United States

Department of

Agriculture

Forest Service

Southern

Research Station

Research Paper

SRS–34

Determining Sample Size for Tree

Utilization Surveys

Stanley J. Zarnoch, James W. Bentley,

and Tony G. Johnson
Stanley J. Zarnoch is a Mathematical Statistician, U.S. Department

of Agriculture, Forest Service, Southern Research Station, Asheville,

NC 28802; and James W. Bentley and Tony G. Johnson are

Foresters, U.S. Department of Agriculture, Southern Research

Station, Forest Service, Knoxville, TN 37919.

October 2004

Southern Research Station

P.O. Box 2680

Asheville, NC 28802

The Authors

DISCLAIMER

The use of trade or firm names in this publication is for reader information and does not imply endorsement

by the U.S. Department of Agriculture of any product or service.
1

Determining Sample Size for Tree

Utilization Surveys

Stanley J. Zarnoch, James W. Bentley,

and Tony G. Johnson

1

Abstract

The U.S. Department of Agriculture Forest Service has conducted

many studies to determine what proportion of the timber harvested in

the South is actually utilized. This paper describes the statistical

methods used to determine required sample sizes for estimating

utilization ratios for a required level of precision. The data used are

those for 515 hardwood and 1,557 softwood trees harvested in east

Texas and classified into 5 product types. Two-stage sampling was

used to collect the utilization data. The primary units were the

logging operation locations and the secondary units were the trees

within locations. The ratio of means estimator was used to calculate

each of three utilization ratios. However, for simplicity, the mean of

ratios approach was used to develop the statistical methodology for

estimating sample sizes for a specified level of precision, defined as

half the width of the 95-percent confidence interval. The infinite

population model was used and variance components for the twostage nested analysis of variance were obtained using PROC MIXED.

The three utilization ratios were computed for all product classes for

hardwoods and softwoods, as were the standard errors and 95-percent

confidence intervals. The variance components were then obtained

and used to develop tables that yield sample size scenarios based on

specified levels of precision.

Keywords: Mean of ratios, PROC MIXED, ratio of means, two-stage

sampling.

Introduction

The U.S. Department of Agriculture Forest Service (Forest

Service) has conducted studies to determine what proportion of the timber harvested in the South is actually utilized.

Logging operations across a State are selected at random, a

sample of the trees cut at each location is selected, and

various utilization proportions are determined. A number

of utilization ratios have been defined and are used in the

calculation of these utilization proportions. Utilization

ratios are ratios of volumes of tree sections used by the

logger to volumes of whole trees or to volumes of various

parts of trees.

This paper has three objectives. The first is to present the

ratio of means estimator, standard error, and confidence

intervals for each of three utilization ratios. The second is to

describe the statistical methods for determining how many

locations and trees per location must be sampled to obtain a

desired error for the utilization ratios. The third is to provide

tables that can be used to determine the sample size options

for estimating the utilization ratios.

Methods

Utilization Ratios

The data used in this paper are taken from a harvest and

utilization study that was performed in eastern Texas by the

Forest Service, Forest Inventory and Analysis Research

Work Unit (FIA) in 2003 (Bentley and Johnson 2004). The

data were those for 515 hardwood and 1,557 softwood trees

that were classified into 5 product types: saw log, veneer,

composite panel, pulpwood, and pole. Tables 1, 2, and 3

show these classifications with the number of locations and

trees per location that were actually sampled. At each

location, one or more product types were sampled.

The three utilization ratios used in this study were defined

based on the following variables that describe the types of

sections obtained from each cut tree:

1 x = saw-log sections utilized (FIA section codes 1 and 4)

2 x = saw-log sections not utilized (FIA section codes 1 and 4)

3 x = upper-stem sections utilized (FIA section codes 2 and 5)

4 x = upper-stem sections not utilized (FIA section codes 2 and 5)

5 x = stump section utilized (FIA section code 0)

6 x = stump section not utilized (FIA section code 0)

7 x = top and limb sections utilized (FIA sections 3, 6, 7, 8, and 9)

8 x = top and limb sections not utilized (FIA sections 3, 6, 7, 8,

 and 9)

The three utilization ratios studied here are based on the

total volume of the cut tree, volume of saw-log and upperstem sections (total merchantable volume as defined by FIA)

or the total used volume.
2

Table 2—Estimates of the utilization ratios, standard error, and 95-percent confidence 

intervals

Trees

Species group

and producta error Lower Upper

Hardwood

 Saw logs 18 179 0.892 0.0129 0.865 0.920

 Veneer logsc 2 20 0.880 0.0019 0.856 0.903

 Composite panelsc 3 61 0.941 0.0087 0.904 0.978

 Pulpwood 28 255 0.950 0.0059 0.938 0.962

 All hardwoods 34 515 0.903 0.0106 0.882 0.925

Softwood

 Saw logs 45 581 0.957 0.0038 0.949 0.964

 Veneer logs 26 417 0.958 0.0036 0.951 0.966

 Composite panelsc 5 91 0.967 0.0199 0.912 1.022

 Pulpwood 37 438 0.976 0.0048 0.967 0.986

 Polesc 1 30 0.933 — — —

 All softwoods 68 1,557 0.958 0.0027 0.953 0.963

— = not calculated. 

a There was no pole product for hardwood. 

b Locations sampled do not total because more than one species group and/or product can be measured at 

an individual location; therefore, double counting of the sample location can occur. 

c Not considered reliable because these estimates are based on very few sample locations.

