MODELING SI NGLE SLASH PINE TREE GROWHTE
AND SPACING RELATIONSHIPS
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.
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.
TABLE OF CONTENTS
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
Data Interpolation ........ .... ... 25
Growth Function ................ 27
Model Developement ............... 29
Field Data Screening ............. 31
Description of the System of Equations . .. 35 Three-Stage Least-Squares. .......... 37
Identification of the System of Equations. . 39 Contemporaneous Correlations. ......... 41
Cross-Sectional Time-Series 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
Model Validation ............ .... 103
Evaluation of the Coefficients. ........ 103 Goodness-of-Fit 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
LIST OF TABLES
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 Cross-sectional time-series adjustments for
diameter breast height ............ 85
10 Cross-sectional time-series adjustments for total tree height. ............... 86
11 Cross-sectional time-series adjustments for cubic foot volume. .............. 87
12 Cross-sectional time-series 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
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
LIST OF FIGURES
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
16 Relative change in the interim multipliers associated with the exogenous variables 1/Age**2
and (I-Alag/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
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
Robin Michael Reich
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, single-equation 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 Three-Stage 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.
The validity of the models was scrutinized following data-splitting (cross-validation) 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 variance-covariance 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.
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.
One of the major disadvantages of modeling the effects of vacancies on individual tree growth has been the use of single-equation 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 single-equation 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).
This approach to estimating the parameters of a system of equations has two main advantages over the single-equation 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.
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.
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.
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 (Ford-Robertson 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
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 old-field 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 density-height 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 height-density 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 height-density 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.
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.
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 old-field 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
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.
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).
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
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.
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
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).
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
surrounding the tree, with the radius being a function of tree size such as volume, bole surface area, crown radius, DBH and total tree
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
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.
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.
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:
A = Hopkins' coefficient of aggregation,
ECl = Sum of squared point-to-tree distances, Em2 = Sum of squared tree-to-tree 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
which has a beta-type distribution with mean 0.5 and variance 1/4(2n+1), where n is the number of point-to-tree and tree-to-tree 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
where Za/2 is the critical Z-value for the standard normal distribution. The population is said to be aggregate if A* is
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 Limited-Information aximum-
Likelihood (LIML) procedure, Full-Information Maximum-Likelihood (FIML) procedure, Zellner's seemingly unrelated least squares procedure, TwoStage Least Squares (2SLS), and Three-Stage 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 L-1 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.
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 one-acre 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.
OL U LU2
~4Q C )
H ~4 p -
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.
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 point-to-tree and 150 tree-to-tree distances. A z-statistic 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).
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
YT. = corrected plot mean for spacing i and replication j, Yij = uncorrected plot mean for spacing i and replication j,
SP = E Yi spacing mean,
REP = z Y..ij replication mean,
1 6 2.
GM = E Y.. grand mean,
Si spacing effect, R = replication effect.
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
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).
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 FP-15. 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.
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.
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.
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
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 + XlYo-X0 Y1
y = where, x0< x < xl x x0
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
P2(x) = E Li(x)yi i=0
2 (x x.)
Li(x) = H ( xx j=0 i j
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
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.
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 Johnson-Schumacher, Gompertz, Gauss, von Bertalanffy, Verhulst and the Pearl-Reed 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:
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
S= Y* (1 Y*) < 0. dY"
The second derivative is given by
d2 = (1 2Y*)(1 Y*) Y* > 0.
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).
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
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.
Field Data Screening
REX (Grosenbaugh 1967), a FORTRAN-IV 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
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
S4 = Ln ( 1)
S4 = SUR/SNAX
SMAX = maximum bole surface area at 50 years of age for a given
DMAX and HMAX
PAR = relative growing space = S
KL = number of nearest neighbors lagged one time period,
KL = 0,1,2
SP = spacing within rows (feet)
A = tree age (years)
A_I = tree age lagged one period
R2 = A_1/A
RIP = (1 R2) PAR
COMP = E DBHk/(2 DBH)
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,
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.
Table 1. Variables used to identify the "best" combination of
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 (1-Alag/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.
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 cross-sectional 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
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
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 = -5-I 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).
