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This is to certify that the
thesis entitled

Evaluation of The Lake States TNIGS
Diameter Growth Model For Upland Hardwoods
In The Lower Peninsula of Michigan

presented by

Patrick James Guertin

has been accepted towards fulfillment
of the requirements for .

M § degree in EQEQSLL

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TO AVOID FINES Mum on or bdoro duo duo.

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MSU is An Afflrmotivo ActioNEqul Opportunity Institution
Wanna

 

EVALUATION OF THE LAKE STATES TWIGS DIAMETER GROWTH MODEL
FOR UPLAND HARDWOODS IN THE LOWER PENINSULA OF MICHIGAN

BY
Patrick James Guertin

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Forestry

1993

ABSTRACT

EVALUATION OF THE LAKE STATES TWIGS DIAMETER GROWTH MODEL FOR
UPLAND HARDWOODS IN THE LOWER PENINSULA OF MICHIGAN

BY

Patrick J. Guertin

Lake States TWIGS, the primary growth and yield model
available in the Lake States, was developed with regional
data. Validation of the model has never been performed
exclusively for Michigan. This study validates Lake States
TWIGS for the Manistee National Forest, and investigates
alternative distance-independent, individual-tree, diameter
growth functions which may improve prediction accuracy.

Diameter growth, basal area, and.mortality were compared
to five-year projections from Lake States TWIGS for five
northern hardwood species in northern lower Michigan.
Although results may have been influenced by the 1988 drought
and infestations of several species of defoliating insects,
Lake States TWIGS appears to accurately project five-year
growth and mortality for most upland hardwoods.

Development of a diameter growth function to improve
prediction accuracy focused on the Chapmann-Richards growth
function, multivariate regression, and other’ established

modeling methods. Results were inconclusive.

ACKNOWLEDGMENTS

First I would like to thank my graduate committee
chairman, Dr. Carl Ramm. His guidance and support proved
invaluable. I would also like to thank Dr. Larry Leefers and
Dr. Scott Winterstein. Their emphasis on the educational
aspect of my efforts made my failures as valuable as my
successes. Thanks are also due to the staff of the Huron-
Manistee National Forest, especially Rose Ingram and Dave
Cleland, their assistance was greatly appreciated during the
summer field season. I would also like to acknowledge the
support of family and friends, especially my parents and Tim,
Carol, Greg and Andy. Finally, I would like to thank Frank
Sapio and Roger Mech of the Michigan Department of Natural
Resources. It is what I learned from working with them that
provided the confidence and understanding to pursue the

forestry profession.

This research was supported by the Agricultural
Experiment Station of Michigan State University and by the
McIntire Stennis Cooperative Forestry Research fund, and in
part by the U.S.D.A. Forest Service, North Central Forest

Experiment Station.

iii

TABLE OF CONTENTS

List of Tables...........................................vi
List of Figures........................................viii
Problem Definition........................................1
Literature Review.........................................2
Objectives...............................................18
Field Methods and Data...................................19
Data............................................... ...19
1991 Field Measurements................................23
Data Processing and Compilation........................24
Model Validation and Development.........................26
TWIGS Validation.......................................27
Data.................. ........ . ..... .... ...... ......27
Validation Statistics...............................28
Error Statistics....................................29
Model Development and Results..........................31
The Multivariate Linear Model.......................34

The Modified Chapmann-Richards Function.............41
Species/ELTP Specific Modeling......................43
Stand Level Variables..... .......... ................49

MOdeling TerminationOOOOO0.........OOOOOOOOOOOOOOOOOSO

iv

Final TWIGS Validation................................Sl
Data................................................51
Validation Statistics...............................53
Error Statistics....................................53
ATEST Validation....................................62

Discussion........................... ........... .........66
Modeling...............................................66
Validation.............................................68
Future Considerations..................................68

Appendix A...............................................72

Appendix B...............................................82

Appendix C.... ........................ ' .......... .........87

Bibliography.............................................92

Table
Table
Table
Table
Table
Table
Table
Table
Table

Table

Table

Table

Table
Table
Table
Table
Table

Table

1.

2.

9.

10.

11.

12.

13.
14.
15.
16.
17.

18.

LIST OF TABLES

Definition of ELTPs............................22
Subplot Distribution per Data Set..............26
Initial TWIGS Validation Results (5 years).....30
Calibration Data: Sample Size by Species Group.32
Variables Used in Model Development............33
Summary Statistics for Calibration Data Set....36
Basal Area C1asses.............................46
Results of BAGSYR Models.......................48
Stand Level Variables..........................49
Summary Statistics for the 156 Sample Points by
Strata (ELTPs).................................52
Calculated Error for Estimated DBH

(5 year projections) by Species and Size C1ass.56
Calculated Error for Estimated BA and TPA by
Species (5 year projections)...................56
SASATEST Calculated Errors for DBH.............64
SASATEST Calculated Errors for Estimated TPA...65
SASATEST Calculated Errors for Estimated BA....65
Northern Red Oak Multivariate Model ANOVA......82
Other Red Oak Multivariate Model ANOVA.........83

White Oak Multivariate Model ANOVA.............84

vi

Table 19. Sugar Maple Multivariate Model ANOVA...........85

Table 20. Red Maple Multivariate Model ANOVA.............86

Figure

Figure

Figure

Figure

Figure

Figure

Figure
Figure

Figure

Figure
Figure

8.

9.

10.

11.

LIST OF FIGURES

Location of TWIGS calibration sites and test
sites in Michigan..............................20
Mean Error (inches) by DBH Class and Species...57
Mean and standard deviation for mean error of
predicted DBH over 5 years (positive values

are underpredictions)..........................58
Mean and standard deviation for mean error of
predicted BA/acre over 5 years (positive values
are underpredictions)..........................59
Mean and standard deviation for mean error of
predicted TPA over 5 years (positive values

are underpredictions)..........................60
Mean Percent errors between observed and
predicted values over 5 years (positive values
are underpredictions) .........................61
Five year change in DBH vs. 1986 DBH...........72
Five year change in DBH vs. log(DBH)...........73
Five year change in DBH vs. 1986 tree

basal area.....................................74

Five year change in DBH vs.DENSE...............75

Five year change in DBH vs. crown ratio........76

viii

Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure

Figure

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

Five
Five
Five
Five

Five

Northern red oak

year
year
year
year

year

change
change
change
change

change

in
in
in
in
in

DBH

DBH

DBH

DBH

DBH

VS.

VS.

V8.

V8.

VS.

site index.........77
BA.................78
1/BA...............'79
QSD................80

ASDOOOOOOOOOOOOOOOOBI

residual plot.................87

Other red oak residual plot....................88

White oak residual plot........................89

Sugar maple residual plot......................90

Red maple residual plot.... ........... .........91

ix

Problem Definition

Michigan has approximately 17.3 million acres of
commercial forest land, including 7.8 million acres of oak-
hickory and northern hardwood forest (Smith and Hahn, 1986).
The increasing public:demand for old growth forests and forest
recreation has put increased, and often conflicting, demands
on these resources and on the silvicultural practices that are
used in their management. To properly manage the state's
forest resources under these pressures, reliable growth and
yield information is a necessity. The key to providing this
data are accurate growth and yield models.

The major growth and yield model available for the Lake
States, TWIGS (Miner et. al., 1988), was based on data
collected to conform to regional rather than individual state
needs. Hardwood data used in the development of the model
consisted primarily of stands from Wisconsin and Minnesota.
These stands are quite different from stands in the Lower
Peninsula, especially in regard to northern hardwood
composition and site quality. Projections of hardwood growth
and yield for the Lower Peninsula of Michigan are therefore
suspect.

TWIGS is widely used as a planning tool by state and
federal agencies in Michigan (Ram and Miner, 1986) . Although
it was designed for comparative studies of alternative

management regimes (Miner et. al., 1988), TWIGS is often used

for straight stand projection. The objective of this study is
to evaluate the accuracy of the TWIGS growth model for

Michigan.

Literature Review

A survey of industry, universities and government
agencies found that TWIGS was the primary growth and yield
model used in the North Central region (Ram and Miner,
1986). TWIGS is an individual tree, distance-independent
system developed by the USDA Forest Service North Central
Forest Experiment Station (Miner et. al., 1988) . The history
of TWIGS is rooted in a general tree growth projection system
developed by the North Central Forest Experiment Station (USDA
Forest Service, 1979). This effort grew into a mainframe
projection system known as STEMS (Belcher et. al. 1982).
STEMS was later updated for the Lake States region in 1985,
this version was named Lake States STEMSBS. The STEMSBS
version incorporated a diameter adjustment factor and a new
mortality function (Holdaway and Brand, 1986) . The adaptation
of STEMSBS to the microcomputer was the final step in the
evolution of TWIGS.

There were two versions of TWIGS developed for the North
Central region. The first is Lake States TWIGS, developed for
use in Wisconsin, Michigan and Minnesota. The second is
Central States TWIGS, developed for use in Indiana, Illinois,

and Missouri. This study focuses on Lake States TWIGS. The

3

growth model for this version was calibrated with 80,000 trees
from 1,500 plots in Wisconsin, Michigan and Minnesota (Miner
et. al., 1988). Representation from Michigan in the STEMS
system was limited to red pine (Pinus resinosa) and jack pine
(Pinus banksiana) plantations in the Lower Peninsula.
Although additional plots were added for the TWIGS model, no
hardwoods were included from the Lower Peninsula» Most of the
data for the oak - hickory and.northern hardwoods timber types
is from Wisconsin (Christensen et. al., 1979). The regional
dimensions of the calibration data set prompted Miner et. a1. ,
(1988) to warn against using TWIGS as a tool for obtaining
exact projections. This restriction limits it's use as an aid
in decision-making.

The regional aspect of TWIGS is the first of many
concerns with the model's predictive power. Both Holdaway
(1985) and Smith (1983) have questioned TWIGS' ability to
accurately represent local growth potentials. In using TWIGS
for projections of northern hardwoods in the Lower Peninsula
of Michigan, the model's predictive abilities have been
extended outside of the geographic range of calibration. If
the model is extended into an area with different
macroclimatic conditions, the regional TWIGS coefficients are
inappropriate (Smith, 1983). The northern hardwood
calibration data obtained in Wisconsin came from areas with
homoclines dissimilar to the Lower Peninsula of Michigan

(Rauscher, 1984). These differences are in average seasonal

4

temperatures, precipitation and solar radiation. The
importance of macroclimatic influences on soil and vegetation
development has been discussed by Spurr and Barnes (1980) and
by Rauscher (1984). Early validation of the STEMS model
showed that its predictive powers decreased as the model
application moved east and south in the Lake States region
(Smith 1983; Leary et. al., 1979).

The STEMS model has been adjusted for sub-regional
variations with positive results. Holdaway (1985) developed
an additive diameter adjustment factor with Wisconsin data,
and reduced the ten year mean diameter growth error from .17
inches to .03 inches. This adjustment factor is currently
incorporated into Lake States TWIGS. Smith (1983) developed
an adjustment factor based on ratios of actual and predicted
diameter at breast height (dbh) . Smith's adjustment was
calibrated with data from Michigan's Upper Peninsula; he
reported a 94 percent decrease in mean annual diameter growth
error from .033 inches to -.002 inches. These adjustments
provide evidence of TWIGS weakness at a sub-regional level.

In addition to its regional dimension, the research plots
used to calibrate TWIGS also raise: questions about its
reliability. Data sources for the growth model included plot
records from cutting experiments, demonstration woodlots,
industrial continuous forest inventory, and personal records
of forest growth. In all cases data was from permanent

research plots (Christensen et. al., 1979). Research plots

5

are usually set up in areas that provide easy access, have
minimal natural damage and are more intensely managed. These
stands are <generally’ more 'uniform. than ‘the majority of
forested lands. When models are developed from such data there
is a tendency for over prediction of individual tree and stand
characteristics (Holdaway, 1985; Bruce, 1977). Ideallywmodels
should be developed to account for the variability of general
forest conditions.

After a model is developed and before it is applied to a
specific use its performance must be validated (Buchman and
Shifley, 1983). This step is even more important if model
performance is suspect. TWIGS' performance has never been
validated on a sub-regional basis in Michigan, however early
tests on a regional level found that STEMS predictive
capabilities diminished as its use moved south and east in the
region (Smith, 1983; Leary et. al., 1979). Twenty-seven year
mean dbh was overestimated by 12% for northern hardwoods in
southern Wisconsin (Leary et. al., 1979). Holdaway and Brand
(1983) validated STEMS' at the regional level and found that
mean ten year dbh predictions for white ash (Fraxinus
americana), sugar maple (Acer saccharum), basswood (Tilia
americana) , and several members of the red oak group had
errors of at least 0.4 inches. Substantial errors were also
found in predictions of red oak mortality. When tested in
Illinois, the Central States version of TWIGS (version 3.0)

overestimated change in sugar maple dbh, trees per acre, and

6
basal area growth by over 20% in 15 years (Kowalski and
Gertner, 1989). The relative. error in survival for all
species, in general, was 22%.