- - - - - number - - - - -

 Locations actually Standard

 sampledb sampled R2

R2

Table 1—Estimates of the utilization ratios, standard error, and 95-percent confidence 

intervals

Trees

Species group

and producta error Lower Upper

Hardwood

 Saw logs 18 179 0.757 0.0165 0.722 0.792

 Veneer logsc 2 20 0.709 0.0002 0.706 0.711

 Composite panelsc 3 61 0.806 0.0196 0.721 0.890

 Pulpwood 28 255 0.807 0.0124 0.781 0.832

 All hardwoods 34 515 0.762 0.0143 0.733 0.791

Softwood

 Saw logs 45 581 0.865 0.0042 0.857 0.873

 Veneer logs 26 417 0.865 0.0045 0.856 0.875

 Composite panelsc 5 91 0.879 0.0082 0.856 0.902

 Pulpwood 37 438 0.883 0.0060 0.871 0.895

 Polesc 1 30 0.844 — — —

 All softwoods 68 1,557 0.866 0.0029 0.860 0.872

— = not calculated. 

a There was no pole product for hardwood. 

b

 Locations sampled do not total because more than one species group and/or product can be measured at 

an individual location; therefore, double counting of the sample location can occur. 

c Not considered reliable because these estimates are based on very few sample locations.

- - - - - number - - - - -

Locations Standard

 sampledb sampled

actually

R1

R1
3

Where the utilization ratio is defined as utilized volume

over total volume of the cut tree, it may be expressed as

Merchantable volume according to FIA standards is based

only on saw-log and upper-stem sections. Where the

utilization ratio is defined as FIA merchantable volume

utilized over total FIA merchantable volume, it may be

expressed as

Some logging operations, however, take timber that does

not meet FIA merchantability standards. This consists of the

stump and the top and limb sections. Where the utilization

ratio is defined as FIA merchantable volume utilized over

total volume used, it may be expressed as

Obviously, the complement of each of these ratios is itself

another utilization ratio as defined by FIA, because it

contains no new information it will not be discussed further.

Utilization Ratio Estimation

A two-stage sampling design (Cochran 1977) was used to

collect the utilization data. The primary units are the

logging operation locations and the secondary units are

the trees within locations. The estimator for any of the

utilization ratios is based on the ratio of means and is

defined as

1357

1

12345678

xxxx R

xx xxxxx x

+++ = +++++++ (1)

1 3

2

1234

x x R

xxxx

+ = +++ (2)

1 3

3

1357

x x R

xxxx

+ = +++ (3)

1

1

ˆ

n

i

i

n

i

i

y

y R

x

x

=

=

= =

∑

∑ (4)

Table 3—Estimates of the utilization ratios, standard error, and 95-percent confidence 

intervals

Trees

Species group

and producta error Lower Upper

Hardwood

 Saw logs 18 179 0.977 0.0024 0.972 0.982

 Veneer logsc 2 20 0.984 0.0011 0.970 0.998

 Composite panelsc 3 61 0.944 0.0167 0.872 1.016

 Pulpwood 28 255 0.927 0.0082 0.910 0.944

 All hardwoods 34 515 0.967 0.0044 0.958 0.976

Softwood

 Saw logs 45 581 0.981 0.0019 0.978 0.985

 Veneer logs 26 417 0.979 0.0035 0.972 0.987

 Composite panelsc 5 91 0.889 0.0152 0.846 0.931

 Pulpwood 37 438 0.914 0.0086 0.896 0.931

 Polesc 1 30 0.981 — — —

 All softwoods 68 1,557 0.974 0.0026 0.969 0.980

— = not calculated. 

a There was no pole product for hardwood. 

b

 Locations sampled do not total because more than one species group and/or product can be measured at 

an individual location; therefore, double counting of the sample location can occur. 

c Not considered reliable because these estimates are based on very few sample locations.

