United States

Department

of Agriculture

Forest Service

Rocky Mountain

Research Station

General Technical

Report RMRS-GTR-126

April 2004

Statistical Techniques for

Sampling and Monitoring

Natural Resources

Hans T. Schreuder, Richard Ernst, and Hugo Ramirez-Maldonado
CORRECTIONS

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Schreuder, Hans T.; Ernst, Richard; Ramirez-Maldonado, Hugo. 2004. Statistical techniques for sampling and

monitoring natural resources. Gen. Tech. Rep. RMRS-GTR-126. Fort Collins, CO: U.S. Department of

Agriculture, Forest Service, Rocky Mountain Research Station. 111 p.

Abstract

We present the statistical theory of inventory and monitoring from a probabilistic point of view. We start with

the basics and show the interrelationships between designs and estimators illustrating the methods with a

small artificial population as well as with a mapped realistic population. For such applications, useful open

source software is given in Appendix 4. Various sources of ancillary information are described and applications of the sampling strategies are discussed. Classical and bootstrap variance estimators are discussed

also. Numerous problems with solutions are given, often based on the experiences of the authors. Key

additional references are cited as needed or desired.

Acknowledgments

We owe a considerable measure of gratitude to reviewers for valuable comments. Chistopher Kleinn, Steen

Magnusson, Ray Czaplewski, Rudy King, and Keith Rennolls reviewed the entire book, Geoff Wood revised

an earlier draft, and Jule Caylor took the time to check the remote sensing section and added some updates

to it. Mike Williams reviewed the coarse woody debris section, Jim Baldwin and Paul Geissler the wildlife

section, Ron McRoberts and Nick Crookston the modeling section, Frank Roesch and Paul Patterson the

section on discrete variable sampling, Charles Scott the section on multilevel sampling, and Gretchen Moisen

the section on small area estimation. Gary Boyak reviewed and partially rewrote the sections on GIS and

GPS, Henry Lachowski reviewed the section on GIS, and Jeff Goebel reviewed the section on sampling for

rare events. Tim Gregoire and Kim Iles made useful suggestions on how to handle boundary trees. Lane

Eskew did an excellent job of reviewing the manuscript from an editorial point of view and making sure that

it conformed with acceptable standards of publication. We owe a considerable debt of gratitude to Ray

Czaplewski for suggesting updating Frank Freese’s book and facilitating the subsequent writing of this book.

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

Rocky Mountain Research Station

Natural Resources Research Center

2150 Centre Avenue, Building A

Fort Collins, CO 80526

The Authors

Hans T. Schreuder is a Mathematical Statistician (retired) with the USDA Forest Service’s Rocky Mountain

Research Station in Fort Collins, CO. Richard Ernst is a Mensurationist with the USDA Forest Service’s

Forest Management Service Center (Washington Office) in Fort Collins, CO. Hugo Ramirez-Maldonado is

a Director General with the National Institute on Forestry, Agriculture and Animal Husbandry Research in

Mexico City, Mexico.
Contents

I. Introduction .................................................................................................................. 1

II. Objectives of Sampling and Monitoring................................................................... 2

Why Sample?.............................................................................................................. 2

Planning Your Survey.................................................................................................. 3

Objectives ............................................................................................................. 3

Information to be collected .................................................................................... 4

Developing the sampling approach....................................................................... 5

III. Sampling Concepts and Methodologies ................................................................. 6

Sampling Frame.......................................................................................................... 6

Purposive and Representative Sampling .................................................................... 6

Populations, Parameters, Estimators, and Estimates ................................................ 7

Bias, Accuracy, and Precision ..................................................................................... 8

Variables: Continuous and Discrete ............................................................................ 9

Distribution Functions ............................................................................................... 10

Tools of the Trade ..................................................................................................... 10

Notation ............................................................................................................... 10

Characterizing a distribution using measures of central

tendency and dispersion ................................................................................. 12

Standard errors and confidence limits ................................................................ 14

Expanded variances and standard errors ........................................................... 15

Coefficient of variation ........................................................................................ 16

Covariance, correlation, and regression ............................................................. 16

Independence ..................................................................................................... 18

Variances of products, ratios, and sums ............................................................. 19

Transformation of variables ................................................................................ 21

IV. Sampling Strategies ................................................................................................ 22

Designs With the Horvitz-Thompson Estimator........................................................ 22

Variance Estimation in General................................................................................. 33

Regression and Ratio Estimators ............................................................................. 34

Some Specific Forestry Sampling Methods .............................................................. 40

Sample Size Determination ...................................................................................... 42

Ground Sampling ...................................................................................................... 44

Edge Effects When Sampling at Stand Boundaries ................................................. 46

Design Issues ........................................................................................................... 47

Instrumentation ......................................................................................................... 48

Sampling for Coarse Woody Debris (CWD) ............................................................. 49

Wildlife Sampling ...................................................................................................... 50

V. Sampling Methods for Discrete Variables.............................................................. 51

Simple Random Sampling (SRS) for Classification Data ......................................... 51

Cluster Sampling for Attributes ................................................................................. 53

Cluster Sampling for Attributes With Unequal-Sized Clusters.................................. 55

Sampling of Count Variables .................................................................................... 57

VI. Remote Sensing and Other Ancillary Information ............................................... 59

Remote Sensing and Photography ........................................................................... 59

Accuracy of Remotely Sensed Information .............................................................. 61

Global Positioning System for Spatial Location Needs............................................. 64

Geographic Information System (GIS) ..................................................................... 64

Small Area Estimation ............................................................................................... 65

i
VII. Sampling for Rare Events...................................................................................... 67

VIII. Multiple Level Sampling ....................................................................................... 68

Multistage Sampling.................................................................................................. 68

Multiphase Sampling ................................................................................................. 69

IX. Monitoring Over Time ............................................................................................. 73

X. Building Models and Cause-Effect ......................................................................... 77

XI. Forest Sampling Situations .................................................................................... 79

Pitfalls........................................................................................................................ 79

Suggestions .............................................................................................................. 80

XII. References .............................................................................................................. 82

XIII. Glossary ................................................................................................................. 85

Appendix 1. Inference................................................................................................... 87

Appendix 2. Distributions ............................................................................................ 90

Continuous Distributions ........................................................................................... 90

Normal distribution .............................................................................................. 90

Gamma distribution ............................................................................................. 91

Multivariate distributions...................................................................................... 91

Discrete Distributions ................................................................................................ 91

Binomial distribution ............................................................................................ 91

Hypergeometric distribution. ............................................................................... 92

Poisson distribution ............................................................................................. 92

Multinomial distribution........................................................................................ 93

Multivariate hypergeometric distribution ............................................................. 93

Laws of large numbers........................................................................................ 93

Appendix 3. Tables........................................................................................................ 95

Appendix 4. Statistical Analysis Worked Examples ................................................ 102

Analysis Software.................................................................................................... 102

Data Sets ................................................................................................................ 102

Results .................................................................................................................... 103

Index ............................................................................................................................. 108

iii
USDA Forest Service RMRS-GTR-126. 2004. 1

I. Introduction

The purpose of this book is to serve as a complete introduction to the statistical techniques of

sampling natural resources starting at a very basic level and progressing to more advanced methods. We describe supplementary tools and materials and identify key references for readers wishing to pursue the subject further. Considerable material is based on direct experiences of the authors. We include introductory material, much of which is taken from the excellent introductory

book of Freese (1962). These sections in Freese’s book are expressed in a compelling and still

relevant manner. A good example is Chapter V: Sampling Methods for Discrete Variables. More

advanced readers can skip these sections. To facilitate reading in general, we dispense with the

proof of properties of estimators, such as their unbiasedness and how the variances of estimators

are derived. Schreuder spent most of his career with the USDA Forest Service working for Forest

Inventory and Analysis Program (FIA). Ernst teaches numerous short courses in forest inventory

and sampling and provides consultation on such methods to inventory and measurement specialists

in the National Forest System (NFS). Ramirez-Maldonado has considerable experience in teaching

courses in forest sampling, inventory, and modeling as well as consulting for Mexican agencies in

forest inventory and monitoring.

There are several good introductory books available on sampling. The book by Johnson (2000)

is very basic and gives extensive information. It is dated, however, in that it does not cover more

recent advances in the field. The books by Freese (1962) and deVries (1986) are still useful in

providing several of the basic concepts, the latter going further afield in what is available. Freese’s

book has the additional advantage that it is available in Spanish. Shiver and Borders (1996) provide

a modernized version of Freese (1962) material with some emphasis on typical forestry methods.

More advanced books are available, too, in forestry. To a large degree this book represents a

simplification of the book by Schreuder and others (1993). The book by Iles (2003) reveals why he

is such a good speaker and writer; it is a delight to read and is worth examining for practical

suggestions. Gregoire and Valentine (2004), as judged from the outline of their book, appear to

have considerable overlap with ours, but it is more likely to be tightly written and aimed at a more

sophisticated audience. It is also more limited in its objectives but helpfully contains numerous

proofs showing the properties of various estimators. Arvanitis and Reich (2004) provide the most

complete description of geostatistical methods in forestry, methods which rely heavily on models

at this time. For readers interested in obtaining a full understanding of how and why probabilistic

sampling methods work, the classical books Sarndal and others (1992) and Cassel and others (1977)

are mandatory readings and surprisingly easy to read given their strong theoretical orientation. The

book by Cochran (1977) is still quite popular with many practitioners and presents the basic sampling theory well (with a few exceptions). It too is available in Spanish.
2 USDA Forest Service RMRS-GTR-126. 2004.

II. Objectives of Sampling and Monitoring

Before discussing the methodology of survey sampling, some brief comments about statistics

are desirable. What is it? Generally, statistics should be thought of as “systematized common sense.”

It protects us from jumping to conclusions. A good example is the classical experiment on the

effect of aspirin on headaches. Initial tests showed that it helped 80 percent of the people who tried

it, certainly a phenomenal success rate. Then someone had the idea of trying a placebo. It showed

a 60 percent success rate, indicating that although aspirin was useful, many people apparently did

not need it to relieve their headache. Because so many things are open to different interpretations

and because the USA is such a litigious society, statistics have become critical in many fields of

endeavor including natural resources. Hence statistically valid sampling methods and surveys have

become important in generating reliable and legally defensible estimates. Surveys, also called inventories, are the basis for strategic, management, and project planning by generating a reliable

data base. Since a census or complete count of resources would be prohibitively costly and time

consuming, sampling of forest resources began around the beginning of the 20th century (Schreuder

and others 1993).

Before designing a sample survey, the objectives must be clearly defined. Many surveys are

started with a single limited objective, e.g., we just want to know how much timber is available for

harvesting from a certain area or what areas may support numerous (often unspecified) rare plant

species. Many of these surveys are subsequently used for other purposes. Often the novice sampler

spends much money collecting data on a large number of items and then cannot answer specific

questions. If a survey is planned, particularly a large-scale one, it is highly desirable to critically

examine the data to be collected to be sure that the survey truly addresses the requirements of the

proposed users. Questions to be asked may be: Are objectives well defined and attainable? Are

measurements on weeds really needed? If tree quality is considered an important variable but cannot be measured reliably, is it still worth measuring? Remember: you may be blamed for failure to

plan ahead even though your users may have assured you that they had only limited objectives or

that timber really was not more important than other information or that they did not have enough

money to finance the survey properly.

Why Sample?

The purpose of sampling is to draw inferences about a population of interest such as what is the

average height of trees in a forest. The overall field of inference is very broad and technical and is

discussed in more detail in the section on inference in Appendix 1. There are many ways of making

inferences and people can differ vehemently on how to get the necessary information and how to

draw inferences/conclusions on the basis of that information. We focus on a limited part of the field

of inference, the drawing of probabilistic samples from finite populations, and the inferences typically made with such data.

Most decisions in life are made with incomplete knowledge. Your physician may diagnose disease from a few drops of blood or microscopic sections of tissue; a consumer judges watermelons

by the sound they emit when thumped; and we select toothpaste, insurance, vacation spots, mates,

and careers with but a fragment of the total information necessary or desirable for complete understanding. Our hope is that the drops of blood or the tissue samples represent the non-sampled

portions of the body, the sounds of the watermelons indicate the maturity of the melon, and that the

advertising claims present an honest representation of the truth.

Partial knowledge is normal. The complete census is rare; the sample is more usual. A ranger

advertises timber sales with estimated volume, estimated grade yield and value, estimated cost, and

estimated risk. Bidders take the accuracy and reliability of this information at their own risk and

judgment. The nursery specialist sows seed whose germination is estimated from a tiny fraction of

the seed lot, and at harvest estimates the seedling crop with sample counts in the nursery beds.

Managers plan the maintenance of recreation areas based on past use and experience.
USDA Forest Service RMRS-GTR-126. 2004. 3

Typically we collect information from a population. We call this a sample. We then summarize

this information in some manner. Probably the most widely used sample summarization is the

sample mean. Assume that we can take samples of 3 units from some population, then our judgment often is based on the mean of the three, i.e., 123

3

yyy y+ + = with i y the value for the variable

on sample unit i, i = 1,2,3.

However desirable a complete census may seem, there are good reasons why sampling is often

preferred. In the first place, complete measurement or enumeration may be impossible, e.g., determining the exact amount of wood in a forest would cost many times its value; the nursery specialist

would be better informed if the germination capacity of all the seed to be sown was known, but the

destructive nature of the germination test precludes testing every seed. Clearly where testing is

destructive, some sort of sampling is inescapable. Use of a recreation area is not known until the

season is over; judging what resources are needed to manage an area has to be based on previous

information.

Sampling frequently provides the essential information at a far lower cost than complete enumeration. For large populations especially, the data collected is often more reliable. There are

several reasons why this might be true. With fewer observations to be made and more time available, crews will become less tired and remain more committed to careful measurement. In addition,

a portion of the savings in cost could be used to buy better instruments and to employ or train

highly qualified personnel. Certainly careful measurements on 5 percent of the units in a population could provide more reliable information than careless measurements on 100 percent of the

units. Finally, sample data can usually be collected and processed in a fraction of the time required

for a complete inventory, so that the information obtained is timely.

Planning Your Survey

Objectives

The first step is to define the objectives, also considering the possibility that they may be amplified, modified, or extended over time. Successful surveys often are continued over time with additional objectives added on. For large-scale forest surveys such as the one in the USA, where a

national survey is conducted by the Forest Inventory and Analysis (FIA) staff of the FS, the objectives have changed over time. This is what one should expect with successful surveys. The objectives of most surveys are covered by the following set of objectives for large-scale surveys such as

FIA:

1. Generate current status estimates such as acreage in forest area, amount of wood volume by

species groups, mortality, timber volume available for harvest, etc.

2. Monitor change in the above and other parameters over time.

3. Establish procedures required for identifying possible cause/effect relationship hypotheses.

4. Establish procedures required to prove or document cause-effect. Since cause-effect can rarely

be established with survey data and usually requires followup experimentation, it is important to indicate what can and cannot be done in this regard.

5. Provide in-place information for managers by proper development of such techniques as using

maps in conjunction with small area estimation.

6. Provide timely information for decision makers.

7. Maintain a reliable database with comprehensive documentation and reliable archiving, and

encourage better and more analyses.

Originally FIA was established for objective #1. Over time as concern for timber supplies became more critical, #2 became as important. #3 became important in the 1980s with the controversy of apparent declining forest growth in the state of Georgia. #4 almost always requires both

survey sampling and experimentation. Much research is being done on #5 right now. #6 will be a

critical one for management areas, and #7 has always been important but will become even more so

with an annualized inventory where industry and the states can and want to analyze the data much
4 USDA Forest Service RMRS-GTR-126. 2004.

more frequently and independently. FIA has changed from a periodic to an annual approach, so

instead of collecting data every 10 years, between 10 and 20 percent of the national plots will be

measured every year with reports on a state basis every 5 years. This was done to meet the increasing need for more current information.

A classic on planning surveys is Hush (1971). This provides useful supplementary reading to

help in such planning. Particularly, his appendix “Sample outline for preparing inventory plans”

would be a very useful starting point for people contemplating a brand new survey.

Information to be collected

For most objectives, existing probabilistic survey designs can be used or modified readily to

satisfy one’s objectives. Cause-effect is a different issue, dealt with in more detail later. Often the

credibility of the results from an inventory and monitoring system is of paramount importance.