When investigating the source of the errors produced
during projection, there are two possible directions to
examine. The first source of error arises from the regional
scope of the projection systems and the research-oriented data
base used in model development. Any inadequacies introduced
by these factors are compounded during the projection process.
In addition, these.errors*will interact with the second source
of error, model structure, to propagate error.

Lake States‘TWIGS uses.a set of three linked equations to

form its annual diameter growth model (Miner et. al., 1988):

Annual Diameter Growth = Potential Growth * Competition
Modifier + Diameter Adjustment Factor [1]
Potential Growth = 1:1 +1320” 4- b,SI*CR*D”’ [1a]
Where:
Potential growth:I potential annual dbh.growth (inches/yr)
D - current tree diameter (dbh in inches)

SI = site index (feet at age 50)

CR = tree crown ratio code
(0-10%=1, 11-2o%=2, ... 81%+=9)
bu”..,tg = species specific equation coefficients

(Hahn and Leary, 1979)

7
Competition Modifier = 1 - e“"""”" ””“m' “V‘” [1b]
Where:
Competition modifier - Competition index bounded by 0 and 1
BAmax = maximum basal area (ftF/acre)
expected for the species
BA -= current basal area (ftZ/acre)
R = ratio of tree dbh to average stand dbh
AD = average stand dbh (inches)

f(R) = a function of characterizing the
individual tree ' 8 relative effect on
the modifier
f(R) - b1[1 - e"“‘]"’3 + b4

g(AD) = a function characterizing the

average stand dbh effect on the
modifier
9(AD) =‘CH(AD + 1)cz
b1, ...,b,,c1,c,_ = species specific equation coefficients
(Holdaway, 1984)
Diameter Adjustment Factor = all) + azn’ + a, [lo]
Where:
Diameter adjustment factor - adjustment in annual diameter gr
growth (inches/yr)
D = current tree dbh (inches)
a1,a;,,a3 = species specific equation
coefficients

(Holdaway,1985)

8
When TWIGS is run over multiple projection cycles the
diameter growth model interacts with the system's mortality
model to project stand dynamics (Miner et. a1. 1988). The
TWIGS mortality function is:
Survival = b1 - [1/(1 + e‘)] [2]

Where:

Survival the tree's annual probability of survival
n = b, + b3DGR“ + b5(D - 1)“ * e'bl‘D'“
DGR = predicted annual diameter growth (inches)
D = current tree dbh (inches)
b1, . . .,b. = species specific equation coefficients

(Buchman, et. al., 1983; Buchman, 1983; Buchman and

Lentz, 1984)

Sources of error related to this model structure are
rooted in suspect variables in the diameter growth model, and
in the way that different functions within the system inter-
relate. For example, TWIGS uses site index (base age 50 yrs.)
as a measure of site quality. Site index only has meaning in
undisturbed, normally stocked, even-aged stands, where the
codominant and dominant trees measured are uninjured and free
growing, and where total age and height has been accurately
determined. Given these stringent standards, and the more
detailed concerns brought up by Carmean (1975) and Monserud
(1984), there are few stands that are eligible to meet the

restrictions of site index.

9

Another model-related problem is TWIGS' use of crown
ratio to measure tree vigor. This variable is seldom measured
during inventory, and when measured is estimated ocularly. If
crown ratio data is not present TWIGS will estimate it (Miner
et. al. 1988), which may induce further error into
projections. Validations of the crown ratio model for the
northern-lower peninsula of Michigan found that the model
overestimated crown ratio codes by 0.32 units (Holdaway,
1985). Crown ratio should therefore be considered another
source of introduced error.

The models used in TWIGS for determining potential
diameter growth and survival are directly or indirectly a
function of current diameter, site index and crown ratio
(Miner et. al., 1988; Hahn and Leary, 1979; Buchman et. al.,
1983) . As noted above, site index and crown ratio are
readily susceptible to measurement error while current
diameter for any period of time after the initial projection
will be biased. Given these circumstances, error in growth
projections over repeated cycles is bound to be magnified.

Given the potential sources of error in TWIGS, it is
clearly necessary to determine if its diameter growth model
provides accurate estimates for upland hardwoods in Michigan's
Lower Peninsula. Buchman and Shifley (1983) have outlined key
areas to examine when evaluating a forest growth projection
system. Validation procedures have been detailed by Burk

(1986), Reynolds (1984) and Snee (1977). These procedures

10
focus on the construction of a validation data base and the
use of validation statistics to determine the accuracy of
model predictions. ATEST, a program developed by Rauscher
(1986) and the SASATEST template written by Gribko and Wiant
(in press) provide a simple means of comparing predictions
with validation data to determine model accuracy.

If the error of estimates produced by the TWIGS
projection system are inordinate, corrections should be made.
These corrections can be achieved through two methods: first,
a sub-regional adjustment to the current model; second,
development of new models with structures that include
variables that better explain environmental conditions and
which are less prone to propagating error.

Two methods have been used to adjust the STEMS regional
model for sub-regional use (Holdaway, 1985; Smith, 1983).
Working with permanent research plot data from Michigan's
Upper Peninsula, Smith (1983) developed a modifier which is
applied to the STEMS projected diameter. This adjustment
factor was based on the ratio of actual to predicted diameter
growth of the individual trees on the study plots. It is
applied by multiplying the adjustment factor to new dbh
predictions. As previously discussed, the results were
lowered errors of estimated dbh. Holdaway ( 1985) developed an
additive adjustment factor for Wisconsin which is now an
integral part of the TWIGS model. Holdaway's adjustment

factor can be calibrated to reflect growth differences in

11
Michigan. This modifier also reduced errors of estimated dbh.

In addition to the methods used by Smith (1983) and
Holdaway ( 1985) , Gertner (1984) proposed localizing a regional
diameter growth model using sequential Bayesian procedure.
This method adjusts regional model parameters in accordance
with the precision of subregional estimates of diameter
growth. Gertner (1984) has outlined examples of how this
procedure could be applied to a variant of STEMS used in
Oregon, however, actual application of the method has not been
done.

If model inadequacies are found, the second method for
correcting TWIGS is to develop a new individual tree,
distance-independent means of projecting diameter growth to
replace the existing diameter growth model. Several methods
are discussed in the literature, which range from basic growth
tables to complex models developed to forecast hardwood
diameter growth.

A simplistic method of projecting diameter growth is a
tabular summarization of growth and other characteristics. An
example of this method is Smith and Shifley's (1984) work
summarizing growth and survival characteristics of Indiana and
Illinois tree species. The authors compiled data from over
15,000 trees by size class (dbh) and species to produce tables
forecasting growth and yield over 10 years.

A more complex, yet more manageable (in terms of use)

solution is to develop a new distance-independent, individual-

12

tree, diameter growth model. Several alternative models are
available, utilizing numerous methods of development (Hahn and
Leary, 1979; Bailey, 1980; West, 1981; Wykoff et. al., 1982;
Mawson, 1982; Shifley and.Brand, 1984; Hilt, 1983; Hilt, 1985;
Harrison et. al., 1986; Shifley, 1987; Hilt and Teck, 1987;
Zeide, 1989; Bolton and Meldahl, 1990; Wykoff, 1990). In
addition, Amateis and.McDill (1989) published a framework for
the use of dimensional analysis in predicting individual tree
growth.

During the initial development stage of what was to
become known as the STEMS model, Hahn and Leary (1979) tested
four potential diameter growth models:

1) AD/At = bISI+bZDb3+b,CR*D

2) AD/At

b1m+bzbb3+b,SI *D

3) AD/At

b1+b2Db3+b,SI*CR*D
4) AD/At - b1+szb3+b,SI*CR*D”5
Where:
AD/At 2 change in dbh over change in time
CR 8 tree crown ratio code (0-10%-1, 11-
2o%=2, ... 81%+=9)
D = current tree dbh (inches)
SI - site index (feet at age 50)

b1. ..bk 8 species specific equation coefficients

All four models predicted change in ddh over time as a

function of crown ratio, site index, and current dbh. The

13

overall best of the four models is used as the potential dbh
growth model in Lake States TWIGS (model 4). The criteria
used for this selection was the ability of the model to
produce accurate predictions, residual plots, and coefficient
of determination (R2). If the model used in TWIGS does not
predict adequately then there is the possibility one of the
other three models may produce better results.

Another alternative is the potential diameter growth
model used in Central States TWIGS (Shifley, 1987; Miner et.
al., 1988). This model is based on the growth form of the
Chapmann-Richards function (Pienaar and Trumball 1973) with
additional variables added to represent the effects of crown

ratio and site index.

POT = (blTBAbz-b3TBA) *(b,+bSSI+b6CR)

Where:
POT = potential annual tree basal area
growth (ftz / yr)
TBA a current tree basal area (ftq)
SI = site index of dominant species on plot
(ft at age 50)
CR = crown ratio class (0-10%=1, 11-

2o%=2, ... 81%+=9)
b1. ..b1 = species specific regression coefficients
The Central States TWIGS model was initially calibrated

with Indiana, Ohio, and Missouri data, with some degree of

14

success (Shifley, 1987). Initial validation of the model
linked with the Central States competition modifier resulted
in dbh prediction errors of less than .02 inches (in a ten
year period) for a majority of the oak species found in the
Central States region. In addition, Shifley (1987) reported
R’s as high as 0.91 during the initial stages of model
development.

Shifley and Brand (1984) have proposed using the
Chapmann-Richards growth function constrained for maximum tree
size for predicting change in individual tree basal area over

time:

ATBA/At= (blTBAbz-b3TBA/A‘1'b2’)
Where:
ATBA/At = Change in tree basal area
(cm’) over time.
TBA = current tree basal area (cmz)
A = Maximum tree basal area (cmz)
b1...b1 == species specific regression
coefficients
When individually fitted to six hardwood species in
Missouri, including several oak species, the model produced
favorable goodness-of-fit statistics. R’s ranging between
0.74 and 0.98, and standard error of estimates between 1.9 and
6.1 cm2 were reported. Four of the six species-specific

models were also significant at an alpha = .01 probability

level. The model was not validated.

15

An additional diameter growth model that has been
incorporated into a computer simulation is in OAKSIM. OAKSIM
is a two-stage tree basal area growth model developed by Hilt
( 1985) with individual tree data from Ohio and Kentucky.
Hilt's (1983) model predicts tree basal area growth as a
function of initial tree diameter, site index, mean stand
diameter, and percent stocking. Hilt published model st
ranged between 0.58 and 0.78 in different stages of the
modeling process. As for model performance, Hilt reported a
.08 ftZ/acre difference between actual and predicted basal
area growth after 15 years.

West (1981) developed another two-stage model for three
species of eucalyptus in Tasmania. The model predicts average
annual diameter increment of an individual tree as a function
of the tree's current dbh, stand age, and stand stocking.
When validated with data from 21 test plots, the model
predicted frequency distributions of tree basal areas and
stocking levels that did not significantly differ from actual
tree basal area and stocking distributions (P = .05).

Several other models have been proposed. One such model
was developed by Hilt and Teck (1987 ) for hardwoods in
northern New England. This model is based on a modified
Chapmann-Richards growth function and projects diameter growth
as a conversion of predicted tree basal area growth.
Validation showed that for a one year projection period the

model consistently underestimated diameter growth by 15 to 30

16
percent for all species involved.

Mawson (1982) proposed three regression models for
predicting diameter growth in small forests. These models
include a linear function, a quadratic function, and a log
transpose model, all of which use current diameter to predict
periodic diameter increment. When tested with hardwood data
from Massachusetts, the log transpose model performed best.
Over an unspecified period of time the model overpredicted
diameter growth of northern red oak and other hardwoods by
less than four percent. Mawson recommends using these
equations for forests under 3,000 acres in size and that
models should be calibrated for each forest.

Bolton and Meldahl (1990) developed a system of
predictive equations for pine, oak, and other hardwoods in
Georgia. These equations included an annual diameter increment
growth model and were developed using a combination of
regression and cluster analysis. No validation statistics
were published.

Harrison ‘et. al. (1986) developed a series of individual
tree basal area increment equations for mixed-hardwoods in the
Blue Ridge areas of Virginia, North Carolina, Tennessee, and
Georgia. These equations predict periodic annual tree basal
area increment using current tree basal area, stand basal area
before and after thinning, and stand age at thinning. As
apparent through the variables used, these models are intended

for use only in managed stands. The published goodness-of-f it

17
statistics for the three models including st of between 0.41
and 0.77 depending on species and model.

Two additional diameter increment models which may be
suitable for use in the TWIGS system were developed for
conifers in the Inland Empire (eastern Washington, northern
Idaho, and western Montana) (Wykoff et. al., 1982 ; Wykoff,
1990). The first is used as the growth function in the Stand
Prognosis Model. This model predicts diameter as a function
of current dbh, stand aspect, stand slope, stand elevation, a
crown competition factor, crown ratio, basal area per acre
greater than the subject tree, a rating for stand habitat, and
a rating for stand location within the Inland Empire. The
second growth function is a variant of the first, which omits
terms for habitat and location.

