during the first 12 to 20 years (Shepard 1974, Jorgensen 1967). At closer spacings, growth is reduced due to increase in competition between trees. The mean periodic annual diameter growth is directly related to stand density and decreases with stand age (Lohrey 1961, Shepard 1974). When the differences in diameter growth between spacings stabilize, competition is severe for all spacings. On poor sites, it takes longer for differences in diameter growth to stabilize over time (Lohrey 1961).
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Cubic Foot Volumes For a given species, volumes of individual trees and stands are directly related to age, site index, and planting density of which only spacing can be controlled by the forester. Maximum yields are realized in higher planting densities and sites. The increase in volume is proportional to stand density (Bennett 1963, Lohrey 1961, Campbell and Mann 1974). On sites below 50, spacing has little or no effect on yields because of the inefficiency of the trees in the stand to fully utilize the site (Bennett 1963). The percentage increase in yields per unit area due to an increase in site is greater for lower sites than for higher sites, but the actual volume increase is larger for the higher sites. Also, yields tend to be higher on oldfield sites than on natural forest sites due to a reduction in competing understory vegetation and residual effects from past fertilization (Bennett 1963). On an individual tree basis, volume growth increases with an increase in spacing. The economic advantage of the wider spacings are due mainly to the increase in wood quality which yields higher monetary returns. However, total wood productivity of the wider spaced stand may decrease. Bennett (1971) indicates that current annual volume increment of slash pine plantations culminates at age 12 years in stands with approximately 200 trees per acre over all sites greater than 50. Culmination occurs later for higher densities but seldom takes place beyond 25 years after planting. Knowledge of the change in the current annual increment is important because it provides the forester with a better understanding of stand growth over time, and a basis for
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10 determining optimal rotation lengths. Culmination of volume growth doe not coincide with peak periods of height and diameter growth because of differences in growth rates. For a given stand, the combination of height and diameter growth results in maximum volume growth per unit land area for that site and stand density. Tree Form The effect of competition and available growing space not only affect the size of individual trees but also the form of the main stem (Lohrey 1961, Horn 1961, Meng 1980). In a study with red pine ( Pinus resinosa Ait.) plantations Horn (1961) observed that stem form improved with competition and age while thinning impaired tree form of dominant and codominant trees. Meng (19bO) found that there was a statistically significant change in stem form following heavy thinning (50 percent of the basal area removed) and fertilization (448 kg/ha) of black spruce ( Picea mariana (Mill.) B.S.P.) stands. The change in the shape of the tree bole is in response to a decrease in competition and an increase in the available growing space (Horn 1961, Husch et al. 1982). The diminution in diameters along the main stem is known as taper and is the fundamental reason for the variation in cubic volumes of trees with the same diameter at breast height and total tree height (Husch et al. 1982, Grosenbaugh 1966). Stem form also varies with species, diameter, age, and the quality of the site. Because of the large variation in stem forms among trees, there rarely exists any resemblance to one of the most common geometric solids (Grosenbaugh 1966, Husch et al. 1982).
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11 Forest mensurationists have sought to discover a simple twovariable function involving only a few parameters which could be used t specify the entire tree profile (Grosenbaugh 1966). For example, Demaerschalk (1972) developed a system in which taper equations can be derived from existing volume equations; both equations yield identical results for total cubic volume if some taper data are available. In a similar study, Bailey et al. (1982) developed taper equations for siteprepaired slash pine plantations. Grosenbaugh (1966) demonstrated that total height (THT), bole surface area (SUR), and volume (VOL) of a given tree are highly correlated with other variables that are affected by the size and form of the tree stem, such as quality or quantity of lumber, plywood, kraft pulp yields as well as specific gravity. Not only are heights, surface areas, and volumes useful in estimating various product yields, but comparisons of ratios involving the three variables THT, VOL, and SUR provide additional information on the form of an individual tree. Grosenbaugh (1966) introduced three measures of tree form based on the ratios of height, volume, and surface areas, namely: Quadratic mean dia. (in.), Q = 13.5406 (VOL/THT) ^ Arithmetic mean dia. (in.), A = 3.8197 SUR/THT, and (Quad. ttean)2/Arith. mean (in.), q2/A = 48.0000 VOL/SUR. The three means coincide if a tree is a cylinder (no taper) and become increasingly different in the presence of taper. The ratio (Arith. Mean)2/ (Quad. Mean)2, f = 0.0796 SUR2/tHT*V0L which will always be less than one, measures the inverse of the relative
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12 variation in diameter along the nsain stem. Zero variation is indicated by a ratio of unity or when the tree is cylindrical. As relative variation increases, the ratio becomes increasingly smaller than unity. Stand Mortality One aspect of forest growth that has not been adequately addressed in the past pertains to the size, spatial distribution, and abundance of vacancies created by mortality in forest stands as well as the anticipated effect on individual tree growth (Bormann and Likens 1981). If vacancies in a given stand are randomly or uniformly distributed in space, adjacent trees may utilize the additional growing space to accelerate their growth. The net result may be an increase in stand productivity because of the reduced competition among neighboring trees. If vacancies are aggregated, the added growing space may not be fully utilized by adjacent trees. In such cases a loss in stand yield may be realized which is proportional to the area occupied by the dead trees (Newnham and Smith 1964, Stauffer 1978). Jones (1977) demonstrated that the spatial distribution of residual trees within a given stand significantly contributed to differences in individual tree volumes as well as total stand volumes. Tree mortality is extremely variable and difficult to predict (Jones 1977). There are two basic types of mortality that occur in forest stands: regular and irregular (Lee 1971). Regular mortality include trees that die due to competition for limited resources such as light, moisture, growing space and nutrients. Mortality caused by insect and disease infestations, as well as windfall and fire, is termed
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13 irregular. Predictions for both types of mortality are cumbersome and highly variable no matter how much mortality data are available (Davis 1966, Mitchell 1969, Hamilton 1974). Competition During the development of a stand, the growth of some trees is accelerated, while in others growth is suppressed or nonexistent. When openings in the stand are created by some kind of nonpersistent disturbances, adjacent trees tend to utilize the vacancies to increase their dominance within the stand (Mitchell 1969). The rate of growth depends upon the vigor of the adjacent trees to successively compete for the additional growing space (Bormann and Likens 1981). A problem arises when analyzing the growth of individual trees due to the difficulty of quantifying the level of competition among neighboring trees (Daniels 1976, Opie 1968, Zedaker 1982). Initially, various measures of stand density in the vicinity of a given tree were used as a measure of competition. Such measures include basal area, angle count, point density, distances to neighboring trees, crown diameter, and the number of primary and secondary competing trees (Bella 1971). The concept of "zone of influence" as a measure of crown competition among neighboring trees has also been advocated. It is defined as the area of crown overlap between neighboring trees (Bella 1971, Venzina 1963). The basic assumptions of various competition indices are a. The ability of a tree to compete for limited resources such as growing space and light can be represented by a circle or zone
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14 surrounding the tree, with the radius being a function of tree size such as volume, bole surface area, crown radius, DBH and total tree height, and b. The competition stress experienced by a tree is a function of the degree to which its influence zone is overlapped by those of its neighbors Several modifications have been made in defining the influencezone, including the use of weighted variables to denote the relative importance of the influence area overlap of a given tree to compete with neighboring trees (Daniels 1976, Venzina 1963). To estimate the growth potential of individual trees, Ek and Monserud (1974) used a weighted competition index based on the ratio of the total height of the subject tree to the average height of its competing neighbors multiplied by the crown radius of the subject tree. Other approaches advocated as a measure of competition are based on the ratio of diameters between the subject tree and its competing neighbors. This approach assigns weights proportional to the size of trees while the effects between trees of similar size remain essentially unweighted (Daniels 1976). Basal area has also been used as a measure of competition among neighboring trees. The basic assumption is that competition among individuals increases proportionally with the basal area of the stand (Bella 1971). Minor (1951) showed a slightly curvilinear relationship between diameter at breast height (DBH) and crown diameter. Minor also demonstrated that the ratio of DBH to crown diameter remains relatively constant once the basal area of a stand is stabilized.
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15 Distance Sampling Competition among individual trees within a stand is also directly related to the spatial distribution and size of the vacancies created by mortality. Therefore, it is important to approximate as closely as possible the spatial distribution of a given tree population in order to make valid inferences. Most sampling techniques and analyses were developed under the assumption of randomly distributed plant populations (Clark and Evans 1954). This assumption may not be valid for many plant communities (Ashby 1935). If the distribution of a population departs significantly from the underlying assumption of randomness, the resulting statistics and the reliability of the inferences may be in serious error (Cole 1946). Distance sampling is one of the methods employed to study spatial patterns of plant populations over a continuous surface. Basically, this method involves distance measurements either from randomly selected points and/or from randomly selected plants to their nearest neighbor. The empirical frequency distribution of such distances and the associated statistics such as means and variances may be used in specific cases to test the degree of departure from randomness. Since random spatial patterns of plants are not very common in the real world, distance sampling methods have been applied to study the degree of aggregation or clustering of plants in a given area. Recent findings (Reich 1980) indicate that Hopkins' modified coefficient of aggregation is an accurate and efficient computer sampling technique capable of analyzing the spatial patterns of natural and planted pine stands.
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16 Hopkins' coefficient of aggregation, A, is based on the ratio of the squared distances between randomly selected points and the nearest tree over the squared distances between randomly selected trees and their nearest neighbor: Z 0), A = Z (o^ where A = Hopkins' coefficient of aggregation, EtOj^ = Sum of squared pointtotree distances, Ziii2 = Sum of squared treetotree distances. Pielou (1977) introduced a modified form of Hopkins' index that is more sensitive in detecting nonrandom spatial patterns. Hopkins' modified coefficient of aggregation is defined as A* = 1 + A which has a betatype distribution with mean 0.5 and variance l/4(2n+l), where n is the number of pointtotree and treetotree distances in the sample. With a large sample size (n>50) Hopkins' index approaches a normal distribution. In such cases Z = 2 (A* 0.5)*(2n +1)^/2 viewed as a standard normal variate. The null hypothesis of a random spatial distribution would be rejected when Z > Z 11 a/z where Z^/^ is the critical Zvalue for the standard normal distribution. The population is said to be aggregate if A* is
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17 significantly greater than 0.5 and uniformly distributed if A* is significantly less than 0.5. System of Equations Most growth simulation models include equations to predict DBH, total tree height, crown width, and volume. They also incorporate some measure of competition to quantify the degree of interaction between neighboring trees. The predictive equations are not independent. When they are combined together they form a system of equations with correlated errors. This interdependence among regressors violates the assumption of a single equation estimation procedure which assumes that no correlation exists among the explanatory variables. The above discussion suggests that studies of single tree growth should not be confined to individual equations but rather be extended to a system of L equations commonly referred to as simultaneous equations. In the development of any tree growth model the most important consideration is the form of the relationship describing the interaction of individual tree parameters with growth. It is sometimes argued that the theory related to forest growth should provide insight into the relationship that the individual equations take on (Fisher 1970). The problem thus is to decide which set of basic characteristics adequately relates individual tree growth to cubic volume, surface area, tree diameter, total tree height, competition and available growing space. Over the years numerous techniques have been developed to estimate the parameters associated with a system of equations. Several of the more common estimation procedures include LimitedInformation Maximum
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18 Likelihood (LIML) procedure, FullInformation MaximumLikelihood (FIML) procedure, Zellner's seemingly unrelated least squares procedure, TwoStage Least Squares (2SLS), and ThreeStage Least Squares (3SLS) to name but a few. The appropriate procedure to use in estimating the parameters of a system of simultaneous equations will depend upon the structure and characteristics of the equations making up the system. For example, LIML takes into account the absence of certain variables from the 1th structural equation but does not assume the absence of any variables from the remaining L1 equations in the system when estimating the parameters of the 1th equation. On the other hand, FIML handles restrictions on the coefficients of all L equations simultaneously (i.e. zero coefficients in the 1th equation). The 2SLS procedure is similar to LIML while the 3SLS procedure is similar to FIML. The FIML and LIML procedures are rarely used because the estimation procedure involves solving a very complicated set of estimation equations (Theil 1971). A recent contribution to the estimation of a set of coefficients for a system of simultaneous linear equations is the 3SLS procedure developed by Zellner and Theil (1962). This procedure yields estimates which are asymptotically at least as efficient as the 2SLS procedure (Madangly 1964). The 2SLS procedure is applied to each of the L structural equations one at a time but computes a joint asymptotic standard error for the system as a whole, while the 3SLS procedure (an extension of the 2SLS procedure) handles the unknown coefficients of all L structural equations at the same time.
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STUDY PROCEDQRES Population The data used In this study were provided by Container Corporation of America. The study area is located near Callahan, Florida, in Nassau County. Slash pine seedlings were hand planted in February 1962 in twelve oneacre plots measuring 2.5x4 chains at six different spacings: 5x10, 7x10, 9x10, 11x10, 13x10, and 15x10 feet. Each plot was replicated twice and consisted of 14 evenly spaced rows. Each plot was surrounded by a buffer strip. Two spacing plots were replanted in February 1963 due to low planting survival (Figure 1). Prior to planting, the study area was part of an improved pasture. Tlie pasture was burned prior to planting in 1962. The site is somewhat poorly drained with a spodic horizon 1.2 to 2.0 meters in depth. All trees were assessed for survival at two years of age and then approximately every two years thereafter, starting at six years of age. Tree measurements were collected on a subsample of trees from each plot. Sample data included the condition of the tree (healthy or dead), the relative location within the study area, diameter at breast height, total tree height, merchantable tree height to a four inch top, tree vigor (position in the canopy) and the number of primary and secondary competing trees. 19
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20 Q < O cr Q > < LO < ro < < ID < (T> CD CD CD ro CD (J) CD ID GD is u to CO u ca 3 U Xi
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21 Prior to the fifth set of measurements, sample trees were selected from two diagonal transects established in each plot. On each transect 14 trees, one from each row, were sampled (28 trees per plot). Commencing from the fifth set of measurements on, four rows were systematically selected from each plot and all trees within the selected rows were measured. The number of sample trees per row varied by spacing and degree of survival. Spatial Analysis Hopkins' modified coefficient of aggregation was used to analyze the spatial pattern of healthy and dead trees in the study area when the trees were four years old. Age four was selected for analysis because almost all mortality in the study area had already occurred by that time. In each plot, ten independent estimates of Hopkins' index A* were obtained based on 150 pointtotree and 150 treetotree distances. A zstatistic was used to determine the degree of departure of the observed spatial distribution from the theoretical (random) one. Analysis of variance was employed to evaluate the magnitude of differences in the spatial pattern of healthy and dead trees among the six spacings. Mean separation was determined using Duncan's multiple range test (a = 0.05).