This requires stringent criteria in the survey. Some or all of the following criteria and thoughts

should generally be considered for any reasonable survey (Schreuder and Czaplewski 1992):

1. Only variables with negligible measurement errors or ones that can be efficiently calibrated

with variables with such negligible measurement errors should be used. Subjective observations have high rates of measurement error and unpredictable biases that can compromise

their utility; objective measurements can readily be justified usually even if more expensive

to collect.

2. Destructive sampling cannot be allowed on permanent sample plots. Off-plot destructive sampling might be acceptable in the immediate vicinity of the plot.

3. Exact locations of permanent sample plots need to be kept secret to avoid biased treatment by

landowners or visitors or excess visitations that damage vegetation or soil and make it unrepresentative of the population.

4. Define all variables so no confusion is possible.

5. Define some variables as truth being measured from remote sensing sources rather than by

ground sampling. Remote sensing interpretation can be more efficient and accurate than field

measurement for some variables, preventing the inadvertent disturbance of plots by field

crews and denying access to plots by landowners. In some cases there is some flexibility in

the definition of a variable of interest; for example, canopy cover measured from low-altitude

photos as opposed to estimated from ground samples.

6. Don’t protect sample plots differently from the remainder of the population as is often done for

growth and yield plots.

Closely related to this is the importance of defining variables with the following characteristics:

1. Those that can be accurately measured on aerial photos so that field sampling is not necessary.

For example, in some cases this may be possible with percent canopy cover or change in area

of mature forest but not in change in commercially suitable forest.

2. Variables that can easily be measured in the field such as tree mortality and number of trees.

Such variables may often be correlated with variables measured on aerial photos.

3. Variables difficult or expensive to measure in the field. Examples are tree volume, tree crown

condition, and horizontal and vertical vegetation profiles. Surrogates should be sought such

as basal area for volume.

4. Variables for which a within-growing season distribution may be desired such as rainfall amounts

and distribution, ozone concentrations, chemical composition of tree components, and symptoms of arthropod and microbial effects on trees. This requires more than one visit in a season,

something we often cannot afford in surveys.

5. Variables for which destructive sampling is required such as soil and needle samples for chemical

composition and tree cores for tree growth and dendrochronological studies. How this may

affect remeasurement over time needs to be considered carefully.
USDA Forest Service RMRS-GTR-126. 2004. 5

The following design objectives are critical:

1. Collect data on explanatory/stress variables such as rainfall deficiency, low soil moisture,

exposure to pollution, etc. This type of data usually cannot be collected on plots yet but are

essential in building reliable models.

2. Simplicity in design. This provides flexibility over time and ease in analysis.

3. Consistency of design over time. This simplifies change estimation and identifying possible

cause-effect hypotheses.

4. Flexibility to address new environmental or management issues while maintaining design consistency.

5. Flexibility to incorporate new measurement technologies while maintaining design consistency.

6. Ability to treat each sample unit as a population. This is important, for example, in classifying

each sample to estimate acreage in forest types. This means, for example, no missing data for

a sample unit because of the design used.

7. Use interpenetrating sampling or similar methods so sampling intensity can be readily increased in time and space if needed. This is a nice feature of annualized inventories if handled

properly.

8. Provide flexibility to accommodate replacement of plots to deal with damage caused by the

measurement process (for example, trampling or destructive sampling) or denial of access to

plots by private landowners—for example sampling with partial replacement.

9. Ability to handle missing data such as plots being inaccessible or landowners denying access

(as noted by C. Kleinn, inaccessibility may also be caused by land mines or wildlife such as

elephants and lions). Inaccessibility is best handled by setting aside a separate stratum for

such plots and clearly stating the estimated size of that stratum and how estimates if any are

generated for it. Implement a strong quality assurance program so that true changes in sample

plots over time will not be confounded with changes in measurement error or subtle details in

measurement protocol.

10. Consider use of several plot designs at the same sample locations. Although this complicates

data collection, it may well be required when a large suite of parameters is of interest. For

example, for number of trees and total basal area of trees, very different plot designs are

efficient (fixed area and variable radius plots respectively).

Developing the sampling approach

Given the objectives of one’s survey, the idea is to develop the most cost efficient approach to

reach those objectives. Most of the remainder of the book is devoted to designing such efficient

sampling approaches by giving the reader insight into methods available and how and when to use

them. Basically what we are developing are sampling strategies, which consist of how to collect the

data—what we call the design and how to use the data to estimate the quantity of interest, i.e., the

estimation process.
6 USDA Forest Service RMRS-GTR-126. 2004.

III. Sampling Concepts and Methodologies

Sampling Frame

We all make inferences about populations based on what is typically a biased sample. Knowing

this often drives us crazy in talking with people. For example: Person A: teenagers are terrible

drivers! Person B: Oh, on what do you base that statement? Person A: Well, when I was driving last

week I got cut off twice by teenagers.

A sampling frame is a complete list of the sample units that can potentially be selected in the

population. To avoid biased inferences such as the one above about the teenagers, make sure that

the population defined for sampling is the one of interest as well as the sample units it consists of.

For example, suppose we are interested in the following two parameters about the city of Colima in

the state of Colima, Mexico, and Fort Collins, Colorado, USA:

1. The average income of each household.

2. The average area of land owned by landowners.

In these examples, a household would initially be the possible sample unit in the first and a

landowner in the second. How do we proceed to list the two populations of interests? Is this important? It is critical that each sample unit in the population has a positive probability of being selected

for the sample and that we know what that probability is. Using a list of people with phones is

clearly not a complete list of all people in either city, but it is certainly less complete in Colima.

Since a list of households may not be available for either city, different sample units may be considered such as city blocks for which there generally would be a list (how to make inferences about

households when the sample unit is a city block will be explained later under cluster sampling).

Obtaining a list of all landowners would probably be fairly easy for both cities since all landowners

presumably pay taxes and hence can be found on a tax listing for the cities.

Selecting a representative sample from a population is easiest when we have a complete list of

all units (sample units) from which to draw a representative sample. For example, assume N1 =

number of ha and N2 = number of trees in the same forest. Clearly we often know N1

 but rarely N2.

If all N1

 ha are listed, we can take a simple random sample (SRS) of n1 ha plots. Then we have a

random sample of plots but with a different number of trees per plot usually. It is generally not easy

(and inefficient) to draw a random sample of trees from a population of trees.

It is possible to draw a random sample without having a sampling frame. Then a list of units is

available only after sampling (see Sect 3.4 p.72-73 in Schreuder and others 1993, where a procedure, described by Pinkham 1987 and Chao 1982, is discussed). However, the procedure is awkward to implement.

Purposive and Representative Sampling

In purpose sampling, also called non-probabilistic or model-based sampling, samples are selected more or less deliberately. This can be done on the basis of the judgment of the sampler of

what is a desirable sample or whatever sample happens to be convenient to collect. This is generally not considered a representative sample of the population of interest.

The idea of selecting a representative sample from a population was extensively discussed in the

literature dating from early in the 20th century (Johnson and Kotz 1988, Vol. 8, p. 77-81). Eight

methods of selection have been described, but the method of random or probability sampling discussed below is generally favored. The basic idea is to select a sample completely by chance selection

to ensure that there is no personal bias involved in selecting it. To do this, we use randomization in

selecting the sample, i.e., select a sample from a deliberately haphazard arrangement of observations.

To implement this, we use probabilistic sampling, which involves sampling in such a way that:

1. Each unit in the population is potentially selected with a known positive probability of selection.

2. Any two units in the population have a joint positive probability of selection.
USDA Forest Service RMRS-GTR-126. 2004. 7

Problem: A property comprises 100,000 ha of forest, range, and water and the owner wishes to find

out what is there. Develop a sampling method that satisfies the two conditions above.

Answer: There are numerous ways of doing so. One approach: divide a map of the property in 100,000

1-ha plots and randomly select 20 of these 1-ha plots for classification into the categories forest,

range, and water. This satisfies the two conditions. Estimation may be difficult because some of the

plots may contain more than one of the classes; but how to deal with that will be covered in the

estimation theory later on.

A sampling strategy is comprised of a sampling design and associated estimation theory where a

sampling design states formally how samples are selected. Potential sample units can have equal or

unequal probabilities and joint probabilities of selection meeting the above two criteria. This flexibility leads us to the designs discussed later, i.e., SRS, stratified sampling, cluster sampling, and

variable probability sampling.

To complete the picture of our sampling strategy, we need estimators accordant with the design

selected. Sampling designs with the simplest estimator, additional estimators, and some general

sampling procedures are discussed below.

Populations, Parameters, Estimators, and Estimates

The central notion in any sampling problem is the existence of a population. It is helpful to think

of a population as a collection of units with values of interest attached. The units are selected in

some way and the values of interest are obtained from the selected ones in some manner, either by

measurement or observation. For example, we may imagine a 40-ha tract of timber in which the

unit being observed is the individual tree and the value being observed is tree height. The population is the collection of trees with heights on the tract. The aggregate number of branches on these

same trees would be another population as would the number of trees with hollows suitable for

animal nesting.

To characterize the population as a whole, we often use certain constants of interest that are

called parameters, usually symbolized with Greek letters. Some examples are the mean number of

trees per plot in a population of plots; the proportion of living seedlings in a pine plantation; the

total number of shrub species in a population; and the variability among the unit values.

The objective of sample surveys is usually to estimate such parameters. In the past, we tended to

estimate the population mean or total of one or more variables. Nowadays, we are often also interested in possible explanations of why a parameter is a certain value. The value of the parameter as

estimated from a sample will hereafter be referred to as the sample estimate or simply the estimate,

symbolized by Roman letters. The mathematical formula generally used to generate an estimate is

called an estimator.

Whenever possible, matters will be simplified if the units in which the population is defined are

the same as those to be selected in the sample. If we wish to estimate the total weight of earthworms

in the top 15 cm of soil for some area, it would be best to think of a population made up of blocks

of soil of some specified dimension with the weight of earthworms in the block being the unit

value. Such units are easily selected for inclusion in the sample, and projection of sample data to

the entire population is relatively simple. If we think of individual earthworms as the units, selection of the sample and expansion from the sample to the population may be very difficult if not

impossible.

Problem: How would you go about sampling an ant nest to estimate the number of ants in it?

Answer: If the nest can be destroyed, one can scoop it up, take samples of a certain volume from it at

random, and count the number of ants in each of these samples. If it cannot be destroyed there is

really no obvious way to take a representative sample from the nest to count the ants.
8 USDA Forest Service RMRS-GTR-126. 2004.

Bias, Accuracy, and Precision

When estimating population parameters, one wishes to obtain good estimates close to the true

values at a reasonable cost. When only a part of a population is measured, some estimates may be

high, some low, some close, and some far from the true value. An estimate that is close to the true

value is considered accurate. If the person selecting or measuring a sample is prejudiced in some

way, then the estimate may be biased. For example, if one was interested in the recreational preferences of visitors to a park and interviewed a sample of 99 women and 1 man, one might feel uneasy

about bias in the results, just as one might if the results for a survey of 50 men and 50 women

showed a heavy preference for fishing in a park not noted for fishing and also knowing that the

interviewer was an avid fisherman.

Though most people have a general notion as to the meaning of bias, accuracy, and precision,

statisticians have well-defined expressions for them because they are crucial in their area of expertise, as follows:

Bias—Bias is a systematic distortion that can arise when selecting a sample, during its measurement, or when estimating the population parameters.

Bias due to sampling selection arises when certain units are given a greater or lesser representation in the sample than in the population. This is not compensated for in estimation. Assume for

example that we are estimating the recreational preferences of visitors to a park and we only interview people on weekends. The results will be biased because weekday users had no opportunity to

appear in the sample.

Measurement bias can result. For example, if seedling heights are measured with a ruler from

which the first half-cm is missing, all measurements will be one-half cm too large and the estimate

of mean seedling height will be biased. In studies involving tree counts, some observers may always include a tree that is on the plot boundary while others may consistently exclude it. Both

routines are sources of measurement bias. In timber cruising, the volume table selected or the

manner in which it is used may result in bias. For example, a volume table constructed from data of

tall trees will give biased results when used without adjustment on short trees. Similarly, if a cruiser

consistently overestimates tree heights, volume tables using heights as input will be biased. The

only practical way to minimize measurement bias is by meticulously training the crews in measurement procedures and the use, care, and calibration of instruments.

The technique used to estimate the population parameters from the sample data is also a possible

source of bias. For example, if the recreation preference on two national forests is estimated by

taking a simple arithmetic average of the preferences recorded on each forest, the result may be

seriously biased if the area of one forest is 500,000 ha and it has a million visitors annually and the

other is 100,000 ha in size with only 10,000 visitors annually. A better overall estimate would be

obtained by weighing the estimates for the two forests in proportion to their sizes and/or their

number of visitors. Another example of this type of bias occurs in the common forestry practice of

estimating the average diameter of trees in a forest from the diameter of the tree of mean basal area.

The latter procedure actually gives the square root of the mean squared diameter, which is not the

same as the arithmetic mean diameter unless all trees are exactly the same size.

Selection and measurement biases are rarely acceptable, particularly if the data are of interest to

several users. However, estimation bias can often be acceptable since some biased estimators are

much better than unbiased ones, the bias being often trivial and the results more precise than those

achieved using the unbiased procedures. Acceptable biased estimators are usually asymptotically

unbiased estimators, defined as follows:

Asymptotically unbiased—If the bias of an estimator approaches 0 as the sample size approaches

the population size, the estimator is considered to be asymptotically unbiased. Such an estimator

used to be called consistent, for example in Cochran (1977).

Precision and accuracy—A biased estimate may be precise but it can never be accurate. Among

statisticians accuracy refers to the success of estimating the true value of a quantity; precision

refers to the extent of clustering of sample values about their own average, which, if biased, cannot

be the true value.
USDA Forest Service RMRS-GTR-126. 2004. 9

+++

++ +

+

+

+

+

+

+

+

+

+

+

++++

+

Unbiased

(Inaccurate)

Biased

(Inaccurate)

Imprecise

Unbiased

(accurate)

Biased

(inaccurate)

Precise

+

+

+

+

+

+

+

+

+

+

A target shooter who puts all of his shots in the inner circle of a target might be considered both

accurate and precise; his friend who puts all of her shots in the outer circle and near the 12 o’clock

position would be considered equally precise but nowhere near as accurate; another friend who

always hits the target at some random location would be unbiased but neither accurate nor precise.

This is illustrated in Figure 1 below.

A forester making a series of careful diameter measurements at a fixed position on the bole of a

tree with a caliper, one arm of which is not at right angle to the graduated beam, would achieve

precise but not accurate results. Since the caliper is not properly adjusted, the measured values will

be off the true value (bias) and the diameter estimate will be inaccurate. If the caliper was properly

adjusted but used carelessly, the measurements would be unbiased but neither accurate nor precise.

Generally we strive to use estimators that predict a parameter more reliably than competing

estimators where reliability is usually measured by the ratio of the mean square errors of the estimators. Such estimators are called efficient estimators.

Variables: Continuous and Discrete

Variation is a fact of life. Coping with some of the sampling problems created by variation is an

important part of making valid inferences. All objects have characteristics such as size, shape, and

color. A characteristic that varies from unit to unit is called a variable. In a population of trees, tree

height is a variable, as are tree diameter, number of cones, volume, form class, and species. The

number of people in each recreation group is a variable, as are their sex, their age, etc.

A variable that is expressed in a numerical scale of measurement, any interval of which may, if

desired, be subdivided into an infinite number of values, is said to be continuous, e.g., time recreating, height, weight, precipitation, and volume. Qualitative variables and those that are represented by integral values or ratios of integral values are said to be discrete. Two forms of discrete

data may be recognized: attributes and counts. An attribute refers to units classified as having or

not having some specific quality; examples of attributes might be species or whether trees are alive

or dead. Results are often expressed as a proportion or percent, e.g., incidence of rust in slash pine

seedlings, survival of planted seedlings, and the percentage of users from a particular country of a

recreation area. A count refers to units described by a number, such as number of people in a

recreation group, number of weevils in a cone, and number of sprouts on a tree stump.

Figure 1. An example of bias, precision, and accuracy if average distance to plot center is used in estimating

distance to center of target for five shots.
10 USDA Forest Service RMRS-GTR-126. 2004.