An alternative to these existing methods is to develop a
new model. Due to its flexibility and ease of development,
the multivariate linear model is preferred. One advantage to
its use is that ordinal data, in addition to continuous data,
may be used. This facilitates the use of ecological land
classification codes as a replacement for site index.
Ecological land classification codes are believed to better
express the biotic and abiotic potential that effect site
quality (Crow and Rauscher, 1984; Pregitzer and Ramm, 1984).
In addition, other variables not found in the existing models

may be used if they were appropriate (i.e. crown class).

Objectives:
Based on the problem definition, the objectives of this

research are to:

1.) evaluate the Lake States TWIGS growth model for

upland hardwoods in Michigan's Lower Peninsula.

2.) modify, if necessary, the existing model or develop

new models to accurately predict the growth

potential of Michigan's hardwood resources.

18

Field Methods and Date:
93:13.

The data used for this study came from 50 northern
hardwood stands on the Manistee National Forest in Manistee,
Wexford, Mason, Lake and Newaygo counties (Figure 1.). These
stands were chosen from seventy-six stands used in the
development of an ecological classification system (ECS) for
upland hardwoods of the Manistee National Forest (Host et. a1.
1988) . The ECS stands were chosen with specific
rejection] acceptance criteria to minimize differences in stand
histories. ‘The criteria set for these stands were as follows:

1) Stand ages between 60-80 years.

2) High stem density ("normal stocking").

3) Minimal evidence of disturbance since stand

establishment.

4) Minimum size of 2.47 acres.

5) Aspen (Populus grandidentata and Populus tremuloides)

accounted for less than 7 nF/ha in stand basal area.

These criterion, along with landform maps, aerial photos,
and the USDA Forest Service database were used to develop a
list of suitable stands. Stands were chosen using a

stratified random design, with strata defined as three

19

20

/ C = Lake Slates TWIGS calibration
T = County with test data

’ B = Both calibration and test data

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

C J
/BB 0 c)
iTB L
B
c
l
/C ‘7 L

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 1. Location of TWIGS calibration sites
and test sites in Michigan

21

major landforms: outwash plains, ice contact hills and
moraines. Once a stand was located, evaluated, and accepted
four prism points were randomly established within each stand.
These points were all referenced to a central soil pit in each
stand. Tree species, dbh (inches), total height (ft.) ,
merchantable height (ft.), crown ratio (%) and crown class
were measured on all tally trees 1" and above dbh. Increment
cores were taken from a sub-sample of trees to estimate an
average age for site index (base age 50 years) and ten year
radial growth. Average species site index for each sample
point was estimated using equations developed by Hahn and
Carmean (1982).

In the summer of 1986, 50 of the original ECS plots were
selected and converted to permanent plots. The method of
selection was a random sampling stratified across the 10
ecological land type phases (ELTPs) developed by the ECS
project (Table 1). These plots fell within the boundaries of
Manistee, Wexford, Mason, Lake and Newaygo counties. In 1986,
selected sites were relocated, mapped, sample points were
permanently marked, and all tally trees 1" dbh and above were
retagged and measured. Methods used in marking the plots
included installing rebar at point centers, tagging tally
trees.with numbered aluminum tags at the base of the trunk,
and marking three witness trees per point so that
triangulation could be used to find plot center. The only

short-coming to this process was that distances and azimuths

22

were not recorded from point centers to witness trees.

 

Table 1. Definition of lLTPe
1. Pin oak-white oak/Deschampsia on excessively well-
drained sands of outwash plains.
10. Black oak-white oak/Vaccinium on excessively well-
drained sands of outwash plains.
12. Black oak-white oak/Vaccinium on sub-irrigated
excessively well-drained sands of outwash plains.
20. Mixed oak-red maple/Viburnum on well-drained sands
on moraines, kames, and outwashed plains.
21. Mixed oak-red maple/Viburnum on well-drained sands
underlain by sandy loam bands on moraines, kames,
and outwashed plains.
35. Red oak-red maple/Viburnum on well-drained sands
with fine loamy substrate on moraines and lacustrine deposits.
37. Red oak-red maple/Desmodium on well-drained sands
over loamy substrate on moraines and lacustrine deposits.
40. Sugar maple-beech/Lycopodium on well drained moraines.
43. Sugar maple-beech/Lycopodium on well drained sands
with fine texture substrate on moraines.
45.

Sugar maple-white ash/Osmorhiza on well-drained sands with fine
textured substrate on moraines.

Data from the 1986 field season was then compiled into a

database.

number,

Information compiled included stand number, plot

subplot (point) number, tree number, tree species,

dbh, total height, merchantable height (to a four inch top for

softwoods and eight inches for hardwoods), crown ratio, and

crown class.

23
W

The 1991 field measurements again focused on the 10 BAF
prism point. All points located within the original 50 stands
were re-tallied from the original point centers. If point
centers could not be located, approximations were made and
noted on point record sheets. Trees found to be in (i.e.
tally trees) in both 1986 and 1991 were measured for dbh,
crown ratio, and crown class“ Trees found to be 1991 ingrowth
(trees which were tallied in 1991 but not in 1986) were
measured for these characteristics as well as total height and
merchantable height. Dbh measurements were made at 4.5 feet
using the same staff used in 1986 as a guide. If cankers or
other deformations interrupted dbh measurements at 4.5 ft.,
adjustments were made and recorded on point record sheets.
These adjustments usually involved moving the point of
measurement above the deformity. All crown ratio and height
measurements were made using the Spiegal Relaskop. Crown
class was ocularly estimated.

In addition to these measurements, observations were also
made on disturbance conditions within each stand. The nature
of these disturbances ranged from forest pest infestations to
human disturbances. All disturbances were recorded on the
1991 point record forms.

White Oaks (Quercus alba), red oak species and Bigtooth
aspen (Popul us grandidenta ta) throughout several of the stands

showed signs of defoliation caused by gypsy moth (Lymantria

24
dispar). American Basswood (Tilia americana) on all plots
were defoliated to some degree by Lindon looper (Erannis
tiliaria). All cases of defoliation were recorded for each
tree on the plot tally sheets. Degree of defoliation was
recorded as an ocular estimate of the percentage of crown
defoliated.

Sources of human disturbance were either active forest
management or vandalism. Stands that had evidence of thinning
or other management practices (i.e. slash piles, etc.) were
noted in plot records. Such stands were dropped from further
analysis, as thinning and other similar silvicultural
practices alter growth potentials in residual trees (Smith
1986). These influences on the natural growth rates of the
research plots were not desired. In addition to management
disturbances, several plots were defaced by vandals. These
plots were reconstructed as accurately as possible using 1986

records and noted in plot records.

D ! E l i g .1 l' .

The database used for the 1986 field data was enlarged
after the 1991 season to incorporate the new field data. In
addition to adding the variables mentioned above, five year
change in dbh and 1991 tree status were also added. The
resulting database consisted of 2,495 tree records from fifty

stands.

25

The process of developing the database into work-sets to
meet research objectives began by eliminating all trees that
were accounted for as mortality from 1986 and all ingrowth
1991 trees. This left tree records with complete five year
growth.data. Stands which had signs of active management were
deleted next. Finally all plots (groupings of 4 to 6 prism
points within a stand) that were disturbed and could not be
reconstructed were deleted. The final dataset contained trees
from 44 of the original 50 stands.

At this point the subplot (or prism point) was adopted as
the sample unit. This provided for a maximum number of
samples which could be used to produce both individual tree
data (i.e. dbh, crown ratio) and stand level data (i. e. basal
area). When split into subplot-units the 44 stands produced
176 sample units.

Next the database was randomly split into a calibration
subset and a validation subset. The calibration data would be
used to construct alternative growth models, if necessary, and
the validation data would be used to validate TWIGS and
compare TWIGS results with the results of any alternative
models. The process used for splitting the database was a
random sample stratified across ELTPs. As for the proportion
to be allocated per subset, Snee (1977) reports that the
common practice involves a fifty percent split between data
sets. However, due to the small sample size (176 subplots),

it was decided to allocate one-third of the data to the

validation subset and two-thirds to the calibration set. The

total number of subplots and their distribution is listed in

 

Table 2:
Table 2. Subplot Distribution per Data Set
Number in Number in
ILTP Total Subplota validation Set Calibration Set
1 16 S 11
10 14 5 9
12 12 4 8
20 18 6 12
21 19 6 13
35 26 9 17
37 20 7 13
4O 22 7 15
43 11 4 7
45 18 6 12
_T0ta1 176 S9 117

 

Model validation and Development:

To fulfill research objectives a predetermined series of
steps were followed. The first of these procedures was to
examine the TWIGS growth model using the validation subset.
After obtaining results, new growth models were developed
using the calibration subset. Validation data was then to be
used to test the new models and results would be compared to
Lake States TWIGS. However, model development was
unsuccessful, so a final validation of TWIGS was performed

using the combined calibration and validation data.

26

27
12195.!ALIDATIQHL
DATA:

Before model validation began, tally tree characteristics
from each point's 1986 measurement were expanded to a per acre
basis using TREEGEN (Miner et. al., 1988) and then five year
projections were made using TWIGS. Projections were made
using TWIGS' "individual tree option" and actual crown ratios
were used to avoid over-prediction in dbh growth as cited by
Holdaway and.Brand (1983). TWIGS' results were grouped in the
five diameter class option (an option in the initial program
setup menu). These diameter classes were 0.0-2.9", 3.0-4.9",
5.0-10.9", 11.0-16.9", and 17.0"+. These sizeclasses were
selected because they produced the most detailed individual
tree output, and because validation results could be directly
related to TWIGS' output without any adjustment. General
recommendations for best results with TWIGS given by Miner et.
a1. (1988) were followed. Each.tree list included at least ten
trees, tree lists were dominated by trees over 2" dbh, and
projection intervals were set to five years.

The five year projections were compared to 1991 field
measurements in the validation process. Evaluations involving
dbh were based on differences between projected versus actual
diameter for individual trees, while basal area and trees per
acre projections were compared to density calculated from the
1991 point sample remeasurements. The methods used to convert

point sample data into pseudo-fixed radius plots for

28
comparisons follow the recommendations of Grosenbaugh (1958)
and.Beers and Miller (1964). Only tree data from the original
1986 tally trees were used for 1991 basal area and trees per
acre expansions. Trees which were too small to be tallied in
1986, but which grew on to the plot in 1991 (ongrowth) were

ignored.

Validation Statistics:

Tree and stand characteristics selected for evaluation
were individual treetdbh, stand basal area (BA), and treestper
acre (TPA). Dbh was chosen to evaluate the performance of
TWIGS' individual tree diameter growth model. TPA was used to
judge TWIGS' projections of tree mortality, and stand BA was
used as an indicator of overall model performance (Kowalski
and Gertner, 1989). The basic sample unit for dbh was the
individual tree, while BA and TPA were evaluated at the stand
level.

TWIGS validation was carried out with five northern
hardwood species: northern red oak (Quercus rubra L.); other
red oak (Quercus velutina Larmarck and Quercus palustris
Muenchh); white oak (Quercus alba L.); sugar maple (Acer
saccharum Marsh.); and red maple (Acer rubrum L.). Sample
sizes varied widely across species. More than a dozen species
were tallied across all plots, but it was felt that these five
species were the only ones with ample representation to allow

testing.

29

TWIGS' volume equation was not tested because the
equation used for acceptable tree class volume (Raile et. al.,
1982) incorporated coefficients developed for Michigan's Upper
Peninsula. These coefficients are the same for northern red
oak, white oak, and other red oak, unlike the coefficients
developed for the Lower Peninsula.

Initial validation procedures (a second validation
followed.in the study) were carried.out using error statistics
described by ‘Kowalski and. Gertner (1989). These error
statistics were selected to have a basis for comparison with

other studies of TWIGS.

Error Statistics:

The error statistics used were (Kowalski and Gertner,

1989):

Mean Error = [8(0£‘P1)1/n (A measure of bias)

Mean Percent Error = lOO[2((O,-P1)/01)/n] (A measure of precision)
Standard Deviation of Error = [[}3(e1--e)2]/(1'1-1)]1/2

Where:
Pi - TWIGS prediction
Oi - Observed value
n - sample size
ei - error, defined (Oi-Pi)

e - mean error, defined as (I ei)/n for all 1

Due to the nature of the equations, overestimations
appear as negative values, while underestimations have
positive values. Sample sizes and results are shown in Table

3.

3O

 

Table 3. Initial TWIGS Validation Results (5 years)
Characteristic Species n Mean Error STD DEV Percent Error
DBH ilndhe§l N. Red Oak 197 .057 .54 .