- - - - - number- - - - -

 Locations Standard

 sampledb sampled

actually

R3

R3
4

where

1

mi

i ij

j

y y == ∑ = the sum of the numerator component of the

utilization ratio for location i

1

mi

i ij

j

x x == ∑ = the sum of the denominator component of the

utilization ratio for location i

n = number of locations sampled

mi = number of trees sampled in location i

The sample variance is

where

2

y S and 2 Sx = the typical sample variances of the ys and xs,

respectively

2

yx S = their covariance

The finite population correction has been ignored because

the number of locations sampled is small with respect to the

total number of locations in Texas. The typical standard error

of Rˆ is the square root of ( ) V R ˆ ˆ , and confidence intervals at

the (1 ) −α level are

where

(1 ),( 1) a n t − − = the two-tailed (1 ) −α value from the t-distribution

with ( 1) n − degrees of freedom

Two-Stage Sampling Design (Finite Population Model)

It is cumbersome to use the ratio of means estimator to

calculate the reliability of the utilization ratios under

alternative sample size scenarios for the two-stage survey

design. Therefore, a mean of ratios approach based on an

alternative estimator to equation (4) was employed to

estimate the utilization ratios and their standard errors.

Although the number of trees that FIA sampled per location

varied, this statistical procedure will first be presented under

the simplifying assumption of equal numbers and then

generalized to the unequal numbers situation. The mean of

ratios estimator for the utilization ratios assuming equal

number of trees per location is

where

ij y = utilization ratio for tree j in location i

n = number of locations sampled

m = number of trees sampled in each location

The estimated variance is based on the finite population

model and is defined as

where

N = the total number of locations in the population

M = the total number of trees in each location

and

Note that 2

1s is commonly referred to as the between primary

variance and that 2

2 s is the within primary variance.

The estimated utilization ratio actually used by FIA is not

based on a mean of ratios approach as in equation (7).

Instead, the ratio of means estimator [equation (4)] is used

and the specific utilization ratio is defined as the average

numerator value of all trees sampled divided by the average

denominator value of all trees sampled. These estimators

may lead to different results and the sample variance

equations are also computed differently. However, the

estimators are very similar in this study, and the standard

errors are also. In addition, as stated previously, the sample

variance for the mean of ratios approach is more appropriate

1 1

ˆ n m ij

MOR

i j

y R = = nm = ∑∑ (7)

2

2 1

1

( )

1

n

i

i

y y

s

n

=

−

= −

∑

(9)

2

2 1 1

2

( )

( 1)

n m

ij i

i j

y y

s

n m

= =

−

= −

∑∑

(10)

( ) 2 2

ˆ ˆ 1 2 1 1 MOR

ns n m s V R

N n N M nm

⎛⎞ ⎛ ⎞ =− + − ⎜⎟ ⎜ ⎟ ⎝⎠ ⎝ ⎠ (8)

( )

⎟

⎠

⎞ ⎜

⎝

⎛ + − − =

= + −

∑∑∑

===

n

i

n

i

n

i

i i i i

y x yx

y R x R y x

n n x

S R S RS

nx

V R

111

2 2 2

2

2 2 2

2

ˆ 2 ˆ

( 1)

1

ˆ 2 ˆ 1 )ˆ (ˆ

(5)

(1 ),( 1) ( ) ˆ ˆˆ R t VR ± − − a n (6)
5

Thus, the infinite population model has estimated variance

defined as

Equations (11) and (15) will yield the same estimated variance when the mis are equal. However, when the number of

trees per location is not a constant, an estimate of the mean

number of trees per location m0 is used for m in equation

(15) and is defined as

(Milliken and Johnson 1984, Montgomery 1976). The

typical standard error of ˆ

RMOR is the square root of ( ) ˆ ˆ V RMOR ,

and confidence intervals at the (1 ) −α level are

where

(1 ),( 1) a n t − − = the two-tailed (1 ) −α value from the t-distribution

with ( 1) n − degrees of freedom

The advantage of equations (15) and (16) is that they are

applicable in the most general situation whether the mis are

equal or not. Equation (11) is a function of the number of

locations ( n ) and is useful for calculating the sample

variance of a given survey. However, for designing a survey,

equation (15) gives the sample variance under any

alternative combination of locations and trees per location.

The variance components in equation (15) could be obtained

by performing a nested analysis of variance using PROC

GLM (SAS Institute Inc. 1989), equating the expected mean

squares to their mean squares from the analysis and solving

for the variance components. Such computations are not

complex but may be tedious if many must be performed or

if there are unequal trees per location, two situations that

were present in this study. Another possibility is to use

PROC VARCOMP (SAS Institute Inc. 1989), which computes

the variance components directly; however, subsequent

calculations must then be performed to get the estimated

variance. To avoid these difficulties, PROC MIXED (SAS

Institute Inc. 1989) can be used to fit a nested analysis of

variance and to obtain an estimate of the ratio, variance

for investigating sample size scenarios based on the number

of locations and trees per location because it is a function of

n and m . The ratio of means approach contains n but not

m explicitly. Therefore, although the ratio of means is used

for FIA utilization ratios (Bentley and Johnson 2004), the

mean of ratios is used here to develop sample size scenarios.