A distinction is made between continuous and discrete variables because the two types of data

may require different statistical procedures. Most of the sampling methods and computational procedures discussed in this book are for use with continuous variables. The procedures for discrete

variables are generally more complex. Often discrete variables can be treated as continuous, especially for larger sample sizes and a large number of classes.

Distribution Functions

Distribution functions for populations show the relative frequency with which different values

of a variable occur. Given such a function, the proportion of units within certain size limits can be

estimated.

Each population has its own distinct distribution function but these can often be approximated

by certain general types of function such as the normal, binomial, and Poisson. The bell-shaped

normal distribution, familiar to most foresters, is often used when dealing with continuous variables. The binomial distribution is used with attributes. The Poisson distribution is used with counts

having no fixed upper limit, particularly if zero or very low counts tend to predominate. Several of

the more important distributions are described in Appendix 2.

The form of the distribution function dictates the appropriate statistical treatment of a set of data

while its exact form will seldom be known. Some indications may be obtained from the sample

data or from a general familiarity with the population. The methods of dealing with normally distributed data are simpler than most of the methods that have been developed for other distributions.

Fortunately, it has been shown that, regardless of the distribution of a variable, the means of

large samples tend to follow a distribution that approaches the normal. This approach to normality

is often used for assessing the reliability of sample-based estimates.

Tools of the Trade

Notation

In describing various sampling methods, frequent use will be made of subscripts, brackets, and

summation symbols. These devices are, like the more familiar notations of +, -, and =, a concise

way of expressing ideas that would be cumbersome if put into conventional language. Using and

understanding them is just a matter of practice.

Subscripts—The appearance of i x , jk z , or ilmn y is annoying to individuals not accustomed to

them. Yet interpreting this notation is simple. In x i , the subscript i means that x can take on different forms or values. Inserting a particular value for i tells which form or value of x we are concerned with. i might imply a particular characteristic of an individual and 1 x might be its height,

2 x its weight, 3 x its age, and so forth. Or the subscript might imply a particular individual. In this

case, 1 x could be the height of the first individual, 2 x that of the second, 3 x that of the third, and so

forth. Which meaning is intended will usually be clear from the context.

A variable (say x) will often be identified in more than one way. Thus, we might want to refer to

the age of the second individual or the height of the first individual. This dual classification is

accomplished using two subscripts. In ik x , i might identify the characteristic (for height, i = 1; for

weight, i = 2; and for age, i = 3) and k could be used to designate the individual we are dealing with.

Then, 2,7 x would tell us that we are dealing with the weight (i = 2) of the seventh (k = 7) individual.

This procedure can be carried to any length needed. If the individuals in the above example were

from different groups we could use another subscript (say j) to identify the group. The symbol

ijk x would indicate the th i characteristic of the th k individual of the th j group.

Summations—To indicate that several (say 6) values of a variable ( i x ) are to be added together

we write 123456 ( ) xxxxxx +++++ or, somewhat shorter, 12 6 (xx x + ++ ... ). The three dots

(...) indicate that we continue to do the same thing for all the values from 3 x through 5 x .

The same operation can be expressed more compactly by 6

i=1

i x .
USDA Forest Service RMRS-GTR-126. 2004. 11

In words this tells us to sum all values of i x , letting i go from 1 up to 6. The symbol is the

Greek letter sigma, indicating that a summation should be performed. The x tells what is to be

summed and the numbers above and below ∑ indicates the limits over which the subscript i will

be allowed to vary. If all of the values in a series are to be summed, the range of summation is

frequently omitted from the summation sign giving i

i x , i x , or sometimes ∑x . All of these

imply that we would sum all values of i x . The same principle extends to variables that are identified by two or more subscripts. A separate summation sign may be used for each subscript. Thus,

we might have 4

1

3

=1 j=

ij

i

x .

This tells us to add up all the values of ij x , j having values from 1 to 4 and i from 1 to 3. Written

the long way, this means ( ) 11 12 13 14 21 22 23 24 31 32 33 34 xxxxxxxxxxxx +++++++++++ . As for

a single subscript, when all values in a series are to be summed, the range of summation may be

omitted, and sometimes a single summation symbol suffices. The above summation might be symbolized by ,

ij

i j

∑x , ij x , or maybe even x . If a numerical value is substituted for one of the

letters in the subscript, the summation is performed by letting the letter subscript vary but holding

the other subscript at the specified value. As an example,

( ) 4

3 31 32 33 34

1

j

j

x xxxx

=

∑ = +++ , and ( ) 5

2 12 22 32 42 52

1

i

i

x xxxxx

=

∑ = ++++

Analogously,

3

i j

i j

y y ≠

∑

indicates that we want to sum both i and j from 1 to 3 but omit the values when i = j.

The sum in long hand then is

12 13 21 23 31 3 2 yy yy yy yy yy y +++++ y .

Brackets—When other operations are to be performed along with the addition, bracketing may

be used to indicate the order of operations. For example,

2

i

i

∑x

tells us to square each value of i x and then add up these squared values. But 2

i

i

x       ∑

indicates to add all the i x values and then square the sum.

The expression

2

ij

i j

∑∑x

tells us to square each ij x value and then add the squares. But 2

ij

i j

x       ∑ ∑
12 USDA Forest Service RMRS-GTR-126. 2004.

indicates that for each value of i we should first add up the ij x over all values of j. Next, this

ij

j

x

      ∑ is squared and these squared sums are added up over all values of i. If the range of j is from

1 to 4 and the range of i is from 1 to 3, then this means

( )( )( )

2 3 4

2 2 2

11 12 13 14 21 22 23 24 31 32 33 34

1 1

ij

i j

x xxxx xxxx xxxx = =

    = +++ + +++ + +++

  ∑ ∑ .

The expression 2

ij

i j

x       ∑∑

tells us to add up the x ij values over all combinations of i and j and then square the total. Thus,

( )

2

3 4

2

11 12 13 14 21 22 23 24 31 32 33 34

1 1

ij

i j

x xxxxxxxxxxxx = =

    = +++++++++++

  ∑∑ .

Where operations involving two or more different variables are to be performed, the same principles apply.

3

11 2 2 33

1

i i

i

x y xy xy x y =

∑ =+ +

but,

( )( )

3 3

12312 3

1 1

i i

i i

x y xxxyy y

= =

   = ++ ++       ∑ ∑ .

Note that 

2

i

i

∑x is not usually equal to 

2

i

i

x       ∑ .

Similarly, i i

i

∑x y is not usually equal to i i

i i

x y          ∑ ∑ .

Factorials—For convenience we use the following mathematical notation for factorials, i.e.,

n! ( 1)( 2)...1 where is an integer and where 0!= =−− nn n n 1.

Characterizing a distribution using measures of central tendency and dispersion

The distribution of values for a population variable is characterized by constants or parameters

such as the mean and the variance. The measure of central tendency gives some idea of the typical

or middle value of the distribution of a variable. The principal measures used are the mean, median,

and mode. Measures of dispersion indicate how much heterogeneity there is in the distribution of

the variable. They summarize the degree to which values of the variable differ from one another.

The most common ones used are the variance or its square root, the standard deviation, and the

range.

Measures of central tendency—Probably the most widely known and used population parameter

is the mean. Given a sample where all units have the same probability of selection, the population

mean is estimated by

1

n

i

i

y

y

n

= =

∑ (1)
USDA Forest Service RMRS-GTR-126. 2004. 13

with sample size n and i y the value for the variable on sample unit i. For example, if we have tree

heights for 5 out of 10 trees with heights 20, 20, 25, 30 and 35 m, then our estimated mean height

for the 10 trees is 20 20 25 30 35 26

5 y

++++ = = m.

Other estimators of central tendency can be useful occasionally too. For example, the median is

the value so that half are larger and half are smaller than this value. In this example the median

would be 25. The mode is the most common value occurring in the data set, which would be 20 in

this case. The median finds some utility in an estimate of central tendency for highly skewed

populations, the classical one being income of people. Clearly the fact that there is a number of

people in the USA, for example, that have incomes of several million dollars a year and others with

less than $10,000 makes the sample mean rather a poor indicator of central tendency; the median

would be more appropriate. Similarly in a stand generated by seed trees, the presence of some huge

diameter seed trees may make the median a more meaningful estimate as a measure of the central

tendency for such a stand. If interest is in identifying beetle-infested stands where only recently

attacked trees may be saved, it may be desirable to identify stands where such trees are the most

common and the mode would be the best indicator of that. Johnson (2000) gives a detailed description of the above three measures of central tendency plus several others. We focus on the mean and

the corresponding total Y = Ny in this book, where N is the total number of sample units in the

population.

A measure of dispersion—Although there are several measures, we will only discuss the variance or its square root, the standard deviation, because it is used by far the most in statistics.

In any population, such as a stand of trees, the characteristic of interest will usually show variation. For example, there will be variation in tree height. Older trees will be considerably taller than

younger ones and both will vary from an overall mean stand height. More observations would be

needed to get a good estimate of the mean height of a stand where heights vary from 2 to 80 m than

where the range is from 10-15 m. The measure of variation most commonly used by statisticians is

the variance.

The variance of a population characteristic such as tree height is a measure of the dispersion of

individual values about their mean. A large variance indicates wide dispersion and a small variance

little dispersion. This variance is a population characteristic (a parameter) and is usually denoted

by 2 σ . Most of the time we do not know the population variance so it has to be estimated from

sample data.

For most types of measurements, the estimate of the variance from a simple random sample is

given by

( )2

2 1

1

n

i

i

y y

s

n

=

−

= −

∑ (2)

where 2

s is the sample estimate of the population variance, and y is the arithmetic mean of the

sample, as defined in (1) above. Sometimes, computation of the sample variance is simplified by

rewriting the above equation as 2

2 1 2 2

2 1 1

1 1

n

n i n

i

i i

i i

y

y y ny

n

s

n n

=

= =

        − −     = = − −

∑

∑ ∑ . (3)

Suppose we have observations on three units with the values 7, 8, and 12. For this sample our

estimate of the variance is

( ) 2

22 2

2

27 7 8 12

3 257 243 7

2 2

s

++ − − = == .
14 USDA Forest Service RMRS-GTR-126. 2004.

Note that the units on variance are the square of the units of the observations. If interest is in

height in meters (m) then the variance will be in 2

m . If interest is in tree volume in 3m then the

variance would be in 3

m squared. To avoid puzzlement we will not show the units of the variances.

Also, we do not distinguish between population values Yi and sample values i y . It has been our

experience that this distinction is unnecessary and confusing for the objectives of this book.

The standard deviation is the square root of the variance and is expressed in the same units as the

mean and the variable. It is symbolized by s, and in the above example would be estimated as

s = 7 = 2.6458. Both the terms “variance” and “standard deviation” are used extensively in statistics.

Standard errors and confidence limits

Sample estimates are subject to variation just like the individual units in a population. The mean

diameter of a stand as estimated from a sample of three trees will frequently differ from that estimated

from other samples of three trees in the same stand. One estimate might be close to the mean but a little

high, another might be much higher, and the next might be below the mean. The estimates vary because

different units are observed in the different samples. And one would expect that generally, a sample of

size six would generate better estimates than a sample of size three. It is desirable to have some indication of how much variation might be expected among sample estimates. An estimate of mean tree diameter that would ordinarily vary between 11 and 12 cm would inspire more confidence than one that might

range from 7 to 16 cm, even though the average is the same. As discussed above, the variance and the

standard deviation (σ = = standard deviation variance ) are measures of the variation among individuals in a population. Measures of the same form are used to indicate how a series of estimates might

vary. They are called the variance and the standard error of the estimate

standard error of the estimate of variance of the estimate of y σ = = y y ). The term “standard error of estimate” is usually shortened to standard error when the estimate referred to is obvious.

The standard error is merely a standard deviation but among estimates rather than individual

units. In fact, if several estimates were obtained by repeated sampling of a population, their variance and standard error could be computed from equation (3) above. However, repeated sampling

is unnecessary; the variance and the standard error can be obtained from a single set of sample

units. Variability of an estimate depends on the sampling method, sample size, and variability

among the individual units in the population, and these are the pieces of information needed to

compute the variance and standard error. For each of the sampling methods described later on, a

procedure for computing the standard error of estimate will frequently be given.

Computation of a standard error is necessary because a sample estimate may be meaningless

without some indication of its reliability. If it takes 100 birds of a rare species to maintain or grow

its population in a forest, we may feel good when the manager tells us that he estimates there to be

150. But how useful is that information? If we subsequently find out that the actual estimate is

between 0 and 300, we have a much more realistic picture of the real situation and realize that we

still do not know whether the population will survive or not and that we have to obtain better

information. Figure 2 from Czaplewski (2003) illustrates the importance of a good sample size in

constructing confidence intervals.

Given the standard error of estimate, it is possible to estimate limits that suggest how close we

might be to the parameter being estimated. These are called confidence limits. For large samples

we can take as a rough guide that, unless a 1-in-3 chance has occurred in sampling, the parameter

will be within one standard error of the estimated value. Thus, for a sample mean tree diameter of

16 cm with a standard error of 1.5 cm, we can say that the true mean is somewhere within the limits

14.5 to 17.5 cm. In making such statements we will, over the long run, be right on average two

times out of three. One time out of three we will be wrong because of natural sampling variation.

The values given by the sample estimate plus or minus one standard error are called the 67-percent

confidence limits. By spreading the limits we can be more confident that they will include the

parameter. Thus, the estimate plus or minus two standard errors will give limits that will include

the parameter unless a 1-in-20 chance has occurred. These are called the 95-percent confidence
USDA Forest Service RMRS-GTR-126. 2004. 15

limits. The 99-percent confidence limits are defined by the mean plus or minus 2.6 standard errors.

The 99-percent confidence limits will include the parameter unless a 1-in-100 chance has occurred.

It must be emphasized that this method of computing confidence limits will give valid approximations only for large samples. The definition of a large sample depends on the population itself

but, in general, any sample of less than 30 observations would not qualify. Some techniques of

computing confidence limits for small samples will be discussed later for a few of the sampling

methods.

Expanded variances and standard errors

Very often an estimate is multiplied by a constant to generate estimates of other parameters, for

example going from an estimate of the mean to an estimate of the total for a population. If a survey

has been made using one-fifth ha plots and the mean volume per plot computed, this estimate

would be multiplied by 5 in order to express it on a per-ha basis, or, for a tract of 800 ha, multiplied

by 4,000 to estimate the total volume.

Since expanding an estimate in this way must also expand its variability, it will be necessary to

compute a variance and standard error for these expanded values. This is easily done. If the variable y has variance 2

s and this variable is multiplied by a constant (say k), the product (ky) will

have a variance of 2 2 k s .

Suppose the estimated mean volume per one-fifth ha plot is 14 3 m with a variance of the mean of

25 giving a standard error of 3 25 5 = m . The mean volume per ha is: 5(14) = 70 3 m and the variance of this estimate is 52 × 25 = 625 with standard error 3 625 25 = m .

Note that if the standard deviation of y is s or the standard error of y is y s , then the standard

deviation of ky is ks and the standard error of k y is y ks . This makes sense since constants have no

variability. So, in the above case, since the standard error of the estimated mean per one-fifth ha is

5, the standard error of the estimated mean volume per ha is 5×5 = 25. A constant may also be added

to a variable. Such additions do not affect variability and require no adjustment of the variance or

standard errors. Thus if z = y + k with y a variable and k a constant, then 2 2

z y s = s . This situation

arises where for computational purposes the data have been coded by the subtraction of a constant.

The variance and standard error of the coded and uncoded values are the same. Suppose we have

the three observations 127, 104, and 114. For ease of computation, these could be coded by subtracting 100 from each, to make 27, 4, and 14. (This was an important advantage in the past when

computers had limited capabilities and had trouble dealing with very large values especially when

used in computing variances.) The variance of the coded values is:

( ) 2

22 2

2

45 27 4 14

3 133

2

s

++ −

= = ,

the same as the variance of the original values

Figure 2. Estimated extent of tropical deforestation with a 10 percent sample of Landsat satellite scenes

(Czaplewski 2003). Frequency

National Scale

(basis for Tucker and Townshend's conclusion)

Frequency

0 2% 4% 6% 8% 10% 12% 14% True value

N = 41

n = 4

Sub-continental Scale

0 2% 4% 6% 14%

N = 124

n = 12

Continental Scale,

Ecofloristic Zones

0 2% 4% 14%

N = 403

n = 40

Global Scale

0 2% 4% 14%

N = 1240

n = 124
16 USDA Forest Service RMRS-GTR-126. 2004.