BA (sq.ft./acre) 35 .71 8.93 .60

TBA 35 21.89 91.48 3.13

DBH (inches) 0. Red Oak 64 -.Ol .79 -.22

BA (sq.ft./acre) 16 5.10 7.26 9.38

TBA 16 16.99 31.81 13.36

DBH (inches) White Oak 98 -.24 .84 -1.88

BA (sq.ft./acre) 32 -2.52 4.68 -8.90

TBA 32 -13.17 92.08 -16.80

DBH (inches) Sugar Maple 53 -.16 .22 -2.i7

BA (sq.ft./acre) 13 -.48 9.25 -10.26

TBA 13 -23.44 115.80 -22.01

DBH (inches) Red Maple 51 .Ol .31 .13

BA (sq.ft./acre) 23 1.11 8.67 .11

TBA 23 8.23 44.33 .32

 

Overall, the validation sample sizes were small for a
comprehensive evaluation of a model so widely used. However,
Lakes States TWIGS did perform well in regards to diameter
projections for all five species. Percent mean errors for 5
year diameter predictions were less than 3% percent for all
species. White oak showed the highest. mean error for
estimated dbh, a .24 inch overprediction in dbh over five
years. BA and TPA results were less accurate. Percent mean
errors were over 8 percent for BA of three of the five species
tested, although northern.red oak and redtmaple were less than
1 percent. Three of the five species tested also had TPA
percent mean errors of over 12 percents ZMean errors for basal
area projections over 5 years ranged from an underestimation
of 5.10 ftF/acre (other red oak) to an overestimation of 2.52
ftFIacre (white oak). Mean errors for trees per acre ranged

from underestimations of 21.89 trees per acre (northern red

31
oak) to overestimation of 23.44 trees per acre (sugar maple)
for the five year period.

Although diameter projections were not grossly
inaccurate, it was thought that a new diameter growth model
would improve accuracy. The primary reason for this focused
on the role of basal area in the Lake States TWIGS diameter
growth. model. Basal area is an integral part of the
competition modifier (equation 1b) (Holdaway, 1985) , which is
multiplied by the product of the potential diameter growth and
the adjustment factor' to jproduce annual diameter' growth
(equation 1) . Given the five year errors in basal area
projections, the possibility exists that longer term
projection errors will be compounded. In addition, errors in
TPA projections suggest a flaw in the Lake States TWIGS
survival model. Given that TPA and dbh are the components of
basal area, the survival model has a direct link to dbh
projections. Considering these circumstances, a new diameter
growth model that avoids the use of the current competition

and survival functions could improve accuracy.

Wits}.

Before model development began one final alteration to
the calibration data set was performed. This alteration was
to eliminate all conifers and noncommercial tree species from
the data set and then split the remaining trees into species-

specific sets. The reasoning for eliminating the conifers and

32
noncommercial tree species is inherent with the goals of the
study: evaluation and development of diameter growth models
for upland hardwoods. Species-specific data sets were
necessary to develop models with unique coefficients for each
species. This is a logical step considering differences in
growth patterns among species (Oliver and Larson, 1990; Godman
et.al., 1990: Walters and Yawney, 1990; Tubbs and Houston,
1990; Schlesinger, 1990; Rogers, 1990; Sanders, 1990a; Sander,
1990b). Splitting the data produced five species-specific

data sets (Table 4).

Table 4. Calibration Data: Sample Size by Species Group

 

___§p_eeis§ MW
Northern Red Oak 457
White Oak 258
0. Red Oak 176
Sugar Maple 157
89

 

The five species data sets contained a combined total of

1,137 trees and the variables listed in Table 5.

33

 

Table 5. variables Used in Model Development
...!eriablei Definition
DBH86 1986 diameter at breast height (inches)
DBH91 1991 diameter at breast height (inches)
DD Five year change in diameter (inches)
BAGSYR Five year change in tree basal area (ft2)
CR Tree crown ratio code (0-10%=1, 11-

20%=2, ... 81%+=9)

CC Crown Class (ordinal variable)

SI Site Index (ft. at base age 50 yrs.)

TBA 1986 tree basal area (ft9/acre)

BA 1986 basal area of stand (ft?/acre)

DENSE 1986 tree basal area divided by 1986 stand
basal area (used to describe tree position
in stand).

-,--7c: a.1o't1-e . as- oaqi a_ a _::.-

In addition to these variables, various transformations
were made during the modeling process to improve the linear
relationships between independent and dependant variables.
These transformations were either a natural log transformation
or using the variable inverse. Any variable altered in either
of these'ways was noted with the prefix "LN" for log transfor-

mation or "INV" for inverse.

It is also important to note that the initial data sets

did not include stand level variables other than BA. This is

34
because 40 of the subplots were missing center points in 1986
(roughly 19%). Basal area measurements were estimated for
these points in 1986, and these measurements were retained.
However, stand level variables including trees per acre,
average dbh, and quadratic stand diameter (QSD) could not be
calculated. Due to the already small sample size, it was not

deemed prudent to eliminate more records.

The Multivariate Linear Model:

Initial attempts at developing alternative diameter
growth models centered on the multivariate linear model. This
format was chosen for ease of development and overall flexi-
bility.

The conceptual model for this approach was:

DGROW = bo+ blTD + szOMP + b3VIGOR + b,QUALITY

Where:
DGROW = 5 year increment of diameter or tree basal
area growth
TD = Current tree diameter or basal area
COMP = A representation of competition in the

stand
VIGOR. = A representation of tree health and vigor

QUALITY = A representation of site quality

35

These factors ‘were. chosen for ‘the conceptual. model
because they are thought to have an effect of diameter growth
(Spurr and Barnes, 1980), are integral components of estab-
lished.growwmodels.(Miner et. al., 1988; Shifley, 1987; Wykoff
et. al., 1982; Hilt, 1983), and.they can be represented in the
database. The TD component can be either dbh, tree basal area
or a derivation of either. Stand basal area, the DENSE
variable and similar variables which represent stand structure
and competition were suitable for the COMP component. Crown
ratio and.class are the only variables that may be represented
as VIGOR from the data set. Site quality variable options
were either SI or ELTP.

Before model development began, assumptions of normality
were tested for the individual variables in each species set.
A series of KolmogorOV' - Smirnov 'tests (Conover, 1980)
indicated that less than 10% of the variables were normally
distributed. Stem-leaf diagrams were also constructed to
produce a graphic view of the distributions. The stem-leaf
diagrams appeared normally or near-normally distributed.
Given these findings and the robustness of ordinary least
squares procedures, model development continued with the
assumption of normally distributed data.

After normality was judged, summary statistics were run
on each species data sets (Table 6). Scatter plots were
constructed with each independent variable graphed versus the

dependant variable (5 year diameter change). The summary

36
statistics showed that variable means and ranges differed
widely between species. This further supported the decision
not to combine species data. The scatter plots, Appendix A,
all showed slight linear relationships between the dependant
and independent variables. For the most part, however, these
relationships provided little evidence that a linear model
would be successful. A number of transformation were also

tried on the variables with little success.

 

 

Table 6. Summary Statistics for Calibration Data Set
Species n Statistic DD DBH86 DBH91 BA CR DENSE
(....incthd
N. Red Oak 457 Minimum -0.20 3.10 3.10 40.00 1.00 .000
Maximum 2.30 33.60 35.10 190.00 5.00 .038
Mean 0.57 12.77 13.35 126.67 3.69 .008
Std Dev 0.32 4.45 4.68 29.59 3.94 .006
O.Red Oak 176 Minimum 0.00 4.20 4.20 40.00 1.00 .000
Maximum 2.22 23.20 23.60 170.00 6.00 .048
Mean 0.53 12.28 12.81 102.77 3.28 .010
Std Dev 0.96 4.17 4.17 22.35 0.95 .008
White Oak 258 Minimum -o.2o 2.60 2.80 40.00 1.00 .000
Maximum 1.00 22.20 22.80 190.00 6.00 .028
Mean 0.23 9.37 9.60 111.63 2.93 .005
Std Dev 0.22 3.58 3.69 28.72 1.13 .005
Sugar Maple 157 Minimum -0.20 2.10 2.10 70.00 2.00 .000
Maximum 1.90 21.10 21.10 190.00 8.00 .016
Mean 0.29 8.07 8.35 145.10 4.17 .003
Std Dev 0.27 3.18 3.28 33.18 1.14 .003
Red Maple 89 Minimum -0.10 2.30 2.40 50.00 1.00 .000
Maximum 1.10 17.10 17.30 190.00 7.00 .014
Mean 0.33 6.57 6.89 126.74 3.75 .003
Std Dev 0.26 3.49 3.62 29.11 1.33 .003

 

Because the independent variables in the data set had
poor linear relationships with five year change in diameter
(and 'with. the log' transpose of diameter change), model

construction focused on trying all possible variable combina-

37
tions versus diameter growth (both transposed and not). It
was not assumed that this approach would yield adequate
results (Coefficient of determination desired above .8) ,
however it was the only reasonable chance of remaining with
the simple model form.
The results of this approach were not satisfactory. The

overall "best" model, based heavily on R”, was:

DD a ho + blLNDBH86 + b2*1/BA + b3DENSE + men +bkELTPk
WHERE: DD = 5 year change in diameter growth
LNDBH86 = Natural log of 1986 DBH
BA = 1986 Basal area (ftF/acre)
DENSE - Ratio of tree BA to stand BA 1986
ELTPt := Dummy variables
representing ELTP membership

b0. ..bk = Regression coefficients

Regression analysis were run for each of the five species
data sets and species-specific models were constructed. The
R2's for these models ranged from 0.194 to 0.507 (Appendisz).
Standard error of the estimated dbh's ranged between 0. 158 and
0.240 inches. Student's t-test varied depending on the
variable and species; only log(dbh) was consistently
significant for all models. The ELTP variable appeared a
combined thirty-three times throughout the five models, and in

only eight occurrences did it test significant (alpha level of

38

0.10 or below). Residual plots for the three oak species
(Appendix C) appeared randomly distributed and suggested no
quadratic or other relationships overlooked. Residual plots
for the two maple species had patterns that suggested
heteroscedasticity; the distribution of residuals around 0
increased as X increased in value. Weighing observations in
regression or transforming the dependant variable can correct
heteroscedasticity (Rawlings, 1988). Because the oak models
had poor fit statistics and any corrections for the maple
models would produce a model inconsistent with the oak models,
corrections for heteroscedasticity were not made.

The results of this approach were unsatisfactory, however
further efforts were made in an attempt to remain with the
mutlivariate linear form. These final efforts centered on
using forward and backward stepwise regression procedures to
find a model format that would best fit all species. .Although
these procedures are often discouraged because they find the
best statistical model and.not.the best.biological.mode1, they
were used because all variables were considered biologically
significant and these methods offered the best option for a
solution. Several regressions were run using these procedures
and the different species data sets. The results produced no
overall model form that would fit all species. The process
was abandoned.

Before moving to other possible models it is important to

discuss why this method provided poor results. The only

39
variable that produced consistent significance of 0.05 or less
was log(dbh). Some form of dbh (or tree basal area) is
present in all published diameter growth models. Given this
fact and that it was successful in the multivariate model,
there is no indication that the poor performance can be
attributed to this variable.

Basal area and DENSE proved erratic in significance
between models. Given the relationship between dependant and
independent variables in the scatter plots this was expected.
There may be several reasons why. The inverse of basal area
was used due to the relationship of this value with change in
diameter. The greater the BA the smaller the value of the
inverse, which should be matched with a smaller diameter
growth (Appendix A). This would coincide with the fact that
a stand with lower density allows the individual tree more
resources to grow. Overall stand basal area as calculated in
prism sampling may not represent the actual density conditions
faced by an individual tree. Using the prism, trees are
tallied as a result of their distance from plot center and
their diameter. Therefore trees are tallied from a variety of
areas within a stand. The stand densities from the micro-
environments that these trees occupy may differ considerably
from the overall stand.

This would also apply to the DENSE variable (ratio of
tree BA (TBA) to stand BA) as stand BA is a component. In

this case, the larger the tree diameter the higher it's basal

40

area (the numerator), therefore the DENSE value is greater.
This provides for a larger DENSE value to coincide with a
greater diameter growth, which would agree with the TBA vs. DD
scatter plot (Appendix A.) Combining the effects of TBA over
stand BA was an attempt to represent the tree's position in
the stand. Modeling results show that this attempt was not
successful.

