United States

Department of

Agriculture

Forest Service

Southern

Research Station

Research Paper

SRS–38

Adjustments to Forest

Inventory and Analysis

Estimates of 2001 Saw-Log

Volumes for Kentucky

Stanley J. Zarnoch and Jeffery A. Turner
The Authors

September 2005

Southern Research Station

P.O. Box 2680

Asheville, NC 28802

Stanley J. Zarnoch, Mathematical Statistician, USDA Forest Service, Southern

Research Station, Asheville, NC 28802; Jeffery A. Turner, Forester, USDA Forest

Service, Southern Research Station, Knoxville, TN 37919.

Cover photo: Logging truck with two deck trailer, Wayah Ranger District, Nantahala

National Forest.

Photo by: Bill Lea.
1

Adjustments to Forest Inventory and Analysis

Estimates of 2001 Saw-Log Volumes for Kentucky

Stanley J. Zarnoch and Jeffery A. Turner

Abstract

The 2001 Kentucky Forest Inventory and Analysis survey overestimated

hardwood saw-log volume in tree grade 1. This occurred because 2001 field

crews classified too many trees as grade 1 trees. Data collected by quality

assurance crews were used to generate two types of adjustments, one based

on the proportion of trees misclassified and the other on the proportion of

saw-log volume misclassified. Measures of variability for the estimated

proportions were based on a cluster sampling design. Both methods

significantly reduced estimated saw-log volume in tree grade 1. We believe

that the saw-log volume approach is superior to the tree approach, but that

both approaches generate improved estimates of tree grade saw-log

volumes. The standard errors of the adjustment proportions are given and

can be used to calculate standard errors of the adjusted values.

Keywords: Cluster sampling, Forest Inventory and Analysis, quality

assurance, saw-log, tree grade.

Introduction

The 2001 Kentucky Forest Inventory and Analysis (FIA)

survey estimated hardwood saw-log volume and classified it

into five tree grades (1 to 5). However, the 2001 FIA volume

estimate for tree grade 1 for Kentucky was more than twice

as large as that for 1988. This caused concern, and FIA

believed that the 2001 field crews might have assigned too

many trees to tree grade 1.

FIA evaluates its tree grade data by comparing it to that

collected in check plots by quality assurance (QA)

personnel. The QA field staff are experienced field crew

members who are responsible for training and evaluating

field production personnel. The QA personnel are

considered to be experts in FIA data collection. The QA

crew collects data in a random sample of the production

plots within a few weeks of the production measurement.

Some of these checks are blind; that is, the QA crew collects

its data without knowledge of the production plot data.

There are also cold checks; in these the QA crew reviews

the initial data, records any inconsistencies with their own

data, and then scores the field crew on accuracy.

The data from the FIA QA database confirmed that trees had

been misclassified as grade 1 but contained only 30

observations for trees that were classified as grade 1—not

enough observations to provide a good basis for an

adjustment. Such a small sample leads to large standard

errors of estimates, large confidence intervals, and poor

precision. For instance, an estimate of the proportion of

trees classified as tree grade 1 by the field crews but as tree

grade 2 by the QA crews was 0.53. A 95 percent confidence

interval for this estimate was 0.35 to 0.72, which was

considered too wide for adjustment purposes.

It was considered desirable that the 95 percent confidence

interval have a half width of no more than 0.10, which

implies that 100 observations are required in each tree grade

to get a good estimate of a proportion and ensure the 0.10

error bound. This sample size was computed under the

worst-case scenario, that is when p = 0.50, so in most

instances a somewhat lower number of observations may be

taken to achieve the same precision. Although the binomial

distribution was assumed here for simplicity when

calculating the required sample size, a more appropriate

method based on cluster sampling was used for subsequent

calculations. This method will be explained later.

The FIA QA data included adequate numbers of

observations for tree grades 2 and 3 but few for tree grades 4

and 5. However, this was not a concern because

misclassification in tree grades 4 and 5 was of no practical

consequence. Thus, the QA crews were sent to grade

additional trees previously classified by the field crews as

tree grade 1 to provide data that would be used to create the

adjustment. Additional trees in the other four tree grades

were graded also, where this was feasible. The QA crews

had no knowledge of the initial field classifications of these

additional trees.