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22 Microsite Variation Swindel and Squillace (1980) demonstrated the existence of microsite variation on the growth of individual trees. To examine this extraneous variation on tree growth over time, the effects of spacing and replications were removed from the survival, DBH, total tree height, cubic volume, and surface area data for each time period. Spacing effects (S^) were estimated as the difference between the spacing means (SP^) and the grand mean (GM); replication effects (R^ ) were estimated as the difference between the replication means (REPj) and the grand mean. Microsite was estimated by subtracting the effects of spacing and replications from the individual plot means: ^i ^^i R. = REPj GM where ^ij corrected plot mean for spacing i and replication j, Y^j = uncorrected plot mean for spacing i and replication j, spacing mean, replication mean, grand mean, spacing effect, replication effect. REP. = 7E Y. 1=1 1 6 2 GM = 77 E E Y. i=l j=l Si =
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23 The effect of microsite on survival and individual tree size was analyzed by (a) comparing the corrected plot means at each age, and (b) comparing corrected plot means over time for a given plot. Effects of Spacing on Individual Tree Growth To estimate the effects of spacing on survival, DBH, total tree height, cubic volume and surface area, individual plot means (Y^j) were corrected for microsite and replication: Y* = Y. M. .R. where M. = Y: GM, Y* = plot mean corrected for microsite and replication, M^. = microsite effect. Plot means for survival, DBH, total height, cubic volume, and surface area corrected for microsite and replication effects were regressed on spacing by age using the standard linear model: Y* = + 6^ (spacing) + e^. The significance of spacing on survival and individual tree size was tested using regression analysis (a = 0.05).
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24 Volume and Surface Area Estimation A sample of 150 standing trees were selected at random for dendrometry measurements in the Spring of 1983. Approximately 12 trees were selected from each spacing x replication plot. Heights and corresponding upper stem diameters of standing sample trees were measured with a magnifying Barr and Stroud optical dendrometer. Model FP15. Measurements were taken at stump height (0.5 feet above ground), breast height (4.5 feet above ground), and various other points along the tree stem. This procedure divides each standing tree into segments that are relatively uniform with respect to taper, quality, and defects. Diameters at breast height were also measured with a diameter tape; bark thickness at breast height was measured with a bark guage. Cubic foot volumes, bole surface area and heights inside and outside bark for each sample tree were computed and accumulated over all measured tree segments. Product yield estimates were computed with Grosenbaugh's (1974) STX computer program. Regression equations were developed to predict total cubic foot volumes (VOL) and bole surface areas (SUR), outside bark, of individual trees as a function of diameter at breast height (DBH) and total tree height (THT) using the functional forms : VOL = a DBH^ THT"^ SUR = a'DBH^'THT^' where a(a'), 6(6'), and y(y') are regression coefficients estimated from the sample dendrometry data obtained from each spacing. The two regression equations were fitted using a standard linear model after a logrithmic transformation.
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25 The regression equations were used to estimate individual tree volumes and surface areas at 6, 8, 10, 12, 14, 16, 19, and 21 years after planting for all sample trees used in the analysis. Tree Form Using the volume and surface area equations described in the previous section, it is possible to express Grosenbaugh s (1966) four ratios of stem form in terms of diameter at breast height and total tree height by substituting the volume and surface area equations into the appropriate ratio. The four ratios will have the same functional form as the volume and surface area equations. Since trees can assume an infinite number of shapes, the coefficients in the above equations would tend to vary from tree to tree. Thus, equations developed for a given population can only predict the average change in tree form, cubic volume and surface area with respect to changes in diameters and heights of the same population. Data Interpolation The proposed growth model assumes a constant time interval (i.e. two years) between current and lagged exogenous and endogenous variables. To satisfy this constraint, three interpolation procedures (linear, Lagrangian, and quadratic) were examined to select an appropriate one to estimate DBH and total tree height at ages 4, 18, and 20 years after planting. The three interpolation procedures are described below
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26 Linear Interpolation If DBH and total tree height (y) are functions of the age of a tree (x), and jtq and are known functional values corresponding to Xq and Xp respectively, the functional value of y can be approximated in a linear manner by (y^Vq) X + x^y^XQ y^ y = X^ Xq where, Xq< x < Xj^ Lagrangian Interpolation Given any three points on a plane (xq, y^) (x^ y^) and (x2, y2) it is possible to evaluate the Lagrangian interpolation of a second degree polynomial P2(x) which passes through the three points. The functional value of x, denoted by P2(x), can be approximated by 2 P2(x) = Z L (x)y. i=0 ^ ^ where 2 (x X.) L^Cx) = n J Quadratic Interpolation The quadratic procedure assumes no functional relationship between X and y. Given two values yQ and y^ the value of y for a given x can be approximated by y = l^^n^ ^ij^^^ where, yg < y < y^ Evaluation of the three interpolation procedures indicated the linear method diameters and total heights during the juvinile growth phase and underestimates during the latter years. The quadratic procedure was not considered because it estimates a single geometric
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27 mean regardless of whether X is much closer to X2 than X^. As a result, the Lagrangian procedure was used to estimate DBH and total tree height at ages 4, 18, and 20 years. The interpolated data was then used to estimate total cubic volume and surface area. Growth Function The cumulative growth of an individual tree over time resembles an S Â— or sigmoid Â— shaped curve. Although the exact form of the cumulative growth curve changes with the dimensions of the tree (diameter, height, volume and surface area) the cumulative growth curve has certain properties that hold for all dimensions of a tree (Husch et al. 1982). Grosenbaugh (1965) investigated numerous sigmoid and nonlinear functions capable of generating a wide variety of sigmoid and other monotonically changing shapes consistent with the growth of individual trees over time. Several nonlinear growth functions were examined to select one that is intrinsically linear with respect to the parameters in the growth model and that are consistent with the growth behavior of individual trees subjected to various competitive stresses. A few of the relationships examined by Grosenbaugh include the JohnsonSchumacher, Gompertz, Gauss, von Bertalanffy, Verhulst and the PearlReed growth functions. Principal features and asymptotic properties of the above growth functions are discussed by Grosenbaugh (1965). The growth function selected to model tree growth in this study was that of Verhulst which has the following form:
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28 Y = A (1 + C exp (BX)) ^ where Y = dependent variable, X = independent variable, A = asymptotic maximum of Y, (YMX) C,B = regression coefficients. The Verhulst growth function was modified to include more than one independent variable in the growth model and to express individual tree growth on a percentage basis. The latter allows for comparison of the effects of competition and relative growing space on diameters, heights, volumes and surface areas over time. The modified Verhulst function has the following form: Y* = (1 + exp(Y'))~^ where, Y* = Y/A or Y/YMAX, 0 < Y* < 1 and Y' = Bq + B^X^ + B2X2 + ... + Bj^Xj^ + e Taking the first derivative of Y* with respect to Y' yields dY* = Y* (1 Y*) < 0. The second derivative is given by d2Y* = (1 2Y*)(1 Y*) Y* ^ 0.
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29 The asymptotic properties of the modified Verhulst function are Lim Y* = 1 < Lim Y* = 0. Y'^00 To utilize the modified Verhulst growth function in modeling individual tree growth over time, the value of YM^X must be defined in terms of the dependent variables being modeled (diameter, height, volume and surface area). Since YMA.X by definition is the maximum size an individual tree can attain, YKAX was equated to the site productivity of the stand. Maximum values for DMAX, HMAX, VM\X and SMX were defined as follows : HMX = site index base age 50, DMAX = maximum tree diameter for a given site index (HMAX), VMA.X = maximum cubic foot volume; f(DMAX, HMA.X) and SMAX = maximum bole surface area; f(DM^^X, HMX). Model Development When one performs several statistical tests to identify the structural form of a regression equation and uses the data to estimate the parameters of the model, certain problems arise which can affect the sampling distribution of the estimators (Theil 1971, Snee 1977). This problem is overcome when the identification and estimation procedures are based on independent data sets. It is usually recommended that the
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30 data set be split into two groups; the first set is used to identify the structural form of the equations, while the second set provides estimates of the coefficients of the model. The main advantage of data splitting is that one can obtain an independent estimate of the prediction accuracy of the model. The disadvantage of data splitting is that the variance of the estimated coefficients in the model increases. Snee (1977) presented a simple formula to approximate the percent increase in the variance when the data are split into two groups of equal size: 1/2 Percent Increase = (2n 1) (n 1) 1 100 where n is the size of the estimation and prediction data sets. With a sample size n = 30 the increase in the variance is approximately 42.6 percent and rapidly approaches 41.4 percent as the sample size goes to infinity. Because of this increase in the variance most researchers use the same data set to identify and estimate the structural form of the equations In this study the sample data were sorted by age, spacing, and replication and numbered from 1 to 5199. The data were sorted in this manner to increase the correlation among the independent variables over the six planting densities and nine time periods. Subsequently, the data were divided into two groups of equal size (Nl = 2600, N2 = 2599). All odd observations were included in data set one; even observations were included in the second data set. The first data set (Nl) was used to identify the structural form of the growth equations using the OLS procedure, while the second data set (N2) was used to estimate the coefficients of the system of equations using the 3SLS procedure.
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31 Field Data Screening REX (Grosenbaugh 1967), a FORTRANIV computer program, was used for combinatorial screening of variables used in the growth model for this study. This approach explores all possible linear combinations of variables to find the "best" combination of independent variables that best explains the variation of the dependent variables. REX uses the mean squared residual, MSE (or variance of the regression estimate), as an indicator of a "best fit" since the NEE does not change monotonically with the number of independent variables in the equation. As opposed to other procedures such as stepwise, the selection of the "best" fitting model by REX is independent of the steps used in the exploratory analysis and provides unbiased estimates of the coefficients, but tends to underestimate the variance of prediction. Fourteen variables were screened through REX for the diameter and height equations; thirteen variables were screened for the volume and surface area equations (Table 1). The final equations for diameter, total height, cubic volume and surface area had the following functional form: r ^lo^^ii^i^n^i^^is^^^ +s^^par2 +0^^r2 n^^A n^^/A^ +^^10^0^? 2= ^20^^21 l"'^22^'l+^23^^^ n^^FAK^ +g^^R2 +6^6^ "^zi^^ +^29^^^ + B211COMP/SP + Â£2 V3= r3^D +r32H +63Q+e33PAR +B3^Par2 +B3^R2 +33^A +638/A2+ 639RIP + B3^^C0MP/SP + Â£3
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32 ^4 = r^iD +r^2 +^40^^43^^^ ^44^^^' +^45^^ Â•'^46^ "^^7^' B^gRlP +0^^^COMP/SP + where Dl = Ln (1^ 1) D* = DBH/DMX DMX = maximum DBH at 50 years of age ^l lagged one period (two years) H* = THT/HMAX Hmx = site index, base index 50 years = H2 lagged one period V3 = (v* 0 V* = VOL/VMX Vmx = maximum cubic volume at age 50 years for a given DM\X and HMX S4 = (If 1] = SUR/SMAX Smx = maximum bole surface area at 50 years of age for a given DMAX and HMA.X SP PAR relative growing space = 3(j^L):yy KL = number of nearest neighbors lagged one time period, KL = 0,1,2
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33 SP = spacing within rows (feet) A = tree age (years) A_j^ = tree age lagged one period R2 = A_^/A RIP = (1 E12) PAR 2 COMP = Z DBH. /(2 DBH) k=l DBHj^ = DBH of neighboring trees within rows (inches) DBH = DBH of subject tree (inches) DMS = diameter microsite (inches) Â€^ = error term for the 1th structural equation Ln = natural logarithm ^ij ^ij ~ parameters of the endogenous and exogenous variables, respectively. The equations for diameter, total height, cubic volume and surface area were fitted using REX to obtain OLS estimates of the coefficients in the model. The coefficients were estimated assuming that diameter and height were exogenous variables in the cubic volume and surface area equations and that the disturbances among equations were uncorrelated The four equations were also fitted using SAS/SYSREG (Helwig and Council 1979) procedure to obtain 3SLS estimates. This procedure assumes a correlation of residuals among the four equations along with the presence of endogenous variables as explanatory ones. Following is a detailed description of the 3SLS procedure.
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34 Table 1. Variables used to identify the "best" combination of explanatory variables. Independent Equation Variable DBH(D) THT(H) VOL(V) C ITT? ( C\ DBH(D) 1 / N.U. 1/ N.U. A A Total tree height (H) N.U. A A DBH Laeeed CD ,) A 2/ A N.U. N.U. Total heieht laeeed (H ,) A c\ A N.U. XT TT N.U. PAR r G G G PAR **2 c G G G COMP *2 G 3/ G G G COMP o S S S COMP/SP s S S S Alag/Age G G G G Age G G (1 1/Age S s S S Age **/ G G G G 1/Age **2 S S S S (lAlag/Age)*PAR G G G G DBH Mcrosite G N.U. N.U. N.U. THT Mcrosite N.U. G N.U. N.U. Not used in the screening process 2/ Variable was automatically included in the model and was not subject to the screening process. G = group; S = set; members of sets within a group are treated as nonfixed, and are included in regressions only when called for by specific combinatorial rules. For example, the competition group contains three sets; only one set will be included in the model if the group is selected in the screening process.
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35 Description of the System of Equations The system of equations described in the previous section contains L linear stochastic structural equations consisting of L jointly dependent variables and K predetermined variables. It is assumed that (a) the system can be solved for the jointly dependent variables and (b) the disturbances of the structural equations have zero mean and (c) the disturbances are serially independent and homoscedastic. The latter implies that their variances and contemporaneous covariances are finite and constant over time. A test for crosssectional and serial correlation will be discussed in a later section as well as the appropriate estimation procedure. If there are n observations then any structural equation, say the 1th equation, can be written in the following form for all n observations combined: where y^^ is a nxl column vector of observations for one of the L jointly dependent variables in the system; Y^^ is a nxL matrix of values taken on by the explanatory dependent variables in the ith equation; is the corresponding Lxl coefficient vector; is a nxK matrix of values taken on by the exogenous variables; is the corresponding Kxl coefficient vector; and is a nxl column vector of structural disturbances. Using matrix notation the system as a whole may be written as follows: Yr +XB = E where Y is a nxL matrix of jointly dependent variables; r is a LxL
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36 matrix of parameters; X is a nxK matrix of predetermined variables; g is a KxL matrix of parameters; and E is a nxL matrix of disturbances. Since there are as many equations as jointly dependent variables, the matrix r is square and nonsingular by definition. Thus, the system may be solved for Y by Y = xgr"'^ + Er~^. This system of equations is referred to as the reduced form. It describes the endogenous variables in terms of the current and lagged independent variables, lagged dependent variables and disturbances. The reduced form coefficients can be estimated indirectly from the structural equations as shown above or directly by the reduced form: Y = Xt + U where t = BF ^ and U = ^f'^. However, if the coefficients of the reduced form equations are estimated directly, it is impossible to obtain estimates of the coefficients associated with the structural equations. The latter approach was used by Sullivan and Clutter (1972) to estimate the coefficients of a simultaneous growth and yield model for loblolly pine stands. Since the reduced form equations do not contain current endogenous variables, as explanatory variables, the L reduced form equations can be considered unrelated even though they still have correlated errors among equations. In such cases Zellner's seemingly unrelated least squares procedure would be the appropriate one to use in estimating the coefficients of the reduced form equations (Theil 1971).