( ) 2

222

2

345 127 104 114

3 133

2

s

++ −

= = .

Coefficient of variation

The coefficient of variation, C, is the ratio of the standard deviation to the mean.

s C

y = , (4)

Thus, for a sample with a mean of y =10 and a standard deviation of s = 4,

4 0.4 or 40 percent

10

C = = .

Variance, our measure of variability among units, is often related to the mean size of the units;

large items tend to have a larger variance than small items. For example, the variance in a population of tree heights would be larger than the variance of the heights of a population of shrubs. The

coefficient of variation expresses variability on a relative basis. The population of tree heights

might have a standard deviation of 4.4 m while the population of shrubs might have a standard

deviation of 0.649 m. In absolute units, the trees are more variable than the shrubs. But, if the mean

tree height is 40 m and the mean height of the shrubs is 5.9 m, the two populations may have the

same relative variability, i.e., a coefficient of variation of C = 0.11.

Variance also depends on the measurement units used. In our example above, the standard deviation of shrub heights was 0.649 m. Had the heights been measured in dm, the standard deviation

would have been 10 times as large (if z = 10y, 10 z y s s = ) or 6.49 dm. But the coefficient of variation

would be the same regardless of the unit of measure. In either case, we would have

0.649 6.49 0.11 or 11 percent 5.9 59

s m dm C

y m dm == = = .

In addition to putting variability on a comparable basis, the coefficient of variation simplifies

the job of estimating and remembering the degree of variability of different populations. In many

of the populations with which foresters deal, the coefficient of variation could be 100 percent or

more. Because it is often possible to guess at the size of the population mean, we can roughly

estimate the standard deviation. Such information is useful in planning a sample survey.

Covariance, correlation, and regression

Covariance and correlation are measures of how two variables vary in relationship to each other

(covariability). In some sampling applications two or more variables are measured on each sample

unit. In measuring forage production, for example, we might record the green weight of the grass

clipped to a height of 1 cm from a circular plot 1 m in diameter. Later we might record the ovendry

weight of the same sample. We would expect that there would be a positive relationship between

these two variables.

Suppose the two variables are labeled y and x. If the larger values of y tend to be associated with

the larger values of x, the covariance will be positive. If the larger values of y are associated with

the smaller values of x, the covariance will be negative. When there is no particular association of

y and x values, the covariance approaches zero. Like the variance, the covariance is a population

characteristic, a parameter.

For simple random samples, the formula for the estimated covariance of x and y ( xy s ) is

( )( ) 1

1

n

i i

i

xy

xxy y

s

n

=

− −

= −

∑ (5)

Computation of the sample covariance is simplified by rewriting the formula
USDA Forest Service RMRS-GTR-126. 2004. 17

1 1

1 1

1 1

n n

n n i i

i i

ii ii

i i

xy

x y

x y x y nxy

n

s

n n

= =

= =

         − −

= = − −

∑ ∑

∑ ∑ . (6)

Suppose a sample of n = 6 units produced the following x and y values, say green weight and

ovendry weight of the forage example above:

“i” 1 2 3 4 5 6 Totals

y 2 12 7 14 11 8 54

x 12 4 10 3 6 7 42

Then, 54 42 (2 12) (12 4) ... (8 7) 6 306 378 14.4

61 5 xy s

×

× + × ++× − − = = =− − .

The negative value indicates that the larger values of y tend to be associated with the smaller

values of x. Clearly we should be dubious about this result and examine more carefully what happened since one would expect larger values of green weight with larger values of ovendry weight.

The magnitude of the covariance, like that of the variance, is often related to the size of the unit

values. Units with large values of x and y tend to have larger covariance values than units with

smaller x and y values. A measure of the degree of linear association between two variables that is

unaffected by the size of the unit values is the simple correlation coefficient. A sample-based estimate of the correlation coefficient, R, is

2 2

covariance of x and y

variance(x) variance(y)

xy xy

xy

x y x y

s s

s s s s

r = == × . (7)

The correlation coefficient can vary between –1 and +1. As in covariance, a positive value

indicates that the larger values of y tend to be associated with the larger values of x. A negative

value indicates an association of the larger values of y with the smaller values of x. A value close to

+1 or –1 indicates a strong linear association between the two variables. Correlations close to zero

suggest that there is little or no linear association.

For the data given in the discussion of covariance we found 14.4 xy s = − . For the same data, the

sample variance of x is 2 12.0 x s = , and the sample variance of y is 2 18.4 y s = . Then the estimate of

the correlation between y and x is

14.4 14.4 0.969

12.0 18.4 14.86 xy r − − = = =− × .

The negative value indicates that as x increases y decreases, while the nearness of r to –1 indicates that the linear association is very close. In this example we would become even more suspicious of the results and hypothesize for example that sample labels were switched somehow, since

we would expect a strong positive relationship between green and ovendry weight.

An important thing to remember about the correlation coefficient is that it is a measure of the

linear association between two variables. A value of r close to zero does not necessarily mean that

there is no relationship between the two variables. It merely means that there is not a good linear

(straight-line) relationship. There might actually be a strong nonlinear relationship.

Remember that the correlation coefficient computed from a set of sample data is an estimate,

just as the sample mean is an estimate. Like the sample mean, the reliability of a correlation coefficient increases with the sample size (see Appendix 3, Table 5).

Regression analysis deals primarily with the relationship between variables of interest and other

variables considered to be covariates. The idea is to use information on the covariates to improve

estimation for the variables of interest either because information on the covariates is available or
18 USDA Forest Service RMRS-GTR-126. 2004.

can be collected more readily/cheaply than on the variables of interest. For this reason we establish

a linear relationship between the variable of interest y and the covariate x such that

y xe i ii =+ + α β , i=1,...,N (8)

where , 1,..., i e i = N are the residuals for the population with the average residual over the population denoted by ( ) E ei , where

( ) 0 E ei =

and the covariate of residuals i and j is denoted by ( ) E eei j where 2 ( ) E eei j i =σ v if i = j or

( ) 0 E eei j = otherwise; α and β are regression coefficients that are estimated from the data so

that we can predict the i y for the i x that were not sampled as well as estimate the mean or total for

the variable y; 2

i σ v denotes the variance of y at i x ( i v is often represented as a function of i x such

as k

i i v x = ). The value k is usually assumed known where k = 0 denotes a constant variance and

k = 1 or 2 are often used when the variance of i y is expected to increase linearly with some

function of i x ). 2 σ is usually estimated from the data.

Independence

When no relationship exists between two variables they are said to be independent; the value of

one variable tells us nothing about the value of the other. The common measures of independence

(or lack of it) are the covariance and the correlation coefficient. As previously noted, when there is

little or no association between the values of two variables, their covariance and correlation approach zero (but keep in mind that the converse is not necessarily true; a zero correlation does not

prove that there is no association but only indicates that there is no strong linear relationship).

Completely independent variables are rare in biological populations, but many variables are

weakly related and may be treated as if they were independent for practical purposes. As an example, the annual height growth of pole-sized loblolly pine dominants is relatively independent of

the stand basal area within fairly broad limits (say 12 to 30 2

m /ha). There is also considerable

evidence that periodic cubic volume growth of loblolly pine is poorly associated with (i.e., almost

independent of) stand basal area over a fairly wide range.

The concept of independence is also applied to sample estimates. In this case, however, the

independence (or lack of it) may be due to the sampling method as well as to the relationship

between the basic variables. Two situations can be recognized: two estimates have been made of

the same parameter or estimates have been made of two different parameters.

In the first situation, the degree of independence depends entirely on the method of sampling.

Suppose that two completely separate surveys have been made to estimate the mean volume per ha

of a forest. Because different sample plots are involved, the estimates of mean volume obtained

from these surveys would be regarded as statistically independent. But suppose an estimate has

been made from one survey and then additional sample plots are selected and a second estimate is

made using plot data from both the first and second surveys. Since some of the same observations

enter both estimates, the estimates would be dependent. In general, two estimates of a single parameter are not independent if some of the same observations are used in both. The degree of

association will depend on the proportion of observations common to the two estimates.

Problem: Two random samples of size n are taken without replacement from a population. By pure

chance, the two samples are identical. Are the 2 estimates independent?

Answer: Yes, they are.

Problem. In the above example, how would you go about combining the two samples?

 Answer. The most sensible solution would probably be to treat it as a sample of size 2n with replacement even though each sample in itself was taken without replacement. The advantage of this is that

the variance estimate would typically be an overestimate of the actual variance.
USDA Forest Service RMRS-GTR-126. 2004. 19

In the second situation (estimates of two different parameters) the degree of independence may

depend on both the sampling method and the degree of association between the basic variables. If

mean height and mean diameter of a population of trees were estimated by randomly selecting a

number of individual trees and measuring both the height and diameter of each tree, the two estimates would not be independent. The relationship between the two estimates (usually measured by

their covariance or correlation) would, in this case, depend on the degree of association between

the height and diameter of individual trees. On the other hand, if one set of trees were used to

estimate mean height and another for estimating mean diameter, the two estimates would be statistically independent even though height and diameter are not independent when measured on the

same tree.

A measure of the degree of association between two sample estimates is essential in evaluating

the sampling error for estimates from several types of surveys. Procedures for computing the

covariance of estimates will be given when needed.

Variances of products, ratios, and sums

Earlier, we learned that if a quantity is estimated as the product of a constant and a variable (say

Q = kz, where k is a constant and z is a variable) the variance of Q will be 2 2 2

Q z s k = s . Thus, to

estimate the total volume of a stand, we multiply the estimated mean per unit ( y , a variable) by the

total number of units (N, a constant) in the population. The variance of the estimated total will be

2 2 N y s . Its standard deviation (or standard error) would be the square root of its variance or N y s .

The variance of a product—In some situations the quantity in which we are interested will be

estimated as the product of two variables and a constant. Thus

Q1 = kyx (9)

where:

k = a constant and

y and x = variables having variances 2

y s and 2

x s and covariance xy s .

For large samples, the variance of Q1 is estimated by

1

2 2

2 2 2 22 22

1 2 2

2

2 y xy x

Q y x xy

s s s

s Q k x s y s xys

y x xy

  = ++ = + +         . (10)

As an example of such estimates, consider a forest survey project which uses a dot count on

aerial photographs to estimate the proportion of an area that is in forest ( p ), and a ground cruise to

estimate the mean volume per ha ( v ) of forested land. To estimate the forested area, the total land

area (N) is multiplied by the estimated proportion forested. This in turn is multiplied by the mean

volume per forested ha to give the total volume. In formula form

Total volume = N p v

where:

N = the total land area in ha (a known constant),

p = the estimated proportion of the area that is forested, and

v = the estimated mean volume per forested ha.

The variance of the estimated total volume would be

2 2

2 2

2 2

2

( ) p pv v s s s

s Npv

p v pv

  = ++  

  .

If the two estimates are made from separate surveys, they are assumed to be independent and the

covariance set equal to zero. This would be the situation where p is estimated from a photo dot
20 USDA Forest Service RMRS-GTR-126. 2004.

count and v from an independently selected set of ground locations. With the covariance set equal

to zero, the variance formula would be 2 2

2 2

2 2 ( ) p v s s

s Npv

p v

  = +  

 .

Variance of a ratio—In other situations, the quantity we are interested in is estimated as the ratio

of two estimates multiplied by a constant. Thus, we have

2

y Q k

x = . (11)

For large samples, the variance of Q2 can be approximated by

2

2 2

2 2

2 2 2

2 y xy x

Q

s s s

s Q

y x xy

 

= +−  

  (12)

as is still often used, for example, by Freese (1962) or Cochran (1977). A more robust estimator is

22 2

( )

1

(1 ) ( 1)

ˆ ( )

n

j

j

J rm

N fX n D

v Y

n

=

− −

=

∑

 , (13)

where f = n/N, X is the population mean for variable x , and for every j in the sample D( j) is the

difference between the ratio ( )

( )

j

j

ny y

nx x

−

− and the average of these n ratios (Schreuder and others

1993).

Variance of a sum—Sometimes we might wish to use the sum of two or more variables as an

estimate of some quantity. With two variables we might have

Q3 11 2 2 = + kx k x (14)

with 1 k and 2 k constants and 1 x and 2 x variables having variance 2

1s and 2

2 s and covariance 12 s .

The variance of this estimate is

3

2 22 22

1 1 2 2 1 212 2 Qs =++ k s k s kks . (15)

If we measure the volume of sawtimber ( 1 x ) and the volume of poletimber ( 2 x ) on the same

plots (and in the same units of measure) and find the mean volumes to be 1 x and 2 x , with variances

2

1s and 2

2 s and covariance 12 s , then the mean total volume in pole-size and larger trees would be 1 x

+ 2 x . The variance of this estimate is

2 22

12 12 s =++ ss s 2 . (16)

The same result would, of course, be obtained by totaling the x and y values for each plot and

then computing the variance of the totals. This formula is also of use where a weighted mean is to

be computed. For example, we might have made sample surveys of two tracts of timber.

Example:

Tract 1

Size = 3,200 ha

Estimated mean volume per ha = 48 3

m

Variance of the mean = 11.25

Tract 2

Size = 1,200 ha

Estimated mean volume per ha is 74 3 m

Variance of the mean = 12.4
USDA Forest Service RMRS-GTR-126. 2004. 21

In combining these two means, to estimate the overall mean volume per ha, we might want to

weight each mean by the tract size before adding and then divide the sum of the weighted means by

the sum of the weights. This is the same as estimating the total volume on both tracts and dividing

by the total acreage to get the mean volume per ha. Thus

3200(48) 1200(74) 55.09

3200 1200

x

+ = = + .

Because the two tract means were obtained from independent samples, the covariance between

the two estimates is zero, and the variance of the combined estimate would be

( ) ( ) 2 2 2 2

2

2

3200 1200 (3200) (11.25) (1200) (12.40) 11.25 12.40 6.8727

4400 4400 (4400) x s   +

=+= =     .

The general rule for the variance 2

Qs of a sum

11 2 2

1

...

n

nn i i

i

Q kx kx kx k x == + ++ = ∑ (17)

is 2 22 22 22

1 1 2 2 1 2 12 1 3 13 1 ( 1)

2 2

1

... 2 2 ... 2 Q nn n n n n

n n

i i i j ij

i ij

s k s k s k s kks kks k k s

k s kk s

− −

= ≠

= + ++ + + ++

= + ∑ ∑ (18)

where:

, 1,..., i k i = n are constants, i x are variables with variances 2

i s and covariance ij s , for

i = 1,...,n and j i ( ) 1,..., ≠ = n .

Transformation of variables

Some of the statistical estimation procedures described already and in later sections imply certain assumptions about the nature of the variable being studied. When a variable does not fit the

assumptions for a particular procedure, some other method must be used or the variable must be

changed to fit the assumptions or, as we say in statistics, transformed.

One of the common assumptions is that variability is independent of the mean. Some variables

(e.g., those that follow a binomial such as proportion of trees that are of a particular species or

Poisson distribution such as a count of number of trees) generally have a variance that is in some

way related to the mean, i.e., populations with large means often having large variance. In order to

use procedures that assume that there is no relationship between the variance and the mean, such

variables are frequently transformed. The transformation, if properly selected, puts the original

data on a scale in which its variability is independent of the mean. Some common transformations

are the square root, arcsin, and logarithm.

If a method assumes that there is a linear relationship between two variables, it is often necessary to transform one or both of the variables so that they satisfy this assumption. For example the

relationship between total tree volume and dbh is usually curvilinear whereas the relationship

between volume and dbh squared is usually linear. A variable may also be transformed to convert

its distribution to the normal on which many of the simpler statistical methods are based. Good

discussions on transformations are given in Kutner and others (2003) and Carroll and Rupert (1988).

Finally, note that transformation is not synonymous with coding (say dividing all numbers by

1,000), which is done to simplify computation. Nor is it a form of mathematical magic aimed at

obtaining answers that are in agreement with preconceived notions. But interpretation of results becomes more complicated with transformations. We might understand a relationship between number

of birds per ha and the density of a desirable plant species, but explaining a linear relationship between log (number of birds) and log (plant density) is hard to grasp even if the latter is required for

valid statistical estimation. When possible, estimates should be transferred back to the original scale

of interest. This is not always straightforward as can be seen in the references cited above.
22 USDA Forest Service RMRS-GTR-126. 2004.

