USDA Forest Service Res. Note RMRS-RN-17. 2004 1

Introduction

Density based sampling is often considered attractive. For example, variable radius plot (VRP) sampling is popular because it avoids the excessive sampling of small

trees. We show here that the procedure of using plots in the field to obtain a fixed

number of trees per plot is undesirable since it unnecessarily violates some key issues

regarding probabilistic sampling. The fact that it may have small bias in some simulation studies does not prove that it is a satisfactory procedure since realistic situations

can invariably be found or constructed where bias will be serious.

Review of Literature

Jonsson and others (1992) propose selecting a fixed number of trees at each primary sampling unit so that each tree has a different probability of selection. Using the

unbiased Horvitz-Thompson estimator requires knowledge of the actual probabilities

of selection of the sample trees. However, these can only be approximated unless the

locations of all trees are known. This is clearly not practical; even then it would be

cumbersome to calculate those probabilities. They approximate the probabilities of

selection of the trees in a given plot by dividing the n trees selected by the local tree

density. They state that the proposed estimator is then approximately unbiased if the

variable of interest is spatially independent of the local tree densities.

Lessard and others (1995) compare this n-tree distance sampling to fixed and variable radius plot sampling in estimating basal area per acre. They conclude that

variations of n-tree sampling, especially 3-tree sampling, are more biased and more

variable than fixed and variable radius plot sampling in all stands examined. They

consider n-tree sampling to be fast and cost-competitive for estimating basal area and

density and have the advantage of being able to provide estimates of spatial pattern

parameters because distance measurements to trees is an integral part of n-tree distance sampling. Lessard and others (2002) compared n-tree distance sampling with

fixed-radius plot sampling for estimating number of trees per ha. They show that ntree distance based sampling estimators are at least as precise as those for plot sampling

generally for both random and clustered forests, provided that the fixed radius plot

size is less than the ratio of (n-2) and less than the expected density where n is the

number of trees included at an n-tree location. They do not break out bias of the estimators in this study.

Lynch and Rusydi (1999) compared n-tree sampling with some other sampling techniques and found that the n-tree sampling underestimated both volume and density

in the plantations sampled in Indonesia.

Sampling Using a Fixed Number

of Trees Per Plot

Hans T. Schreuder1

Abstract—The fixed number of trees sample design proposed by Jonsson and

others (1992) may be dangerous in applications if a probabilistic framework of

sampling is desired. The procedure can be seriously biased. Examples are given

here.

United States

Department

of Agriculture

Forest Service

Rocky Mountain

Research Station

Research Note

RMRS-RN-17

August 2004

1 Hans T. Schreuder

is a Mathematical

Statistician (retired)

with the Rocky

Mountain Research

Station in Fort Collins,

CO.
2 USDA Forest Service Res. Note RMRS-RN-17. 2004

The Horvitz-Thompson estimator is an unbiased estimator of Y:

1

ˆ / ,

n

HT i i

i

Y y π == ∑ (1)

where i y is the attribute of interest and π i is the first order inclusion probability.

The variance is given by:

2 1/2 ˆ ( ) [ ( )( / / )

N

HT i j ij i i j j

i j

V Y yy ππ π π π

≠

= −− ∑ (2)

or

2 1/2 ˆ ( ) [(/ / ) V Y wy y HT ij i i j j = − π π (2a)

with wij i j = − ππ π ij .

Unbiased variance estimators are:

2

1 1/2 ˆ ( ) [ {( ) / }( / / )

n

HT i j ij ij i i j j

i j

v Y yy ππ π π π π ≠=− − ∑ (3)

and

2 2

2

1

ˆ ( ) [ {(1 ) / } {( ) / }( / )]

n n

HT i i i ij i j ij i j i j

i ij

v Y y yy π π π ππ π ππ = ≠=− + − ∑ ∑ (4)

(Schreuder and others 1993).

Methods

If the estimated probabilities of selection of trees are * π i , (i N =1,... ) with N the

number of trees in the population, then

with bias:

*

1

(1 / ) .