Crown ratio also proved erratic in significance. For the
maples and white oak, crown ratio variables were significant
at levels of at least 0.10, while for the red oaks they were
not significant at any reasonable level. Possible
explanations for this include crown structure and.neasurement
error. Crown ratio measurements indicate that the proportion
of the tree in foliage but does not account for the actual
leaf area and structure of the crown. These differences in
leaf area and structure were most prevalent among red oaks,
varying with site quality. On the more mesic, higher quality
sites, red oak crowns were full and had extended forms. On
the drier, poor sites where red oak species dominated the
hardwoods, crowns were more cylindrical in form and gaps in
foliage were prevalent. Trees with identical crown ratio
values occupying these different sites would not necessarily
have the same leaf area and photosynthetic potentials.
Another reason for the differences in CR significance among
species models may be measurement error. Of all the data

collection techniques used, crown ratio measurement was the

41

most variable in terms of accuracy. Even with the aid of a
clinometer or other similar device, crown ratio is an ocular
estimation. In addition, crowns that have gaps and other
irregularities often have their crown ratio values adjusted to
compensate. These factors leave room for guess work and
bias. The result is a variable open to a large degree of
error» Given the circumstances of red oak crowns on the study
plots, this may explain part of negative results.

The final variable in question is ELTP. This variable
was rarely significant in the models, and when significant
there was no discernable pattern between species and site
qualityu This would suggest that either the use of an ordinal
variable:does not fully capture the influences of the site, or
that the differences between ELTPs do not fully differentiate

between site quality potentials for these stands.

The Modified Chapmann-Richards Function:

The next approach to developing a diameter growth model
followed the work of Hilt and Teck (1987). Using data on
5,313 northern hardwood trees from 1/5-acre forest inventory
and analysis plots in New England, Hilt and Teck developed a
diameter growth model based on a modified Chapmann-Richards

function. Their's was a three stage model as outlined below:

42
POTBAG = b1*SI* (1. 0-exp(-b2*DBH) )
BAG = po'raacv: (exp (-b3*BAL) )
DGROW - scam (DBH2*C+BAG) /c) -DBH
Where: POTBAG a Potential basal area growth
BAG 8 Basal area growth

DGROW

Diameter growth
SI = Site Index

DBH - Diameter at breast height

BAL - Basal area greater
than tree size
class

C = Constant (.005454)

b1. . .b3= Unique coefficient

The potential growth function (POTBAG) was calibrated
using the top 10% of trees in terms of basal area growth per
species. To repeat this procedure on this study's data would
involve relatively small sample sizes for model development.
Using the sample sizes shown in Table 2, the 10% level used in
POTBAG calibration would produce sample sizes from 44 to 3.

Although there were concerns over sample size, model
calibration for POTBAG was done for northern red oak data.
The adjusted R? for this model was 0.20. With so little of
the variation in.Y being explained by the model, and.the small

sample sizes, it was decided to abandon this model.

43
Species/ELTP specific Modeling:

The poor performance of the initial modeling approaches
lead to a reexamination of the data sets. ELTPs were initial-
ly represented as dummy variables, with little success. These
results were not expected. ELTPs are delineated by landform,
climate, geology, and vegetation; therefore, they are
associated with site quality. This association with site
quality can be seen in the field by comparing stand
composition and condition between the high and low range
ELTPs. ELTP 45 stands are composed of well stocked sugar
maple-ash stands, while ELTP 1 typed stands have a major
(poorly stocked) jack pine - black oak component. This
correlation with site quality was expected to appear in the
models, however ELTP variables were rarely significant.

The fact that ELTPs performed poorly as variables led to
the reconstruction of scatter plots. In the process the
species-specific data sets were further divided into
ELTP/species-specific data sets. In addition, the diameter
growth variable was converted to five year basal area growth
(individual tree) or BAG. When BAG was plotted versus DBH86
the resulting plots appeared to have a slight J-shaped curve.
This curve was further exaggerated when the independent
variable was changed to DBH862. This pattern was very close
to the plots Hilt (1983) described, 'Thus, the next attempt at
producing a diameter growth model focused on Hilt's conceptual

model:

44
BAGSYR = f(DBH s1 D p5)
Where:
BAGSYR - five year basal area growth (ft?)
DBH a diameter at breast height (inches)
SI - site index (ft. at base age 50 yrs.)
D - mean stand diameter (inches)

PS a percent stocking

(Calculated with Gingrich' s stocking equation
(1967) :PS,,--.005066 + .016977DBH1,+

.003168iDBHulz)
Hilt worked with tree data from seventy-seven 0.25 to 1. 0
acre permanent growth and yield plots. The following equation

was fit to each stand using individual tree data:

BAGSYR = 1::1 * DBH2
Where: iBAGSYR.= 5 year tree basal area growth (ft?)
DBH = diameter at breast height squared
(inches)
b, = unique coefficient per stand
He then fitted the b,s to the following equation:
bi '3 913192 * exp(g3D+g,PS)
Where: tn = unique coefficient per stand
SI = site index (ft. at base age 50 yrs)
D = mean stand diameter (inches)

PS - percent stocking

(Calculated with Gingrich ' s stocking equation
(1967) :PSU=-.005066 + .016977DBH15+
. 003168 (DBH,,)2)

g1. . .g, = unique coefficients

45

The result is a two stage model where site and stocking
effects are considered in the second equation, which in turn
is used for basal area growth projection. in the first
equation.

Initial work with this model focused on fitting the white
oak data sets to the BAGSYR model. White oak was chosen at
random from the five species available. Results were
positive, with st ranging between 0.518 and 0.772. Problems
arose, however, when the second stage model was to be fitted.

The stage two model is composed of stand level
independent variables. The trees in the ELTP/species data
sets came from several stands brought tOgether to form a
composite stand with similar characteristics. This grouping
provided for the largest. possible sample size involving
species from the same ecological environments. It does not,
however, provide for stand level variables. This prevents the
second equation from being used.

There were two possible solutions to this problem. The
first was to redefine the data sets to be species-specific
sets with trees only from individual stands. The second
solution was to find a replacement for the "tn" equation. As
the first solution would produce a large number of relatively
small data sets, it was deemed impractical. Further work was
then directed towards finding possible substitutes for the
second equation.

In pursuing a substitute for the second equation,

46
reasonable alternatives had to be found to capture the
functions of the variables in ‘that equation. The first
variable examined for replacement was percent stocking (PS).
1986 stand basal area (BA86) was thought to be a substitute
for PS, a stand density variable. Scatter plots and new
models, however, failed to demonstrate any relationships
between the dependant variable (in any form) and the
independent variable BA86 or any of its transformations.
Considering the success of the ELTP/SPP - specific data and
equation one, the data sets were further stratified by basal
area. This was accomplished by developing three basal area
classes that could be associated with stocking levels. These

classes and their associated stocking levels are shown in

Table 7.
Table 7. Basal Area Classes
Eggs; Area glasses (ztzlagrg) Associated fitgcking Level
Basal Area > 130 Highly stocked
Basal Area 90 to 130 Normally stocked
M < 90 UWG

 

 

After the data sets were split the BAG equation was
fitted with the new white oak data. The resulting st ranged
from 0.426 to 0.828. This increase in the coefficient of
determination prompted further calibration of the northern red

oak data sets.

47

The product of this line of modeling was a simple
equation that explained a large portion of variation in the
dependant variable (BA65YR) . Instead of expressing stand
density as a independent variable, basal area characteristics
are included as an integral part of the data set. This same
concept is further brought into play with the site quality
variable. Initially site index was represented as an
independent variable in the second stage equation. However,
by breaking the data sets down across ELTPs, the maximum range
in site index between sites per data set was 15 feet, with a
majority of the ranges being within 10 feet. Therefore, the
need for an independent variable was circumvented by
expressing site quality in the data set.

Although initial results seemed positive, there were two
principle shortcomings to this model. The first is the small
sample size involved with the calibration of each model (Table
8). The initial species-specific data sets were relatively
small to begin with (Table 2). After stratifying each
species-specific data set further down to the SPP/ELTP/BA
level, and.after'removing outliers, the final data sets ranged
in size from 5 to 69 trees. It is doubtful that models
calibrated with so few trees have any practical use.

The second shortcoming was the large number of model
coefficients needed to represent all the specific tree
groupings. This unmanageable aspect resulted in halting the

modeling process to look at still other options.

48

 

Table 8. Results of BAGSYR Models
QAIA £3! E BETA BQQQARED gs:

GZOB 25 1.4E-05 0.696 0.001354
F20A 7 5E-07 0.478 9.37E-05
F208 12 1.3E-06 0.622 0.000169
F20T 19 1E-06 0.518 0.000158
E20A 8 1.9E-06 0.828 6.89E-05
E20B 10 5E-06 0.798 0.000299
E20T 20 3.9E-06 0.755 0.000222
D20A 7 3E-06 0.892 0.000106
020B 28 1.9E-06 0.68 0.000114
020T 37 2.1E-06 0.702 0.000119
C20B 39 5.2E-06 0.679 0.00034
C20C 12 4.6E-06 0.426 0.000777
C20T 48 4.8E-06 0.656 0.000351
820A 16 1.5E-06 0.717 7.52E-05
B20B 14 1.5E-06 0.428 0.000115
BZOC 6 3.7E-06 0.659 8.42E-05
B20T 38 1.6E-06 0.563 9.44E-05
A208 14 5.7E-06 0.805 0.000184
A20C 23 8.7E-06 0.748 0.000371
A20T 24 7.9E-06 0.772 0.00033
J21A 5 1.26E-05 0.898 0.002394
121A 5 2.04E-05 0.632 0.011288
1213 28 9.8E-06 0.879 0.00117
H21A 8 2.8E-06 0.899 0.0001
H213 44 1.15E-05 0.892 0.000532
621A 13 1.43E-05 0.756 0.001768
621B 69 1.52E-05 0.738 0.002299
G21C 15 1.71E-05 0.894 0.001396
F21A 46 1.39E-05 0.803 0.000988
F21B 47 1.37E-05 0.841 0.00103
F21C 42 7.8E-06 0.712 0.001245
E21B 15 5.7E-06 0.68 0.000795
E21C 5 1.22E-05 0.701 0.000582
D21B 28 8.9E-06 0.757 0.000862
D21A 8 1.4lE-05 0.921 0.000584
C21B 8 1.64E-05 0.982 0.000663
B21A 17 1.73E-05 0.823 0.000408
3218 20 1.24E-05 0.919 0.000357
D223 397°W
KEY: G = ELTP: 45 43 40 37 35 21 20 12 10 1
JIHGFEDCBA

## = Species: 22 = O.Red Oak

21 = N.Red Oak

20 = White Oak

B = BA (ft2/acre) range: A.= 130)
B = 90 to 130
C = <90

T

Full Range

49
Stand Level Variables:

The last effort in the modeling process was to re-examine
the parent database and calculate stand level variables.
Because several of the prism points were missing plot center
in 1986, stand.basal area was estimated.and several trees were
unaccounted for. Therefore, there was initial concern that
calculating these variables would require eliminating some of
the 40 sample 'points, thus decreasing' an already' small
database. It was decided that this was the last chance at
developing an alternative diameter growth model. The

variables calculated are defined in Table 9.

Table 9. Stand Level variables

 

Variable Formula
Average Stand Diameter A5086 = EdbhilN
Where: dbhi = Individual tree diameter
N = sample size
Quadratic Mean Diameter QSD = SQRT(1/N * Bdbhfl)

Where: dbhi = Individual dbh
N = sample size

Basal area greater than tree class GBA = Stand BA - EAL
Where: BAL = Basal area of current
size class and less.

Trees per acre An expansion of point data

The quadratic stand diameter and trees per acre (TPA)
variables should reflect stand density. The basal area
greater than tree class variable should determine a tree
class' position within the stand. ASD86 represents the
average tree diameter within a stand and therefore may be

useful in explaining aspects of stand structure.

50
Once derived, all stand level independent variables were
plotted versus the dependent variable and its transformations.
These plots were made at all levels of the data sets (i.e.
species - specific sets, SPP/ELTP sets and SPP/ELTP/BA sets.) .
Inspection of the scatter plots revealed no distinct patterns

at any level.

Modeling Termination:

The choices in model development to this point were three
models that were thought to be best able to provide an
alternative to Lake State TWIGS diameter growth model. Poor
results and. sufficient. data (i.e. sample size, variable
selection) resulted in no adequate replacement developed.
Other possibilities still exist, such as Mawson's (1982)
models and the predecessors to the current TWIGS' model (Leary
et. a1. , 1979) , however, time constraints limited their
exploration. Since the small sample sizes resulted in the
initial validation of the TWIGS' model to be far less than
comprehensive, it was decided to use all available data to
perform a larger scale validation of the Lake States TWIGS

growth model.

51

2° JIEIIEEVJ'Z !° .

DATA:

Originally one-third of the database was set aside for
validation. However, after the modeling phase failed to
produce satisfactory results the calibration and validation
data sets were combined into one validation set. Summary
statistics for the combined data are presented in Table 10.
The initial sample unit for validation remained the plot
(prism point), with the individual tree being used for
diameter comparisons. As with the initial validation, all
plots were expanded to the per acre basis using TREEGEN and
five year projections were run with TWIGS. The same methods
were used in each validation in regards to TREEGEN and TWIGS

options and data manipulation.