IV. Sampling Strategies

Designs With the Horvitz-Thompson Estimator

We discuss only single-phase probability sampling designs in this chapter, i.e., we assume that

there is a sampling frame available from which we can select a sample directly. This could be a

sampling frame of trees, field plots, recreation users, campgrounds, or sampling days for recreation

use. Recall that a sampling strategy comprises both the sampling design and the estimator(s) used.

For clarity of understanding, we start with the simplest probability design: simple random sampling (SRS) to illustrate the concepts underlying probabilistic sampling. This is combined with the

simple estimator of the total or the mean of the variable of interest to give us a simple sampling

strategy. This allows us to point out that the simple mean and total estimators are special cases of

the general unbiased unequal probability sampling estimator, called the Horvitz-Thompson estimator. We then go on to look at the general case of unequal probability sampling and note how SRS,

stratified sampling, cluster sampling, sampling with probability proportional to size (pps), and to

some degree systematic sampling with a random start are special cases and why they are good

sampling designs to use under specific circumstances.

 In the text a small population of size 10 is used with the data shown in Table 1. For computer

oriented readers, Appendix 4 uses a more realistic large mapped population called Surinam with

some worked examples. Readers can use the examples in the text and others directly with program

R as shown in that Appendix. This data set consists of a 60 ha stem-mapped population of trees

from a tropical forest in Surinam used and described by Schreuder and others (1997). This population of 6,806 trees has the relative spatial location of the trees and can be used to illustrate the

efficiency of several sampling strategies.

Simple random sampling (SRS)—This is the simplest probabilistic approach. All samples of size

n (samples including n sample units) have the same probability of selection. All sample units have

probability of selection n/N and each set of two units have joint probability of selection ( 1)

( 1)

n n

N N

−

− in

the most usual situation of sampling without replacement. This may appear to be difficult to implement because there are 

!

!( )!

N

nN n − possible samples if sampling is without replacement (so that all n

units are distinct). For example, for a small population of size 10 as in Table 1 with two units

selected, there are 10!/(2! 8!) possible distinct samples. Selecting distinct units is more efficient

than selecting with replacement, where a unit can be selected and measured more than once. This

should be intuitively reasonable, since remeasuring a unit already in the sample does not provide us

any new information as would the measurement of a new unit for the sample. Tied to this is the

concept of the finite population correction (fpc) = 1 N n n

N N

− = − , based on the sampling fraction

(n/N) = f. The fpc is usually part of the variance estimate and indicates that as sample size n goes to

Table 1. A small population used for illustration of some of the ideas discussed, where y = variable of interest

and x1 and x2 are covariates.

Unit Age y = tree volume x1 = basal area (ba) x2 = remotely measured ba

15 1 1 2

25 2 2 2

33 3 3 2

46 4 4 2

57 5 5 2

6 9 10 10 4

7 10 10 10 4

8 12 10 10 4

9 12 10 20 4

10 15 20 20 4

Y = 75 X1 = 85 X2 = 30
USDA Forest Service RMRS-GTR-126. 2004. 23

population size N, the variance estimate becomes zero. This is true because basically we are measuring the entire population as the sampling fraction goes to one, or stated in another way, as the

fpc goes to zero. We often ignore the fpc because many populations are quite large and sample sizes

are small so that the fpc is essentially 1.

 SRS is not hard to implement conceptually if there is a list of the population units available.

One only has to make sure that the selection of any one unit is not influenced by the others selected

or to be selected. For example, one can give each of the units a distinct number from 1 to N and then

select n distinct random numbers between 1 and N. Traditionally one could use a random number

table (Appendix 3, Table 1) but it is often more convenient now to use a random number generator,

also given in the Appendix.

SRS sampling also has the advantage that since all units have the same probabilities of selection,

applicable analysis techniques are easy to implement and estimation is straightforward and understandable, e.g., when estimating the mean μ or total Y of a population. The unbiased estimator of

the total is:

1

n

i

i

N y

Y

n

= =

 ∑ (19)

with sample size n and i y the value for variable of interest on sample unit i. The variance of the

sample mean is: 2

2 22 2

1 2

( ) () () ( ) (1 ) ( 1) n

N

i

i

y Y N N n N N nS S V y Nf Nn N N n

=

− − − = × = =− −

∑

. (20)

An unbiased estimator of the variance of the estimated total is:

2

2 22

1

( ) () () ˆ ( ) ( 1)

n

i

i

y y N N n N N ns

v Y

Nn n N n

=

− − − =× = −

∑ (21)

where:

N = number of sample units in the population and

2

s is the sample variance.

An estimator of the mean μ , y would be obtained by dividing Yˆ by N, so ˆ y Y = / N and its

variance would be 2 ˆ v( ) ( )/ y vY N = .

Example:

Assume we have the small population shown in Table 1 and are interested in estimating either

the average tree volume, μ = Y , or total volume Y, for this mini-forest. A possible sample of size

n = 4 is:

Sample 1

Unit 1234

Value 1234

Then the estimated average tree volume for the population of trees is:

(1 2 3 4) 2.5

4 y

+++ = = and the variance is:

2 2 2 2 2 22 2

2 (1 2.5) (2 2.5) (3 2.5) (4 2.5) ( 1.5) ( .5) .5 1.5 1.67

(4 1) 3

s

− + − + − + − − +− + + = == −

and 10 4 ( ) 1.67 0.2505

10 4

v y − = ×=

× .

If interest is in the total Y, our estimate would be ˆY =× = 10 2.5 25 with estimated variance

ˆ v( ) 100 0.2505 25.0 Y =× = 5 .
24 USDA Forest Service RMRS-GTR-126. 2004.

Note that this is not a good sample since the actual Y = 75. But for all SRS samples, the average

value of Yˆ would be 75. To illustrate how sampling estimates can vary dramatically with SRS, take

another random sample of size n = 4 from this population, say (1, 2, 9, 10).

Sample 2

Unit 1 2 9 10

Value 1 2 10 20

Then the estimated average timber volume for the population of trees is:

1+2+10+20 33 = =8.25

4 4 y =

22 2 2

2 (1 8.25) (2 8.25) (10 8.25) (20 8.25) 77.58

(4 1)

s

− +− + − + − = = −

and 10-4 ( ) = 77.58=11.64 10 4

v y

× .

Yˆ= 10 8.25= 82. × 5 and ˆ v( ) 100 11.64 116 Y =× = 4 .

Therefore, the first, inaccurate estimate shows a small estimated variance whereas the second

estimate is much more accurate but shows a large estimated variance. This is something that can

happen with probability sampling, especially with SRS, which is why we have other designs that

typically perform much better on average.

Problem: What are the advantages of SRS? Identify at least one key drawback.

Answer: The overriding advantage of SRS is the simplicity in analysis. The equally serious disadvantage is that it often is quite inefficient in estimation since more reliable and informative probabilistic

samples can usually be collected.

Note that for the simple population of size 10 above, there are 

10! 210

4!6! = without-replacement

samples of size 4, but 715 with-replacement samples (ignoring the order of units selected). Clearly

it would be advantageous if we can improve the chances of favoring the selection of some of those

samples over others in the probabilistic sampling context if more is known about the population.

For example, it makes sense to have the units selected be different to gain maximum information

about the population. Hence selecting a without-replacement sample is clearly better than a withreplacement sample if we note that for samples of size four there are only 210 completely distinct

samples out of 715 with-replacement samples, 360 with three distinct units, 135 with two distinct

units, and 10 with only one distinct unit. Hence only 210/715 = 0.34 of the with-replacement

samples contain the maximum of information for 4 units in them.

Problem: Show that for large populations with small sample size, it does not make a difference whether

or not with or without replacement sampling is used.

Answer: Especially for small sample sizes, the maximum information is desired for the sample taken.

So a sample consisting of all different units is better than one containing duplicates. Then the

probability of n distinct units in a sample of n units out of a population of N units is P(n out of n distinct)

= N(N  1)(N  2)...(N  n + 1)/Nn. For example, for a population of 10 units with a sample size of 4 this

is: 5,040/10,000 = 0.504. For a population of 20 units with n = 4, this becomes: 0.727. For a population

of 100 with a sample of n = 4, this becomes: 0.941. Clearly this probability is essentially 1 for large N

holding n = 4.
USDA Forest Service RMRS-GTR-126. 2004. 25

Problem: A fuels researcher comes to you rather upset. He measured the depth of the organic matter

layer on both a burned and an unburned stand. For random samples of size 5 on each stand, he

obtained the following results: for the burned plot, average depth was 10 cm; for the unburned plot it

was 8 cm. Could you explain what might have happened?

Answer: Encourage him to compute the standard errors of estimates for both estimates! It is likely

that the variability was so high that one or both of the samples were unlucky samples in the sense that

they either yielded estimates too high or too low for the sampled plots. The confidence limits for the

two estimates are likely overlapping considerably indicating that the sample estimates are not statistically different—which would make the researcher feel somewhat better, at least, since he would certainly expect the burned area to have smaller average depth than the unburned area.

We can often do even better than simple random sampling without replacement. Sometimes, we

may have complete knowledge on a covariate associated with the variable of interest for which we

know all the values in the population; or we can often get these with relative ease. This information

when combined with the information on the variable of interest measured on a sub-sample of the

units can be used in various ways in sample selection and estimation.

Denoting by y = variable of interest and x = covariate, numerous sample selection schemes and

estimators are possible.

Unequal probability sampling—One big advantage of unequal probability sampling is that, for a

fixed sample size, it is a generalization of the other single-phase probabilistic procedures. Understanding the concept of unequal probability sampling greatly facilitates understanding of the other

procedures and why it is advantageous to use them in certain circumstances. Let π i be the probability of selecting unit i and π ij the joint probability of selecting units i and j. Then the HorvitzThompson estimator of the population parameter Y is:

1

n

i

HT

i i

y Y = π = ∑ . (22)

ˆYHT is an unbiased estimator of Y with variance:

( )

2

1 ( ) 2

N j i

HT i j ij

i j i j

y y V Y ππ π

≠ π π

  = −−  

  ∑ 

(23)

or 2

1 ( ) 2

N j i

HT ij

i j i j

y y V Y w ≠ π π

  = −  

  ∑ 

(24)

with π i the probability of selecting unit i, π ij the probability of selecting units i and j, and

wij i j = − ππ π ij .

Note that (19), (20), and (21) are special cases of (22), (23), and (24), respectively. In the following we will not give the actual variance for the different sampling strategies since they are all

special cases of (24).

Unbiased variance estimators based on a sample are:

( )

2

1

1

2

n i j ij j i

HT

i j ij i j

y y

v Y

ππ π

≠ π ππ

−   = −  

  ∑  (25)

and

( ) 2

2 2

1

(1 ) n n ij i j i j i

HT i

i ij i ij i j

y y

v Y y π π ππ

= ≠ π π ππ

− − = + ∑ ∑ 

. (26)

If sampling is with replacement (wr) and the probability of sample unit i at a single draw is i p ,

then the estimated total Y wr is:

1

1 n

i

wr

i i

y Y

n p =

= ∑ 

(27)
26 USDA Forest Service RMRS-GTR-126. 2004.

with variance

( ) 2

1

1 N

i

wr i

i i

y V Y pY

n p =

  = −     ∑ 

(28)

and unbiased variance estimator

( ) 2

1

1

( 1)

n

i

wr wr

i i

y v Y Y

nn p =

  = −   −   ∑  

. (29)

Note that if all the i

n

N

π = and all ( 1)

( 1) ij

n n

N N

π − = − then equation (22) reduces to the simple mean in

(19) for SRS and, similarly, unbiased variance estimators in (23) and (24) reduce to the unbiased

variance estimator for SRS in (21). Let us examine equations (22) and (24) in more detail.

If i i π = ky with k a constant, then ˆYHT is constant, actually Y, and clearly ˆ ( ) V YHT would be

0, the ideal situation. This is only an idealized condition that won’t happen in practice but we

can approximate it. For example, in the small population shown in Table 1, we are interested

in total volume. If we can select trees proportional to their basal area then the ratios

( i y = volume for tree i)/( i x = basal area for tree i) are essentially constant over the 10 trees so that 2

j i

i j

y y

π π

 

  −

 

 is close to 0. Since we are approximating volume rather well with basal area, such a

procedure should be efficient, and this is confirmed in the variance computation since the variance

estimates will be close to 0. Similarly, if our interest is number of trees, giving each tree an equal

weight in selection is efficient and the procedure of selecting proportional to basal area would not

be. Selecting as closely as possible with probability proportional to the variable of interest is the

idea behind unequal probability sampling.

Stratified sampling—In this method, the population of interest is divided into subpopulations or

strata of interest. In this case, the covariable x represents strata, e.g., say x = 1 represents an oldgrowth stratum, x = 2 pole tree stratum, x = 3 clearcut areas, and x = 4 agricultural/wooded lands.

This is a simple but powerful extension of SRS. We simply implement SRS within each stratum.

The idea behind stratification is fourfold:

· To provide information on subpopulations as well as the total population.

· To divide the population into more homogeneous subpopulations or strata and improve the

efficiency in estimation by a more effective distribution of the sample.

· To enable one to apply different sampling procedures in different strata; e.g., sampling in the

Amazon jungle is likely to be very different from that done in the pampas or other less forested areas.

· For convenience; e.g., sampling may be done from different field stations.

In situations where a population is relatively homogeneous, SRS may be more economical than

stratified sampling.

An unbiased estimator of the population mean is

1

1 k

st h h

h

y N y N == ∑ (30)

with estimated variance of the mean

2

2

2

1

( ) ( )

k

h hh

st h

h h

NNn

v s y = N N

− = ∑ (31)

where:

h y = sample mean for stratum h,

k = number of strata,
USDA Forest Service RMRS-GTR-126. 2004. 27

and Nh and nh are number of sample units in the population and sample respectively in

stratum h.

In Table 1, if we stratified on the basis of the remote sensing variable, 2 x , we might put the first

5 units in stratum 1 and the last 5 in stratum 2. It is clear that the within-strata variability is much

less than that between strata. Then suppose we decided to select a sample of size 4, 2 samples in

each stratum, such as units 1, 3, 8, 10. Thus we have:

Stratum 1:

2 2

2

11 1 1

(1 3) (1 2) (3 2) 2, 2, 2, 5

2 (2 1)

n ys N + − +− ==== = = −

Stratum 2:

2 2

2

22 2 2

(10 20) (10 15) (20 15) 2, 15, 50, 5

2 (2 1)

n ys N + − +− == = = = = −

and

( )( ) 5 2 5 15

8.5

10 st y

×+× = = .

Then

( ) 2 2

2

1 5 2 5 2 780 5 2 5 50 7.8

10 5 5 100 st v y   − − = + ==    

Then ˆ 10 8.5 85 Yst =× = with variance estimate ˆ ( ) 100 7.80 780 st v Y =× = .

Problem: Where would you and would you not use stratified sampling?

Answer: Use if interest is in different subpopulations (strata) or if strata are more homogeneous than

the overall population; also, use if different sampling schemes are desirable for different parts of the

population. Do not use if simplicity is desired, for example when differences in probabilities of selection are not desired. Generally stratification is desirable.

In stratified sampling, different sampling intensities can be used in each stratum. In proportional

sampling the sampling intensity is proportional to the number of units in each stratum. In optimal

allocation, sampling the overall variance estimated by (31) or the overall sampling cost, C, is

minimized. Clearly this requires knowledge or an estimate of the within-strata variances and a cost

function, so optimal allocation is usually at best an approximation (see Schreuder and others 1993

for details on proportional and optimal allocation).

Cluster sampling—This is another extension of SRS in that now clusters of (say) trees are sampled

by simple random sampling. Cluster sampling is useful when no list of sample units is available or a

list is costly to obtain, which is often the case with trees; and it is usually cheaper to visit and measure

clusters of trees rather than individual trees as in SRS. In cluster sampling there are actually two

covariates, for example, the area of each cluster which is kept equal (say 1-ha plots) and the number

of trees in each cluster which is rarely known and becomes known only for the clusters sampled.