N

i ii

i

Bias y π π == − ∑ Thus, the degree of bias of the estimator depends on how

well we estimate π i , the true probabilities of selection for the trees.

Specific situations:

Assume that we have a grid sample of n locations imposed on the population of

interest and that we have a consistent bias in the sense that in the top and

bottom half of the population, respectively, the π i (i=1...N) are over- and

underestimated by 10 percent. Then we have:

1

2 12

1 1

*

1 1

1 11

1.1

0.9 1.1 0.9

ˆ ( ) ( / ) { /( )}

{ /( )} / / .

N N

HT i i i i i i

i i

N NN

i ii i i

iN i iN

EY y y

y yy

ππ π π

π π

= =

=+ = =+

= =

+ =+

∑ ∑

∑ ∑∑

Hence the bias is 

2 1

1 1 1

0.1 1.1 / 0.9 /

N N

i i

i N i

y y = = =

    −

  Σ ∑ . Clearly this bias can be quite serious.

It is easy to construct realistic cases where estimating πi

 by πi

* can be seriously in error.

See figure 1 for an example.

2 3 4 6 7 8

5 1 9

Figure 1—An illustration of a possible scenario with

numbered tree locations, where probabilities of

selection will be seriously underestimated.

This original page has missing equations. Please see the 

following scanned page for complete information.
The Horvitz-Thompson estimator YHT is an unbiased estimator of Y: n 

where yi is the attribute of interest and xi is the first order inclusion probability. 

The variance is given by: 

N 

~(i&) 2 = 1 /2[X (n;n, - nv)(yi In; - y, I n,) 

Unbiased variance estimators are: n 

vI(fHT)=l/2[x{(n;nj -~~)I~~}(~; /xi -y, Inj) 2 

and 

i=l i# j 

(Schreuder and others 1993). 

Methods 

If the estimated probabilities of selection of trees are r,* , (i = 1, ... N) with N the 

number of trees in the population, then - - 

n N 

E(fH~) = EX yi 1 nl* = C (Y; / < )xi 3 with bias: 

N 

Bias = X (1 - ? 4 )Y;. Thus, the degree of bias of the estimator depends on how 

i=l 

well we estimate xi, the true probabilities of selection for the trees. 

Specific situations: 

Assume that we have a grid sample of n locations imposed on the population of 

interest and that we have a consistent bias in the sense that in the top and 

bottom half of the population, respectively, the xi (i=l.. .N) are over- and 

underestimated by 10 percent. Then we have: 

Nl 

Hence the bias is 0.1 yi / 0.9 - yi / 1.1 . Clearly this bias can be quite serious. 

i=l 

It is easy to construct realistic cases where estimating ni by nl* can be seriously in error. 

See figure 1 for an example. 

Figure I-An illustration of a possible scenario with 

numbered tree locations, where probabilities of 

selection wjil be seriously underestimated. 

USDA Forest Service Res. Note RMRS-RN-17. 2004 

This page is a scanned duplicate and is complete with equations. 

However, it may not be as clean as the original.
USDA Forest Service Res. Note RMRS-RN-17. 2004 3

In this example for n=5, tree 1 will be selected for any sample location falling in the

rectangle. Hence πi

 =1 and πi

*

 will never be close to 1 for this tree for any sample falling

in the rectangle.

The following illustrates another problem with fixed number of trees per plot sampling: Assume that a plot location falls in part (a) and another in part (b) (figure 2). For

n=5, plot size for (a) will obviously be much smaller than for (b) and measuring plot

size for (b) would take time.

Sample location (a):

Sample location (b):

What are we gaining by this approach relative to a fixed area plot sample? Clearly in

the latter case we would have several samples with 1 or even 0 trees in it. Certainly it is

easier to obtain the information required for the fixed area plots. Are we gaining in information with the fixed number of tree plots? It is not clear that we are unless we are talking

about variables keyed to specific trees requiring a large or guaranteed number of sample

trees. If distances to trees are required, they could easily be measured with either the fixed

or variable radius plots.