Two-Stage Sampling (Infinite Population Model)

Although the utilization estimator and sample variance

could be obtained by the finite population model approach,

it is often easier and more informative to assume the infinite

population model (Marcuse 1949). This is justifiable when

n N is negligible as is the case for the Texas survey, where

the number of locations sampled is very small in relation to

the total number of logging locations in Texas. Under the

infinite population model the estimator is the same as in the

finite population model [equation (7)] but the sample

variance is now simply

which is obtained from equation (8) by allowing n N to

approach 0. Furthermore, it is helpful to realize that this

could be equated (after taking the expectation) to the

typical variance components in a nested general linear

model because

where the mean square for treatments in a nested analysis of

variance is

and it’s expected mean square is

where

2 σ1 = the variance component for the primary units

2 σ 2 = the variance component for the secondary units

( ) 2

ˆ ˆ 1

MOR

s V R

n = (11)

2

1

( )

1

n

i

i

T

m yy

MS

n

=

−

= −

∑

(13)

2 2

2 1 ( ) E MS m T = + σ σ (14)

(1 ),( 1) ( ) ˆ ˆˆ R t VR MOR a n MOR ± − − (17)

( ) 2 2

ˆ ˆ 1 2 V RMOR n nm

σ σ

= +

(15)

2

1

0

1

1

1

1

n

n i

i

i n

i

i

i

m

m m

n

m

=

=

=

⎛ ⎞ ⎜ ⎟

= − ⎜ ⎟ − ⎜ ⎟

⎝ ⎠

∑

∑

∑ (16)

mn n nm (12)

m

mn

MS

mn n

m y y

n n

y y

n

s

T

n

i

i

n

i

i

2

2

2

1

2

1

2

2

1

2

1

2

2

1

( 1)

( )

( 1)

( )

σ σ σ σ = +

+ = =

−

−

= −

−

=

∑ ∑ = =
6

components, and standard error directly. PROC MIXED is a

relatively new procedure in SAS and is much more general

than the older PROC GLM and PROC VARCOMP.

Specification of the Error

Information from the Texas survey could be used to determine the intensity of sampling that is required to achieve a

specified error in future utilization studies. In the present

study, the error is defined as half the width of the 95-percent

confidence interval [equation (17)], specifically

where the components are as defined in equations (15) and

(17) for the infinite population model. This is used as an

approximation for the error under the ratio of means approach, as previously discussed. The error is a function of

the two variance components and the number of locations

( n ) and trees per location ( m ). Hence, given the estimated

variance components from the Texas survey, the error could

be determined for various combinations of n and m .

Results

Utilization Ratios

The three utilization ratios were computed for all product

classes for hardwoods and softwoods using the ratio of

means estimator equation (4) (tables 1, 2, and 3). The R1

utilization ratio was 0.762 for all hardwoods combined and

0.866 for all softwoods combined, and indicates the amount

of total tree volume (merchantable and nonmerchantable)

that was utilized for these two species groups. Similar results

were obtained for products within species groups, with

utilization about 10 percent higher for softwoods than for

hardwoods.

When only FIA-defined merchantable volume is considered,

the R2 utilization ratios were 0.903 for hardwoods and

0.958 for softwoods, which indicates that most of the

merchantable volume was being utilized, especially for

the softwoods. Similar trends were found for each product.

The R3 utilization ratios differed little by species group,

which shows that the FIA merchantable proportion of timber

taken by the logger was about 0.967 for hardwoods and

0.974 for softwoods.

In addition to the utilization ratios, tables 1, 2, and 3 give

the standard errors computed from equation (5) and 95-

percent confidence intervals computed from equation (6).

The width of these confidence intervals is a measure of the

reliability of the utilization ratios and, thus, an indication

of the adequacy of the initial survey.

Although these utilization ratios were computed for all

products for both species groups, they are not considered

reliable if they were based on very few sample locations.

Thus, the hardwood veneer and composite panel and softwood composite panel and pole estimates should be viewed

with caution.

Sample Size Options

To facilitate the calculation of sample sizes based on specified errors for future surveys, variance components were

computed with PROC MIXED for both species groups and

all products (table 4). The mean of ratios estimator was

employed. Again, the hardwood veneer and composite

panel and softwood composite panel and pole figures are

given for completeness and should not be relied upon for

future surveys. The sample size options are shown for R1 in

table 5, R2 in table 6, and R3 in table 7. These tables were

constructed based on the error defined in equation (18) with

a range of sampling locations ( n ) from 10 to 100 and a

range of trees per location ( m ) from 10 to 30, which seems

reasonable from an operational sampling perspective.

These tables can be used to find a sampling option that

gives a specified error. For instance, if we are interested in

the hardwood saw-log utilization ratio R1 within 0.020 error

we have to sample 50 locations and 10 trees per location

(500 trees). This is found from table 5 by glancing down the

hardwood saw-log column until an error of 0.020 or smaller

is found. Often several alternatives are available with each

alternative specifying a different number of locations and

trees per location. In this example, a slightly smaller error

(0.018) would require 60 locations and 10 trees per location

(600 trees). Obviously, this slight decrease in the error is not

worth the cost of sampling 10 more locations.