For maximum efficiency, clusters should be heterogeneous rather than homogeneous as with

strata. Cluster sampling is most useful when no list of sample units is available or is very costly to

obtain and the cost of obtaining observations increases as the distance between them increases. A

mechanism for randomly selecting and locating the clusters must be available.

Suppose we select n out of N clusters at random for sampling and each cluster is measured

completely for the variable of interest. Then for clusters of different sizes a biased estimator, cl y , of

the mean per unit is: .

1

1

n

i i

i

cl n

i

i

M y

y

M

=

=

=

∑

∑ (32)
28 USDA Forest Service RMRS-GTR-126. 2004.

where Mi is the number of units in cluster i, with an estimator of the variance:

2

2

2 .

1

( )

( ) ( ) ( 1)

n

i

i cl

i n

cl

M

y y

N n M

v y Nn n

=

−

− = −

∑

(33)

with N = number of clusters in the population, n = number of clusters selected by SRS, 1

M

i

i

n

M

M

n

= =

∑ ,

the average number of units per cluster in the sample, and i. y = the total for all observations in cluster

i. This estimator is asymptotically unbiased, which means that as n N → , the bias goes to 0.

Using the data in Table 1 let us take a cluster sample. This is not something that can usually be

done in practice but we assume it can be done here to illustrate a point. Let us first do it in an

undesirable way, i.e., have minimal variability in the clusters. If we put units 1-2 in cluster 1, 3-4 in

cluster 2, ..., and 9-10 in cluster 5, we would have little within-cluster variability and considerable

variability between clusters. To implement cluster sampling with n = 4, we set up 5 clusters of 2

units each as indicated above, and select 2 of those clusters at random. If the following clusters

were selected

Cluster i Sample units Volume i. y

1 1, 2 1.5

5 9, 10 15

then ( )( ) 5 1.5 5 15

8.25

10 cl y

× +× = =

and ( ) 10 2 25 25 2 2 (1.5 8.25) (15 8.25) 18.225 10 2 25 25 cl v y −   = −+− =   ×   .

Then ˆ 10 8.25 82.5 Ycl =× = and ˆ ( ) 100 18.225 1822 cl v Y =× = .

Recall that for cluster sampling we would like considerable variability within clusters. If we put

units 1 and 10 in cluster 1, 2 and 9 in cluster 2, ..., and 5 and 6 in cluster 5, we would have maximum

variability within the clusters. Suppose the following two clusters are now selected:

Cluster i sample units volume i. y

1 1,10 10.5

5 2, 9 6

Then ( )( ) 5 10.5 5 6

8.25

10 cl y

× +× = =

and ( ) ( )2 10 2 25 25 2 10.5 8.25 (6 8.25) 4.55 10.2 25 25 cl v y −   = − +− =     .

Then ˆ 10 8.25 82.5 Ycl =× = and ˆ ( ) 100 4.55 455 cl v Y =× = .
USDA Forest Service RMRS-GTR-126. 2004. 29

Clearly, based on the results of the two samples, the second set of clusters was much more

effective than the first one in efficient estimation of the mean or total volume.

Problem: Assume you wish to estimate the average age of the 10 trees in Table 1. You are allowed to

core one tree in each of three clusters to determine age and you can set up the clusters as you like.

How would you go about assigning trees to the three clusters? How would you assign the trees to

three strata selecting one tree from each stratum randomly?

Answer: To maximize the information collected it would be best to group the 10 trees to maximize the

variability within clusters for cluster sampling and to minimize the variability within groups for stratified

sampling. Although no information is given on the age of the trees, it is most reasonable to assume

that age is positively correlated with either volume or basal area. This means that for cluster sampling

cluster 1 might be (1,2,9,10), cluster 2: (3,4,8) and cluster 3: (5,6,7). For stratified sampling: stratum 1:

(1,2,3), stratum 2: (4,5,6) and stratum 3: (7,8,9,10).

pps sampling—In sampling with probability proportional to size (pps sampling), we sample

proportional to the covariate (or independent variable). This is efficient when y and x are highly

positively and linearly correlated. For example, basal area, 1 x , is an excellent covariate when sampling for total tree volume, y. In Table 1, tree 10 would have 20 times the probability of selection of

tree 1 if trees were selected proportional to basal area. The information collected on the covariate

and on the variable of interest is then combined in the unbiased Horvitz-Thompson estimator to

generate an estimate of, say, total volume.

It is usually best to sample without replacement rather than with replacement. The problem with

pps sampling without replacement is that when the sample size is larger than 2, the joint probabilities of selection needed for variance estimation are usually not computable. There are also questions of ease of implementation, fixed sample size, and selection probabilities exactly proportional

to size. Many procedures have been developed to avoid such problems in pps sampling, e.g., Brewer

and Haniff (1983) discuss 50, and more have been developed since. Some of the difficulties and

some of the key methods are also discussed by Schreuder and others (1993, p. 57-62). One advantage of pps sampling is that the other procedures discussed (SRS, stratified sampling, cluster sampling) are special cases of it.

An unbiased estimator of the population mean is:

1

1 n

i

HT

i i

y y N = π = ∑ (34)

with an unbiased variance estimator:

( )

2

2

1

n i j ij j i

HT

i ij i j

y y

v y N

ππ π

= π ππ

−   = −  

  ∑ (35)

where:

n = number of units in sample and

N = number of units in population.

To illustrate pps sampling, assume using Table 1, that the sample consists of units 3, 6, 9,10

selected proportional to basal area, x1

. Then:

85 3 10 10 20 85 3.5 7.437

4 10 3 10 20 20 40 HT y   = +++ =× =   ×   and ˆ 74.37 YHT = .

We have not computed the variance estimate in (35) because it requires the joint probabilities of

selection for the four units selected. We can compute that in this case but it is not easy. We did

compute the bootstrap variance estimate as shown in Table 2.
30 USDA Forest Service RMRS-GTR-126. 2004.

Problem: Think of a situation in natural resources sampling where pps sampling would really be

efficient!

Answer: The classical answer is the selection of trees proportional to basal area if interest is in volume. This is done using a prism currently.

Generally, we would not recommend pps sampling in actual practice. In multivariate inventories, it is unlikely that there is a covariate that is positively correlated with all or several variables.

Even when interest is only in one variable, often times stratified sampling can guarantee us an

efficient allocation of sample units to different sizes of units. On the other hand, with pps sampling

even the less desirable samples consisting of the n smallest or largest units are probabilistically

possible.

Connectivity of the above designs—To get a feeling for when to use the above sampling strategies from a theoretical point of view, consider the variance in (24) called V here for convenience.

If all units have the same probability of sampling and all sets of n sample units have equal

probability of selection, then with all joint probabilities of selection being equal, i.e.,

( 1)

( 1) ij

n n

N N

π − = − ,

the weights wij are 2

( )

( 1) ij i j ij

nN n

w

N N

ππ π − = −= − for all units i and j so that all 1⁄2 N(N-1) terms in the

summation contribute to the variance in (24). As noted earlier, this is simple random sampling (SRS). For SRS using the data in Table 1 with n = 4, all 12 / 90 2 /15 0.133 π ij = == and

all 4 / 25 2 /15 2 / 75 0.027 wij =−= = .

In general, assuming all the π i are equal to n/N and making some of the π ij equal to 

2

2

n

N

so that

the corresponding 0 wij = , implies that those i and j have to be selected independently. For such

units i and j, the π ij increased from 

( 1)

( 1)

n n

N N

−

− , so some of the other π ij have to be reduced correspondingly because the sum of all joint probabilities is ( 1)

N

ij

i j

π n n ≠

∑ = − . To reduce the variance V, it

would be advantageous if 0 wij = for large values of 

2

j i

i j

y y

π π

 

  −

  or equivalently i j y y − for equal

probabilities of selection even if this increases wij for small values. This is the idea behind stratified sampling, where we try to put units that are dissimilar into separate strata to maximize i j y y −

and similar units into the same ones. For example, in Table 1 if interest is in volume, we can use

remotely sensed basal area 2 x as a covariate. It then makes sense if with two strata we put units 1-

5 into stratum 1 and 6-10 in stratum 2 because (say) for n = 4, with 2 units per stratum, then

4312

N

ij

i j

π

≠

∑ =×= with the joint probabilities of selection of 2 units within each stratum being

Table 2. Comparison of results from sampling the small population in Table 1 using five sampling methods

with a sample size of 4 units.

Method Estimated total Estimated variance of the total

SRS case 1 25 25

case 2 82.5 1164

Stratified sampling 85 780

Cluster sampling case1 82.5 1822

case 2 82.5 455

pps sampling 74.4 585

Systematic sampling 45 245
USDA Forest Service RMRS-GTR-126. 2004. 31

π ij=(2/5)*1/4=1/10=0.10 and the joint probability of 2 units in different strata being

π ij= (2/5)(2/5) = 4/25= 0.16. Then 0 wij = for units in different strata and 3/ 50 0.06 wij = = for units

in the same stratum. Clearly this is an effective strategy relative to SRS because we have attached

the higher joint probabilities of selection to units in the same stratum (which are quite homogeneous) and the lower probabilities to units in the two separate strata.

The ideal in cluster sampling is that negative weights wij should be attached to larger values of

2

j i

i j

y y

π π

    −

 

for reductions in V. No π ij can exceed π i orπ j , so that for example if all / i π = n N

then all / ij π ≤ n N . / ij π = n N implies that if i is selected then so is j. Thus all sample units for

which π ij = n/N are all selected together. This is the idea of a cluster. To make some of the wij < 0,

we want the π ij that are equal to n/N to be attached to the largest differences 

2

j i

i j

y y

π π

    −

 

, which

implies that the members within each cluster ideally should vary as much as possible. For example,

in Table 1 for estimating volume assume there are 5 clusters of size 2 each and take a sample of

n = 4. Then, as one good option, put units 1 and 10 in cluster 1; 3 and 9 in cluster 2; 3 and 8 in

cluster 3; 4 and 7 in cluster 4; and 5 and 6 in cluster 5. Then the probabilities of selection for each

unit is 2/5 = 0.40 but now the joint probability of 2 units in the same cluster is 2 / 5 0.40 π ij = = and

in separate clusters is π ij= (2/5)*1/4 =1/10 = 0.10 so that 1/ 25 0.04 wij =− =− for units in the same

clusters and 3/ 50 0.06 wij = = for units in different clusters.

For the example of n = 4 above using the data in Table 1, we have for SRS that all 0.133 π ij = with

0.027 wij = . For stratified sampling we have 0.40 π ij = with 0 wij = for units in different strata and

π ij=0.10 with 0.06 wij = for units within the same stratum. For cluster sampling, 0.40 π ij = and

0.04 wij = − for units in the same clusters and π ij= 0.10 and 0.06 wij = for units in different clusters.

As the results in Table 2 show, stratification and cluster sampling can reduce the variance of the

estimates dramatically relative to SRS.

The idea behind cluster sampling is antithetical to the idea behind stratification. Cluster sampling is more risky than stratified sampling. There will be sharp gains if the clusters are chosen well

but sharp losses if the negative wij are associated with small values of 

2

j i

i j

y y

π π

    −

 

. In stratified

sampling the wij are changed much less typically than in cluster sampling because few sample units

will be selected with joint probability 1 as in the latter. This is all elegantly explained in Stuart (1964).

In probability proportional to size (pps) sampling, a version of unequal probability sampling, it

is assumed that there is a covariate that is positively correlated with the variable of interest, the

ultimate dream being that y and x are essentially the same so that V is essentially 0. We do reasonably well in that respect with 1 x in regards to estimating volume y in Table 1 as noted before. Pps

sampling is even more risky than cluster sampling. For example, if the wij are held constant, it is

clear that 

2

j i

i j

y y

π π

 

  −

  can be very large if the probabilities π i are negatively correlated with the i y .

Systematic sampling with a random start—In this type of sampling, a random sample unit is first

selected as a starting unit and then every kth unit thereafter is selected. Systematic sampling assumes that the population can be arrayed in some order, which may be natural—such as days of the

week in recreation sampling—or artificial, such as numbered plot locations on a map. The ordering

must be carefully considered in the first case but may be haphazard in the latter. For example, when

sampling the use of a recreation area, we probably would not want to sample every seventh day, say
32 USDA Forest Service RMRS-GTR-126. 2004.

every Sunday. In the past systematic sampling has not been generally endorsed by theoretical statisticians but practitioners and applied statisticians have prevailed because it is a practical way of

collecting information in the field and avoids the problem of poorly distributed samples as can

happen in random sampling. In general, SRS estimation procedures are used in systematic sampling (with a random start), the assumption being that the variance estimate for SRS should generally give an overestimate of the variance actually achieved with systematic sampling.

Systematic sampling with a random start should not be used when the population is distributed

in a grid pattern and the sample pattern may match it! For example, in sampling recreation use of an

area it may not be desirable to select every seventh day since clearly a sample consisting of every

Monday could yield quite different results from a sample of every Sunday.

Problem: What is a practical situation in forestry where systematic sampling would really be efficient?

Answer: In most field situations, it is usually practically more efficient to put in a grid of plots or select

a systematic sample of trees in the forest.

An unbiased estimator of the population mean is:

1

n

i

i

syst

y

y

n

= =

∑ (36)

with biased variance estimator:

( ) 2

syst

N ns

v y N n

− = . (37)

Note that these formulas are the same as for SRS.

From Table 1, assume we decide to select our starting point at random from units 1, 2, and 3 and

unit 2 is selected. Then, if n = 4, we would take units 2, 5, 8, and 1. We do this by using mode 10

numbering, i.e., we select units 2, 5, 8, and 11 so that 11 becomes 1. Then our estimate would be:

1+2+5+10 = =4.5 

4 syst y

with

222 2

2 (1 4.5) (2 4.5) (5 4.5) (10 4.5) 49 16.33

3 3

s

− +− +− + − = ==

so ( )

10 4 16.33

10 2.45

4 syst v y

− ×

= = .

Then

ˆYsyst = 45 and ˆ ( ) syst v Y = 245.

Problem: Assuming the population is visited in the above order with a systematic sample of size three,

starting with unit 2, what samples of size three cannot occur?

Answer: One example: units 2, 3, and 4 cannot occur together.

In Table 2 we show the results from the examples above for the various sampling methods. It is

clear that both estimated totals and their variance estimates vary considerably from sample to
USDA Forest Service RMRS-GTR-126. 2004. 33

sample. Being quite inefficient in this case, one would expect SRS to vary much more than the

others, and the table certainly indicates tremendous differences in the results for the two SRS

samples. One would expect the other methods to vary much less. The cluster sampling results show

dramatically the differences between effective clustering, as in case 2, vs. poor clustering, as in

case 1 in table 2. Pps sampling should be especially efficient here since we are sampling proportional to basal area, which is quite closely linearly related to volume in this small population.

Problem: Show how stratified sampling with “optimal” allocation is an unequal probability sampling

procedure. Show that even with proportional allocation it should be considered such.

Answer: In optimal sampling the units in different strata would have different probabilities of selection.

In proportional allocation two units in the same stratum would typically have different joint probabilities

of selection than two units in different strata.

Problem: Assume in a population of 25 grizzly bears that the meat consumption for bear 13 is typical.

Bear 1 eats only 1⁄2 the average of the 25 bears, and bear 24 eats as much as the other 24 combined,

according to a local wildlife specialist. She is willing and able to give you good estimates of the amount

eaten by each bear. If due to a limited budget, we can sample the actual consumption of only 1 bear

and we need to make sure that enough meat is provided to minimize mauling of customers, how would

you pick the sample bear?

Answer: If you decided to sample proportional to estimated consumption as given by the wildlife

specialist, you failed! It is clearly best to select bear 13. This is an example of using common sense

rather than applying theory. One has to make an immediate decision and selecting either bear 24 or

bear 1 for example would yield useless results for making such a decision. This example is a modification of a circus elephant example given by Basu (1971) to illustrate the blind use of probabilistic

sampling. In his example, the statistician recommended using pps sampling and was promptly fired by

the circus director for giving such bad advice.

Problem: Show how systematic sampling with a random start can be considered a special case of:

stratified sampling

cluster sampling

Answer: It is stratified sampling where one unit is selected per stratum or it can be considered cluster

sampling where all selected units form a cluster.