Methods

The field and QA tree grade classifications for an individual

hardwood tree can be compared only if both crews agree

that the tree has diameter at breast height (d.b.h.) > 11

inches and is a growing-stock tree (tree class 2) .1 Tree class

is an initial measure of tree quality and indicates whether a

tree has a log meeting certain size, soundness, and grade

requirements. The trees that do not have a log meeting these

requirements are classified as cull trees and are not graded.

1 U.S. Department of Agriculture Forest Service. 2001. Forest Inventory

and Analysis, Southern Research Station field guide, Volume 1: field data

collection procedures for phase 2 plots. [Not paged]. Unpublished report.

On file with: USDA Forest Service, Southern Research Station, Forest

Inventory and Analysis, 4700 Old Kingston Pike, Knoxville, TN 37919.
2

Gij

Pij

T

PT =

0 42353 0 32941 0 16471 0 05882 0 02353

0 042

.....

. 74 0 72650 0 13675 0 08547 0 00855

0 00000 0 035

....

. .35 0 82323 0 11616 0 02525

0 03846 0 11538 0 115

...

. . .38 0 65385 0 07692

0 05263 0 15789 0 15789 0 210

. .

. . . .53 0 42105 .

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⎥

PT

11

PT

V PV A

T T

U = ′

PT ′ PT

Pij

S

PS =

0 49595 0 30127 0 13890 0 04426 0 01962

0 078

.....

. 92 0 64954 0 13905 0 12214 0 01034

0 00000 0 049

....

. .32 0 81728 0 09680 0 03660

0 09569 0 17541 0 040

...

. . .46 0 65230 0 03614

0 13594 0 11704 0 07101 0 182

. .

. . . .99 0 49301 .

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36 28 14 5 2

5 85 16 10 1

0 7 163 23 5

1 3 3 17 2

1 3 3 48

Vu =

Trees that do have a log meeting the requirements are

classified as growing-stock trees and are graded.

A summary of the field crews’ and the QA crews’ tree grade

classifications for 445 hardwood trees distributed across

Kentucky that were jointly classified as tree class 2 and had

a d.b.h. of > 11 inches is shown in the matrix below:

 G =

where

 = the number of trees that the field crews classified as

tree grade i that should have received tree grade j

according to the QA crews’ data for row i = 1, 2, 3, 4, and

5 and column j = 1, 2, 3, 4, and 5

For instance, out of the 85 trees (36 + 28 + 14 + 5 + 2) that

the field crews classified as tree grade 1, only 36 were

similarly classified as tree grade 1 by the QA crews. This is

an indication that the field crews may have been classifying

too many trees as tree grade 1, and thus overestimating the

increase in the total saw-log volume in tree grade 1 for the

2001 survey.

The matrix G is easier to interpret if it is converted to a

probability matrix that reflects the proportions of the field

crews’ trees that are correctly and incorrectly classified

based on the QA data. Let:

 = the proportion of the field crews’ tree grade i trees that

should have been tree grade j according to the QA crews’

data.

This yields a probability matrix based on trees as follows:

Note that the probabilities in each row sum to 1 because

they are conditional on the field crews’ tree grade

classification. Obviously, is the proportion of tree

grade 1 trees that was classified correctly by the field crews.

This is 0.42353 and indicates that only approximately 42

percent of the field crews’ tree grade 1 trees were actually

tree grade 1 and that about 58 percent of the field crews’ tree

grade 1 trees were classified incorrectly. This is much too

large an error to be tolerated and a correction is needed.