Generally, the error decreases most as the number of locations increases and decreases little with increases in the

number of trees per location. For instance, consider the

hardwood saw-log R1 utilization ratio and assume that 1,000

trees will be sampled. It is found from table 5 that n = 40

locations with m = 25 trees per location gives an error of

0.021. However, if n = 100 locations with m = 10 trees per

location, the error is only 0.014. It should be realized that

the cost of traveling to 60 more locations has not been taken

into account. The second option has a smaller error for the

same number of trees, but it involves additional travel costs

and may be inefficient for this reason.

(1 ),( 1) ( ) ˆ ˆ Error t V R = − − a n MOR (18)
7

These tables are presented for convenience, but it should be

realized that the error could be computed from the statistical

formulas previously given for any sample size option. As an

example, to compute the hardwood saw-log R1 utilization

ratio where n = 50 and m = 10, obtain the variance components 2

1 σˆ = 0.00426 and 22 σˆ = 0.00422 from table 4.

Then, using equations (15) and (18) and the t-distribution

value at the 0.95 level with 49 degrees of freedom, the error

is

which is identical to that in table 5.

( )

0.00426 0.00422 2.00958 0.019

50 50 10

Error = +=

Table 4—Estimates of the variance components for the utilization ratios computed with the mean 

of ratios approach using PROC MIXED 

Species group

and product

Hardwood

 Saw logs 0.00426 0.00422 0.00428 0.00555 0.00004 0.00017

 Veneer logs

a 0.00000 0.00011 0.00000 0.00161 0.00002 0.00002

 Composite panelsa 0.00088 0.00655 0.00000 0.00445 0.00141 0.00684

 Pulpwood 0.00373 0.00700 0.00002 0.00489 0.00255 0.00733

 All hardwoods 0.00371 0.00612 0.00142 0.00514 0.00359 0.00473

Softwood

 Saw logs 0.00039 0.00200 0.00042 0.00237 0.00012 0.00011

 Veneer logs 0.00020 0.00172 0.00017 0.00206 0.00023 0.00036

 Composite panelsa 0.00000 0.00723 0.00069 0.00948 0.00330 0.01570

 Pulpwood 0.00049 0.00267 0.00005 0.00125 0.00252 0.00859

 Polesa 0.00000 0.00332 0.00000 0.00386 0.00000 0.00001

 All softwoods 0.00042 0.00249 0.00033 0.00246 0.00258 0.00410

a Not considered reliable because these estimates are based on very few sample locations.

R 1 R 2 R 3

2

1 σˆ 22 σˆ 21 σˆ 22 σˆ 21 σˆ 22 σˆ

The error computed in tables 5, 6, and 7 is based on the

0.95 confidence level. However, if a lower or higher level

is desired, the t-value that corresponds to this significance

level would be used in equation (18). All other equations

would remain the same and the error would be computed

as explained in the above example.

When a survey is being designed, tables 5, 6, and 7 can be

used to determine an appropriate sample size for the desired

level of precision. However, the error and associated standard error is only an approximation that is based on the

means of ratios approach. The utilization ratios should be

obtained by the ratio of means method using equation (4)

and the standard error and confidence interval from equations (5) and (6). All calculation should be based on only