Variance Estimation in General

Classical variance estimation was discussed earlier. The variances are typically derivable and

usually give unbiased or at least consistent estimates of the actual variance. In many cases, however, the actual sampling strategy used is quite complex and such “classical” variance estimators

cannot be derived and, hence, variance estimates cannot be computed. For such situations and even

in cases where the actual variances can be derived and computed, other methods are available, the

two best known being jackknifing and bootstrapping. We only discuss bootstrapping since it is the

easiest to implement in most situations.

Bootstrapping is a clever technique taking full advantage of the computing power that we now

have worldwide. This computer-based method allows one to calculate measures of precision to

statistical estimates. Confidence intervals can be constructed without having to make normal theory

assumptions. The basic concept is most easily understood for simple random sampling. Suppose

we have a sample of n units of y, with sample mean y and variance v( y) . Bootstrapping is accomplished by taking a sample of n units with replacement from the n sample units. This is done B
34 USDA Forest Service RMRS-GTR-126. 2004.

times. Then for each of the B samples we compute , 1,..., b y b = B with average B y

 . The variance

between these bootstrap estimates is: 2

1

( )

( ) 1

B

b B

b

B

y y

v y B

=

−

= −

∑ 

. (38)

This variance estimator can also be used for y . In addition the B sample estimates generate a

distribution of estimates for easy confidence interval construction. The selection of the bootstrap

samples should mimic the actual sample selection method used. Using simple random with

replacement bootstrap sampling from a sample selected by unequal probability sampling is unacceptable. So is the application of bootstrapping to a purposive sample. There are various ways of

bootstrapping described, for example, in Schreuder and Williams (2000). When both the bootstrap

and classical variance estimates can be computed it is not yet clear which is best to use. The

bootstrap method yields immediate, non-symmetric confidence intervals whereas the classical variance is easier to compute.

Regression and Ratio Estimators

Although the Horvitz-Thompson estimator is efficient in many situations, it can be quite unreliable in some. For ease of understanding we limit ourselves to one covariate; those interested in

several covariates should consult Sarndal and others (1992). Consider a population where some of

the covariate values, x, are quite small relative to the values of the variable of interest, y. It is clear

that if some of the sample units contain y and x values where x is quite small, these ratios in the

estimator, y/x could be quite large. For example if x = 0 for one or more units, its ratio would be

undefined. Units with x = 0 would not be selected by pps sampling (causing bias in the estimation)

but would be with SRS. Having extreme ratios can cause serious problems with the mean-of-ratio

estimators (only the Horvitz-Thompson one was discussed here) and their use is generally not

recommended at all with SRS.

Regression and ratio-of-means estimators, like stratification, were developed to increase the

precision or efficiency of a sample by making use of supplementary information about the population being studied. The critical difference of when to use the regression or the ratio-of-means

estimator is illustrated in Figure 3. Consider the linear relationships between two variables x and y

shown with the line marked A passing through the origin and the one marked B intersecting the

ordinate y.

A

A

B

B

X

Y

Figure 3. Postulated relationships between variables y and x.
USDA Forest Service RMRS-GTR-126. 2004. 35

If line B is the relationship expected between the variables, so that clearly the relationship does

not go through the origin, one should use regression. With relationship A through the origin, ratio

estimation is indicated. Mathematically both regression and mean-of-ratio estimators are based on

the following model being reasonable for the data

, 1,... i ii y x ei =+ + = α β N where ( ) 0 E ei = and 2 ( ) E eei j i = σ v if i=j

and ( ) 0 E eei j = otherwise. (39)

Here ( ) E ei indicates the average error for the regression model over the population of y and x

values, ( ) E eei j denotes the covariance of the errors given that the average error is zero and

2

i σ v denotes the variance of y at i x ( i v is often represented as a function of i x such as ki i v x =

where k = 0 denotes a constant variance and k = 1 or 2 are often used when variance of i y is

expected to increase with i x ).

Then if α ≠ 0 use a regression estimator and if α = 0 approximately use a ratio estimator.

When in doubt, it is generally better to use the regression estimator. Ordinarily the question is

answered based on our knowledge of the population and by special studies of the variability of y at

various values of x. If we know the way in which the variance changes with changes in the level of

x, a weighted regression procedure may be used by setting k to known values such as k = 1 or 2.

Regression estimation—Assuming a straight line relationship between y and x with constant

variance (i.e., νi

 = 1, i = 1,..., N) is still the most generally accepted approach at this time. The

equation for the line can be estimated from

y y bX x a b R =+ − =+ ( ) x (40)

where:

R y = the mean value of y as estimated at a specified value of the variable x, x .

y = the sample mean of y,

x = the sample mean of x,

1

2

1

( )( )

( )

n

i i

i

n

i

i

y yx x

b

x x

=

=

− −

=

−

∑

∑ , the linear regression coefficient of y on x, and

a = − y bx = the intercept of y on x.

As noted in Sarndal and others (1992), the regression estimator is equal to the Horvitz-Thompson estimator plus an adjustment term. The regression estimator works well when the adjustment

term is negatively correlated with the error of the Horvitz-Thompson estimator. For large errors in

the Horvitz-Thompson estimator, the adjustment terms will be about equal to the errors but of the

opposite sign for large samples with a strong linear relationship between the variables y and x.

Standard error for regression—In computing standard errors for simple random sampling and

stratified random sampling, it was first necessary to obtain an estimate 2 ( ) y s of the variability of

individual values of y about their mean. To obtain the standard error for a regression estimator, we

need an estimate of the variability of the individual y-values about the regression of y on x. A

measure of this variability is the standard deviation from regression , ( ) y x s computed by

2

.

( )

2

xy

y

x

y x

SP

SS

SS

s

n

−

= −

(41)
36 USDA Forest Service RMRS-GTR-126. 2004.

where 2

1

( )

n

y i

i

SS yy == − ∑ , 2

1

( )

n

x i

i

SS xx == − ∑ , and 1( )( )

n

xy i i

i

SP y yx x == −− ∑ .

Then the standard error of R y is 2

,

1( )

Ry yx

x

Xx N n

s s

n SS N

  − − = +     . (42)

So for y = volume and 1 x = basal area in Table 1 for a sample of n = 4 with observations (1,2,9,10)

we have:

1 Y x = + 0.401 0.73 so our estimated mean volume is

0.401 0.73 10.75 8.25 R y = +× = and the estimated total volume is ˆ 10(8.25) 82.5 YR = = with standard deviation from regression

. 5.0 y x s = and standard error from regression: 2.5 Ry s = .

It is interesting to compare Ry s with the standard error that would have been obtained by estimating the mean volume by SRS from the y-values only. An estimated mean volume per tree is

y = 8.25 with standard error of s = 8.8 , and standard error of the estimate of 4.4 y s = .

The family of regression estimators—The regression procedure in the above example is valid

only if certain conditions are met. One of these is, of course, that we know the population mean for

the supplementary variable (x). As will be shown in a later section (double sampling for regression), an estimate of the population mean can often be substituted. Often the x variable can be

measured on a much larger sample than the y-variable so that our estimate for the x-variable is

much better and can be used to improve the estimate for the y-variable.

The linear regression estimator that has been described is just one of a large number of related

procedures that enable us to increase our sampling efficiency by making use of supplementary

information about the population. Two other members of this family are the ratio-of-means estimator and the mean-of-ratios estimator. The Horvitz-Thompson estimator can be considered an example of the mean-of-ratios estimator. It is very dangerous to use with equal probability sampling

such as SRS, and we will only discuss ratio-of-means estimation here.

The ratio-of-means estimator is appropriate when the relationship of y to x is in the form of a

straight line passing through the origin and when the standard deviation of y at any given level of x

is proportional to the square root of x. This means that in equation (39) we assume that α 0 and

that i i v x = approximately for all i = 1,...,N units in the population. The ratio estimate ( ) rm y of

mean y is

ˆ rm y R = × X (43)

where

Rˆ = the ratio of means obtained from the sample y

x = = x

y

and

 X = the known population mean of x.

The standard error of this estimate can be reasonably approximated for large samples by the

jackknife variance estimator:

( )

2

( )

ˆ 2 2 1 (1 ) ( 1)

n

j

i

J rm

D

v Y N fX n

n

= =− −

∑ , (44)

where for every j in the sample, D( j) is the difference between the ratio j

j

ny y

nx x

−

− and the average of

these n ratios. This robust estimator often provides an overestimate of the actual variance (Schreuder

and others 1993).
USDA Forest Service RMRS-GTR-126. 2004. 37

It is difficult to say when a sample is large enough for the standard error formula to be reliable,

but Cochran (1977) has suggested that n must be greater than 30 and also large enough so that the

ratios y s

y

 and x s

x

 are both less than 0.1.

From this sample the ratio-of-means using the same sample of four trees as for the regression

estimator is:

ˆ

R = = 33/ 43 0.77 .

The ratio-of-means estimator is then

_

rm y R

∧

= X = 0.77*8.5 = 6.52 and the standard error of the estimated total is ( ) ˆ 1.5 J rm v Y = .

This computation is, of course, for illustrative purposes only. For both the regression and the

ratio-of-means estimators, a standard error based on less than 30 observations is usually of questionable value.

Warning! The reader who is not sure of his knowledge of ratio and regression estimation techniques would do well to seek advice before adapting regression estimators in his sampling. Determination of the most appropriate form of estimator can be very challenging. The ratio estimators

are particularly troublesome. They have a simple, friendly appearance that beguiles samplers into

misapplications. The most common mistake is to use them when the relationship of y to x is not

actually in the form of a straight line through the origin (i.e., the ratio of y to x varies instead of

being the same at all levels of x or α ≠ 0 ). To illustrate, suppose that we wish to estimate the total

acreage of farm woodlots in a county. As the total area in farms can probably be obtained from

county records, it might seem logical to take a sample of farms, obtain the sample ratio of mean

forested acreage per farm to mean total acreage per farm, and multiply this ratio by the total farm

acreage to get the total area in farm woodlots. This is, of course, the ratio-of-means estimator, and

its use assumes that the ratio of y to x is a constant (i.e., can be graphically represented by a straight

line passing through the origin). It will often be found, however, that the proportion of a farm that

is forested varies with the size of the farm. Farms on poor land tend to be smaller than farms on

fertile land, and, because the poor land is less suitable for row crops or pasture, a higher proportion

of the small-farm acreage may be left in forest. The ratio estimator may be seriously biased.

The total number of diseased seedlings in a nursery might be estimated by getting the mean

proportion of infected seedlings from a number of sample plots and multiplying this proportion by

the known total number of seedlings in the nursery. Here again we would be assuming that the

proportion of infected seedlings is the same regardless of the number of seedlings per plot. For

many diseases this assumption would not be valid, for the rate of infection may vary with the

seedling density.

In general, more complex but also more robust estimators should be used. The following generalized regression and ratio-of-means estimators are generalizations of the above simple linear and

ratio-of-means estimators. There are of course other estimators possible, for example regression

estimators based on nonlinear relationships between y and x, but those are only applicable in very

specific situations—especially since transformations may often make the relationship between

variables linear so that linear regression estimation can be used on the transformed scale.

A very general efficient estimator, the generalized regression estimator (Sarndal 1980), is:

1 1 1 11

1 ˆ ˆ n n n Nn

i ii

gr gr gr i

i i i ii ii i i

y xe Y aN bX y = = = == ππ ππ

   = + − + − =+       ∑ ∑ ∑ ∑∑ (45)

where:

ˆ , ˆ i gr gr i i i i y a bxe y = + =− y ,
38 USDA Forest Service RMRS-GTR-126. 2004.

1 1

1

1

n n

i i

gr

i i ii ii

gr n

i i i

y x b

v v

a

v

π π

π

= =

=

−

=

∑ ∑

∑

11 11

2 2

11 1

1

1

nn nn

ii i i

ii ii ii ii i i i i

gr nn n

i i

ii i ii i i i i

xy y x

vvvv b

x x

vv v

ππ ππ

ππ π

== ==

== =

−

=

 

−    

∑∑ ∑∑

∑∑ ∑

with variance:

( ) ( )

2

1 ˆ

2

N j i

gr i j ij

i j i j

e e

V Y ππ π

≠ π π

  = −−  

  ∑ (46)

and two possible variance estimators

( ) ( ) 2

1

1 ˆ

2

n i j ij j i

gr

i j ij i j

e e

v Y

ππ π

≠ π ππ

−   = −  

  ∑ (47)

and

( ) 2 ' '

2

1 ( ) ˆ

2

n i j ij j i

gr

i j ij i j

e e

v Y

ππ π

≠ π ππ

−   = −  

  ∑ (48)

where:

( ) i i s gr i s e =−− − y y bx x   ,

2

2

' 1 1

1 1

2 2

11 1

ˆ ˆ () () 1 ˆ ˆ {( ) ( ) }

1

1

n n

l l

n n l l l l ll l i

i ii

l l i l l ll i

nn n

i i

ii i ii ii ii

x x NN XX

vv x x

e ee NN XX

v v vv

x x

vv v

π π

π π

ππ π

= =

= =

== =

     − −−     =− +− − + −               

 

 

×

 

−      

∑ ∑

∑ ∑

∑∑ ∑

, 1

11 1

1 1 ˆ ˆ , ,,

n

i

nn n

i ii i

s s

ii i i ii i s

x

x v NN Xx

v N

π

ππ π

=

== =

= = ==

∑

∑∑ ∑ , and 1

n

i

i i i

s

s

y

v

y N

= π =

∑ .

Schreuder and others (1993) give some alternative variance estimators.
USDA Forest Service RMRS-GTR-126. 2004. 39

Problem: Show how the widely used simple linear regression estimator in (40):

ˆ ˆˆ () ( ) Ylr = + =+ − N a bX Y b X X with 

1

2

1

( )( )

( )

n

i i

i

n

i

i

y yx x

b

x x

=

=

− −

=

−

∑

∑ and a = − y bx is a special case of ˆYgr .

Answer: Set all 1 i v = and select units by SRS, i.e., all / i π = n N .

The generalized regression estimator in (45) takes into account both the probabilities of selection and the variance structure in the relationship between y and x. The latter is usually not known,

but can often be approximated based on existing knowledge.

A generalization of the ratio-of-means estimator is:

( ) 1 1

ˆ ˆˆ n n

i i

grm HT HT

i i i i

y x Y XYX X = = π π

  = =     ∑ ∑ (49)

with approximate variance

2 ˆ ˆ ˆˆ ˆ ( ) ( )2 ( , ) ( ) V Y V Y RCov Y X R V X grm HT HT HT HT =− + . (50)

There is a good discussion on variance estimators for this ratio-of-means estimator in Schreuder

and others (1993).

We recommend the use of the bootstrap variance estimator for both the generalized regression

estimator in (45) and the generalized ratio estimator in (49). Both the generalized regression and

the ratio-of-means estimators are biased but asymptotically unbiased in the sense that when n N → ,

the bias goes to 0.

Problem: Show that both the generalized regression and ratio-of-means estimators are biased but

asymptotically unbiased.

Answer: Proving that the estimators are biased is not easy. It requires deriving approximate formulas

for the bias, something beyond the capabilities of most readers. The easiest way is to look at the

formulas for the bias in books like Schreuder and others (1993). Proving that the estimators are asymptotically unbiased can be shown by letting n N → in (45) and (49). Then the sample estimators

become the population parameter.

Problem. In the state of Jalisco, Mexico, all farmers of agave have to register with an industry cooperative in terms of acreage grown, when agave is planted and at what density. The cooperative wants

to find out how much dies each year for each age and how much is stolen each year from the fields

(agave is a very lucrative crop and each head on a harvestable plant is worth quite a bit of money).

Present two alternatives to the cooperative.

Answer: We actually have a complete sampling frame of the population of interest and the solution is

straightforward. We offer two possibilities:

We can stratify the population into age classes of agave and select a random sample from each

stratum. Since theft should only be a problem in harvestable agave, we should take a larger sample

from the harvestable age classes. In addition to the stated objectives, we might ask the cooperative if

they may want the information by size of ownership too. If yes, we might impose additional stratification based on ownership and take a random sample from all such strata. Note that the disadvantage is

that number of strata could easily get out of hand. If we have 9 age classes and 5 ownership size

classes, we already have 45 strata. So we have a tradeoff between information by strata, each of

which is presumably of interest, and possible limitations on sample size. Note that in both cases we

could also use pps sampling, such as pps sampling proportional to age of the fields or size of ownership. We prefer the stratified sampling generally because the pps sampling can give undesirable

sample size allocation due to random chance. We may also be able to use regression estimation

rather than the Horvitz-Thompson estimator if we think some registered variable such as size of ownership might be linearly related to either mortality or incidence of theft.
40 USDA Forest Service RMRS-GTR-126. 2004.