52

 

Table 10. Summary Statistics for the 156 Sample Points by Strata (ELTPs)
Site IndexLL
ELTP n2 STT TPA: BA AVEDBHE W0 NRO BO R_M fl WA AGE
I 12 mean 2I8 86 8.1 ‘44 -51 74
min 113 50 7.2 37 42 71
max 321 120 10.4 53 63 75
10 13 mean 514 108 6.3 40 53 48 79
min 94 50 4.3 27 45 44 70
max 1113' 170 10.6 55 62 54 95
12 11 mean 299 95 7.8 49 59 71
min 126 60 5.3 40 52 61
max 601 140 10.9 63 70 84
20 19 mean 328 114 7.7 54 59 58 84
ndn 189 70 4.3 48 45 40 77
max 788 160 9.9 63 75 70 93
21 17 mean 503 114 6.6 49 64 60 48 73
min 149 70 3.4 38 55 47 48 64
max 1302 180 10.8 57 71 73 48 83
35 24 mean 347 138 8.6 66 80 73
min 62 50 5.9 59 63 63
max 809 220 16.2 77 106 84
37 19 mean 146 114 11.9 64 87 67 75
min 76 80 8.3 51 72 44 66
max 289 150 16.1 73 103 88 79
40 17 mean 384 107 7.3 78 66 60 70 62
min. 146 70 3.6 65 52 49 50 59
max 785 150 11.7 83 80 67 82 65
43 8 mean 391 128 7.5 90 71 63
min 159 80 4.6 79 66 61
max 804 160 10.4 100 76 68
45 16 mean 424 139 7.7 77 74 84 62
min 123 80 4.8 77 57 73 56
Overall Mean 352 116 8.1 72
Std. Dev. 229.6 32.4 2.5 937

a WO - white oak; NRO - northern red oak; BO - black & pin oaks; RM

 

maple; SM -sugar maple; and WA - white ash.
b Number of sample points per ELTP
c trees/acre, calculated for all trees > 1.0" DBH

d.AVEDBH = arithmetic mean dbh

- red

53
Validation Statistics:

Tree and stand characteristics selected for evaluation
were again individual tree dbh, stand BA, and stand TPA. Dbh
was chosen to evaluate the performance of TWIGS' individual
growth model. TPA was used to judge TWIGS' projections of
tree mortality, and BA was used as an indicator of overall
model performance (Kowalski and Gertner, 1989). The basic
sample units also remained the same. Due to small sample
sizes, the same five species were selected for validation.

Final validation procedures were carried out in two
phases. The first phase was the use of error statistics
outlined.byflKowalski and.Gertner (1989). The second.phase‘was
characterized by the use of ATEST, developed by Rauscher
(1986) for testing prediction accuracy. This method was
chosen because it provides a means of constructing confidence

intervals around future prediction.

Error Statistics:

Depending on the characteristics judged, statistics were
calculated at two levels. Diameter growth comparisons were
made at an individual tree level with results aggregated by
TWIGS diameter classes within species (Table 11, Figures 2, 3
and 6). Errors for BA and TPA were calculated at the plot
level for each species. In this case the stand (expanded point
sample) was the sample unit. These errors were then compiled

by species to produce a summary (Table 12, Figures 4 - 6).

54

Mean errors for five year predictions of dbh varied by
species and size class (Table 11). In general, TWIGS
overestimated dbh growth across species and diameter classes.
All white oak and sugar maple size classes were overestimated,
while predictions for the other three species varied with size
class. When totaled across all size classes, northern red
oak, white oak and sugar maple dbh were overestimated and
other red oak and red maple dbh were underestimated. Overall,
red maple, other red oak, and northern red oak predictions
were the best (in descending order) with percent errors under
1% in all cases. These results were similar to Brand and
Holdaway's (1986) validation tests on Manistee data, white
oak and sugar maple had the largest over-predictions of dbh
growth. Prediction intervals for future projections of single
trees are given in Table 12.

Mean errors for TPA projections (Table 12) were all
within +35 trees per acre. White oak and northern red oak
performed best, with errors of +12 trees per acre
respectively. The validation of STEMSSS with Manistee data
(Holdaway and Brand, 1986) found that both species were over-
predicted by at least 20 trees per acre. Overall percent
errors were all under 20%. White Oak performed poorest with
an over-prediction of 18.6%. Northern red oak had the best
results with an underestimation of 3.9 trees per acre.

Errors for mean basal area were all within +5 ft2/acre

(Table 11) . Other red oak showed the most bias with an

55
underestimation of 4.4 ft2/acre. All other species were within
+3 ft2/acre. In terms of percent error, all species had less
than 12% in projection bias. Northern red oak had the lowest
percent error, 2.2% (underestimation). In lHoldaway and
Brand's (1986) validation of STEMSSS with Manistee data (10
year growth), northern red oak was shown to be overpredicted
by 4.7 ft2/acre; this study found an underprediction of 2.2
ft2/acre Over 5 years. Overall, percent errors were
relatively high for BA projections in comparison to dbh
projections. As noted by Holdaway and Brand (1986), slight
overpredictions in dbh growth will lead to overpredictions in
BA growth due to the nature of the relationship between the

two variables.

56

Table 11. Calculated Irror for lstissted.Dll (5 year projections) by Species and Size Class

 

S ecies n‘ Diameter Class Mean Error STD DEV 9 Error
N. R33 Dak T5 3.0 - 74:9 -0.15 0.10 -3.36
209 5.0 - 10.9 -0.20 0.92 -1.61
300 11.0 - 16.9 0.02 0.58 0.01
109 17.0 + 0.07 0.36 0.28
633 TOTAL -0.05 0.83 -0.56
Other Red Oak 11 3.0 - 4.9 0.03 0.02 0.55
87 5.0 - 10.9 -0.02 0.68 -0.11
95 11.0 - 16.9 0.07 0.20 0.45
28 17.0 + 0.12 0.30 0.06
221 TOTAL 0.04 0.46 0.25
White Oak 23 3.0 - 4.9 -0.17 0.11 -4.14
196 5.0 - 10.9 -0.26 0.76 -2.46
95 11.0 - 16.9 -0.08 0.23 -0.64
7 17.0 + -0.40 0.25 -2.01
321 TOTAL -0.20 0.61 -2.03
Sugar Maple 26 3.0 - 4.9 -0.25 0.88 -2.68
118 5.0 - 10.9 -0.23 0.27 -3.10
32 11.0 - 16.9 -0.15 0.26 -1.08
0 17.0 + -- -- --
176 TOTAL ~0.22 0.42 -2.67
Red Maple 42 3.0 - 4.9 -0.03 0.18 -1.04
60 5.0 - 10.9 0.09 0.46 1.22
17 11.0 - 16.9 -0.23 0.39 -1.88
6 17.0 + 0.12 0.32 0.64
125 TOTAL 0.01 0.38 0.01

a - number of sample trees

Table 12. Calculated Error for Estimated IA.and TlA.by Species (5 year projections)

 

 

Basal Area ftZ/acre Trees/Acre

Species Mean STD 4 Mean STD 9 Number

Error DE! Error Error DE! Error Points‘
N. Red Oak 2.2 7.6 2.2 12 43 3.9 97
Other Red Oak 4.4 5.8 8.6 33 87 14.4 56
White Oak -2.7 6.0 -11.5 -12 50 -18.6 65
Sugar Maple -1.0 10 5 -3.0 -16 78 -lO.9 38
Red Maple 2.2 5.0 8.4 21 45 7.7 95

a - number of sample points, out of 156, used for error calculations.

57

Figure 2 . Moan orror (Inchos) by DBll class and species.

125

0.75

a:
8

Mean Error 08H

Norlhorn Rod Oak

 

 

-125

I

b

l I

underprediction

 

 

i.

l l

MG’CGH

—
.—
h.

khan

 

 

125

'025

Meana'rorOBt-L

-0.76

-125

125

0.7 5

8 025
g ~025

-0.75

-125

 

 

 

 

 

 

 

 

 

 

A B C D
DBH Cbss
Other Red Oak
I I f T
underprediction
l.
- 4
° 0 i
l. q
+- A
l l l l
A B C 0
mi Class
White Oak
I l r
underprediction
A 4
l- 4 o a
l l L l
A B C 0

Sugar Maple

 

 

 

 

 

 

 

 

1.25 .
underprediction
0.75 L
'r
0.25 r
~025 - 0
-0.76 l-
J.
_125 l l J
A B C
DBH Class
Red Maple
‘25 I T I
underprediction
0.75 -
025 —
{ ii
-025 -
is
-0.75 '-
_125 l l L
' A B C
DBH Class

DBH Classes
A = 3.0 - 4.9 Inches
8 = 5.0 - 10.9 Inches
C = 11.0 - 16.9 Inches
0 = 17.0 + inches
(ranges are standard deviation)

 

 

58

 

‘ ‘L() l l l l l

underprediction

T

 

 

 

 

05 ’7‘ T .—
I
m
D “r
.‘9— . l
ULJ 00 (f 9 ~
c
g; t) 0
E
Jr.
'05 - _ _

 

 

 

 

._ 1.0 l l l l l
NRO ORO WO SM RM

Species ,

Figure 3. Mean and standard deviation for mean error of
predicted DBH over 5 years (positive values are underpredictions)

59

 

22(3 1 l l l I

underprediction

 

 

 

Mean Error BA/acre
o
i
i

 

 

 

 

 

-20 l l l l l
NRO ORO WO SM RM

Species

Figure 4. Mean and standard deviation for mean error of predicted BA/acre
over 5 years (positive values are underpredictions)

60

 

140 1 l i l l

undorprodicllon

7o~ _ a

 

 

 

T .
T
<( 1-
O. ( )
i“ (D
a 9
t O L ..
LU r) (5
C ..
<0 ..
o)
E
b i

 

 

 

 

—140 l l i L I
NRO ORO WO SM RM

 

Species

Figure 5. Mean and standard deviation for mean error of predicted TPA over 5
years (positive values are underpredictions)

Percent Error

61

30 .

 

20-

   

 

   

ITPA
' SBA

-30 I l l l l E DBH
NRO ORO RM SM WO

 

 

 

Species

Figure 6. Mean percent errors between observed and predicted values over 5

years (positive values are underpredictions)

62
ATEST Validation:

Rauscher (1986) developed ATEST, a computer program
written in Basic, as a tool to simplify the testing of
prediction accuracy. The program determines prediction bias
as the difference of prediction minus observed values.
Results are presented on both an absolute and percentage
basis. In addition, confidence intervals are calculated for
future predictions. The methods used for constructing these
intervals depend on thetdistribution of sample errors. If the
errors are normally distributed Student's t is used for
placing the intervals. If errors are not normally
distributed, a trimmed mean and a jackknifed variance is used
to determine the intervals.

Rauscher's ATEST requires programming before use and
demands a large quantity of user input while running. Gribko
and'Wiant (in press) have converted Rauscher's program into a
template for SAS statistical software, SASATEST. The template
requires minimal input from the operator: an ASCII file
containing'prediction.data and information on the level of the
desired confidence intervals. It was this version of ATEST
that was used in this study.

SASATEST validations were run using the same validation
data set used in calculation of the validation statistics.
Results of these runs included both percent and absolute bias
and standard deviations, as with the previous error

statistics. The only note here is that the ATEST results have

63

opposite signs (+/-) in comparison to previous statistics, due
to the nature of the calculations. In addition, prediction
intervals were calculated on future samples of size one, to
allow for some idea of how accurate a projection of a single
stand or tree would be. .Although.most of the work of SASATEST
duplicates previous efforts, the value of the future
prediction intervals warranted its use.

Results of dbh comparisons are presented in Table 13, TPA
comparisons are in Table 14, and Table 15 provides BA results.
Prediction interval values are interpreted as a 95%
probability that a mean of one future error will fall within
the interval presented (Gribko and Wiant (in press)). The
level of the interval was set at 95% because of the relatively
large range of the intervals. Another important note is that
data with error distributions judged non-normal (indicated
with "*" in the tables) :may have some discrepancies in sample

size. This is due to the jackknife procedures.