the new sample data.
8

Table 5—Error associated with sample size options for the utilization ratios 

n m Total All All

10 10 100 0.049 0.048 0.047 0.017 0.014 0.020 0.019

10 15 150 0.048 0.046 0.046 0.016 0.013 0.018 0.017

10 20 200 0.048 0.046 0.045 0.016 0.012 0.018 0.017

10 25 250 0.048 0.045 0.045 0.016 0.012 0.017 0.016

10 30 300 0.047 0.045 0.045 0.015 0.011 0.017 0.016

20 10 200 0.032 0.031 0.031 0.011 0.009 0.013 0.012

20 15 300 0.032 0.030 0.030 0.011 0.008 0.012 0.011

20 20 400 0.031 0.030 0.030 0.010 0.008 0.012 0.011

20 25 500 0.031 0.030 0.029 0.010 0.008 0.011 0.011

20 30 600 0.031 0.029 0.029 0.010 0.008 0.011 0.010

30 10 300 0.026 0.025 0.025 0.009 0.007 0.010 0.010

30 15 450 0.025 0.024 0.024 0.009 0.007 0.010 0.009

30 20 600 0.025 0.024 0.024 0.008 0.006 0.009 0.009

30 25 750 0.025 0.024 0.023 0.008 0.006 0.009 0.009

30 30 900 0.025 0.024 0.023 0.008 0.006 0.009 0.008

40 10 400 0.022 0.021 0.021 0.008 0.006 0.009 0.008

40 15 600 0.022 0.021 0.021 0.007 0.006 0.008 0.008

40 20 800 0.021 0.020 0.020 0.007 0.005 0.008 0.007

40 25 1,000 0.021 0.020 0.020 0.007 0.005 0.008 0.007

40 30 1,200 0.021 0.020 0.020 0.007 0.005 0.008 0.007

50 10 500 0.019 0.019 0.019 0.007 0.005 0.008 0.007

50 15 750 0.019 0.018 0.018 0.007 0.005 0.007 0.007

50 20 1,000 0.019 0.018 0.018 0.006 0.005 0.007 0.007

50 25 1,250 0.019 0.018 0.018 0.006 0.005 0.007 0.006

50 30 1,500 0.019 0.018 0.018 0.006 0.005 0.007 0.006

60 10 600 0.018 0.017 0.017 0.006 0.005 0.007 0.007

60 15 900 0.017 0.017 0.017 0.006 0.005 0.007 0.006

60 20 1,200 0.017 0.017 0.016 0.006 0.004 0.006 0.006

60 25 1,500 0.017 0.016 0.016 0.006 0.004 0.006 0.006

60 30 1,800 0.017 0.016 0.016 0.006 0.004 0.006 0.006

70 10 700 0.016 0.016 0.016 0.006 0.005 0.007 0.006

70 15 1,050 0.016 0.015 0.015 0.005 0.004 0.006 0.006

70 20 1,400 0.016 0.015 0.015 0.005 0.004 0.006 0.006

70 25 1,750 0.016 0.015 0.015 0.005 0.004 0.006 0.005

70 30 2,100 0.016 0.015 0.015 0.005 0.004 0.006 0.005

80 10 800 0.015 0.015 0.015 0.005 0.004 0.006 0.006

80 15 1,200 0.015 0.014 0.014 0.005 0.004 0.006 0.005

80 20 1,600 0.015 0.014 0.014 0.005 0.004 0.006 0.005

80 25 2,000 0.015 0.014 0.014 0.005 0.004 0.005 0.005

80 30 2,400 0.015 0.014 0.014 0.005 0.004 0.005 0.005

90 10 900 0.014 0.014 0.014 0.005 0.004 0.006 0.005

90 15 1,350 0.014 0.014 0.013 0.005 0.004 0.005 0.005

90 20 1,800 0.014 0.013 0.013 0.005 0.004 0.005 0.005

90 25 2,250 0.014 0.013 0.013 0.005 0.003 0.005 0.005

90 30 2,700 0.014 0.013 0.013 0.004 0.003 0.005 0.005

100 10 1,000 0.014 0.013 0.013 0.005 0.004 0.005 0.005

100 15 1,500 0.013 0.013 0.013 0.005 0.004 0.005 0.005

100 20 2,000 0.013 0.013 0.013 0.004 0.003 0.005 0.005

100 25 2,500 0.013 0.013 0.012 0.004 0.003 0.005 0.005

100 30 3,000 0.013 0.012 0.012 0.004 0.003 0.005 0.004

Hardwoods Softwoods

Saw logs Pulpwood Saw logs Veneer logs Pulpwood

R1
9

Table 6—Error associated with sample size options for the utilization ratios

n m Total All All

10 10 100 0.050 0.016 0.031 0.018 0.014 0.009 0.017

10 15 150 0.049 0.013 0.030 0.017 0.013 0.008 0.016

10 20 200 0.048 0.012 0.029 0.017 0.012 0.008 0.015

10 25 250 0.048 0.011 0.029 0.016 0.011 0.007 0.015

10 30 300 0.048 0.010 0.029 0.016 0.011 0.007 0.015

20 10 200 0.033 0.011 0.021 0.012 0.009 0.006 0.011

20 15 300 0.032 0.009 0.020 0.011 0.008 0.005 0.010

20 20 400 0.032 0.008 0.019 0.011 0.008 0.005 0.010

20 25 500 0.031 0.007 0.019 0.011 0.007 0.005 0.010

20 30 600 0.031 0.006 0.019 0.010 0.007 0.005 0.009

30 10 300 0.026 0.008 0.016 0.010 0.007 0.005 0.009

30 15 450 0.025 0.007 0.016 0.009 0.007 0.004 0.008

30 20 600 0.025 0.006 0.015 0.009 0.006 0.004 0.008

30 25 750 0.025 0.005 0.015 0.008 0.006 0.004 0.008