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37 Threestage LeastSquares Procedure The 3SLS procedure uses the results of the 2SLS procedure to estimate the coefficients of the entire system simultaneously. The matrix E(Y^) is a submatrix of the matrix E(Y) which corresponds to the jointly dependent variables that occur on the right hand side of the equation. The Â£th structural equation may then be expressed in the following form: The first stage of the 3SLS procedure yields an ordinary least squares (OLS) estimate of E(Y^) which is given by where \ = \ M = I X(X'X)~^X' E(Y^) = X(X'X)"^ X'Y^ In the second stage OLS is used to provide an estimate of the coefficient vector 6^. Zellner and Theil (1962) and Theil (1971) have shown that the 2SLS estimate of 6^ is given by d^ = [Z* X(X'X)"^ X(X'X)"4' Y^
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38 where = [y^ U^: X^] The same authors discuss in detail the estimation procedure in the two cited references. An estimate of the variancecovariance matrix S^^^ is given by X(X'X)"^X'Zj from which asymptotic standard errors can be computed by taking the square roots of the diagonal elements. The scalar S^j^, an estimate of the variance a^^ which is defined as the mean square of the associated residuals To obtain the third stage estimates, the L structural equations, for all n observations, are stacked in the following manner: or simply 0 0 ^2 Â• Â• Â• 0 0 0 .. Z, y = Z6 + E Premultiplying the above equation by the transpose of a LnxKL Kronecker product of the form:
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39 X = X 0 ... 0 0 x' 0 0 0 ... X where X''is an Kxn matrix of explanatory variables yields (I X') y = (I X') Z6 + (I X')E where y is a Lnxl vector of endogenous variables. Applying the Generalized Least Squares (GLS) procedure to the above system of equation leads to the following estimate of 6: 6 = (Z'[j:"W(X'X)~4']Z)~^ Z'["^ X(X'X)"^X']y. If the variancecovariance matrix ^ is replaced by S = (Sjj)> the matrix of mean squares and products of the 2SLS residuals leads to: 6 = (Z'[X"^ X(X'X)"^ X'jZ)"^ Z'[s"^X(X'X)~^X'ly the threestage least squares (3SLS) estimator. Identification of the System of Equations To obtain an efficient estimate of the coefficients of the structural system of equations and an estimate of the variancecovariance matrix, the sytem of equations must satisfy an order condition of identif lability (Theil 1971). A necessary condition for
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40 the existence of the 3SLS estimator is that K > where K is the total number of predetermined variables in the system and is the number of parameters to be estimated in the Ith structural equation. Three possible conditions exist K ^ N. If K < N, the Ith equation is said to be under identified and the 2SLS estimator does not exist. This implies that at least one row or column of the variancecovariance matrix j: cannot be estimated. In other words the parameters of the Ith equation are not estimable because of a linear combination of the other equations that contains only variables which do not occur in the ith equation. In such a situation, Zellner and Theil (1962) recommend the elimination of all under identifiable equations from the system and confinement of the 3SLS procedure only to identifiable equations. If K = N the ith equation is said to be just identified. If all the equations in the system are just identified or if the variancecovariance matrix ^ is diagonal, the 3SLS estimator provides no gain over the 2SLS procedure (Theil 1971, Zellner and Theil 1962). The 3SLS procedure provides an asymptotic gain over 2SLS when the structural disturbances have nonzero contemporaneous covariances and at least some of the equations are overidentif ied, K > N (Zellner and Theil 1962). When the above conditions hold, the asymptotic covariance matrix of the 2SLS estimator d^^ will exceed that of the corresponding subvector of 6 by a positive semidefinite matrix. The asymptotic superiority of the 3SLS procedure is due to the optimality of the GLS procedure which underlies the joint estimation procedure (Theil 1971, Madangly 1964, Zellner and Theil 1962).
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41 Conteiaporaneous Correlations If the disturbances of the growth model are correlated across equations, a systems approach is required to assure asymptotically efficient parameter estimates. If the assumptions of the standard linear model hold for all equations, the elements of the disturbance vector e have zero mean but with unequal variances for each equation in the system. In general, there are three types of disturbances associated with a system of simultaneous equations. The first E(e e ) a n and E(e^,,e^^), for a n concerns disturbances of different years but the same equation (serial correlation). In the standard linear model these covariances are assumed to be zero. The second type of covariance ^^^a'^n''^ for a 5* n pertains to the disturbances of both different years and different equations. It is also assumed that these covariances are equal to zero. The third type of covariance E(e ,e ^) deals with a a disturbances of different equations but of the same year. This type of covariance is referred to as the contemporaneous covariance. The disturbances associated with the four structural equations, diameter, total height, cubic volume and surface area are assumed to be correlated. There are several reasons for suggesting the contemporaneous nature of the disturbances. If two variables are correlated with a third common variable, they will also appear to be correlated with each other. Since both tree height and tree diameter are correlated with age, height also appears to be correlated with diameters (Husch et al. 1978). Since tree diameters and tree heights are commonly used to predict volumes and surface areas, it would be natural to assume that volumes and surface area are correlated with diameters and heights as well as with age.
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42 The contemporaneous covariance matrix for a given observation has the following form: E (Â£ ,e J = a a 21 ^22 Â• Â• Â• "li \i 'IL '2L LL The diagonal elements o^^ are the variances associated with the structural equations and the offdiagonal elements a^^ are the contemporaneous covariances in the ath time period. The condition that (e^j = Ej^) and the independence of the observations over time is an extension of the homoscedasticity constraint on the standard linear model. It also indicates that the disturbances are random variables from a multivariate population with zero mean and constant covariance matrix. The disturbance vector e implies that the covariance matrix is block diagonal with ndiagonal submatrices all equal to . 0 0 V(e,) = 0 0
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43 The offdiagonal submatrices are all zero since their elements are of the first E(e ,e ), E(e ,,e and second type E(Â£ ,e ^) for a r]. The a n a n an inverse of V(ej^) is also block diagonal with ^ in the diagonal blocks. Estimates of the parameters of the 1th equation in the system are obtained by the use of an Aitkens' estimator: where 6^ is the coefficient vector for the 1th equation; X is a n x K matrix of explanatory variables; and y^ is an n x 1 vector of dependent observations for the 1th equation. The difficulty in applying the Aitken estimator is that the matrix j: is usually unknown and must be estimated from the sample data. However, the variancecovariance matrix j: under 3SLS will be misspecified if a cross sectional and/or time series correlation exists among the equations in the system. Cross Sectional Time Series Analysis Data obtained by repeated measurements on an individual will contain correlated errors (Sullivan and Clutter 1972). Measurements taken closer together in time are more highly correlated than measurements taken further apart in time. Furthermore, unequal variances may be present in the data (McHugh and Wall 1962). In practice serial correlations and heterogeneous errors are frequently ignored and parameter estimates are obtained by ordinary least squares (OLS). The OLS estimates of the coefficients are unbiased but have inflated errors due to a specification error. Also, the estimate of the residual mean
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44 squared error is biased downward (Sullivan and Clutter 1972). The presence of nonzero covariances and nonhomogeneous variances provide a basis for considering the inclusion of this additional information into the estimation procedure to obtain a more efficient estimate of the error components of the model (McHugh and Wall 1962). Consider the following crosssectional time series model. Suppose N individual trees, n = 1,2,...,N have been sampled over T periods of time, t = 1,2,...,T. The variable to be explained is denoted v The nt dependent variable is explained by K truly exogenous variables, and L lagged values of the dependent variable. The model may be expressed in the following manner: y = r + xg + e where ^11 IT y = ^Nl ^NT an NT x 1 vector; ^11 X' 11 X = ^NT ^NT
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45 an NT X K matrix of exogenous variables; ye .1 L y NT0 y NT1 Â• Â• an NT X L matrix of lagged values of the dependent variable; Â•11 e = Â•IT Nl NT an NT X 1 vector of residuals. Also, 3 is a K x 1 vector of constant coefficients of the exogenous variables, and r is a L x 1 vector of coefficients for the lagged endogenous variables. The model can be simplified by letting Z = [X: Yg ] an NT x K + 0 matrix, and 6 = [grF] a K + 9 X 1 vector of coefficients: y = Z5 + e. Rao (1959) considered the estimation of the parameters of a linear model for data containing correlated errors and derived estimators requiring an independent estimate of the variancecovariance matrix. Elston and Grizzle (1962) compared Rao's GLS procedure with that of OLS and suggested that OLS might in some cases provide unbiased estimates
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46 because of its simplicity. Sullivan and Clutter (1972) suggested that OLS should not be ignored in the analysis of growth and yield data with correlated errors because of its simplicity and when parameter estimates rather than confidence intervals are desired for the parameters and predictions A large proportion of the theory relating to the growth and yield of individual trees is aimed at explaining the effects of various stand factors (i.e. stand density, age, and competition) on tree growth over time. However, the majority of the growth models currently in use have frequently been developed from data covering a large crosssection of stand conditions within a given population and the specification of the model is based on the overall response observed in the aggregated sample data. For example, the growth of an individual tree is dependent upon the initial planting density, level of competition, the age of the tree, and productivity of the site. Thus, for a given site and initial planting density, a theoretical growth function can be developed to explain the growth as a function of stand age and the level of competition. However, the actual specification and subsequent estimation of the model parameters are based on data consisting of a large crosssection of individual trees from different sites and stand densities making up the population. It is not selfevident that an individual growth function could be developed to model the growth of individual trees over a wide range of stand conditions (Theil 1971, 3ehr 1981). For a given level of competition, site productivity, and age it would be reasonable to assume that the growth response would differ over a wide range of planting densities. It would be unrealistic to think that one variable such as planting density could be used to estimate the
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47 growth response of an individual tree for various levels of competition over time. In such a case, all sample observations are grouped together and the estimation of the model parameters are performed on the combined sample of crosssectional and timeseries data (Balestra and Nerlov 1966). Often in pooling data across a wide range of planting densities, it is difficult if not impossible to capture all of the important crosssectional effects. There may be specific factors associated with each crosssection effect that help explain the observed variation in the dependent variables but are unavoidably excluded from the model (Behr 1981, Balestra and Nerlove 1966). Inferences on the estimated model parameters and any confidence statements are based on the assumptions pertaining to the properties of the residual vector e. The specific ignorance due to the exclusion of specific crosssectional relationships results in overestimating the disturbance term. Each residual e^^ for the nth observation during time period t, may be decomposed into two statistically independent parts: ^nt = + where y^^ is the residual variation due to general ignorance (sampling error), and u^^ is the residual variation due to specific crosssectional ignorance. It is assumed that y^^ has zero mean and variance and covariance matrix o^l. It is further assumed that is a random variable with mean zero and variance Thus the residual e has zero U nt mean and variance = + and covariance p of y and v for e Y M e 'nt 'n't' any pair of observations (nt,n't') where nt^^n't' is equal to Hence, p = a^/{a^ + a^] It is further assumed that the crosssectional
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48 disturbances are serially independent and that the disturbances are independent from one observation to the next. In other words: 0^ n = n' M 0 otherwise 'a^ n = n", t = t 0 otherwise 'nt "^n^ 0 for all n,t. The assumption that the crosssectional disturbances ^re independent from one observation to the next needs mentioning. If the sample trees are selected at random and are independent of each other, then this assumption can readily be satisfied. However, if the trees are systematically selected as in this study, this assumption may not be well satisfied and must be taken into consideration when interpreting the results of the analysis. From the above discussion the variancecovariance matrix can be written as follows: 0 0 0 E(e,e'] = Q = a2 0 0 A N where
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49 p p p 1 p Â• p Â• Â• Â• Â• Â• Â• Â• Â• Â• Â• Â• Â• Â• p p 1 a2 + a2 Y U a2/a2 In a single equation model with a crosssectional timeseries component, OLS estimates of the coefficient vector are inefficient. Consequently, GLS is required to obtain efficient parameter estimates: However, since Q, is usually not known, it must be estimated from the sample data. A technique often employed to solve for the parameter vector 3 is to search over all values of p and choose the corresponding parameter set which maximizes the likelihood function (Behr 1981). If e is distributed according to a multivariate normal distribution with zero mean and covariance matrix f2 then the probability density function P(e) may be written as follows: P(e) = (2n)"^^/2s^"^f/2exp(L/2c'J^lÂ£] where e = y Xg The logarithmic likelihood function for the parameters p, y and may be written as follows: L(p,Y,a2) = ^ Ln(2n) V2Ln$^"^ ^2^^^'^^. 1 P ^ n n* 02a Â£ = n2 is a T X T matrix, and t2 = P =
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50 The maximum of the likelihood function L(p,Y,a^) is obtained when the e quadratic form z^oT^e = [y XY]fi~^[y Xy] is minimized. This occurs when The value for may be found by differentiating L(p,Y,a^) with respect to a and setting the results to zero. Behr (1981) presented an alternative maximum likelihood estimator called the concentrated likelihood estimator which is given by N N N 1p + T.p 2Lfp,Y,a21 = E T.Ln(2n) + E T Lnf ( 1p ) 1 + E Lnf ^1 ^ 1 i^Â£^ ^ 1p ^ Using the GLS procedure, estimates of a are obtained for each value of p along with the corresponding estimates of y The value of p that maximizes the concentrated maximum likelihood function is the maximum likelihood estimate of p which implies that the corresponding estimates of Y and a are also maximum likelihood estimates (Maddala 1977, Behr 1981, Balestra and Nerlove 1966, Theil 1971). Since p varies over the closed interval, zero to one, L(p,Yja^) must reach a maximum within the e interval or at one of the boundaries. If the maximum likelihood estimator of p is zero, this implies that the cross sectional error is equal to zero = 0 and that the total variability of the model = is due to general ignorance and not to a specification error in the model In the presence of a crosssectional timeseries error component, the normal 2SLS procedure will be inefficient. The suggested procedure
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51 is to apply the ML estimator to each of the four first stage structural equations (Maddala 1977, Behr 1981). For each equation, search over the values of p and choose the p value and corresponding parameter set that maximizes the concentrated likelihood function. Thus, for each equation a set of modified 2SLS estimates and corresponding p values are obtained. The third stage is then applied to the data using the variancecovariance matrix obtained in the second stage. Model Validation The efficiency of the OLS and 3SLS estimation procedure was evaluated by four goodnessoff it statistics (Murphy 1983): Fit Index, Mean Difference, D n Root Mean Square Error, RMSE D% = Z [ n n Y.Y. Percent Mean Difference, ] 100 n Y. Y. 1. Percent Root Mean Square Error, RMSE% = [ ^ E [Â— ]^] ^100 i where
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52 n = number of observations = observed value Y = mean of observed values Y^ = predicted value. Other checks on the validity of the growth model included (a) An examination of the model coefficients for wrong signs and associated variance inflation factors (VIF). The VIF is defined as the inverse of partial correlation coefficients associated with the dependent variables in the model, (b) Comparison of the correlations of errors across equations for the 3SLS procedure, and (c) The efficiency of the 3SLS procedure relative to the OLS estimation procedure. An Ftest was used to examine the equality of variances of the two estimation procedures: ^ ''OLS(B) ^ij Â— r where F^^ is the Fstatistic for the ith equation and jth parameter.