64

 

Table 13. SASATEST calculated Errors for DES
CCIFTDEWEE PREDICTION
INTERNAL INTERVAL
SPECIES DEE CLASS ERROR n BIAS (BIAS +le) (BIAS +le) STD DEV
. a . - . 5 .15 .06 .22 .10'
PCT 15 3.38 1.31 5.22 2.36‘
5.0 - 10.9 A83 209 .20 .13 1.82 .92
PCT 207 1.61 .96 13.83 7.00
11.0 - 16.9 A88 300 -.01 .07 1.14 .58
PCT 300 -.00 .50 8.69 4.41
17.0 + A85 327 -.07 .04 .71 .36
PCT 327 -.28 .18 3.31 1.68
Total ABS 633 .05 .05 1.35 .69
PCT 631 .56 .40 10.15 5.16
0. Red Oak 5.0 - 10.9 ABS 87 .02 .15 1.37 .68
PCT 87 .11 1.48 13.91 6.96
11.0 - 16.9 A83 95 .07 .04 .39 .19
PCT 95 .47 .28 2.77 1.39
17.0 + A88 28 -.12 .12 .62 .30*
PCT 28 -.58 .60 3.21 1.54'
Totalfi ABS 220 -.O4 .06 .91 .46
PCT 220 -.25 .60 8.93 4.52
White Oak 3.0 - 4.9 ABS 23 .17 .05 .23 .11
PCT 23 4.27 1.22 5.96 2.81‘
5.0 - 10.9 A88 196 .26 .11 1.49 .76
PCT 194 2.46 .37 5.11 2.58
11.0 - 16.9 A88 95 .09 .05 .45 .23
PCT 95 .64 .34 3.33 1.67
Total ABS 321 .21 .07 1.20 .61
PCT 319 2.03 .28 5.01 2.54
Sugar Maple 3.0 - 4.9 ABS 26 .25 .36 1.84 .88
PCT 25 2.68 3.68 18.74 8.90
5.0 - 10.9 ABS 118 .23 .05 .54 .27
PCT 118 3.10 .72 7.81 3.93
11.0 - 16.9 A85 32 .15 .09 .53 .26
PCT 32 1.01 .67 3.84 1.85:
Total ABS 176 .22 .06 .82 .42
PCT 175 2.67 .71 9.39 4.74
Red Maple 3.0 - 4.9 A88 42 .03 .06 .37 .18‘
PCT 42 .86 1.42 9.29 4.55'
5.0 - 10.9 A85 60 -.09 .12 .92 .46
PCT 60 -1.21 1.72 13.43 6.65
11.0 - 16.9 A88 17 .23 .20 .86 .39
PCT 17 1.88 1.62 6.88 3.15
Total A88 7125 -.01 .07 .76 .38
PCT 125 -.01 .98 11.03 5.55

 

 

 

 

 

8 - non-normally distributed errors

65

 

 

 

Table 14. SASATEST Calculated Errors for Estimated TPA
CONFIDENCE PREDICTION
INTERVAL INTERVAL
SPECIES ERROR n BIAS (BIAS +/-) (BIAS +/-) STD DEV
N. Red Oak ABS 9r -11.51 8.71 86.20 43.19
PCT 96 -3.90 3.47 34.30 17.20
0. Red Oak ABS 56 -33.16 21.57 175.22 87.02
PCT 56 -14.44 8.67 70.41 34.97
White Oak ABS 95 11.99 10.26 100.57 50.38
PCT 94 18.77 13.61 132.64 66.43
Sugar Maple ABS 38 15.59 26.04 160.55 78.10
PCT 38 10.94 16.94 104.43 50.80
Red Maple ABS 65 -21.11 11.15 90.56 44.98
PCT 65 ~7.74 2.48 20.17 10.02
Table 15. SASATEST Calculated Errors for Estimated BA
CONFIDENCE PREDICTION
INTERVAL INTERVAL
SPECIES ERROR n BIAS (BIAS +/-) (BIAS +/-) STD DEV
N. Red Oak ABS 97 -2.17 1.53 15.12 7.58
PCT 97 -2.22 2.57 25.42 13.74
0. Red Oak ABS 56 -4.35 1.45 11.77 5.85
PCT 56 -8.64 3.15 25.67 12.90
White Oak ABS 95 2.73 1.22 11.95 5.99
PCT 94 11.59 5.71 55.68 27.89
Sugar Maple ABS 37 1.01 3.56 21.67 10.52
PCT 37 3.05 6.35 35.58 18.77
Red Maple ABS 65 -2.15 1.24 10.07 5.00
PCT 65 -8.45 5.68 46.15 22.92

 

Discussion:
Medellin;

The failures and shortcomings of each model are discussed
in the methods section following the model in question. The
only model which merits extended discussion is Hilt's.

Rejection of Hilt's model as a candidate for an
alternative diameter growth model was based on the small
sample sizes of data possible for calibration, the large
number of models needed to represent all species/site
conditions, and the lack of sufficient data to calibrate the
second stage model. In respect to variation explained in the
dependent variable, a majority of the R’s ranged between .6
and .8. In addition, all independent variables were
significant at a .10 level. These results were the most
promising of all methods examined.

Although Hilt's two stage model format was rejected, it
provided important insights on how a simple model form can
account for a large portion of variation.in.diameter (or basal
area) growth. Hilt (1983) reported coefficients of
determination in excess of 0.7 for stand data fitted to the
BAGSYR equation. These stands were composed mainly of white
oak, chestnut oak, black oak, scarlet oak, and northern red
oak; and in all cases growth patterns between species over-
lapped. In this study the first stage model was fitted with
white oak and northern red oak data. This produced results

that mirrored those of Hilt's. The major difference between

67

the data sets used was that Hilt's tree data was grouped by
stand, while data for this study was grouped in sets with
common ELTPs, stand basal areas, and species. ‘The grouping of
trees under similar natural conditions in essence form
artificial stands similar to Hilt's stand data sets. With
density and site quality conditions at a near constant within
these "stands", a large portion of diameter growth can be
explained by tree dbh.

The second stage ‘model, which. provides a 'means of
calibrating the first stage model for any general stand, was
not fitted in this study. Given the assumption that stands
representative of all conditions can be compiled, this
equation may not be needed. The limiting factor of this
assumption is the extreme amount of data needed. Another
approach that could be used, which would negate the second
equation, is similar to that which Mawson (1982) describes.
Mawson presents a simple diameter growth equation which relies
only on dbh as an independent variable. This equation is
fitted to stands with either increment core data or extended
growth record data.

Mawson's model:

lnDG = lna + b(1/dbh)

Where: DG a periodic diameter increment (inches)

dbh = diameter at breast height (inches)

a and b = unique coefficients

 

68
V 'd 'o '

As with the initial validation, dbh results were judged
acceptable. Of the nineteen species-size class combinations
evaluated only five exceeded a mean error of 0.20 inches,
while percent errors were under 3% in all but three cases.
Again, the highest errors in terms of percent resulted in
stand basal area and trees per acre projections. This would
make the Lakes States mortality equation suspect. It is
therefore suggested that future research be focused on
developing an alternative survival model. Possibilities for
such a model include the function used in Central States TWIGS
(Miner et. al., 1988) and several functions developed by
Monserud (1976) and Buchman (1979) . These alternatives
utilize several methods of development including discriminant

analysis, probit analysis, and logit analysis.

s t S'

If the current research plots are to be used for future
evaluations of TWIGS' diameter growth or mortality functions,
several considerations should be made in regards to sample
size, environmental conditions, and plot design.

It was apparent in this study that sample size was
insufficient to produce both a validation and calibration data
set. In the future there is a possibility that the database
will further decrease in size due to human disturbance and

gypsy moth. Between 1986 and 1991 one stand and several plots

69

were lost due to change in forest boundaries and human
disturbance. These and similar phenomenons are sure to claim
more trees in the future. Additionally, the Manistee National
Forest is beginning to experience wide-scale gypsy moth
infestations. Past experiences on the Huron National Forest
and adjoining state forests have shown that until the gypsy
moth population establishes itself (which takes roughly twenty
years), varying degrees of annual defoliation can be expected
(Sapio, 1992 personal communications). This repeated stress
on the trees will probably increase mortality (Nichols, 1968)
and decrease the data base size. If continued study is to be
done in this area a possible alternative to this data base is
the USDA Forest Service continuous forest inventory data.

In addition to reducing sample size, gypsy moth and
other similar natural disturbances may effect individual tree
growth and mortality. The Manistee National Forest
experienced forest tent caterpillar (Malacosoma disstria)
outbreaks and drought in the mid-1980's, is currently having
Lindon looper problems throughout the forest, and will
continue to experience large scale gypsy moth defoliation
throughout the 1990's. Defoliation and drought stress trees,
resulting in the introduction of secondary pathogens,
decreased radial growth, and increased mortality (Sapio, 1992
personal communications; Nichols, 1968). Lake States TWIGS'
models were calibrated with data that was collected over

periods of thirty years or more (Christensen et. al., 1979).

70

In this period of time, calibration stands were most likely to
have experienced some degree of insect and drought problems,
but probably not to the degree that now exists on the Manistee
National Forest. This raises the question of how such
disturbance affected data used in this study, if relationships
between dependant and independent variables were affected; and
if so could this explain poor modeling results, and if similar
validation. results can. be expected in ‘the future ‘under
different environmental conditions. .Additionally, gypsy moth
defoliation and its affects can be expected to influence
future measurements and the results of future examinations of
the Lake Sates TWIGS' growth model and mortality function.

A final concern is the design of research plots used in
this study, especially in regards to the use of variable
radius plots. Concern has already been expressed as to the
ability of stand basal area calculated with the aid of a prism
to represent the conditions under which individual trees are
growing; In addition, Zeide (1992) has raised question to the
appropriateness of using point sampling in research oriented
inventories. Because of the reliance on human vision and the
lack of precision inherent in point sampling, he has expressed
concern that the accuracy of estimating stand basal area may
be substantially less than expected. If the current research
plots are to be used in the future, it is suggested they be
converted to fixed radius plots.

In summary, results of this study are limited by time (5

71
years), sample size, and the environmental conditions which
existed during the period of measurements. These factors may
have biased both model development and validation results. If
future research is to follow, it would be advised to explore
the possibility of using an alternate data source, examine the
Lake States TWIGS' mortality function in addition to the
diameter growth function, and explore the accuracy of variable

radius plots in comparison to fixed radius plots.

APPENDICES

 

If?

APPENDIX A

72

White Oak

Scatter Plot

 

N2

.m.‘u.l.‘ 'vtfv'fi. .-’ _f.~‘.~¥. ‘ .:' '. ..

 

 

 

1.4
1.24
1. 246
249K 46
0.8: 9* *6
3K :46 )K 2%
0,6 x as as 3K 3K . x
A 0.4. 3% xxx )EEJGKBK xxx PKBK 9K
3 xx motels mmaeexxxk 3K
1: 024 BK maxemmmm 5K :46 )K
C O- ..................... WM'WXW X ‘
C" EKBK 3K K
C) -024 3+: :46 3K )K
D _O.4l
-O.6-*
-O.8+ .
-1-
-1 .27
-1a4 I I I I
O 5 10 15 20
‘ DBH86 (inches)
Figure 7. Five year change in DBH vs. 1986 DBH

 

25

73

White Oak

Scatter Plot

 

 

 

 

 

 

Figure 8.

1.4
1.24
1‘ ' as
3K]\ 3K
0.8a as as
0.64 as as asas
x x xxxx
8 x x xxx axaaxxx
‘5 0,24 as asas asasasasasasasasasas as
0 x xxaxxxxxxxxxx x
Q -O.2‘ as asas
D -o.4a
-O.6~
-O.Ba
-1-
-1.24
‘1.4 I I I I
0.5 1 1.5 2 2.5

Log(DBH86)

Five year change in DBH vs. log(DBK)

3.5

74

White Oak

Scatter Plot

 

 

 

 

 

Figure 9.

TBA86 (sqr. ft.)

5 year change in growth vs. 1986 tree basal area

104
12*
1 x as
0.8‘ as as
06 as as x x
' xx xxxx x x
8 xxx xxxxx xxx x
‘1: 02‘ xxxxxxxxx x . x as
0 xxxxxx xx xx
C: (3- ......................... ¥*8%¥%9K% X
D -O,2a xx x- x
D -0.4“
-O.6-
-O.8‘
-11
-1.2‘
'1.‘l I I l T l
-O.5 O 0.5 1 1.5 2 2.5

 

 

75

White Oak

Scatter Plot

 

ax x

0.8a ..as as
ax x ~ as
0.6a xas x as x
0,4— aemeaxsx as x x x x as x
axxxaeasmmassxaaeasxis x xx
agar/ems: assets as . as x

0mm --------- xx

.02 x xx x

 

DD (inches)

 

 

 

0 0.005 0.01 0.015 0.02 0025‘ 0.03
DENSE

Figure 10. Five year change in DBH vs. DENSE

 

76

White Oak

Scatter Plot

 

 

 

 

 

 

 

 

1.4
1.2
1. as
0.8‘ 3*
x as x
0.61 as x as
A 0.4a ‘1‘ 3K 3K >16 X
g x as as x
Q 1‘ as x x x , as
c: O as as x x as x
2:. as as x
Q -O.2‘ ‘1‘ 3K 31‘
D -O.4‘
-O.6~
30.8“
. -1J
-1.22.
"1.4 l i I l i
-1 O 1 2 3 4 5
Crown Ratio

Figure 11. Five year change in DB“ vs. crown ratio

77

White Oak

Scatter Piot-

 

 

 

 

1.4
1.2*
1‘ as
0.8‘ x
PK as x x
0.6‘ x x x x as
A 0.44 aaeas aerasxaasaerx x
8 axis axxxaerxrs x x
,c: 02‘ as aersx xaersxaasm as as asx as
O 46 319K xxx asaeasx x as as
t: 0 axis ------------ aaas asas as xx 46*
::> 3% X x
Q -O.2‘ * x xx
D -O.4‘
-0.6"
-O.8‘
-14
-1.2~
-1s4 I I I n]. I

 

 

 

20 30 4O ‘ 50 60 7O 80
Site Index (base age 50 yrs)

Figure 12. Five year change in DBH vs. site index

..