Three-stage Least-Squares 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'
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 .
where Z. = [Yk Uk: XZ]. The same authors discuss in detail the estimation procedure in the two cited references. An estimate of the variance-covariance 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 .
Pre-multiplying the above equation by the transpose of a LnxKL Kronecker product of the form:
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 variance-covariance matrix t is replaced by S = (sj), the matrix of mean squares and products of the 2SLS residuals leads to:
6 = (Z'[X-l X(X'X)- X']Z)- Z'[S-1X(XX)-1 'y the three-stage 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
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 variance-covariance 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 non-zero 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).
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 )
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.
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 off-diagonal 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 n-diagonal submatrices all equal to .
1 o O 0 0 0 f2 0
0 0 .n TIn
The off-diagonal 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 variance-covariance 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
squared error is biased downward (Sullivan and Clutter 1972). The presence of non-zero covariances and non-homogeneous 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 cross-sectional 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
an NT x 1 vector;
X1 XK 11
an NT x K matrix of exogenous variables;
1 L y NT-1 y NT-e an NT x L matrix of lagged values of the dependent variable; 611
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 variance-covariance 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
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 cross-section 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 cross-section of individual trees from different sites and stand densities making up the population. It is not self-evident 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
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 cross-sectional and time-series 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 cross-section 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 cross-sectional 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 cross-sectional ignorance. It is assumed that ynt has zero mean and variance a2 and
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
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 cross-sectional
P' Y W'
disturbances pn are serially independent and that the disturbances are independent from one observation to the next. In other words: G2 n = n'
2 n = n', t = t' E(y nt'Y't') =
E(Ynt'in = 0 for all n,t.
The assumption that the cross-sectional 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 variance-covariance matrix can be written as follows: Al 0 0 0 A2 O E(,c'] = a2
0 0 AN
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 cross-sectional time-series component, OLS estimates of the coefficient vector are inefficient. Consequently, GLS is required to obtain efficient parameter estimates: ^ x -1 -1 -1
= (X-- X) x-0-Y.
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/2-1 /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
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'a-X)- 1 -ly.
The value for a2 may be found by differentiating L(p,y,a2) with respect
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 1-p + T.p
-2L(p,y,2) = E T.Ln(2H) + E T.Ln(a2(1-p)) + E Ln( )
E 1 1 E I-p
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 cross-sectional time-series error component, the normal 2SLS procedure will be inefficient. The suggested procedure
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 variance-covariance matrix obtained in the second stage.
The efficiency of the OLS and 3SLS estimation procedure was evaluated by four goodness-of-fit statistics (Murphy 1983):
n (Yi Yi)2
Fit Index, R2 = 1 E
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
Percent Root Mean Square Error, RMSE% = ( Y Y)2] I2 100
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 F-test was used to examine the equality of variances
of the two estimation procedures:
where Fij is the F-statistic for the ith equation and jth parameter.
RESULTS AND DISCUSSION
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
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
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 point-to-tree and 150 treeto-tree 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.
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.
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
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
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 east-west 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 east-west 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
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.
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 11.3-16.0 Ilex glabra (L.) Gray 6.6 4.9- 8.0 Myrica cerifera L. 15.5 5.0-25.0 Persea borbonia L. 10.9 6.8-15.5 Quercus incana Bartr. 15.0 15.0 Quercus laurifolia Michx. 10.2 4.8-18.0 Quercus nigra L. 13.8 6.8-25.5 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
0 u 8
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.
LLLLJ < 40
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.
-- 8 16
o 66 4
7B 15B 98 13B 11B 5B 9A 15A 7A 13A IIA 5A
Figure 4. Cubic volume corrected for spacing and replication by
stand age and plot location.
L, 100 t
4 0 LU0
<' 40 -12
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.
O 8 LL 10
o 0 908 0
Z 19 ...0 2I
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.
(D 21 W 66
78 158 98 IBE lIB 58 9A 15A 7A !3A 11A 5A
Figure 7. Diameter breast height by stand age and plot location.
GOGE LL 501
Fiure 8. Total tree height by stand ae and lot location.