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RESULTS AND DISCUSSION Spatial Analysis Hopkins' modified coefficient of aggregation was used to depict the spatial pattern of healthy and dead trees in the study area at four years of age. Hopkins' index yields a point estimate of the spatial distribution ranging from 0.2 to 0.87 (fairly uniform to highly aggregated). A value of 0.5 corresponds to a random spatial pattern. A population is said to be distributed at random if the relative location of an individual tree is independent of the location of any other tree within an infinitely large area. An aggregated population is one in which the trees occur in clumps of varying densities and sizes. A uniform spatial distribution is one in which the trees are evenly distributed over a given area on the vertices of a triangular, square or hexagonal lattice (Reich 1980). Two transition levels, occurring at approximately 0.4 and 0.6, separate the three basic spatial patterns, uniform, random and aggregated. Values near the transition levels are not clearly discriminating. Following is a discussion of the spatial patterns of healthy and dead trees observed in the study. At four years of age healthy trees were uniformly distributed throughout the study area (Table 2). But significant differences were observed in the spatial patterns of healthy trees among spacings and 53
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54 replications (Table 3). The significant interaction suggests a lack of consistency of spacing and replication effects. However, the effect was very small. All but one of the spacings had higher coefficients in replication A than in replication B. The significant replication suggests a location effect (Table 2). In general, the spatial patterns of healthy trees were significantly more uniformly distributed at wider spacings. It should be noted that Hopkins' index is independent of tree density and that the index is actually measuring the distribution and size of the vacancies created by mortality. Since the survival rate increased with spacing, the wider spacings retained more of the original planting configuration than the smaller spacings which had a lower survival rate. The spatial patterns of dead trees were randomly distributed throughout the study area, but varied significantly among spacings and replications (Table 2 and 4). No spacing (x) replication interaction was observed. In five of the 12 plots a transition was observed among spatial patterns for the dead trees. Four of the transitions were from random to uniform while one was from random to aggregate. The transitions between spatial patterns are indicated by a two letter code: U/R, R/U, A/R, and R/A. The sequence of letters is based on the frequency of the observed spatial pattern. For example R/U indicates that at least 50 percent (or five out of ten) of the observed spatial patterns were random while the remaining estimates indicated a uniform spatial pattern. As mortality in the stand continues over time, the spatial distribution of the healthy and dead trees will change with time. In general, healthy trees will tend to converge from uniform to more of a
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55 Table 2. Spatial analysis for healthy and dead trees at four years of age. Spacing Rep Healthy^/ Trees Spatial^^ Pattern Deadi'^ Spatial Trees Pattern 5 A .3277^'^ 11 U .4899'^'^'^ R B .3352^'^ U ^4335cde,l R 7 A .2829^'^ U ^5255abc,l R B .2387'^'2 U .3966'^^'1 R/U 9 A .2161^'^ U .5085^^'^ R B IT U ^5202^^,1 R 11 A .2337^'2 U ^45Q3cde,l R/U B .2008'^^'^ u .4011'^<^'1 U/R 13 A .1002^'^ u .6247^'^ R/A B .0994^'^ u .6240^^'^ R 15 A .2359*^''^ u .4442'^'^^'^ R B .2266*^'^ u ^33926,1 U/R Mean value for Hopkins' modified coefficient of aggregation, A* based on ten independent samples of 150 pointtotree and 150 treetotree distances. Â— Means with different letter superscripts indicate significant differences among spacing and replications. Means with different number superscript indicate significant difference between healthy and dead trees (a = 0.05). 3/ Â— U uniform spatial pattern; R = random spatial distribution A = aggregated spatial distribution.
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56 Table 3. Analysis of variance of the spatial distribution of healthy trees in the McArthur spacing study at age four years. Source df SS MS F Spacing (S) 5 .541 .108 173.06** Replication (R) 1 .009 .009 14.29** S X R 5 .009 .002 2.92* Error 98 .061 .001 TOTAL 109 .620 ** Significant at the a = 0.01 level. Significant at the a = 0.05 level.
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57 Table 4. Analysis of variance of the spatial distribution of dead trees in the McArthur spacing study at age four years. Source df SS F Spacing (S) 5 .614 .123 11 .02** Replication (R) 1 .079 .079 7.07** S X R 5 .067 .013 1.21 Error 98 1.080 .011 Total 109 1.840 Significant at the a = 0.01 level
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58 random spatial distribution while the dead trees will converge toward a uniform or aggregated spatial pattern (Kent and Dress 1979, Kent and Dress 1980, Reich 1980). The convergence of spatial patterns over time are evident in certain stands infected with fusiform rust and pitch canker. Effects of Microsite Differences on Individual Tree Growth Analysis of variance was used to study the effects of spacing on survival, DBH, total height, cubic volume and bole surface area at 2, 6, 8, 10, 12, 14, 16, 19 and 21 years after planting. Results indicate that survival and tree size (DBH, total height, cubic volume, and surface area) are significantly influenced by spacing. No significant differences were observed between replications; however, there was a significant spacing (x) replication interaction. The significant interaction suggests a lack of consistency of spacing and replication effects and suggests that the location of the plot played an important role in the growth of the individual trees in this study. Even though the 12 acre study area appears to be homogeneous, in reality there seems to be an intricate pattern between "good" and "poor" microsites within the study area. After correcting DBH, survival, tree heights, cubic volumes, and surface area data for spacing and replication effects and plotting the corrected values over stand age and plot location, a definite microsite pattern emerge which help explain the significant spacing X replication interaction. The effects of microsite on survival, DBH, total height, cubic volume, and surface area are depicted in Figures 2 through 6 by
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59 plot location and stand age. Uncorrected plot means are shown in Figures 7 through 10. In the center of the study area there is a clearing with very little understory vegetation. East and west of the clearing the understory vegetation increases both in density and height. The dominant understory species are wax myrtle ( Myrica cerifera L.) which averaged 15.5 feet in height (Table 5). At four years of age, survival, DBH, and tree heights were somewhat larger in the center of the study area and decreased in size and numbers along the eastwest direction. The observed microsite pattern corresponds approximately to the distribution of the understory vegetation of the study area. The microsite effect on tree heights decreases with age and diminishes by the age of 21 years (Figure 3). Microsite noticeably influenced DBH, cubic volume, surface area and survival over the 21 year period (Figures 2,4,5, and 6). Over time, the microsite pattern changed; diameters, volumes, surface area and survival rates were higher in the eastwest portion of the study area and decreased toward the center. No silvi cultural practices, such as burning, have been conducted on the study area since establishment in 1963. Most likely the observed microsite pattern is a biological response caused by microtopography and underground drainage variation. A recent study by Permar and Fisher (1983) indicated that wax myrtle is an important species for fixing nitrogen in southern flatwoods. It has been shown that nitrogen fertilizer can result in substantial increases in volume growth of slash pine on spodosal soil sites in Southeastern U.S. (Fisher and Garbett 1980). These flatwood soils are generally low in available nitrogen and respond to added
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60 nitrogen with increased growth. Stands fertilized with nitrogen have a gain of approximately 4.5 /ha/year compared to unfertilized ones (Fisher and Garbett 1980). Also, the nitrogen content of both the litter and soil tend to be greater beneath wax myrtle than pines. Average annual accumulation of nitrogen under wax myrtle was 2.4 g N/m^ /year (Permar and Fisher 1983). It may be assumed that the understory vegetation became established shortly after or during the planting of the stand. During the first four to six years, understory species (wax myrtle, in particular) competed with the young pine seedlings in the east and west portion of the stand. This competition resulted in a decrease in both diameter and height growth compared to trees located in the center of the study area where there was not competing understory vegetation. When the pine trees were able to compete with the understory vegetation for growing space, nutrients, light and water, they were able to efficiently utilize the additional nitrogen fixed by the wax myrtle. The net result is a higher survival rate and larger than average trees in the east and west portion of the stand compared to trees in the center where little or no additional nitrogen from the understory vegetation was available. This hypothesis may explain the microsite patterns observed in survival and tree size, but it still needs to be tested with a study designed specifically to analyze the effects of microsite and/or nitrification of slash pine by wax myrtle on the growth of individual trees over time.
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61 Table 5. Average height of major understory species for Che McArthur study area at 21 years of age. Understory Average Species Height (FT) Range Acer rubrum L. 1.4 11.316.0 Ilex glabra (L.) Cray 6.6 4.98.0 Myrica cerifera L. 15.5 5.025.0 Persea borbonia L. 10.9 6.815.5 Quercus incana Bartr. 15.0 15.0 Quercus laurifolia Michx. 10.2 4.818.0 Quercus nigra L. 13.8 6.825.5 Quercus virginiana Mill. 6.4 6.4 Sabal palmetto (Walt.) Lodd. ex. Sholt. 5.7 4.38.0 Serenoa repens (B.) Small 4.2 3.54.7 Vacciniura stamineum L. 11.2 11.2
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62 QlOr I< o I 7B I5B 98 I3B IIB 5B 9A !5A 7A I3A A 5A PLOT LAYOUT Figure 2. Diameter breast height corrected for spacing and replication by stand age and plot location.
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63 70r PLOT LAYOUT Figure 3. Total tree height corrected for spacing and replication by stand age and plot location.
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64 PLOT LAYOUT Figure 4. Cubic volume corrected for spacing and replication by stand age and plot location.
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65 1000' 1 ^ 1 1 1 1 1 1 III. 7B I5B 98 I3B IIB 5B 9A I5A 7A i3A IIA 5A PLOT LAYOUT Figure 5. Surface area corrected for spacing and replication by stand age and plot location.
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66 no r 10090 80 70 60 7B I5B 9B 138 116 58 9A ISA 7A I3A IIA 5A PLOT LAYOUT Figure 6. Survival corrected for spacing and replication by stand age and plot location.
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67 10 r 7B I5E SB 13B MB 53 9A I5A 7A !3A IIA 5 A PLOT LAYOUT Figure 7. Diameter breast height by stand age and plot location.
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68 70 r 73 I5B 98 I3B I IB 5B SA ISA 7A i3A !IA 5A PLOT LAYOUT Figure 8. Total tree height by stand age and plot location.
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69 7B I5B 9B 138 MB 5B 9A ISA 7A 13A IIA 5A PLOT LAYOUT Figure 9. Cubic volume by stand age and plot location.
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70 0' 1 ^ 1 1 7B I5B 9B I3B IIB 5B 9A ISA 7A I3A IIA 5A PLOT LAYOUT gure 10. Surface area by stand age and plot location.
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71 Effects of Spacing on Individual Tree Growth Regression analysis indicates that spacing had no significant effect on height growth after correcting for microsite and replications (Figure 12). However, spacing significantly affected survival rate and size of individual trees, excluding height (Figures 11 and 15). The effects of spacing on survival, DBH, cubic volume and surface area increased with age. Several studies (Bennett 1974, Shepard 1974) have reported both positive and negative correlations between height growth and spacing. These discrepancies may be in part due to the exclusion of microsite effects from the analysis of the data. In this study conventional analysis of variance indicated that spacing had a significant effect on tree heights. After correcting for microsite and replication effects, regression analysis indicated that spacing had no significant effect on tree heights. Volume and Surface Area Regression analysis was used to develop volume and surface area equations to predict cubic foot volumes and bole surface area in square feet for the sample trees used in the study. Regression equations were developed for both inside and outside bark. An exponential equation was used to estimate total cubic volume and surface area as a function of diameter at breast height outside bark (DBH) and total tree height (THT). The form of the equations was based on the residuals and resulting coefficient of multiple determination. The final form of the
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72 2I i5 SPACING (FT) Figure 11. Effects of spacing on diameter breast height corrected for microsite and replication by stand age.
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73 70 o uj5 q: H Â— uj J o < cr. 60 50 403020^ 10 r2 = 0.18 R^ = 0.04 r2 = 0.20 R^'O.Q! r2 = 0.0I R^0.15 AGE 21 !9 Â•12 7 !l SPACING (FT) Figure 12. Effects of spacing on total tree height corrected for microsite and replication by stand age.
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74 SPACING (FT) Figure 13. Effects of spacing on cubic volume corrected for microsite and replication by stand age.
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75 AGE J I I I I 1 5 7 9 II 13 15 SPACING (FEET) Figure 14. Effects of spacing on surface area corrected for microsite and replication by stand age.
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76 7 9 II 13 15 SPACING (FT) Figure 15. Effects of spacing on survival corrected for microsite and replication by stand age.
PAGE 89
77 equation used to estimate cubic volume and surface area is Y = aDBH^ THT^ where a.g and y are regression coefficients estimated from the sample dendrometry data using log transformation. Volume VOB = exp[5. 3036+2. 1963Ln(DBH)+0.7492Ln(THT)] R2 = 0.970 SLn(voe) = 0.098 Relative variation: 0.91 to 1.10 VIB = exp[5. 9612+2. 3004Ln(DBH)+0.7928Ln(THT)] r2 = 0.964 Sl^^v^) = 0.114 Relative variation: 0.89 to 1.12 Surface Area SOB = exp[l. 4180+1. 1376Ln(DBH)+0.8358Ln(THr)J R2 = 0.969 3^^(303) = 0.060 Relative variation: 0.94 to 1.06 SIB = exp[l .7201+1 .1859Ln(DBH)+0.8559Ln(THT)] = Ln(SIB) = 0066 Relative variation: 0.94 to 1.07 where
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78 VOB = cubic foot volume outside bark VIB = cubic foot volume inside bark SOB = bole surface area outside bark SIB = bole surface area inside bark exp = exponential Ln = natural logarithm. Cubic volume and bole surface area outside bark by DBH and total height are given in Tables 6 and 7. Tree Form Equations Substituting the cubic volume and surface area equations into Grosenbaugh's (1966) four ratios involving solids of revolution yields the following relationship between tree form, diameter at breast height and total height for the sample trees used in this study: Outside Bark Qq = 0.9549(DBH 2.196 THT ,0.250) 1/^ A = o 0.9251 DBH 1.138 THT 0.164 Qq/Aq = 0.9857 DBH 1 .059 THT 0.087 0.927 DBH ,0.079 THT 0.076
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79 ON o ON U1 00 o CO u1 o u1 o u1 u1 O ui iri o <3u1 O ui CM CO S CO /s m to z 41 j: M Â•H 0) a: o H CM CO CO <3N CNI u1 ro CNI ON u1 u^ o U^ O ON aO UI p^ UI 00 p~. o(N NO CM CO UI \o CO \0 ON CM u1 ON CM ON CM ui CO 00 CM UI 3NO UI u1 ^O ON U1 00 CM CO
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80 to (U u CO >> 3 U 3 j: u I 4J O c CO CO u o ll CO n Â•H CO 4J 3 O CO (U li CO !U CJ CO 141 il 3 CO o o H XI CO H 33 O H ON o in o 00 o u1 o U1 in o m in S3o <3CO X CO PQ CO Q rH 00 o CM m a\ CM x> 1^ 00 CM 00 o ON x> m in 00 in 00 CSl x> in o in in CO CM ON X) x> o X XI CM CM ON in in CM CJN ON o CM X) CM CM CM CM 00 CM 00 00 CM 00 00 00 X) x> x> CM 00 CO 1 Â— 1 X) o ON in O o CM X) CO in X) CO 00 ON ON vO CM 00 00 m CO CO O in o 3in CO ON CO ON in Â•. X CM in ON X) 00 X5 00 CM 00 r~ 00 in 00 p^ o CM ON CO ain X) 00 O CM in X3 00 rH o o o 00 00 o o o P^ 00 CM in C3N in X) ON CO o P. x> P^ 00 ON o CM X 00 Â•x ON CM CO x> 00 ON o CO in ON CNJ X o\ 00 CM o m CO x> m o X in ps x> r CO ON ON O CTN CO ON CO O x> CO CM CJN 00 ON P. CO X5 ON CM 00 in CO p~ in 00 CM m CO CO p^ CM in CO m P~. CO CO H 00 00 m p^ o CO m CM Â—I X 1^ Ov X5 P00 00 o o 00 p^ ON ON ON ON o 00 in ON m CO CO CO Â• Â• Â• Â• Â• Â• in CM o 00 X5 CM CO in x> ON O o x> in 00 in CO CNl CO Â• Â• CM ON x> CO CN CO CM Â— ( o ON ON CO Â• Â• Â• o in CM CM CO P^ in CM CM CM Â• Â• CM CM M m X3 p in in in in x> H in X o CM H Â— 1 Â• Â— 00 Â—I CM CO u (Si Â§ o u T3 e CO 10 u o 1.1 S.I w a c T) U CO CO c CO CO O) CO 4J 4J a CO CO 0) TD o u ON >N X U ON 4J Â• qj o s O II u a XI T3 CM o a c U rH 3 to pa 2 0) a cs
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81 Inside Bark A J = 0.6839 DBH^^^^ ^^^0.144 Qj/Aj = 0.6908 DBH^^^^ ^^^^0.063 Fj. = 0.9880 DBH'^^^ r^^^0.081. Table 8 compares average tree form, inside and outside bark, by diameter and total height. Parameter Estimation The growth model for diameter, total height, cubic volume and surface area was fitted to a sample of 2600 crosssectional and timeseries observations using a 3SLS procedure. Although the thirdstage estimates are the parameters of interest, the results of the first and second stages will also be discussed. First Stage Estimates In the first stage, estimates are obtained for the endogenous variables (diameter and total height) which appear as explanatory ones in the volume and surface area equations. This is accomplished by regressing all predetermined variables on diameter and height using OLS. The estimated equations represent the unrestricted reduced form parameter estimates for diameter and total height. The predicted values for diameter and total height are uncorrelated with the disturbances associated with the cubic volume and surface area equations.