To adjust for the misclassification, the matrix could be

used to assign the unadjusted saw-log volume in each tree

grade obtained from the 2001 survey to an adjusted volume

for each tree grade. Let the unadjusted saw-log volume in

each of the five tree grades be defined by the vector

12,236

16,319

22,022

3,937

3,079

These values are the actual results from the 2001 survey and

are in millions of board feet. Then, an adjusted volume

based on the comparison of the field to QA crews’ data is

defined as:

 (1)

where

 = the matrix obtained by transposing

When the matrix and the uncorrected volume vector are

multiplied together, we get a corrected volume vector that

redistributes the uncorrected volume. The adjusted volume

vector sums to the same total volume. In this method, it is

assumed that the QA data classifies trees by tree grade

correctly. However, it must be emphasized that this has not

been validated in the field. Thus, the accuracy of the volume

correction depends on the correctness of the QA

classifications, which could only be assumed.

Although trees were classified by tree grade by the field

crews and QA crews, a more appropriate adjustment method

may be to determine the proportion of volume misclassified

instead of the proportion of trees misclassified. The data

provided volume on each saw-log. Let = the proportion

of the volume of the field crews’ tree grade i trees that

should have been volume of tree grade j trees according to

the QA crews’ data.

Using computations similar to those used for the tree

approach, we obtained the following saw-log probability

matrix.
3

V PV A

S S

U = ′

PS′ PS

PS

PT

PT PS

PT

aijk

ni

mik

yijk

PS

xik

ni

x

x

n i

ik

k i

ni

= =

∑1

ni

aijk yijk mik xik

ni

An adjusted volume based on the comparison of the field

crews’ to QA crews’ data is defined as:

 (2)

where

 = the matrix obtained by transposing

This matrix based on saw-log volume exhibits trends

similar to those found in the matrix based on trees. The

differences may reflect a relationship between tree size and

the probability that field crews will classify trees correctly.

For instance, consider those trees classified as grade 1 by

both the field and QA crews. Forty-two percent are

classified as grade 1 trees by both field and QA crews when

the tree approach is employed, and 50 percent of the volume

is classified correctly when the tree volume approach is

taken. This suggests that larger trees may have a higher

likelihood of being jointly classified as tree grade 1, which

is reasonable because they are the best saw-log volume

producing trees. In addition, since volume is to be adjusted,

it’s reasonable to use volume as a criterion in the matrix.

Another characteristic of the adjustment is the precision

associated with the and matrices, which are not

known quantities but are only estimated based on survey

data. The variability of each element of these matrices is

based on a sample of trees within a sample of plots, and

hence calculation of the variance of each proportion cannot

be based on the assumption of simple random sampling.

Thus, variability is based on a cluster sampling design

(Cochran 1977). For the tree approach, the estimated

proportions shown in the matrix are defined as:

 (3)

where

 = number of trees on plot k that are classified as tree

grade i by the field crews and tree grade j by the QA

crews

 = total number of trees on plot k that are classified as

tree grade i by the field crews

 = number of plots that had trees classified as tree grade

i by the field crews

The typical sample variance of a proportion in cluster

sampling is defined approximately (ignoring the finite

population correction and the sampling variability of ni) as:

(4)

where

 , which is the average number of trees

classified as tree grade i by the field crews on the plots

The standard error is simply the square root of .

The saw-log volume approach utilizes a slightly different

form for the estimators and is based on the ratio of means

(Cochran 1977). The proportions shown in the matrix are

defined as:

 (5)

where

 = total saw-log volume of all trees on plot k that are

classified as tree grade i by the field crews and tree grade

j by the QA crews

 = total saw-log volume of all trees classified as tree

grade i by the field crews on plot k

 = number of plots that had trees classified as tree grade

i by the field crews

The typical sample variance for the ratio of means is defined

approximately (ignoring the finite population correction and

the sampling variability of ni) as:

 (6)

where

 , the average total saw-log volume for all trees

classified as tree grade i by the field crews on the plots.

The standard error is simply the square root of .

Note that the equations for both estimated proportions are

identical, except for the tree or saw-log proportion, as are

those for the sample variances. This is easily verified by

setting = and = , when it is easy to see that

equation (3) becomes equation (5) and equation (4) becomes

equation (6).
4

SE PT ( )

.....