78

White Oak

Scatter Plot

 

 

 

 

 

1.4
1.24
1. x
0.8a 9*
0,6 as x x x as
g; x x x x x x x _ x x
50.21 x xxxxxxxxxxx
0 xxxxxxxxxxx x
D ~O.2‘ 3K BK 3K 9K
D 04‘
OS“
-O.8-*
-1a
-1.2-
-104 I

Figure 13.

20 40 6'0 8'0 1001é012101é01s0 200

Basal Area (sqr. tt./acre)

Five year change in DBH vs. BA

 

79

White Oak

Scatter Plot

 

 

 

 

 

 

1.4
1.2%
1‘ as
0.8“ 3*
0.6a as x x x x
g xasaeasxxx as x
.c: 024 xxreasxxxx as x as
O WXXX 36 )K
C O WWEKéKX ....... 9K gag
Q -O.2-a 3K xxx
0 43.44
-0.6‘
-0.84
_1.
-1.2‘
-1,4 I I I I W
0 0.005 0.01 0.015 0.02 0.025 0.03
1/BA
Figure 14. Five year change in DBH vs. 1/BA

 

80

White Oak

Scatter Plot-

 

 

 

 

 

1.4
1.2
1‘ x
0.84 x
as x as x
a xaxx x x
0'6 xx xxx aaass x x
A 04-4 as xaeasaers asaeasaaszix x
3’3 x xxsasaetsaeas areas as ,
£- 0.25 aasasasx aeteaasasas asaassersaasxas x
0 x asasarasaaeaaeaerersasassassaas x x
C O- ........................... 3K ....... W006 ..... aemw..Wxg*ng
22> J x X xx
[3 -O.2 x x x x
D -0.4‘
-0.6a
-0.8‘
_1.
-1.2-*
-104 I T I I I I
4 6 8 10 12 i 14 16

Figure 15.

Quadratic Stand Diameter

Five year change in DBH vs. Q80

18

 

81

White Oak

Scatter Plot

 

08*

0.6a
7,7 0.4“
2’; 0.24
.t-i 0‘
Q -0.2~
D

as x x x
xxssxaetsasxxexxxsxs
x memes xxaxx ax x

 

.................. 9K¥€4€WKW9§3€
‘ 9K ax :x x
x x age

 

 

Figure 16.

I l I

2. 4 e 8 1'0 1'2 1‘4 1'6

Average Stand DBH (inches)

Five year change in DBH vs. ABD

 

18

APPENDIX B

 

82

Table 16. Northern Red Oak Multivariate Model ANOVA.

N.RED OAK

l

DEP VAR: DD D“ 457 MULTIPLE R: 0.722 SQUARED MULTIPLE R: 0.521
ADJUSTED SQUARED MULTIPLE R: .507 STANDARD ERROR OF ESTIMATE: 0.224

 

VARIABLE COEFFICIENT STD ERROR T P(2 TAIL)
CONSTANT -0.292 0.158 0.064
LNDBH 0.352 0.064 0.000
INVBA -14.252 6.652 0.033
DENSE 16.122 3.977 0.000
CR ‘ 0.000 0.003 0.966.
E10 0.275 0.615 0.656
E12 0.026 0.105 0.803
E20 -0.072 0.076 0.342
E21 -0.129 0.081 0.114
E35 -0.047 0.068 0.492
E37 0.067 0.072 0.352
E40 -0.068 0.073 0.347
E43 -0.044 0.079 0.582
E45 -0.023 0.124 0.854
ANALYSIS OF VARIANCE
SOURCE SUM-OF-SQUARES DP MEAN-SQUARE F-RATIO P
REGRESSION 24.268 13 1.867 37.081 0.000
RESIDUAL 22.302 443 0.050

83

Table 17. Other Red Oak Multivariate Model ANOVA.

O. RED OAK
DEP VAR: DD In 176 MULTIPLE R: 0.497 SQUARED MULTIPLE R: 0.247
ADJUSTED SQUARED MULTIPLE R: .201 STANDARD ERROR OP ESTIMATE: 0.211
VARIABLE COEFFICIENT STD ERROR T P(2 TAIL)
CONSTANT -0.383 0.267 0.153 .
LNDBH 0.284 0.089 0.002
IstA 0.383 7.787 0.961
DENSE 1.570 4.654 0.736
CR 0.015 0.019 0.415
E1 0.101 0.129 0.433
E10 0.069 0.136 0.609
E12 0.103 0.129 0.425
E20 0.031 0.127 0.808
E21 0.104 0.139 0.454
E35 -0.062 0.173 0.722
ANALYSIS OF VARIANCE
SOURCE SUM-OP-SQUARES DF MEAN-SQUARE F-RATIb P
RECRESSION 2.404 10 0.240 5.401 0.000
RESIDUAL 7.343 165 0.045

84

Table 18. White Oak Multivariate Model ANOVA.

WHITE OAR
DEP VAR: DD In 258 MULTIPLE R: 0.702 SQUARED MULTIPLE R: 0.493
ADJUSTED SQUARED MULTIPLE R: .470 STANDARD ERROR.OE ESTIMATE: 0.158
VARIABLE COEPPICIENT STD ERROR T Ps2.TAIL)
CONSTANT —0.258 0.128 0.045
LNDEM 0.164 0.055 0.003
IstA 1.749 4.883 0.721
DENSE 8.128 4.863 0.096
CR 0.041 0.011 0.000
E1 0.347 0.267 0.196
E10 -0.088 0.043 0.041
E12 -0.039 0.047 0.401
E20 -0.098 0.043 0.025
E21 -0.031 0.049 0.527
E35 -0.223 0.049 ~0.000
E37 0.112 0.049 0.024
ANALYSIS OF VARIANCE
SOURCE SUM-OE-SQUARES DP MEAN-SQUARE F-RATIO P
RECRESSION 5.958 11 0.542 21.709 0.000
RESIDUAL 6.138 246 0.025

85

Table 19. Sugar Maple Maltivariate Model ANOVA.

SUGAR MAPLE

DEP VAR: DD PH 157 MULTIPLE R: 0.480 SQUARED MULTIPLE R: 0.230
ADJUSTED SQUARED MULTIPLE R: .194 STANDARD ERROR.OF ESTIMATE: 0.240
VARIABLE COEFFICIENT STD ERROR T P(2 TAIL)

CONSTANT -0.274 0.244 0.264

LNDBM 0.352 0.093 0.000

INVBA 23.559 12.302 0.057

DENSE -22.840 15.257 0.137 ,

CR -0.042 0.018 0.022

E35 -0.220 0.283 0.437

E40 -0.112 0.059 0.060

E45 -0.089 0.056 0.117

ANALYSIS OF VARIANCE
SOURCE SUM-OF-SQUARES DF MEAN-SQUARE F-RATIO P

REGRESSION 2.561 7 0.366 6.367 0.000
RESIDUAL 8.562 149 0.057

 

Table 20. Red Maple Multivariate Model ANOVA

RED MAPLE

DEP VAR: DD

ADJUSTED SQUARED MULTIPLE R:

VARIABLE

CONSTANT
LNDBH
INVBA
DENSE
CR
E10
E12
E20
E21
E35
E37
E40
E43

SOURCE

REGRESSION
RESIDUAL

86

N: 89 MULTIPLE R: 0.690 SQUARED MULTIPLE R: 0.476

.393
COEFFICIENT STD ERROR
-0.403 0.222
0.367 0.106
10.753 12.480
-22.349 15.269
0.047 0.022
1.161 0.693
-0.398 0.136
-0.108 0.104
-0.087 0.098
-0.190 0.095
-0.019 0.119
-0.064 0.107
-0.204 0.168

ANALYSIS OF VARIANCE

SUM-OF-SQUARES DF MEAN-SQUARE

2.785 12
3.070 76

STANDARD ERROR OF ESTIMATE:

T P(2 TAIL)

0.073
0.001
0.392
0.147
0.036
0.098
0.004
0.304
0.379
0.049
0.876
0.552
0.227

F-RATIO

0.201

0.000

 

APPENDIX C

87

N. filed Oak (Residual Plot)

Multivariate Linear Model

 

 

 

 

 

Standardized Residuals
O

 

 

 

 

 

-4 I I I I I I I I
~0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4' 1.6 1.8

00 Estimate (inches)

Figure 17. Northern red oak residual plot

 

88

0. Red Oak (Residual Plot)

Multivariate Linear Model

 

 

 

 

 

 

 

 

 

3 I
I
w 2 I III I I .
T3 'l
g 1. 2"!-
(D
D:
U
E O ' .
P.
8
c: '1‘
.53
C0
-24
'3 l I 1 I 1 I
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

00 Estimate (inches)

Figure 18. Other red oak residual plot

 

89

White Oak (Residual Plot)

Multivariate Linear Model

 

 

 

 

 

 

Standardized Residuals
O

 

 

 

 

0.2-0.1 0 0:1 .012 0:3 0:4 015 0.76 017 0.8
00 Estimate (inches)

Figure 19. White oak residual plot

 

90

 

Sugar Maple (Residual Plot)

Multivariate Linear Model

 

 

5

41
m 3. I .I
“g I
g 21 I
g I II I II
C: 1 n I. luau: I I

I

Ea (3 ........................... :flirl' !4'“£::::3!-Ii.i:-WI:LI
'9 _1. I'liusm I I
c0 . "I fl I I .
P: 24 I
E ..
(D .

-3 I

-44

 

 

 

 

020.1 0 011 02 0:3 0:4 0:5 0:6 017 0.8
00 Estimate (inches)

Figure 20. Sugar maple residual plot

 

91

Red Maple (Residual Plot)

Multivariate Linear Model

 

 

 

 

 

 

 

 

 

3 . ._ .,
2.57 _ I
52 2. _ I .
g 1.5 - ' ' '
U)
85 1‘1 I I’I _.' I. L I
8 0.52 I II . I I
.E I II... I
'0
166. O I ‘ IL I .
I: Ii III I.
% "O.5d I: H I . a
5 I III I ' i
'1‘ 2 . I I
I II '
-1.5‘ I
-2 , '

 

0.2-0.1 0 0:1 0:2 0:3 0:4 015 01.6 017 0.8
00 Estimate (inches)

Figure 21. Red maple residual plot

BIBLIOGRAPHY

 

BIBLIOGRAPHY

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and yield models using dimensional analysis. Forest Science.
35(2): 329-337.

Bailey, R.L. 1980. Individual tree growth derived from
diameter distribution models. Forest Science 26(4): 626-632.

Beers, T.W., and C.I. Miller; 1964. Point sampling:
research results, theory, and application. Resource Bulletin
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Lafayette, IN. 56p.

Belcher, D.M., M.R. Holdaway, G.J.*Brand. 1982. STEMS
- the Stand and Tree Evaluation and Modeling System. Gen.
Tech. Rep. NC-79. USDA Forest Service North Central Forest
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Bolton, R.I<. and R.S. Meldahl. 1990. Design and
development of a multipurpose forest projection system for
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Experiment Station, Auburn University: 51p.

Bruce, D. 1977. Yield differences between research plots
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Buchman, R.G. 1979. Mortality functions. IN: A
generalized forest growth projection system applied to the
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Paul, MN.

Buchman, R.G. 1983. Survival predictions for major
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Forest Experiment Stations. 7p.

Buchman, R.G., and E.L. Lentz. 1984. More Lake States
tree survival predictions. Res. Note. NC-312. St. Paul, MN:
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92

 

93

Buchman, R.G., S.P. Pederson, and N.R. Walters 1983. A
survival model with application to species of the Great Lakes
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Buchman, R.G. and S.R. Shifley. 1983. Guide to
evaluating forest growth projection systems. Journal of
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Burk, T.E. 1986. Growth and yield model validation:
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in forestry. Forest Resources Systems Institute, 98p.

Carmean, W.H. 1975. Forest site quality evaluation in
the United States. Advances in Agronomy 27:209-269.

Christensen L., J.T. Hahn, and R.A. Leary. 1979. Data
base. IN: A generalized forest growth projection system
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Tech. Rep. NC-49. pp.16-21 North Central Forest Experiment
Station, St. Paul, MN.

Conover, W.J. 1980. Practical Nonparametric
Statistics, second edition. John Wiley and Sons, Inc. New
York. 493p.

Crow, T.R. and H. M. Rauscher. 1984. Forest growth
models and land classification. pp 190-204. IN(J.G.
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