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82 Table 8. Comparison of tree form inside (Fj) and outside (F ) bark by diameter and total height. Total Height Uutl Inside Bark (Fj) Outside Bark (F ) o /U yu 50 70 90 5 81 79 77 79 77 75 6 82 80 78 80 78 76 7 83 81 79 81 79 77 8 84 81 80 82 80 78 9 84 82 80 82 80 79 10 85 83 81 83 81 79 11 86 83 82 84 82 80 12 86 84 82 84 82 81 13 87 84 83 85 83 81 14 87 85 83 85 83 82 15 87 85 83 86 84 82
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83 Second Stage Estimates In the second stage, maximum likelihood estimates of the structural parameter vector and variancecovariance matrix are obtained based on the assumptions regarding the crosssectional and timeseries characteristics of the disturbances. The endogenous explanatory variables are replaced by the predicted values for diameter and total height, which are obtained from the first stage. The GLS procedure was used to estimate the structural parameters of the growth model for the second stage. The variancecovariance matrix was estimated iteratively. Because of the large data set (Nl=2600) (requiring excessive time and space to invert a 2600 square matrix) the total data set (Nl+N2=5199) was divided into 52 groups with approximately 100 observations per group. A random sample of 15 groups was selected to obtain a maximum likelihood estimate of p. Searching over all values of p within the interval 0 to 1 in increments of O.I, a unique set of structural parameters and model variances was obtained for each equation and sample group. The ML estimate of p, and the corresponding model variance and concentrated likelihood function are given in Tables 9 through 12. The parameter p is the ratio of the crosssectional and timeseries error component to the total model variance. All ML estimates of p for the four equations and 15 samples were zero. A p value of zero implies that the model variance arises from random sampling error with no crosssectional or timeseries error component. Table 13 shows the absolute value of the concentrated likelihood function for the four equations over the range of p. Table 14 depicts the relationship between the error sum of squares and the ML estimate of p. The error sum of squares
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84 increases exponentially as p approaches one, while the concentrated maximum likelihood function decreases in a somewhat linear fashion. The estimated structural parameters and model variance are very stable near the region of the ML value of p. The changes in the estimated parameters for the diameter equation are given in Table 15 for selected values of p. Although the ML estimator has many desirable properties, there have been questions concerning its use in the variancecomponents models (Behr 1981). Monte Carlo simulations have shown that the estimated p values were zero when the true value was actually larger. It was argued that the serial nature of the exogenous variables interacted strongly with the autoregressive nature of the problem under study, causing problems in estimating p. This was not considered a problem in this study Third Stage Estimates In the third stage an asymptotically efficient estimate of the growth model is obtained by regressing the four equations together as a system. Since the ML estimate of p was zero, the standard 3SLS procedure was used to obtain the third stage estimates of the parameters. Since the disturbances from the four equations are contemporaneously correlated with variancecovariance matrix Q, and since a is not known, it is estimacea from the second stage using data set Nl. In the second stage each equation is transformed in accordance with the GLS procedure when Q, is unknown.
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85 Table 9. Grosssectional timeseries adjustments for diameter at breast height RUN OBS P CONLK^'^ RSO VAR 1 1 iUU U 1 .329 0.974 0.013 1 .196 1 fif> LXJyJ Q U 1 O "7 O 1.0/0 0.964 0.019 1.690 J 1 nn Q U I mill 0 .965 0.021 1 .909 4 1 nn Q A U 0 hi ~l lm"tll 0.956 0.024 2.229 J 1 nn Q A u 1 n o 1 1 .yol 0.965 0.020 1 .783 5 1 nn q U o n A o 2.943 0.029 2.612 1 1 nn Q A 0 4 .073 0.920 0.041 3.666 H u 1 nn Q y A 0 2 .859 0.934 0.029 2.573 Q 1 nn y A u 3.518 0.935 0.035 3.167 1 nn y 0 1 .405 0.971 0.014 1.265 11 100 9 0 4.293 0.935 0.043 3 .863 12 100 9 0 1.413 0.974 0.014 1 .272 13 100 9 0 2.800 0.937 0.028 2.520 14 100 9 0 1.860 0.958 0.019 1.674 15 100 9 0 4.191 0.909 0.042 3.772 1/ Number of time periods 2/ Â— Concentrated likelihood value 3/ Â— Error sum of squares
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86 Table 10. Crosssectional timeseries adjustments for total tree height RUN DBS p CONLKÂ— ''^ RSQ VAR ^qq3/ CiO ^Â— 1 100 9 0 1.031 0.985 0.010 0.928 I 100 9 U 0.845 0.988 0.008 0.761 3 100 9 0 1.353 0.981 0.014 1.218 4 100 9 0 1.373 0.982 0.014 1.235 5 100 9 0 1 .096 0.985 0.011 0.986 6 100 9 0 3.412 0.947 0.034 3.071 7 100 9 0 3.151 0.953 0.032 2.836 8 100 9 0 0.798 0.988 0.008 0.718 9 100 9 0 3.208 0.955 0.032 2.887 10 100 9 0 0.833 0.987 0.008 0.750 11 100 9 0 3.342 0.956 nn J uuo 12 100 9 0 1 .230 0 .983 1 1 n9 13 100 9 0 0.968 0.985 0.010 0.871 14 100 9 0 0.914 0.985 0.009 0.823 15 100 9 0 4.178 0.930 0.042 3.760 1/ Number of time periods 2/ Â— Concentrated likelihood estimate 3/ Â— Error sum of squares
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87 Table 11. Crosssectional timeseries adjustments for cubic foot volume RUN OBS TL/ p CONLK^'^ RSQ VAR poc3/ \U \J O 1 1 1 r\f\ 100 9 0 3.494 0.986 0.035 3.144 L 100 9 0 4.626 0.983 0.046 4.164 100 9 0 9.248 0.968 0.092 8.323 <* lUU 0 6.907 0.974 0.069 6.216 c 100 9 0 8.283 0.970 0.083 7.455 c. 0 lOU 9 0 13.562 0.945 0.136 12.205 7 1 nn iUU /> y 0 20.176 0.921 0.202 18.158 Q O 100 9 0 5.391 0.974 0.054 4.852 Q 100 9 0 17.360 0.933 0.174 15.624 iU 100 9 0 4.718 0.980 0.047 4.246 11 100 9 0 21 .632 0.934 0.216 12 100 9 0 6.026 0.978 0.060 Â• *+ ^ J 13 100 9 0 5.347 0.976 0.053 4.812 14 100 9 0 5.327 0.974 0.053 4.795 15 100 9 0 17.511 0.921 0.175 15.760 11 Number of time periods 2/ Â— Concentrated likelihood estimate 3/ Â— Error sum of squares
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88 Table 12. CrossSectional timeseries adjustments for bole surface area RUN OBS P RSQ VAR 1 100 9 0 1.459 0.989 0.015 1 .313 2 100 9 0 1.829 0.988 0.017 1.556 3 100 9 0 3.747 0.975 0.037 3.372 4 100 9 0 3.031 0.979 0.030 2.728 5 100 9 0 3.302 0.978 0.033 2.972 6 100 9 0 6.672 0.949 0.067 6.005 7 100 9 0 9.098 0.933 0.091 8.188 8 100 9 0 2.604 0.978 0.026 2.344 9 100 9 0 8.880 0.937 0.089 7.992 10 100 9 0 1.825 0.985 0.018 1.643 11 100 9 0 9.181 0.946 0.092 8.263 12 100 9 0 2.346 0.984 0.023 2.111 13 100 9 0 2.134 0.982 0.021 1 .920 14 100 9 0 2.328 0.980 0.023 2.096 15 100 9 0 8.659 0.927 0.087 7.793 1/ Number of time periods Â— Concentrated likelihood estimate Error sum of squares
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89 Table 13. Concentrated maximum likelihood values for DBH (D), total tree height (H), cubic volume (V), and surface area (S). Equation P value D H V S 0 1.9811 1 .0955 8.2831 3 ini ft .1 8.1684 7.2392 14.3705 Â• Z 11 .6820 10.7407 17.8556 12.9275 .3 14.3836 13.4356 20.5429 15.6189 .4 16.7478 15.7965 22.8980 17.9766 .5 19.0006 18.0474 25.1439 20.2247 .6 21.3093 20.3552 27.4470 22.5300 .7 23.8691 22.9149 30.0016 25.0873 .8 27.0328 26.0792 33.1602 28.2491 .9 31.8420 30.8897 37.9641 33.0570
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90 Table 14. Error sum of squares for DBh (D), total tree height (H), cubic volume (V), and surface area (S). Equation p value D H V S 0 1 .7830 0.9860 7.4548 2.9717 .1 1.9641 1.0349 8.1662 3.2283 .2 2.2014 1.1418 9.1467 3.6026 .3 2.5099 1.2911 10.4290 4.0982 .4 2.9228 1 .4958 12.1481 4.7659 .5 3.5017 1.7859 14.5597 5.7051 .6 4.3703 2.2236 18.1800 7.1170 .7 5.8178 2.9552 24.2150 9.4724 .8 8.7116 4.4204 36.2850 14.1850 .9 17.3893 8.8187 72.4880 28.3240
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91 Table 15. Comparison of the estimated coefficients for the DBH equation at three levels of p. p value COEFFICIENT .0 .5 .9 INTERCEPT DBH LAGGED THT LAGGED PAR PAR** 2 ALAG/AGE AGE (1/AGE)**2 COMP. DBH MICROS ITE 1.7216 0.8446 0.3196 0.3201 0.0950 2.3077 0.3117 58.2769 0.0021 0.3624 3.7653 0.8338 0.3259 0.3032 0.0885 5.1239 0.0581 70.0511 0.0042 0.0342 4.4884 0.8278 0.3291 0.2783 0.0815 6.3334 0.0791 74.1267 0.0096 0.0339
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92 The third stage parameter estimates along with the second stage and the unrestricted OLS estimates of the structural equations are given in Tables 16 through 19. The model has a reasonable level of explanatory power with a weighted r2 value for the system of approximately 93 percent Reduced Form Equations The reduced form of the system of equations describes the endogenous variables in terms of the current exogenous variables and lagged endogenous and exogenous ones. The coefficients associated with the reduced form equations are given in Table 20 for the individual tree growth model. The reduced form coefficients for the diameter and height equations are identical to the coefficients in the structural equations. This is due to the fact that the two equations do not contain endogenous variables as explanatory variables. Final Form Equations The final form describes the endogenous variables in terms of the current and lagged exogenous variables. If we let y^. be the vector of endogenous variables in period t, and the vector of exogenous variables for the same time period, the reduced form can be expressed in the following form: ^t = ^o + Dl\1 + h\ + ^3\l + ^t where d^ is a L x 1 element vector of constant terms in the reduced form
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93 Table 16. Estimates of the model parameters and standard errors for the DBH equation by OLS, 2SLS, and 3SLS. Estimate (S.E. x 10^) PARAMETER OLS 2SLS 3SLS R**2 0.937 0.934 0.933 N 2599 2600 2600 TNTFRfFPT O Â• ODH (57.82) (57.89) (49.60) UO ll TOTAL HEIGHT DBH LAGGED 0.903 0.829 0.862 (1.40) (1.43) (1.39) THT LAGGED 0.277 0.328 0.289 \L mOO ) PAR 0.232 0.307 0.296 (2.86) (2.84) (2.82) PAR**9 IT AiX C. U U J / f\ no "5 U .UBj (1 .16) (1 .17) (1 .16) ALAG/AGE 8.047 6 .822 1 1 34fi (71.19) (71.25) (60.93) AGE 0.071 0.066 0.091 (0.45) (0.45) (.40) AGÂ£**2 (1/AGE)**2 89.584 79.830 104.668 (404.49) (403.37) (339.38: (1ALAG/AGE)*PAR COMP. 0.020 0.020 0.029 (0.75) (0.78) (.058) COMP. /SPACING DBH MICROS ITE 0.035 0.013 0.029 (1.26) (1.28) (0.70)
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94 Table 17. Estimates of the model parameters and standard errors for the total height equation of OLS, 2SLS, and 3SLS. Estimate (S.E. x 10^) PARAMETER OLS 2SLS 3SLS R**2 0.959 0.964 0 .964 2i)99 ^ f r\r\ 2600 2600 INTERCEPT 1.305 1.387 1.430 ^ 16 .99 ; (15.47) (14 .20) DBH TOTAL HEIGHT DBH LAGGED 0.131 0.134 0.157 (1.26) (1.20) (1.13) THT LAGGED 0.744 0.725 0.701 (1.75) (1.58) (1.54) PAR 0.079 0.057 0.114 (3.29) (3.03) (2.85) PAR**2 0.017 0.013 0.019 (1.14) (1.05) (1.04) ALAG/AGE 1.775 2.008 2.199 (32 .75) (30.20) (25.85) AGE 0.044 0.052 0.067 (1.62) (1.49) (1.25) AGE**2 0.001 0.001 0.001 (0.04) (0.04) (0.03) (1/AGE)**2 (1 ALAG/AGE)* PAR 0.140 0.070 0.244 (7.70) (7.10) (5.42) COMP. COMP. /SPACING 0.123 0.104 0.154 (6.25) (5.50) (4.74) DBH MICROS ITE
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95 Table 18. Estimates of the model parameters and standard errors for the cubic volume equation by OLS, 2SLS and 3SLS. Estimate (S.E. x lO^) PARAMETER OLS 2SLS 3SLS R**2 N INTERCEPT DBH TOTAL HEIGHT DBH LAGGED THT lAGGED PAR PAR**2 AGE ALAG/AGE AGE**2 1/AGE**2 (1 ALAG/AGE)* PAR COMP. COMP./SP DBH MICROS ITE 0.996 2599 0.790 (30.67) 1.338 (00.001) 0.744 (1.21) 0.119 (2.09) 0.029 (0.72) 0.023 (0.25) 2.370 (37.96) 28.048 (197.41) 0.138 (4.86) 0.308 (4.01) 0.993 2600 2.924 (38.07) 1.063 (1.57) 1.135 (2.37) 0.185 (2.72) 0.053 ().90) 0.032 (0.31) 4.636 (46.68) 36.817 (237.65) 0.189 (5.91) 0.306 (4.75) 0.993 2600 1.428 (16.46) 1.088 (1.49) 1.096 (2.28) 0.213 (2.66) 0.063 (0.90) 0.068 (0.21) 0.756 (19.99) 9.074 (89.84) 0.136 (5.55) 0.146 (4.30)
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96 Table 19. Estimates of the model parameters and standard errors for the surface area equation by OLS, 2SLS and 3SLS. Estimate (S.E. x lO^) PARAMETER OLS 2SLS 3SLS R**2 0.998 0.997 0.997 N 2599 2600 2600 INTERCEPT 2.052 1.728 1.584 (3.27) (4.41) (4.19) DBH 0.780 0.649 0.661 (0.42) (0.75) (0.72) TOTAL HEIGHT 0.734 0.924 0.903 (0.58) (1.14) (1.09) DBH LAGGED THT LA.GGED PAR 0.040 0.106 0.116 (0.996) (1.31) (1.28) PAR**2 0.010 0.029 0.033 (0.34) (0.43) (0.43) AGE 0.076 0.107 0.048 (0.41) (0.51) (0.21) ALAG/AGE 2.162 2.136 1.373 (6.88) (8.28) (5.45) AGE**2 0.001 0.002 0.0002 (0.01) (0.01) (0.01) 1/AGE**2 (1 ALAG/AGE) PAR 0.033 0.117 0.096 (2.33) (2.84) (2.70) COMP. COMP./SP 0.113 0.102 0.043 (1.85) (2.22) (2.08) DBH MICROS ITE