Conclusions

Using designs with fixed number of tree plot sampling can be dangerous. The method is

biased and can be hard to implement in practice. In addition it is not fulfilling a need that

cannot be met by traditional sampling designs.

Acknowledgments

The author would like to thank Drs. Tom Lynch and Veronica Lessard for their

valuable comments.

Literature Cited

Jonsson, B.; Holm, S.; Kallur, H. 1992. A forest inventory method based on densityadapted circular plot size. Scandinavian Journal of Forest Research. 7: 405-421.

Lessard, V. C.; Reed, D. D.; Drummer, T. D. 1995. N-tree distance sampling compared

to fixed radius plot and variable radius point sampling in forest inventory estimation of basal area per acre. In proceedings Monte Verita Conference on Forest Survey Designs; “Simplicity versus Efficiency” and Assessment of Non-Timber Resources.

M. Kohl and P. Brassel, Eds. May 2-7, 1994. Monte Verita Ascona, pp 81-90. Swiss

Federal Institute of Technology. Section of Forest Inventory and Planning. Zurich,

Switzerland.

Lessard, V. C.; Reed, D. D.; Drummer, T. D. 2002. Precision of density estimates from

fixed-radius plots compared to N-tree distance sampling. Forest Science. 48: 1-6.

Lynch, T. B.; Rusydi, R. 1999. Distance sampling for forest inventory in Indonesian

teak plantations. Forest Ecology and Management 113:215-221.

Schreuder, H. T.; Gregoire, T. G.; Wood, G. B. 1993. Sampling methods for multiresource

inventories. New York: John Wiley and Sons. 446 p.

1 2 3 4 6 7

 5 8

 9 10

Figure 2—An illustration of

a possible scenario with

widely varying plot sizes.

1 2 3

 4

 5
The Rocky Mountain Research Station develops scientific

information and technology to improve management, protection,

and use of the forests and rangelands. Research is designed to

meet the needs of the National Forest managers, Federal and State

agencies, public and private organizations, academic institutions,

industry, and individuals.

Studies accelerate solutions to problems involving ecosystems,

range, forests, water, recreation, fire, resource inventory, land

reclamation, community sustainability, forest engineering

technology, multiple use economics, wildlife and fish habitat, and

forest insects and diseases. Studies are conducted cooperatively,

and applications may be found worldwide.

Research Locations

Flagstaff, Arizona Reno, Nevada

Fort Collins, Colorado* Albuquerque, New Mexico

Boise, Idaho Rapid City, South Dakota

Moscow, Idaho Logan, Utah

Bozeman, Montana Ogden, Utah

Missoula, Montana Provo, Utah

Lincoln, Nebraska Laramie, Wyoming

*Station Headquarters, Natural Resources Research Center, 2150

Centre Avenue, Building A, Fort Collins, CO 80526.

The U.S. Department of Agriculture (USDA) prohibits discrimination

in all its programs and activities on the basis of race, color, national

origin, sex, religion, age, disability, political beliefs, sexual orientation,

or marital or family status. (Not all prohibited bases apply to all

programs.) Persons with disabilities who require alternative means

for communication of program information (Braille, large print, audiotape, etc.) should contact USDA’s TARGET Center at (202) 720-

2600 (voice and TDD).

To file a complaint of discrimination, write USDA, Director, Office of

Civil Rights, Room 326 W, Whitten Building, 1400 Independence

Avenue, SW, Washington, D.C. 20250-9410 or call (202) 720-5964

(voice and TDD). USDA is an equal opportunity provider and

employer.

You may order additional copies of this publication by sending your mailing

information in label form through one of the following media. Please specify the

publication title and series number.

Fort Collins Service Center

Telephone (970) 498-1392

FAX (970) 498-1396

E-mail rschneider@fs.fed.us

Web site http://www.fs.fed.us/rm

Mailing address Publications Distribution

Rocky Mountain Research Station

240 West Prospect Road

Fort Collins, CO 80526

Federal Recycling Program Printed on Recycled Paper