LL 10 AGE
7B 158 98 13B 118 58 9A 15A 7A 13A IIA 5A PLOT LAYOUT
Figure 9. Cubic volume by stand age and plot location.
16 L) 4
7B 15B 9B 13B 118 5B 9A 15A 7A 13A IIA 5A PLOT LAYOUT
Figure 10. Surface area by stand age and plot location.
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
0 21 8-9
-R2=0.1 8 Z 4
5 7 9 II 3 5
Figure 11. Effects of spacing on diameter breast height corrected
for microsite and replication by stand age.
RZ= O.I 21 60 R2 =0.04 9
LL R2=O20 6 R 2=0.20
O R2= .20 I 30= C !-Z
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.
5 7 9 1i 5
Figure 13. Effects of spacing on cubic volume corrected for
microsite and replication by stand age.
0 100 ,6 21
O -l 0. I6
r 10 LLJ
U 0 Rz= 0.00 () R2=0.04
5 7 9 II1 13 15 SPACING (FEET)
Figure 14. Effects of spacing on surface area corrected for microsite
and replication by stand age.
;~ (D. 16
R2 0.10 2= 0.1
-- I -s ~ ~ '
Figure 15. Effects of spacing on survival corrected for microsite
and replication by stand age.
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
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
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:
Q = 0.9549(DBH2196 THT-0250) 1/2
Ao = 0.9251 DBH1.138 THT-0.164
Qo/Ao = 0.9857 DBH1.059 THT-0.087
Fo = 0.927 DBH0079 THT-0.076
cn Lna-~00 -30 M 0 CD
Ch 4 N M 1 I V
N 0 r
00NO> 0 o0 N w
C4 C 00 W 0 r- 10
00 0 O o
cd vi 0 N' 0 c0 C CO 01 N N. -.
N u' u -< 14 0 N 4 O m cc Lncc~hr
0 ui r V Ln Ln
. . S t i NN C Cr) UO
-T O N CO 0 C a r. N 0
0 4 N o i "0 N
N0 -4 4 00 N r CA tC 4
Ul N- '.0 It) r- M N n N 0 \ 1OI If Cr -#0 N- CO .7 '0 Cr LI) C
-T o cr% N ) a' -T a' in 0
--4 N C14.
C4t Ln 0 I I M7 -I N. N
CD V a%' N h i -Z
& On N140 ON OO O 4tO~r hCO~9ml u l I ~ -~rl U0) -;
b IN 0 C O 0 -4 O 01 c a L4 41-4 C-1 4 m (n Cr I
Tt e1. CO a' N LO ) C* CO C) o C, 4 0 0 '.1 Cr) CO CO -4( O a'- CO a'1 11 '.0 N- Cr -.7 1
'.0 .r c. .. . .
o N ) I "o COO M. (I- CO -. -% r C Z
--4 N ) ) N I
b O 0 N N O O *o O . ) (d be N Cr LI N- 0 C) '00 hit a' L) 0f cl() a *** N N N c '4
w o CO o e
0 C14 a' CO 0 M 0 CO w. -41 )-4 -#' 5 -7 *-1 -Z
0 m Ch L-4 ; N N O m II Oq W'0 0 N- ON r- NI LI) '0
0 0a N 4 '0 a' '0 '00 O
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-:T o14 WNO. -4 '0-j-a' -# a0 ,c
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un tr 4 so. 4 bj
0 -# '0** *
N 0 Co Cr 00
S c~) 110 co0'
-4 '0 0 *'0 ar c W1 v" 0 'n- 0 = T C *o
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0- 0 C .O m W "
- : r I I00 N Cr w w
I) If -# '0 N. .Z'.
C.) Cr4 W J U. .1.1-4 co 1 0n w "o m9c~rl L)C
*4 N r- LI -L 0 U 0 0
Er) w J)a C. r a0 0) w tor 2 nv i r )
n C: uI .0 ':14U1- 0
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- C. -- 4 C)
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'0 CX -40, 1 0 a% 0 q cn U. E. 4 UI --4 J4> i- I Cl .-' r hi mI II ~ r
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a' CO C 0 N
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c,4 00 r- 0 c4 00 D N o 0
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co. .4 -4 4 N
I- a c 00 N M N D a
0ON C C I ON ON "
C O C N N a' -O a a U
00 0 N 04 O ON
ol a 0
00 T O) -1 N0C C 1 % D00 ON N- 0 -11 ON 'C "o V) n 9 3 cu c ) L co -4*4*4* 4 --4
V 4 CD 00 0) (O O .