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97 Table 20. Reduced form coefficients for the DBH (D), total height (H), cubic volume (V), and surface area (S) equations. EQUATION VARIABLE D H V INTERCEPT 8.864 1.430 9.519 6.150 DBH LAGGED 0.862 0.157 1.111 0.711 THT LAGGED 0.289 0.701 1.083 0.824 PAR 0.296 0.114 0.411 0.209 PAR**2 0.079 0.019 0.127 0.067 ALAG/AGE 11.346 2.199 10.708 6.884 AGE 0.091 0.067 0.094 0.047 AGE**2 0.0 0.001 0.002 0.001 1/AGE**2 104.668 0.0 104.957 69.167 (1 ALAG/AGE)* PAR 0.0 0.244 0.131 0.124 COMP. 0.029 0.0 0.031 0.019 COMP. /SPACING 0.0 0.154 0.315 0.182 DBH MICROS ITE 0.029 0.0 0.031 0.190
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98 while Dj^ and are matrices of multiplicative reduced form coefficients (D^ is a L x L square matrix and D2 and D3 are both of the same order, L x K) and e* is the L element vector of reduced form disturbances in time period t. The final form of the system is obtained by eliminating Yj._^ from the right hand side of the equations by substituting y^ lagged one period. Upon substitution, the limit of approaches : y, = (IDi)ld^ + D2X^ + Z U '^(D D + D )X + Z d\ e* t=l ^ t=0 t The sequence Z ^j^^^l ^^^'^^'^S^^ to zero while the sequence t=o 00 Z ^^1*^2"'" ^3^ converges to (ID^"^ (D2 + D3). The latter sequence is referred to as the "total multipliers" of the system. The elements of D2 describe the immediate impact of a unit change in the exogenous variables and are known as "impact multipliers". The coefficients 00 associated with the sequence Z ^ describe the effects t=l during a latter period and are referred to as "interim multipliers". An approximation of the limiting distribution of the final form may be expressed as follows: y^ = (I op ^d^ + D2X^. + (I D^)"^ (D^D2 + D3)X^_ 1 Â• The final form coefficients of the system are given in Table 21. Figures 16 through 18 depict the relative change in the interim multipliers over time.
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99 Table 21. Final form coefficients for the DBH (D), total height (H), cubic volume (V), and surface area (S) equations. EQUATION VAR TART u U Â£1 \r V o o INTERCEPT 562.695 300.102 940.395 641.36 PAR 13.999 7.728 23.505 16.116 PAR**2 4.525 2.439 7.540 5.162 ALAG/AGE 11.346 2.199 10.708 6.884 AGE 0.091 0.067 0.094 0.047 AGE**2 0.0 0.001 0.002 0.001 A/AGE**2 104.668 0.0 104.957 69.167 (1ALAG/AGE)*PAR 0.0 0.244 0.131 0.124 COMP. 2.152 1.130 3.582 2.443 COMP./ SPACING 11.178 5.353 17.896 12.179 DBH MICROS ITE 0.029 0.0 0.031 0.190
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100 Figure 16, Relative change in the interim multipliers associated with the exogenous variables 1/Age**2 and ( 1Alag/Age) PAR.
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101 Figure 17. Relative change in the interim multipliers associated with the exogenous variables PAR and Comp.
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102 Figure 18. Relative change in the interim multipliers associated with the exogenous variables Alag/Age and Age.
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103 Model Validation Regression analysis has widely been used In data analysis and the development of empirical models. Once an adequate model Is found, one proceeds to use the model for prediction purposes, process control, or investigation of the mechanisms involved in the developmental or growth processes being studied. For whatever purpose a model is used, some checks on its validity should be made (Brand and Holdaway 1983, Luppold 1982, Reynolds et al. 1981, Snee 1977). A few useful procedures to check the validity of a regression model Include: (a) Comparison of the signs of the estimated coefficients with the theoretical expectations, (b) Comparison of the results with theoretical models and simulated data, and (c) Use of an Independent data set to obtain a measure of the prediction accuracy of the model. Evaluation of the Coefficients An examination of the model coefficients and associated variance inflation factors (VIF) provides a helpful insight into the validity of a model. Coefficients with wrong signs are indications of an inappropriate or poorly estimated model and/or lack of understanding of the system being modeled. The VIF's, the diagonal elements of the inverse of the correlation matrix, measure the collective effect of the correlations among the independent variables on the variance of the estimated coefficients. In general, a VIF larger than ten is an indication that the associated coefficient is likely to be poorly
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104 estimated because of correlations among independent variables. Variables that are highly correlated, generally produce coefficients that are too large and often have wrong signs. Upon examination of the estimated coefficients for the 3SLS model only two coefficients in the equation for diameters (D) had a wrong sign. The sign of the coefficients indicates that competition has a positive effect on diameters while an increase in microsite had a negative impact on diameters. Competition had a negative effect on height, volume and surface area. The relative spacing of a given tree had a positive effect on diameters, cubic volume and surface areas and a negative impact on heights. Both diameters and heights lagged one period (i.e. two years) had positive effects on diameters and heights. Diameter and total tree height had a positive correlation with cubic volumes and surface areas. Correlation coefficients among regression coefficients for the four equations are given in Tables 22 through 25; their respective VIF factors are presented in Table 26. The largest VIF factors were associated with the intercept and exogenous variables that included age while the smallest VIF factors were associated with competition and microsite. All but six of the VIF factors were larger than ten, which indicates the coefficients may be poorly estimated because of the existing correlation among independent variables. The OLS and 3SLS estimates were consistent with respect to the signs of the coefficients for the diameter and height equations. Differences were observed in the magnitude of the coefficients estimated by the two procedures. In the volume and surface area equations there were inconsistencies in the signs as well as differences in magnitude of
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105 the estimated coefficients. The inconsistencies do not indicate an error in the two estimation procedures but rather indicate differences in the underlying assumptions of the two procedures. Estimation of the regression coefficients depends upon the assumptions made about the variancecovariance matrix of the residuals. If the variancecovariance matrix is misspecif ied, the resulting estimates are unbiased but not efficient (Amemiya 1966). It should be noted that the inconsistencies in signs between the OLS and 3SLS procedures occurred in the two equations which contain current endogenous variables as explanatory variables Goodnessoff it Statistics A preferred method of model validation is to collect a new set of data which can be used to measure the prediction accuracy of the model (Reynolds et al. 1981, Snee 1977). The validity of the mathematical and theoretical assumptions used in developing and estimating the coefficients of the model is less open to question when the model gives precise estimates of the new data. In many instances the collection of new data is not practical or possible. An alternative procedure is to split the data into two independent data sets. The first data set is used to estimate the model coefficients, while the second data set is used to verify the validity of the model (Theil 1971, Snee 1977). The OLS and 3SLS procedures were evaluated by calculating various goodnessoff it statistics based on the estimated coefficients from each procedure. Each procedure was applied to both data sets which are assumed to be independent. The goodnessoff it statistics included a fit index (r2), mean difference (D) root mean square error (RMSE), percent mean difference (D%), and percent root mean square error (RMSE%).
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106
PAGE 119
107 Â• z o < CO I o c o Â•H U (d 9 cr 0) Â•H J3 CO u o u o u 3^ K ai < Pi < SQ Q H to a^ o XI 00 o n >^ CM o sr CM 1Â—4
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108 Â• z z u o Â— ( o CM CM o ro CM CN o m o O 1 O 1 o 1 o o 1 O o 1 o 1 o OS ON ir, CM o 00 ON O o a^ CM CM o o ON o o o 1 O 1 O o o 1 O o 1 o o iH vO o m o ON CM CN CM o CM ON 00 Â•>r CN 00 O IH CM O O o 1 o o 1 o o 1 uO 00 CN o CN o CM o o o Â• Â• Â• Â• o 1 o 1 o o 1 o 1 CO CO CO CM o o rsi 00 o Â• Â• Â• Â• Â• o O 1 o o 1 CN
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109 o < CO CM HC K Cd <: d o Â•H CM CM ul r> o o o o o CM cn o o 00 o O O 1 o o o 1 o 1
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110 Table 26. Variance inflation factors for the 3SLS growth model. VARIABLE D H V S INTERCEPT 8443.0 DBH THT DBH LAGGED 67.1 THT LAGGED 87.1 PAR 36.2 PAR**2 18.4 ALAG/AGE 5402.6 AGE 51.4 AGE**2 1/AGE**2 789.7 RIP COMP. 2.4 COMP./SP DBH MICROS ITE 1.1 484.5 1049.1 346.0 18.8 23.5 58.5 78.6 30.4 38.3 31.5 24.1 39.4 13.5 12.9 14.3 1058.8 1196.7 764.4 764.5 37.9 985.9 210.7 9.4 274.4 12.4 203.0 13.2 3.2 3.1 3.6
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Ill Table 27 provides information on the performance of the four equations in predicting growth. The 3SLS procedure consistently performed better than the OLS procedure in estimating volumes and surface areas. The average difference for the 3SLS procedure was 0.07 cubic feet for volume and 0.13 square feet for surface area while the average error for the OLS procedure was 0.13 cubic feet and 0.17 square feet. On the average the mean difference for diameter and total height was less for the 3SLS procedure than for the OLS. The root mean square error, RMSE gives more weight to large deviations from the observed value more than does average differences. There was no noticable difference in the RMSE for the OLS and 3SLS procedure in the diameter and height equations. The RMSE was larger for OLS compared to the 3SLS procedure in estimating volume while the opposite was observed for the surface area equation. Percentage of mean differences, D%, and percentages of root mean square error, RMSE%, depict the error as a proportion of the observed values. In all four equations the percentage of mean differences and percentages of root mean square error were larger in absolute value for the 3SLS procedure than for OLS The goodnessoffit statistics indicate that all equations explained the variation in the data quite well using either the OLS or the 3SLS procedure.
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112 Table 27 Evaluation of cubic volume, prediction equations for DBH, and surface area estimated by total height, OLS and 3SLS. DATA ESTIMATION FIT STATISTICS SETÂ— ^ PROCEDURE r2 D RMSE D% RMSE% DBH EQUATIONS (IN.) 1 OLS 0.948 0.03 0.50 0.91 24. 10 1 3SLS 0.946 0.01 0.50 1.39 24.70 2 OLS 0.945 0.01 0.51 1.24 17.46 2 3SLS 0.943 0.01 0.52 1.74 18.00 TOTAL HEIGHT EQUATION (FT.) 1 OLS 0.954 0.22 3.52 0.29 13.16 1 3SLS 0.954 0.04 3.52 0.87 13 .47 2 OLS 0.945 0.06 3.84 0.96 14.02 2 3SLS 0.945 0.13 3.85 1.52 14.33 TOTAL VOLUME EQUATION (CU.FT.) 1 OLS 0.978 0.13 0 .74 0.41 18.28 1 3SLS 0.981 0.08 0.68 0.69 19.50 2 OLS 0.980 0.12 0.69 0.31 12.34 2 3SLS 0.982 0.06 0.65 0.60 14.32 TOTAL SURFACE AREA EQUATION (SQ.FT .) 1 OLS 0.996 0.18 1.73 0.02 5.80 1 3SLS 0.995 0.15 1.99 0.06 6.35 2 OLS 0.997 0.15 1.65 0.00 4.91 2 3SLS 0.995 0.11 1.99 0.06 5.72 '^ Data set 1 (N = 2600) was used to estimate the 3SLS coefficients and data set 2 (N = 2599) was used to estimate the OLS coefficients 2/ I2=l Y.]2/j:rY. Y.]2; D = i 1^ n RMSE = / n 1 D% = = ZfY.Y.l/Y. ; RMSE% = /; / n C((Y.Y.)/Y.)2
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113 Cumulative Distribution of Errors Knowledge of the accuracy and the efficiency of a model is often not enough. There is also a need to know the variability or precision of the estimates. Precision is measured by the standard deviation of the errors. One way to demonstrate the variability of a model is to graph the cumulative distribution of errors (Brand and Holdaway 1983). The cumulative distribution of errors for the diameter (DBH), total height (THT), cubic volume (VOL), and surface area (SUR) are depicted in Figures 19 through 22, respectively. The cumulative distribution of errors may be used to assess the likelihood of underestimating or overestimating tree variables such as diameters or total heights by a certain amount. For example, Figure 19 shows that 97.8 percent (99 minus 1.2 percent) of the errors associated with estimating diameters were within one inch and 87 percent of the errors were within a half inch. All cumulative error distributions were skewed to the right which suggests that there is a higher likelihood of underestimating than overestimating diameters. For example, the likelihood of underestimating diameters by more than a half inch is ten percent while there is only a three percent likelihood of overestimating diameters by more than a half inch. Similar interpretations may be made for the total height, cubic volume and surface area equations.