=

0 05939 0 04520 0 04404 0 03058 0 01654

0. . . . .

.

02142 0 05530 0 03180 0 03170 0 00826

0 00000 0. . . .

. .

01257 0 03265 0 02749 0 01320

0 03524 0 06708 0. . .

...

06240 0 12975 0 05500

0 05208 0 09561 0 10205 0. . 07655 0 11788

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SE PS ( )

.....

=

0 05563 0 04752 0 03842 0 02533 0 01447

0. . . . .

.

04657 0 07504 0 03387 0 04498 0 00975

0 00000 0. . . .

. .

01871 0 03564 0 02601 0 02152

0 08328 0 08202 0. . .

...

02350 0 16528 0 02986

0 12177 0 07474 0 05259 0. . 07119 0 15328

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VA

T VA

S

VU

increased the saw-log volume most in tree grade 4. The

standard error matrices reflect the variability in the

proportion estimates for both approaches. They are useful in

setting confidence intervals around the proportions and

determining the errors. Recall that an approximate

confidence interval is plus or minus two times the standard

error. Thus, since most of the standard errors are 0.05 or

less, the error bounds for most proportions are within

2(0.05) = 0.10. This gives confidence that the adjustments

are reasonably well behaved.

Conclusion

FIA QA data was used in developing an adjustment method

to correct unusually high tree grade 1 saw-log volumes

generated by the 2001 Kentucky survey. We believe that the

saw-log volume approach is superior to the tree approach

but that both approaches yield improved estimates of treegrade saw-log volumes. The standard errors of the

adjustment proportions are given and can be used to

calculate standard errors of the adjusted values. These

results depend on the assumption that the QA data is

accurate and thus may contain an unknown amount of error.

Acknowledgment

We would like to thank the FIA field crews from the

Kentucky Division of Forestry (Frankfort, KY) and the

USDA Forest Service (Knoxville, TN) for collecting the

Kentucky 2001 survey data. In addition, we thank David L.

Gartner and the QA crews from the USDA Forest Service

(Knoxville, TN) who obtained the QA data for this report. In

addition, we are appreciative to Bernard R. Parresol and

Frank A. Roesch for reviews of this manuscript.

Literature Cited

Cochran, W.G. 1977. Sampling techniques. 3rd ed. New York: John Wiley.

428 p.

VA

T =

VA

S =

Results

The adjusted volume based on the tree approach was

obtained from equation (1) and results in

6,193

17,606

23,317

7,895

2,583

In addition, using equation (4) and taking square roots, the

standard errors of the proportions are:

The adjusted volume based on the saw-log approach was

obtained from equation (2) and results in

8,152

16,424

22,345

7,798

2,875

In addition, if we use equation (6) and take square roots, the

standard errors of the proportions are:

Both and significantly reduced the saw-log volume in

tree grade 1 from that in the original FIA 2001 inventory

estimates in . To compensate for this, both methods
Zarnoch, Stanley J.; Turner, Jeffery A. 2005. Adjustments to Forest Inventory

and Analysis estimates of 2001 saw-log volumes for Kentucky. Res. Pap.

SRS–38. Asheville, NC: U.S. Department of Agriculture, Forest Service,

Southern Research Station. 4 p.

The 2001 Kentucky Forest Inventory and Analysis survey overestimated hardwood saw-log volume in tree grade 1. This occurred because 2001 field crews

classified too many trees as grade 1 trees. Data collected by quality assurance

crews were used to generate two types of adjustments, one based on the proportion of trees misclassified and the other on the proportion of saw-log volume

misclassified. Measures of variability for the estimated proportions were based

on a cluster sampling design. Both methods significantly reduced estimated sawlog volume in tree grade 1. We believe that the saw-log volume approach is

superior to the tree approach, but that both approaches generate improved

estimates of tree grade saw-log volumes. The standard errors of the adjustment

proportions are given and can be used to calculate standard errors of the adjusted

values.

Keywords: Cluster sampling, Forest Inventory and Analysis, quality assurance,

saw-log, tree grade.
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