N t 0' N u u 0 N 0
0 Il 1; 11 Cm 1:0 : :
T U o r =- C) -4 N -T Ul 0 :)O
.- .-4 .-4 -. -4 --4'
0 pa 0 0 0 0 P -, 0 a, ~ ~ ~ ~ ~ ~ O O *)L) 1 l a O a* NW vi00
400 I0 CO C 0 N-ON- CO N
44 un : tt -l '0 4 Nl O N* *4 -40 0 UIII amZQM
.i 4 CN CLfl0 ) () 00 L) C C1 U
4. V. C 0'c ON '0 CO C -4 _: C0V4 Nr 0
M4- '-T c n %0 r-C a' O N- C4 Mt L C Nco 1-- -.4 -4 .4 .4 .-4
w 0 0 0f L0 r.0 N-M 0 C) 'T~ LNLt' Li) '0 N a ") -t 7c rl N &I cn -.T3 O 0 Ln 1.0 00 r
44 T Ln o 3 00 01% 0 cq 0 I
-4, 4 L --4 (7 "'4 O-( "o r c
0J 000 U N-, 04 N- 4r -It4 co "V) I4 I -t 0 0' -t ON L' L) '
,a n N 0'0 a N 0
-4) -44 -4w
IT CD Olo N r cn -T 00 1.0 w1
7 I'4 -4 a' M IS + '0 N.0 *T 'oN a N C f) '7 7C
(1)0 0' CO CO N- N- N- N--aS~ 0 C co"4 -T U'S '0 N- 'C 00 CO 4
co ~ ~~~ (O (7 L0io11 i
U'D N' (-7% N- m" 0 0 +1 C) 7 0 m T IT.1 a
M c 4.
CO 00 ("S N 0 -S 0
aO )% a: 0 N 0 C '-'
0 'S a' Li a) CO3 CN 0 C -0 CO N CO -7 u' '0 -4 0
14.4 N- $4J u 0O
0- CO Ci : 4J 0/' C4 rn CO N *
0c N W CO 0 11
-4 N N CO -n +1 W UCO a.J
'- 0 0o 0' C: ic 'O
J14 0 4 4 1 4 1' c
~u I I I 1 I i S ,
1-4 = m 1Li' N N X 0\3 3 9Ic 1
N- N 0) o 02) ~f i ) C/ -4 N I I CL C a)
co -4 1 --T U' N- CO 0' 0 n NT O -7 U'S E 4 -4 -4 -4 C./
Q = 0.6873(DBH2.300 TH-0.207) 1/2
AI = 0.6839 DBH186 THT-0.144
QI/AI = 0.6908 DBH1.114 THT-0.063
F = 0.9880 DBH0071 THT-0081"
Table 8 compares average tree form, inside and outside bark, by diameter and total height.
The growth model for diameter, total height, cubic volume and
surface area was fitted to a sample of 2600 cross-sectional and timeseries observations using a 3SLS procedure. Although the third-stage 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.
Table 8. Comparison of tree form inside (FI) and outside (Fo)
bark by diameter and 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
Second Stage Estimates
In the second stage, maximum likelihood estimates of the structural parameter vector and variance-covariance matrix are obtained based on the assumptions regarding the cross-sectional and time-series 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 variance-covariance 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 cross-sectional and time-series 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 time-series 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
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 variance-components 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 variance-covariance 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.
Table 9. Cross-sectional time-series 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
Table 10. Cross-sectional time-series adjustments for total tree
RUN OBS T--/ p CONLK2/ RSQ VAR ESS-3/ 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
Table 11. Cross-sectional time-series adjustments for cubic foot
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
Table 12. Cross-Sectional time-series adjustments for bole surface
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