30 30 900 0.025 0.005 0.015 0.008 0.006 0.004 0.008

40 10 400 0.022 0.007 0.014 0.008 0.006 0.004 0.008

40 15 600 0.022 0.006 0.013 0.008 0.006 0.004 0.007

40 20 800 0.022 0.005 0.013 0.007 0.005 0.003 0.007

40 25 1,000 0.021 0.005 0.013 0.007 0.005 0.003 0.007

40 30 1,200 0.021 0.004 0.013 0.007 0.005 0.003 0.006

50 10 500 0.020 0.006 0.012 0.007 0.006 0.004 0.007

50 15 750 0.019 0.005 0.012 0.007 0.005 0.003 0.006

50 20 1,000 0.019 0.005 0.012 0.007 0.005 0.003 0.006

50 25 1,250 0.019 0.004 0.011 0.006 0.005 0.003 0.006

50 30 1,500 0.019 0.004 0.011 0.006 0.004 0.003 0.006

60 10 600 0.018 0.006 0.011 0.007 0.005 0.003 0.006

60 15 900 0.018 0.005 0.011 0.006 0.005 0.003 0.006

60 20 1,200 0.017 0.004 0.011 0.006 0.004 0.003 0.005

60 25 1,500 0.017 0.004 0.010 0.006 0.004 0.003 0.005

60 30 1,800 0.017 0.003 0.010 0.006 0.004 0.002 0.005

70 10 700 0.017 0.005 0.010 0.006 0.005 0.003 0.006

70 15 1,050 0.016 0.004 0.010 0.006 0.004 0.003 0.005

70 20 1,400 0.016 0.004 0.010 0.006 0.004 0.003 0.005

70 25 1,750 0.016 0.004 0.010 0.005 0.004 0.002 0.005

70 30 2,100 0.016 0.003 0.010 0.005 0.004 0.002 0.005

80 10 800 0.015 0.005 0.010 0.006 0.004 0.003 0.005

80 15 1,200 0.015 0.004 0.009 0.005 0.004 0.003 0.005

80 20 1,600 0.015 0.004 0.009 0.005 0.004 0.002 0.005

80 25 2,000 0.015 0.003 0.009 0.005 0.004 0.002 0.005

80 30 2,400 0.015 0.003 0.009 0.005 0.003 0.002 0.005

90 10 900 0.015 0.005 0.009 0.005 0.004 0.003 0.005

90 15 1,350 0.014 0.004 0.009 0.005 0.004 0.002 0.005

90 20 1,800 0.014 0.003 0.009 0.005 0.003 0.002 0.004

90 25 2,250 0.014 0.003 0.008 0.005 0.003 0.002 0.004

90 30 2,700 0.014 0.003 0.008 0.005 0.003 0.002 0.004

100 10 1,000 0.014 0.004 0.009 0.005 0.004 0.003 0.005

100 15 1,500 0.014 0.004 0.008 0.005 0.003 0.002 0.004

100 20 2,000 0.013 0.003 0.008 0.005 0.003 0.002 0.004

100 25 2,500 0.013 0.003 0.008 0.005 0.003 0.002 0.004

100 30 3,000 0.013 0.003 0.008 0.004 0.003 0.002 0.004

Softwoods 

Saw logs Pulpwood Saw logs Veneer logs Pulpwood

Hardwoods

R2
10

Table 7—Error associated with sample size options for the utilization ratios

n m Total All All

10 10 100 0.005 0.041 0.046 0.008 0.012 0.042 0.039

10 15 150 0.005 0.039 0.045 0.008 0.011 0.040 0.038

10 20 200 0.005 0.039 0.044 0.008 0.011 0.039 0.038

10 25 250 0.005 0.038 0.044 0.008 0.011 0.038 0.037

10 30 300 0.005 0.038 0.044 0.008 0.011 0.038 0.037

20 10 200 0.004 0.027 0.030 0.005 0.008 0.027 0.026

20 15 300 0.003 0.026 0.029 0.005 0.007 0.026 0.025

20 20 400 0.003 0.025 0.029 0.005 0.007 0.025 0.025

20 25 500 0.003 0.025 0.029 0.005 0.007 0.025 0.025

20 30 600 0.003 0.025 0.029 0.005 0.007 0.025 0.024

30 10 300 0.003 0.021 0.024 0.004 0.006 0.022 0.020

30 15 450 0.003 0.021 0.023 0.004 0.006 0.021 0.020

30 20 600 0.003 0.020 0.023 0.004 0.006 0.020 0.020

30 25 750 0.003 0.020 0.023 0.004 0.006 0.020 0.020

30 30 900 0.003 0.020 0.023 0.004 0.006 0.020 0.019

40 10 400 0.002 0.018 0.020 0.004 0.005 0.019 0.017

40 15 600 0.002 0.018 0.020 0.004 0.005 0.018 0.017

40 20 800 0.002 0.017 0.020 0.004 0.005 0.017 0.017

40 25 1,000 0.002 0.017 0.020 0.004 0.005 0.017 0.017

40 30 1,200 0.002 0.017 0.020 0.004 0.005 0.017 0.017

50 10 500 0.002 0.016 0.018 0.003 0.005 0.017 0.016

50 15 750 0.002 0.016 0.018 0.003 0.005 0.016 0.015

50 20 1,000 0.002 0.015 0.018 0.003 0.004 0.015 0.015

50 25 1,250 0.002 0.015 0.017 0.003 0.004 0.015 0.015

50 30 1,500 0.002 0.015 0.017 0.003 0.004 0.015 0.015

60 10 600 0.002 0.015 0.016 0.003 0.004 0.015 0.014

60 15 900 0.002 0.014 0.016 0.003 0.004 0.014 0.014

60 20 1,200 0.002 0.014 0.016 0.003 0.004 0.014 0.014

60 25 1,500 0.002 0.014 0.016 0.003 0.004 0.014 0.014

60 30 1,800 0.002 0.014 0.016 0.003 0.004 0.014 0.013