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114 DBH ERROR (IN.) Figure 19. Cumulative distribution of errors associated with the diameter breast height equation.
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115 Figure 20. Cumulative distibution of errors associated with the total height equation.
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116 100 UJ O (r LU Q. UJ > < I o 10 5 0 5 10 TOTAL VOLUME ERROR (CU. FT) Figure 21. Cumulative distribution of errors associated with the cubic volume equation.
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117 Figure 22. Cumulative distribution of errors associated with the surface area equation.
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118 Correlation Among Equations Estimation of the correlations of errors across equations for the 3SLS procedure is given in Table 28. Ttie highest correlation occurred between the volume and surface area equations (r = 0.99). This high correlation is due to the geometric relationship between surface area and cubic volume. The errors associated with the diameter equation were positively correlated with total height, cubic volume and surface area, while the errors for the total height equation were negatively correlated with cubic volume and surface area. The negative correlation associated with the height equation does not mean that total height is negatively correlated with cubic volume and surface area. Rather the height equation overestimates total height when the volume and surface area equations underestimate volume or surface area and viceversa. Efficiency of the OLS and 3SLS Procedures In comparing goodnessoff it statistics for the OLS and 3SLS procedure it is rather difficult to infer that the 3SLS Is more precise than the OLS procedure in predicting diameters, total heights, cubic volumes, and surface areas. Even though both procedures (OLS and 3SLS) provide unbiased estimates of the coefficients this does not mean that the variances associated with the coefficients are efficient with respect to the GaussMarkov Theorem.
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119 Table 28. Correlation across equations. Equation Equation DBH THT VOL SUR DBH 1.00 0.65 0.34 0.31 THT 1.00 0.26 0.29 VOL 1.00 0.99 SUR 1.00 4
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120 In addition to having a model that is accurate it is also desirable that the model be efficient with respect to the estimation of the variancecovariance matrix. Since the variances associated with the estimated coefficients in the OLS and 3SLS models are independent of each other, the ratio of variances for the OLS and 3SLS procedures can be taken as an Fstatistic with = 2589 and V2 = 2590 degrees of freedom, respectively. The null hypothesis Ho<^OLS(e) ^3SLS(0)' rejected when ^ ^ij ^ ^la/2^i^2 ^^^^^ ^a/2^i^2 ^ upper a/2 and lower la/2 percentage points of the Fdistribution, with v^ and V2 degrees of freedom, and F^^ is the ratio of variances associated with the jth coefficient in the ith equation. The relationship between the upper and lower tail percentage points is given by: I la/2v,vÂ„ F 1 2 a/lv^v^ Since there are no Ftables available for large sample sizes (n > 500) a conservative estimate of the rejection region (v=500) was used for hypothesis testing. In the diameter (D) and height (H) equation all but four of the varianceratios were significantly greater than one at the 0.10 level (Table 29). If Ftables were available for large sample sizes (n > 500] at least two of the nonsignificant varianceratios would be significantly greater than one since F^/2v v ^PP^^^^hes one for all levels of a as n goes to infinity. The results indicate that 3SLS is more efficient in estimating the coefficients of the diameter and height equations than OLS.
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121 In the volume (V) and surface area (S) equations all variance ratios were significantly different from one at the 0.02 level; six and seven of the variance ratios, respectively, were significantly less than one. At first glance it would seem that OLS in more efficient than 3SLS in estimating certain parameters, but in fact the results are due to a specification error in the variancecovariance matrix. The OLS procedure does not take into consideration the correlation of errors among equations nor the presence of endogenous variables as explanatory ones I
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122 Table 29. Efficiency of the OLS and 3SLS estimation procedure for DBH (D), total height (H), cubic volume (V), and surface area (S) equations PARAMETER ^OLS / 3SLS D H V S INTERCEPT 1.36** 1.43** 3.47** 0.61** DBH Â— Â— 0.35** 0.35** TOTAL HEIGHT Â— Â— 0.28** 0.28** DBH LAGGED 1.02 1.24** Â— Â— THT LAGGED 1.08 1.29** Â— Â— Â— PAR 1.02 1.34** 0.62** 0.60** PAR**2 1 .00 1.21* 0.64** 0.63** ALA G/ AGE 1.36** 1.60** 3.61** 1.59** AGE 1.31** 1 .69** 1.45** 3.85** AGE**2 Â— 1.70** Â— 4.84** 1/AGE**2 1.42** 4.83** (1ALAG/AGE)*PAR 2.02** 0.77** 0.74** COMP. 1.69** COMP./SP 1.74** 0.87** 0.80** DBH MICROS ITE 3.22** significantly different (a=0.10) according to a twotailed Ftest; v^= = 500. ** = significantly different (a=0.02) according to a twotailed Ftest; v^= = 500.
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SUMMARY AND CONCLUSIONS A growth model is presented to depict the effects of competition and available growing space on the growth and development of individual slash pine trees over time. The model predicts the cumulative growth of individual tree diameters, total height, total cubic volume, and total bole surface area using a m.odified form of the Verhulst growth function. Two estimation procedures, (1) Ordinary Least Squares (OLS) and (2) Three Stage Least Squares (3SLS) are considered under two assumptions concerning the residual errors (contemporaneous covariance and serial correlation) The OLS procedure assumes that the variables in the model simultaneously satisfy certain assumptions regarding the equality of variances among regressors as well as the independence of the explanatory variables from the residual errors. Likewise, the procedure does not take into consideration the interrelationship among equations. The 3SLS procedure, on the other hand, assumes a correlation among equations (contemporaneous covariance) and simultaneously estimates the coefficients associated with the system of equations. To assess the effects of size and distribution of vacancies on individual tree growth, Hopkins' modeified coefficient of aggregation was used to analyze the spatial distribution of healthy and dead trees four years after planting. This age was selected because most of the mortality had occurred by this time. Healthy trees were regularly 123
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124 distributed throughout the study area while dead trees resembled a random spatial pattern. The spatial distribution of healthy and dead trees varied significantly according to spacing and plot location. The results of the spatial analysis indicate no adverse affects on the growth and development of the remaing trees in the study area due to mortality. Often, in pooling data across a wide range of planting densities, it is difficult, if not impossible to capture all of the important crosssectional effects. Furthermore, data obtained by repeated measurements will contain correlated errors. Unequal variances may also be present in the data. To test the assumptions regarding the crosssectional and timeseries error component of the growth model, Maximum Likelihood (ML) estimates of the parameter vector and variancecovariance matrix were obtained by searching over all values of p (0 < p < 1) and choosing the corresponding parameter set which maximizes the likelihood function. Here, p is the ratio of the crosssectional and timeseries error component to the total model variance. All ML estimates of p for the four equations were zero, which implies that the model variance arises from random sampling errors and not from a crosssectional or timeseries error component. As a result, the standard 3SLS procedure was used to estimate the parameters associated with the growth model. The growth model was evaluated using data from a spacing study established in 1962. Slash pine seedlings were hand planted in twelve oneacre blocks measuring 2.5 x 4 chains at six different spacings; 5x10, 7x10, 9x10, 11x10, 13x10, and 15x10 feet. Tree measurements were collected on a subsample of trees, from each plot, approximately every
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125 two years starting six years after planting. Measurements included diameter at breast height, merchantable height to a four inch top, tota height, position in the canopy, and number of primary and secondary competing trees. Volumes and surface areas were estimated from regression equations developed from dendrometry data obtained from the study area. The data were split into two sets; the first set (N=2599) was used to identify the structural form of the growth model and obtain OLS estimates of the coefficients, while the second data set (N=2600) was used to estimate the coefficients associated with the structural system of equations using 3SLS. The validity of the two models were assessed by comparing datafit and prediction accuracy. The efficiency of the two procedures were evaluated by taking the ratio of variances for the OLS and 3SLS procedures as an Fstatistic. The signs of the coefficients were also compared with theoretical expectations. The OLS and 3SLS estimates were consistant with respect to the signs of the coefficients for the diameter and height equation. However differences were observed in the magnitude of the coefficients. Two coefficients in the diameter equation had wrong signs. The signs indicate that increasing competition increases diameter growth, while an increase in site decreases diameter growth. In the volume and surface area equations inconsistencies were observed in the signs of the coefficients as well as differences in the magnitude of the estimated coefficients. These inconsistencies do not indicate an error in the estimation procedures, but rather indicate differences in the underlying assumptions regarding the structure of the variancecovariance matrix used in the estimation procedure.
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126 The 3SLS procedure consistently performed better than OLS in estimating volumes and surface areas. For example, the average difference for 3SLS was 0.07 cubic feet for volume and 0.13 square feet for surface area, while the average error for OLS was 0.13 cubic feet and 0.17 square feet. On the average, the mean difference for diameters and total height was less for 3SLS than for OLS. The cumulative distribution of errors were skewed to the right for the four equations; this suggests that there is a higher likelihood of underestimating than overestimating. For example, the likelihood of underestimating total height by three feet is ten percent while there is only a three percent likelihood of overestimating total height by more than three feet. In comparing goodnessoff it statistics for the OLS and 3SLS procedures it is rather difficult to infer that 3SLS is more precise than OLS in estimating diameter, total height, total cubic volume, and total surface area. Even though both procedures yield unbiased estimates of the coefficients, this does not ensure that the estimates are efficient. Results of this study indicate that 3SLS was more efficient than OLS in estimating diameter and total height. In the volume and surface area equations OLS was more efficient than 3SLS in estimating certain parameters. This difference is due to a specification error in the variancecovariance matrix and the presence of endogenous variables as explanatory ones. In conclusion, it is suggested that OLS though quite adequate for estimating model parameters, should not be used in constructing confidence intervals or hypothesis testing with data containing contemporaneous covariances and/or autocorrelated errors. To ensure
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127 asymptotically efficient parameter estimates, 3SLS is a better procedure to simultaneously estimate the coefficients associated with a system of equations.
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FUTURE BUESEARCH An important consideration in stand management decisions is the variability and uncertainty in the growth and development of a given stand over time (Lembersky and Johnson, 1975). Stand management in terms of thinning or harvest option at one point in time influence the alternatives feasible in the future. Random or chance events (e.g. levels of disease infection and mortality) will influence present and future decisions. In the past the development of management strategies for infected slash pine plantations has been based primarily on the level of infestation (hazard rating), level of stocking, and the age of the stand. Two aspects of this problem that have not been addressed pertain to the spatial distribution of the disease and the economic implication of managing infected stands. If the spatial distribution of the disease related mortality is random or uniformly distributed throughout a given stand, the additional growing space would be similar to a random or systematic thinning which may increase the productivity of the stand over time. If the spatial distribution of the mortality is aggregated the residual trees may not be able to utilize the additional growing space and may result in a decrease in the productivity of the stand (Reich 1980). 128
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129 The development of management strategies for infected stands should be evaluated and based on an individual stand basis. Management decisions should not only take into consideration the level of infestation but the economics of managing the stand (e.g. interest rate, management costs, and timber prices), management objectives (pulpwood and/or sawtimber), planting density, site index, and management alternatives (e.g. row thinning, as is, thinning from below, and sanitation thinning). All of these factors play a significant role in determining the optimal rotation age and present net value of a given stand. In recent years, interest has developed in defining and solving sequential decision problems associated with forest management. The process is a multistage or sequential decision problem characterized by a sequence of decisions made chronologically. Following each decision a chance event occurs which will, in turn, influence the next decision (Bubnick et al. 1977). A widely used approach for solving sequential decision problems is dynamic programming for solving joint optimization of the thinning plan and rotation length. Recently the impacts of regeneration cost, initial stocking level, site, quality premiums and variable logging costs on optimal thinning and harvest scheduling have been solved via this approach (Brodie et al. 1978). Most applications of dynamic programming for forest stand analysis have incorporated standlevel growth models which implicitly assume some type of diameter distribution structure. This is an important assumption since a stand's diameter distribution defines both its current value and expected growth.
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130 If an optimal thinning and final harvest scheduling model is to truly assist in stand management, the model should allow for disease infection and growth losses to affect the marginal value growth of the stand. A Dynamic Simulation Model (DSM) incorporating an individual tree growth model can be utilized to develop alternative management strategies correlated to a diseased infection level and its spatial distribution. Such a model can develop growth estimates which incorporate individual tree competition and the effect of individual tree removals (either through cut or mortality) especially as the spatial distribution is varied. Until the effects of various management practices on growth, yield and susceptibility of a stand are quantified, realistic assumptions will have to be made to adjust certain stand growth coefficients accordingly. Depending upon the degree of sophistication of the growth and yield models, they may include such variables as the growth rate of individual trees, site competitive status of trees, one or more random tree components representing the microsite and/or genetic variability, or susceptibility to certain forest pests. The information obtained would not only assist land managers in developing rational and contingency plans, but would also give them an estimate of the expected volume loss for a given set of conditions.