70 10 700 0.002 0.014 0.015 0.003 0.004 0.014 0.013

70 15 1,050 0.002 0.013 0.015 0.003 0.004 0.013 0.013

70 20 1,400 0.002 0.013 0.015 0.003 0.004 0.013 0.013

70 25 1,750 0.002 0.013 0.015 0.003 0.004 0.013 0.012

70 30 2,100 0.002 0.013 0.015 0.003 0.004 0.013 0.012

80 10 800 0.002 0.013 0.014 0.003 0.004 0.013 0.012

80 15 1,200 0.002 0.012 0.014 0.003 0.004 0.012 0.012

80 20 1,600 0.002 0.012 0.014 0.002 0.004 0.012 0.012

80 25 2,000 0.002 0.012 0.014 0.002 0.003 0.012 0.012

80 30 2,400 0.002 0.012 0.014 0.002 0.003 0.012 0.012

90 10 900 0.002 0.012 0.013 0.002 0.003 0.012 0.011

90 15 1,350 0.002 0.012 0.013 0.002 0.003 0.012 0.011

90 20 1,800 0.001 0.011 0.013 0.002 0.003 0.011 0.011

90 25 2,250 0.001 0.011 0.013 0.002 0.003 0.011 0.011

90 30 2,700 0.001 0.011 0.013 0.002 0.003 0.011 0.011

100 10 1,000 0.001 0.011 0.013 0.002 0.003 0.012 0.011

100 15 1,500 0.001 0.011 0.012 0.002 0.003 0.011 0.011

100 20 2,000 0.001 0.011 0.012 0.002 0.003 0.011 0.010

100 25 2,500 0.001 0.011 0.012 0.002 0.003 0.011 0.010

100 30 3,000 0.001 0.010 0.012 0.002 0.003 0.011 0.010

Pulpwood

Hardwoods Softwoods

Saw logs Pulpwood Saw logs Veneer logs

R3
11

Literature Cited

Bentley, J.W.; Johnson, T.G. 2004. Eastern Texas harvest and

utilization study, 2003. Resour. Bull. SRS–97. Asheville, NC:

U.S. Department of Agriculture, Forest Service, Southern

Research Station. 28 p.

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

John Wiley. 428 p.

Marcuse, S. 1949. Optimum allocation and variance components

in nested sampling with an application to chemical analysis.

Biometrics. 5: 189–206.

Milliken, G.A.; Johnson, D.E. 1984. Analysis of messy data.

Volume I: designed experiments. New York: Van Nostrand

Reinhold Co. 473 p.

Montgomery, D.C. 1976. Design and analysis of experiments.

New York: John Wiley. 418 p.

SAS Institute Inc. 1989. SAS/STAT user’s guide. Version 6. 4th ed.

Cary, NC: SAS Institute Inc. 846 p. Vol. 2.
12

Zarnoch, S.J.; Bentley, J.W.; Johnson, T.G. 2004. Determining sample size for tree utilization surveys. Res. Pap. SRS–34. Asheville, NC: U.S. Department of Agriculture, Forest

Service, Southern Research Station. 11 p.

The U.S. Department of Agriculture Forest Service has conducted many studies to determine

what proportion of the timber harvested in the South is actually utilized. This paper describes

the statistical methods used to determine required sample sizes for estimating utilization ratios

for a required level of precision. The data used are those for 515 hardwood and 1,557 softwood trees harvested in east Texas and classified into 5 product types. Two-stage sampling

was used to collect the utilization data. The primary units were the logging operation locations and the secondary units were the trees within locations. The ratio of means estimator

was used to calculate each of three utilization ratios. However, for simplicity, the mean of

ratios approach was used to develop the statistical methodology for estimating sample sizes

for a specified level of precision, defined as half the width of the 95-percent confidence

interval. The infinite population model was used and variance components for the two-stage

nested analysis of variance were obtained using PROC MIXED. The three utilization ratios

were computed for all product classes for hardwoods and softwoods, as were the standard

errors and 95-percent confidence intervals. The variance components were then obtained and

used to develop tables that yield sample size scenarios based on specified levels of precision.

Keywords: Mean of ratios, PROC MIXED, ratio of means, two-stage sampling.
13

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