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LITERATURE CITED Allen, H.L., and Duzen, H.W., Jr. 1980. What measure of stand density is best for growth predictions in loblolly pine plantations. In ; Proceedings of the first biennial southern silvicultural research conference. Atlanta, Q^. USDA Gen. Tech. Rep. SO34, Southern For. Exp. Stn., New Orleans, LA. 375p. Amemiya, T. 1966. Specification analysis in the estimation of parameters of a simultaneous equation model with autoregressive residuals. Econometrica 34:283306. Amidon, E.L., and Akin, G.S. 1968. Dynamic programming to determine optimal levels of growing stock. For. Sci. 14:287291. Ashby, E. 1935. Quantitative analysis of vegetation. Ann. Bot. 49: 779802. Bailey, R.L., Pienaar, L.V., Shiver, B.D., and Rheney, J.W. 1982. Stand structure and yield of site prepared slash pine plantations. University of Georgia. Res. Bull. 291. 83p. Balestra, P., and Nerlove, M. 1966. Pooling cross sectional and time series data in the estimation of a dynamic model: The demand for natural gas. Econometrica 34:585612. Behr, R.M. 1981. A simultaneous equation model of futures market trading activity. University of Florida. Ph.D. dissertation. 262p. Bella, I.E. 1971. A new competition model for individual trees. For. Sci. 17:364372. Bennett, F.A. 1963. Growth and yield of slash pine plantations. USDA For. Serv. Res. Pap. SE1 Southeast. For. Exp. Sta., Ashville, NO. 23p. Bennett, F.A. 1971. The role of thinning and some other problems in management of slash pine plantations. USDA For. Serv. Res. Pap. SE86, Southeast. For. Exp. Stn., Ashville, NC. 14p. Bennett, F.A. 1974. Slash pine growth and yield relationships. In ; Symposium on management of young pines. Southeastern Area, State and Private Forestry, Southern and S outheast. For. Exp. Stn., USDA For Serv. 349p. Bormann, F.H., and Likens, G.E. 1981. Pattern and process in a forested ecosystem. SpringerVerlag, Inc., New York. 253p. 131
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132 Boyce, S.G., McClure, J. P., and Sternitzke, H.S. 1975. Biological potential for slash pine ecosystem. USDA. For. Serv. Res. Pap. SE141, Southeast. For. Exp. Stn., Ashville, NC. 29p. Brand, G.J. and Holdaway, M.R. 1983. Users need performance information to evaluate models. J. of For. 81:235237, 254. Brodie, J.S., Adams, D.M., and Kao, C. 1978. Analysis of economic impacts on thinning and rotation for Douglasfir using dynamic programming. For. Sci. 24:513522. Budnick, F.S., Mojena, R., and Vollmann, T.E. 1977. Principles of operation research for management. Irwin, Inc. Chicago, IL. 756p. Campbell, T.E., and Mann, W.F.,Jr. 1974. Growth in a loblolly pine spacing study in southwest Louisiana. In : Symposium on management of young pines. Southeastern Area, State and Private Forestry, Southern and Southeast. For. Exp. Stn., USDA For. Serv., Ashville, NC. 349p. Clark, P.J., and Evans, F.C. 1954. Distance to nearest neighbor as a measure of spatial relationships in populations. Ecology 35:445453. Clutter, J.L., and Jones, E.P.,Jr. 1980. Prediction of growth after thinning of old field slash pine plantations. USDA For. Serv. Res. Pap. SE217, Southeast. For. Exp. Sta., Ashville, NC. 14p. Cole, L.C. 1946. A theory for analyzing contagiously distributed populations. Ecology 27:329341. Daniels, R.F. 1976. Simple competition indices and their correlation with annual loblolly pine tree growth. For. Sci. 22:454456. Daniels, R.F., Burkhart, H.E., Spittle, G.P., and Someus, G.L. 1979. Methods for modeling individual tree growth and stand development in seeded loblolly pine stands. Sch. For. and Wildlife Res., Virginia Polytechnic Institute and State University FWS179, Blacksburg, Virginia. Davis, K.P. 1966. Forest management: Regulation and valuation. McGrawHill Book Co., New York. 519p. Demaerschalk, J. P. 1972. Converting volume equations to compatible taper equations. For. Sci. 18:241245. Devine, O.J. 1982. Joint estimation of mortality for infected and uninfected trees in slash pine plantation subject to fusiform rust infection. University of Georgia. M.S. thesis. Ek, A.R., and Monserud, R.A. 1974. Forest: A computer model for simulating the growth of reproduction of mixed species forest stand. Univ. Wis., Coll. Agric. Life Sci. Res. Rep. R2635. 85p.
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133 Elston, R.C. and Grizzle, J.E. 1962. Estimation of timeresponse curves and their confidence bands. Biometrics 18:148159. Fisher, F. 1970. A correspondence principle for simultaneous equation models. Econometrica 38:7392. Fisher, R.F., and Garbett, W.S. 1980. Response of semimature slash and loblolly pine plantations to fertilization with nitrogen and phosphorus. Soil Sci. Soc. Am. J. 44:850884. FordRobertson, F.C., editor. 1971. Terminology of forest science, technology, practice and products. Society of American Foresters, Washington, D.C. 349p. Grosenbaugh, L.R. 1965. Generalization and reparameterization of some sigmoid and other nonlinear functions. Biometrics 21:708714. Grosenbaugh, L.R. 1966. Tree form: Definition, interpolation, extrapolation. For. Chron. 42:444457. Grosenbaugh, L.R. 1967. REX Â— Fortran4 system for combinatorial screening or conventional analysis of multivariate regression. Berkeley, Calif., USDA For. Serv. Res. Paper PSW44, Pacific SW. Forest and Range Exp. Stn., Berkeley, CA. 47p. Grosenbaugh, L.R. 1974. STX3373: Tree content and value estimation using various sample designs, dendrometry methods, and VSL conversion coefficients. USDA For. Serv. Res. Paper SE117, Southeast. Forest Exp. Stn., Ashville, NC. 112p. Haavelmo, T. 1943. The statistical implications of a system of simultaneous equations. Econometrica 11:112. Hamilton, D.A., Jr. 1974. Event probabilities estimated by regression. USDA For. Serv. Res. Pap. INT152, Intermountain For. and Range Exp. Stn., Ogden, UT. 18p. Harms, W.R. and Langdon, O.G. 1977. Competitiondensity effects in a loblolly pine seedling stand. USDA For. Serv. Res. Pap. SE164, Southeast. For. Exp. Stn., Ashville, NC. 12p. Harms, W.R. 1982. An empirical function for predicting survival over a wide range of densities. In: Proceedings of the second biennial southern silvicultural research conference, Atlanta, GA. USDA Gen. Tech. Rep. SE24, Southeast. For. Exp. Stn., Ashville, NC. 514p. Helwig, J.T. and Council, K.A., eds. 1979. SAS USER'S GUIDE. SAS Institute, Inc., Raleigh, NC. 494p. Hock, I. 1958. Simultaneous equation bias in the context of the CobbDouglas production function. Econometrica 36:568578. Horn, A.F. 1961. Changes in form class in a red pine plantation. J. For. 89:181183.
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134 Husch, B., Miller, C.I. and Beers, T.W. 1982. Forest mensuration. John Wiley & Sons, New York. 402p. Jones, E.P., Jr. 1977. Precoramercial thinning of naturally seeded slash pine increases volume and monetary returns. USEA For. Serv. Res. Pap. SE164, Southeast. For. Exp. Stn., Ashville, NC. 12p. Jorgensen, J.S.A. 1967. The influence of spacing on the growth and development of coniferous plantations. In: International review of forestry research, Vol. 2. Academic Press, New York. Kent, B.M., and Dress, P.E. 1979. On the convergence of forest stand spatial pattern overtime: The case of random initial spatial pattern. For. Sci. 25:45451. Kent, B.M. and Dress, P.E. 1980. On the convergence of forest stand spatial pattern over time: The case of regular and aggregated initial spatial patterns. For. Sci. 26:1022. Lee, Y. 1971. Predicting mortality for evenaged stands of lodgepole pine. For. Chron. 47:2932. Lembersky, M.R., and Johnson, K.N. 1975. Optimal policies for managed stands. An infinite horizon Markov decision process approach. For. Sci. 21:109122. Lloyd, F.T., and Jones, E.P., Jr. 1982. Density effects on height growth and its implication for site index prediction and growth projections. _In: Proceedings of the second biennial southern silvicultural research conference. Atlanta, GA. USDA Gen. Tech. Rep. SE24, Southeast. For. Exp. Stn., Ashville, NC. 514p. Lohrey, R.E. 1961. Stand density influences stem taper in a thinned red pine plantation. USDA For. Serv. Tech. Note 607, Lake States For. Exp. Stn., St. Paul, MN. 2p. Luppold, W.G. 1982. An economic model of the hardwood lumber market. USDA For. Res. Pap. NE512, Northeast. For. Exp. Stn., Broomall, PA. 15p. Madangly, A. 1964. On the efficiency of threestage leastsquares estimation. Econometrica 32:5156. Maddala, G.S. 1977. Econometrics McGrawHill, New York. 401p. McHugh, R.B., and Wall, F.J. 1962. Estimating the precision of time period effects in longitudinal models with serially correlated and heterogeneous errors. Biometrics 18:520528. Meng, C.H. 1980. Detection of stem form change after stand treatment. Can. J. For. Res. 11:105111. Minor, CO. 1951. Stemcrown diameter relations in southern pine. J. For. 49:490493.
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135 Mitchell, K.J. 1969. Simulation of the growth of evenaged stand of white spruce. Yale University, New Haven, CT. Ph.D. dissertation. Murphy, P. A. 1983. A nonlinear timber yield equation system for loblolly pine. For. Sci. 29:582591. Nance, W.L., Froelic, R.C., Bell, T.R., and Shoulders, E. 1982. A growth and yield model for unthinned slash pine plantations infected with fusiform rust. _In_: Proceedings of the second biennial southern silvicultural research conference; Atlanta, GA. USDA Gen. Tech. Rep. SE24. Southeast. For. Exp. Stn., Ashville, NC. 514p. Newnham, R.H., and Smith, H.G. 1964. Development and testing of stand models for Douglas Fir and Lodgepole pine. For. Chron. 40:491502. Opie, J.E. 1968. Predictability of individual tree growth using definitions of competing basal area. For. Sci. 14:314323. Permar, T.A., and Fisher, R.F. 1983. Nitrogen fixation and accretion by wax myrtle (Myrica cerifera ) in slash pine (Pinus elliottii ) plantations. For. Ecol. Mgmt. 5:3946. Pielou, E.G. 1977. >lathematical ecology. John Wiley & Sons, New York. 385p. Rao, C.R. 1959. Some problems involving linear hypotheses in multivariate analysis. Biometrika 46:4958. Reich, R.M. 1980. An evaluation of distance sampling and ratio estimation: A case study in slash pine plantations infected by fusiform rust. University of Florida. M.S. thesis. 140p. Reynolds, M.R., Jr., Burkhart, H.R., and Daniels, R.F. 1981. Procedures for statistical validation of stochastic simulation models. For. Sci. 27:349364. Shepard, R.K., Jr. 1974. An initial spacing study. _In_: Symposium on management of young pines. Southeastern Area, State and Private Forestry, Southern and Southeast. For. Exp. Stn., USDA For. Serv., Ashville, NC. 349p. Snee, R.D. 1977. Validation of regression models: Methods and examples. Technometrics 19:415428. Stauffer, H.B. 1978. Aggregation points to fit Pielou 's index of nonrandomness. Can. J. For. Res. 8:355363. Sullivan, A.D., and Clutter, J.L. 1972. A simultaneous growth and yield model for loblolly pine. For. Sci. 18:7686. Swindel, B.F., and Squillace, A.E. 1980. Measuring treatment effects through comparisons among plot boundaries. For. Sci. 26:704709.
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136 Theil, H. 1971. Principles of econometrics. John Wiley & Sons, New York. 736p. Venzina, P.E. 1963. More about the "crown competition factor". For. Chron. 39:313317. Wensel, L.C. 1975. Computer generation of points on a plane. Treatment of boundary line overlap in a forestsampling simulator. Hilgardia 43:131159. Wright, J.W. 1976. Introduction to forest genetics. Academic Press, New York. 463p. Zedaker, S.M. 1982. The competitionrelease enigma: Adding apples and oranges and coming up with lemons. In ; Proceedings of the second biennial southern silvicultural research conference. Atlanta, GA. USEA Gen. Tech. Rep. SE24, Southeast. For. Exp. Stn., Ashville, NC. 514p. Zellner, A., and Theil, H. 1962. Threestage least squares: Simultaneous estimation of simultaneous equations. Econometrics 30:5478.
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BIBLIOGRAPHY Basmann, R.L. 1980. On the asymptotic distribution of generalized linear estimators. Econoraetrica 28:97107. Beck, D.E., and Trousdell, K.B. 1973. Site index: Accuracy of prediction. USDA For. Serv. Res. Pap. SE108, Southeast. For. Exp. Stn., Ashville, NC. 7p. Bentley, W.R., and Kaiser, H.F., Jr. 1967. Sequential decisions in timber management a Christmas tree case study. J. For. 65: 714719. Berkey, C.S. 1982. Bayesian approach for a nonlinear growth model. Biometrics 38:953961. Boyce, S.G., McClure, J. P., and Sternitzke, H.S. 1975. Biological potential for slash pine ecosystem. USDA For. Serv. Res. Pap. SE141, Southeast. For. Exp. Stn., Ashville, NC. 29p. Buchman, R.G., and Shefley, S.R. 1983. Guide to evaluating forest growth projection systems. J. For. 81:232234. Fair, R.C. 1970. The estimation of simultaneous equation models with endogenous variables and first order serially correlated errors. Econometrica 38:507516. Glahe, F.R., and Hunt, J.G. 1970. The small sample properties of simultaneous equation least absolute estimates visavis least squares estimators. Econometrica 38:742753. Goulding, C.J., and Murray, J.C. 1976. Polynomial taper equations that are compatible with tree volume equations. N.Z.J. For. Sci 5: 313322. Griliches, Z. 1961. A note on serial correlation bias in estimates of distributed lags. Econometrica 29:6573. Hamilton, D.A., Jr., and Edwards, B.M. 1976. Modeling the probability of individual tree mortality. USDA Forest Serv. Res. Paper INT185, Interraountain Forest and Range Exp. Stn., Ogden, UT. 21p. Keister, T.D. 1972. Predicting individual tree mortality in simulated southern pine plantations. For. Sci. 18:2113217. 137
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138 Kozlowski, T.T. 1963. Growth characteristics of forest trees. J. For. 61:655662. Little, S.N. 1983. Weibull diameter distributions for mixed stands of western conifers. Can. J. For. Res. 13:8588. Nagar, A.L. 1961. A note on the residual variance estimation in simultaneous equations. Econometrica 29:238243. Schreuder, H.T., and Hafley, W.L. 1977. A useful bivariate distribution for describing stand structure of tree heights and diameters. Biometrics 33:471478. Watson, G.S. 1955. Serial correlation in regression analysis. Biometrica 42:327341.
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BIOGRAPHICAL SKETCH Robin Michael Reich was born October 24, 1954, in Bayreuth, Germany, while his father was stationed in Germany with the Second Cavalry Regiment, U. S. Army. His younger years reflected moves to Nurnberg, Stuttgart and Berlin. In 1960 his family returned to the U. S. to Fort Sam Houston, San Antonio, Texas, followed by another transfer to Fort Benning, Georgia, in 1964. His family settled in Winter Park, Florida, in 1966 where Robin graduated from high school and received his AA degree from Seminole Community College. Robin entered the University of Florida in September, 1975, where he received his Bachelor of Science in June, 1978, and his Master of Science in August, 1980, in forest resources and conservation. Robin received an award of excellence for graduate research in October, 1980, and a certificate of recognition from MSA for creative development of a technical innovation in remote sensing, June, 1981. Robin is a member of Phi Kappa Phi and Xi Sigma Pi, honorary societies at the University of Florida. 139
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Loukas G. Arv Professor of and Conser is, Chairman st Resources ion I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Anthony E^Squillajfe Professor of Forest Resources and Conservation I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Jahes. E. Keesling Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Lewis R. Grosenbaugh Professor of Forest Resources and Conservation
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Mark C. K. Yang Professor of Statistics This dissertation was submitted to the Graduate Faculty of the School of Forest Resources and Conservation in the College of Agriculture and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August, 1984 Director, School of Forest Resources and Conservatio Dean for Graduate Studies and Research