Some Specific Forestry Sampling Methods

Almost all sampling methods that have proved useful in other disciplines have been used in

forestry. However, only three methods unique to or of considerable interest to natural resources

inventories are discussed here. For other methods see Schreuder and others (1993, 1990), and

Hajek (1957). The three methods are variable radius plot sampling (VRP), fixed area plot, and

Poisson sampling:

VRP sampling—This method was introduced in forestry by Bitterlich (1947) to estimate total

basal area, G, of a forest by a simple counting technique variously known as angle count sampling,

point sampling, plotless cruising, and Bitterlich sampling. The method works as follows: An assessor

visits a number of locations, m, in the forest and counts the number of trees at each which, when

viewed at a given height on a tree, usually breast height, subtend an angle greater than some fixed

critical angle α generated by an angle gauge. This gauge could be one’s thumb held at a given distance from one’s eye, a simple rod with a cross piece, or for precise work, a Spiegel Relaskop or a

prism. Trees are selected proportional to their cross-sectional area at the sighted point. If interest is in

basal area, the trees are viewed at breast height. Since the trees are selected proportional to the variable of interest, a simple count of those selected multiplied by a known constant gives an estimate of

the total basal area in the forest. In general, in analogy with equation (34), the estimator is

1

1 1

ˆ 1 k

m

m N k

ki k

HT

k i ki

Y

y Y

m π m

=

= =

= =

∑

∑∑  (51)

where πki = gki / (FA) with gki the basal area of tree i at point k, F the basal area factor that determines

the size of the angle α, A the area of the population of interest, and Yk

∧

 the estimated total basal area

from point k. The variance is:

2

1 2 ( )

N

i N i j ij i

HT

i j i j

y y y V Y Y m

m

π

π π

=

≠

   

=+ −

 

∑

∑ 

(52)

with an unbiased variance estimator:

( ) ( )2

1

1

ˆ ˆ

( 1)

m

k HT

k

HT

Y Y

v Y

m m

=

−

= −

 ∑

. (53)

For volume estimation the general recommendation is to select a prism (or basal area factor)

resulting in a count on average of 6-10 trees at each sample point. VRP sampling has the big

advantage especially to timber-oriented people that trees are selected proportional to their size and

so minimizes the selection of numerous small trees.

Problem: If in VRP sampling interest is in basal area, why is the variance, V, not zero?

Answer: Because the sample size is random so that the variance in sample size is not zero. The

variance of the basal area estimate is a combination of the variability in basal area estimates and

variability in sample size. The first part is zero but the second one is not.

Problem: Several people had the idea of taking prisms of different sizes to the field and then selecting

the one that gave them the desired number of trees at each point. What is wrong with this procedure?

(See Schreuder and others 1981.)

Answer: It can be seriously biased. In fact, that is how it came to the authors’ attention. Estimates

based on the approach were so much larger than previous estimates that estimates of growth were

clearly unrealistic and forest managers suspected something had gone wrong.
USDA Forest Service RMRS-GTR-126. 2004. 41

The basic principle used in VRP sampling is applicable in other forestry disciplines, e.g., in

sampling an area for amount of recreational use. An instant count of the number of users at random

times during the day gives an estimate of the amount of use for that day since users are selected

proportional to their use. For example, a fisherman who is there the whole day would be counted

every time a sample is taken whereas a family who spends only a few minutes would most likely be

missed. Clearly, if we are interested in number of users, we need to adjust the estimated count of

people by their use (i.e., their probability of selection).

Fixed area plot sampling—This procedure is usually applied using circular plots and subplots.

With the general interest now in ecological as opposed to timber information, it is difficult to

optimize for any specific variable in sample selection as one does with VRP sampling for volume.

Because of its simplicity, fixed area plot sampling is easy to understand and implement relative to

VRP sampling. In tropical areas, long rectangular plots are still often used because of ease of

establishment in dense forest and rough terrain (Wood 1990).

Poisson sampling—This form of sampling, developed by Hajek (1957), was introduced into the

forestry literature as 3-P sampling by Grosenbaugh (1964). Grosenbaugh proposed the method for

timber sales where trees to be cut must be selected and marked and some of them can be sampled

for volume at that time too. In the original application, sampling was done proportional to a covariate,

which could be the ocularly estimated basal area or volume of a tree. To be efficient, the cruiser

needed to be skilled. One way to implement Poisson sampling is to visit every unit i in the population and while doing that obtain the covariate value i x for each tree (say ocular estimate of volume). Each estimate x is then compared to a random number generated between 0 and / X nt where

X is the population total for the population and nt is the target sample size. If the random number

for unit i is less than or equal to i x , the unit is part of the sample to be measured; otherwise it is not.

Clearly if / i t x X > n , the unit is selected with certainty. In actual implementation, X is not known

and has to be estimated beforehand by * X so random numbers have to be used between 0 and * / L X t = n . Here L is set by estimating X by * X and then determining the desired sample size nt .

Wood (1988) clarifies procedures for how to select Poisson samples. Note then that the achieved

sample size na is a random variable with variance:

2

1 1

( )

N N

a ii

i i

V n π π

= =

= − ∑ ∑ .

Hajek (1957) introduced the unbiased Horvitz-Thompson type estimator:

*

1

/

n na

i

u ii

i i t i

y Y y X

n x

π = =

  = =     ∑ ∑ 

. (54)

The variance of ˆYu is:

2

1

(1 ) ˆ ( )

N

i i

u

i i

y V Y π

= π

− = ∑ (55)

and an unbiased variance estimator is:

2

2

1

(1 ) ˆ ( ) a n

i i

u

i i

y v Y π

= π

− = ∑ (56)

where na is the achieved sample size. ˆYu is unbiased but can be a spectacularly inefficient estimator.
42 USDA Forest Service RMRS-GTR-126. 2004.

Grosenbaugh (1967) suggested a slightly biased but generally more efficient estimator for Poisson sampling called the adjusted estimator, ˆYa , where:

ˆ ˆ u e

a

a

Y n Y

n = (57)

with / ne = X L , the expected sample size. An approximate variance of ˆYa is:

( ) 2

2

1

( ) ˆ / 1

N

i a

ai e

i i e

y Vn V Y p Yn = p n

         − +         

 ∑ (58)

where / i i p x = X .

A reliable variance estimator is:

( )

2

2 1 1 ˆ

1

a a n n

j i

i j i aj

a

e a

y y

X x nx

v Y

n n

= =

    −

  = −

∑ ∑

(59)

(Schreuder and others 1993). A special case of this where every unit has an equal probability of

selection is called binomial sampling.

Problem: Show how the unbiased Poisson estimator can be very inefficient and unreliable.

Answer: Substituting 

*

t

X L

n = into *

1 11

aa a a nn n n

ii i i

u t

ii i i i ti ti i

yy y y Y X nL L == = = π nx nx x == = = ∑∑ ∑ ∑ 

 shows that when

i i y = x for all i = 1,...,N units, our estimate can still be far from Y since substitution yields

*

1

a n

i a

u

i i e

y nX Y L = x n = = ∑ 

. Clearly even if on average e a n n = , our initial guess X * of Y can often be

pretty rough.

Problem: A land management agency sampled a large forest area for volume using several strata

based on expected timber volume in the strata. Ten years later they wanted to resample the forest for

volume and change in volume but had lost track of the probabilities of selection used earlier. They

would like to treat their original sample as a simple random sample from the forest and remeasure

those same plots for both volume and change in volume. Is this advisable? (See Schreuder and

Alegria 1995.)

Answer: No! The referenced paper derives a formula for the bias of this procedure. It can be quite

severe. An important lesson is to save the probabilities of selection of units for future use in case a

random sample of these plots are to be revisited for remeasurement.

Sample Size Determination

The most frequently asked statistical question by users of sample surveys is, what sample size

do I use? A first step is to specify well-defined objectives for the sampling. More money has been

wasted because a person has poorly defined objectives. This often leads to unmet objectives with

the sample collected. Once clear objectives are specified, the decision about sample size is much

easier to make. In general, the recommendation will be to take the largest sample possible consistent with the money available. If this is not a satisfactory answer, a systematic statistical approach

is called for. Typically one wants confidence intervals of a certain acceptable width to estimate a

parameter Y, i.e., we would like a confidence interval:

ˆ ˆ 1 y y zS zS

PY Y Y

n n

α     − ≤ ≤ + =−

 
USDA Forest Service RMRS-GTR-126. 2004. 43

where z is the standard normal percentile, to ensure a high probability (1−α) and y S

n

is the standard error of estimate of the estimate Yˆ we would like to generate. This equation implies that the

parameter of interest Y is likely to be within the interval on average (1 ) 100 − × α % of the time. The

problem is that usually we do not know what y S is and since we also do not know the sample size,

the t distribution rather than the z distribution should be used. To estimate sample size, do as

follows for SRS:

• Develop an equation that expresses n in terms of the desired precision of estimate. For SRS, 2 2

2

y

t

t s

n α

λ

≥ where n is the desired sample size, t

α is the 1−α / 2 quantile of the central t distribution with n-1 degrees of freedom that can readily be found in t-tables (Appendix 3, Table 2),

2

y s is the estimated variance for variable of interest y, usually based on a preliminary sample

of some sort, and 

2

2 y

t

t s

n

α λ = is the desired width of the confidence interval specified.

• Estimate the unknown population parameters in the equations used to estimate the desired

sample size. If this is not possible, a rule of thumb is to take a sample of size 50.

• Set priorities on the objectives of sampling. For example, if you have more than one characteristic of interest in the population, compromise is probably required to determine the “optimal”

sample size desired to satisfy the different requirements. Is tree mortality as important as

volume, etc.?

• Ensure that the value of n chosen is consistent with the resources available to take the sample.

Often n is determined solely on this basis and it may well be that if one goes through the

above exercise, one may recommend not sampling at all because the feasible sample size is

too small. Usually this recommendation is ignored.

Problem: A research group wants to sample pollutants in the air above a fire using an airplane. The

group has a budget of $2,000. You estimate that to make a reliable estimate, it takes a sample of size

50 to sample carbon dioxide and 60 to sample nitrogen. The group can only afford a sample of 1 to

sample both carbon dioxide and nitrogen and is also interested in another 5 chemicals. What would

you recommend?

Answer: The sensible answer is to recommend not sampling at this time until more money is available. The more likely outcome is that the group will actually do the sampling. A situation very similar to

this was actually encountered by the senior author. One could argue that with the tremendous variability one can expect in this situation that a sample of size one could be worse than not sampling at all

since the sample of size one could often generate a very misleading estimate of the actual parameters

to be estimated.

Example: We are interested in estimating needle length on a tree with a confidence interval of

no more than 10 mm at the 95 percent confidence level. Based on a small sample from another tree

nearby we estimated mean leaf length to be y =19.8 and s = 4.1 mm. To achieve our objective then

we need 

2 2

.05

2 2.69

(10 / 2)

t s

n = = . Hence we would probably take a sample of 3 needles from the tree to

ensure that the sample obtained is sufficient and hope that the preliminary sample on which we

based our sample size determination was valid for our tree of interest.

Problem. An organization tells you that for a population of 100,000 ha it found that a sample of size 40

ha was enough to give a reliable estimate for a given variable. It wants you to sample 10,000 ha for this

variable and wants you to take a sample of size 4 since it is 1/10 of the original population and hence

in its opinion should give an equally precise estimate for the smaller population. Do you agree?

 Answer: No, you should not! The result is liable to be much less reliable for the smaller population.

See for example Czaplewski (2003) for an actual example of a similar situation. See also Table 2.
44 USDA Forest Service RMRS-GTR-126. 2004.

Ground Sampling

What approach of locating plots and what types of plots should be used? The aim in sampling is

to obtain a representative sample of the population of interest. Frequently in large-scale surveys,

sampling is based on a grid sample with a random start. Strictly speaking, this is not a random

sample since some locations will have joint probabilities of selection of 0. But it is justified as

being SRS since the estimator is unbiased and the variance estimator for systematic sampling with

a random start will usually be an overestimate of the variance assuming SRS.

It is likely that at some point in the future, multi-resource inventories will require different plot

sizes and shapes for different variables of interest but sharing the same plot centers. But this is not

true currently where often sampling for resources other than timber is superimposed on traditional

timber surveys. For example, in the USA, FIA uses four circular 0.017 ha (1/24 acre) subplots

sampling a 1-ha (2.5-acre) plot for most ecological tree variables and use transects for down woody

materials and understory variables.

Plot and transect sampling techniques—Unbiased estimates of forest population parameters

can be obtained from any combination of shape and size of sample units if done properly but the

optimum combination varies with forest condition. The shapes of fixed area plots in forestry are

commonly rectangular, square, circular, and narrow-width rectangular with the circular plot being

by far the most common. Clusters of plots are often more efficient than single plots and are used

commonly in forestry. If there is a clear gradient, rectangular plots laid out across it are efficient

(remember that cluster sampling is more efficient if clusters are heterogeneous) but orientation

should be decided in the office prior to sampling.

Rectangular and square plots are usually laid out by starting with a corner point located by

survey (compass and tape) using an aerial photo or map. The second corner is then located and at

both corners, right angles are established to locate corners three and four.

Circular plots are defined by the plot center and radius. Establishing a circular plot is usually

simpler than other plot types because distances from the plot center have to be checked only for

those trees within a peripheral strip of width approximately 1.5 to 3.0 m. The length of the strip and

hence the number of boundary trees increases with increase in the radius of the plot. Sometimes

exact measurements are needed to determine whether a tree is in or out of the plot.

Narrow rectangular plots are most convenient if information on topography and forest composition is also required as part of the survey and if dense undergrowth or difficult terrain necessitates

spending a large amount of time on plot establishment. The width of the strips, determined beforehand in the office, usually ranges from 5 to 40 m depending on sampling intensity, topography,

forest composition, density of undergrowth, and variability and value of the forest.

For a given sample intensity, a strip survey may be faster than a survey based on plots because

the ratio of working time on the units to traveling time between them is greater for strips. Strips and

plots may be combined in what are called “line plots.” With these, topographical and forest-type

data are gathered on the strips and quantitative information on the forest is obtained from plots

located at intervals along their length.

In forestry three procedures have been popular for sampling timber attributes such as volume,

growth, mortality, etc.:

• Variable radius plot (VRP) sampling usually consisting of a cluster of four or five VRP subplots

sampling a certain area such as an acre or ha. This is a version of unequal probability sampling where trees are selected proportional to basal area. It is efficient for sampling for volume

and basal area, since tree basal area is of course highly correlated with volume. VRP sampling

was invented by W. Bitterlich, an Austrian forester, in the 1930s although he did not publish

his work until the 1940s presumably because of the intervening war. This method is still used

in quite a few European countries. In the USA, the Chief of the FS mandated that the procedure not be used anymore by FIA. But this is clearly still a highly desirable procedure for a

timber sale and for some other uses.
USDA Forest Service RMRS-GTR-126. 2004. 45

• Fixed area plot sampling. Generally a large plot is subsampled by a cluster of small circular

plots. Trees are selected with equal probabilities. This is now used by FIA and NFS of the

USFS and by several European countries. Rectangular plots could also be used but are not

popular at this time although they might be highly desirable in tropical regions or in conjunction with remote sensing.

• Line intercept or line intersect sampling. This is used often for down woody material on the

ground and understory vegetation. For the former, the inclusion probability is sin / i i l w L where

i l is the length of the log, wi the acute angle between the log and the survey transect, and L

the spacing between the lines.

FIA and the current vegetation system (CVS) plots used by Region 6 (Oregon and Washington)

of the USFS (Max and others 1996) are compact, sampling a circular 1-ha plot. Although they can

be established in the field faster than long rectangular plots, they are less efficient for estimation

because of spatial correlations and the similarity of adjacent compact subplots. Measuring them

duplicates much of the work already done and yields relatively little new information. Long subplots spread out over the observation area reduce the effect of spatial correlation relative to circular

or square subplots sampling the same size area.

To increase the precision of the estimates for large areas, one seeks to make the plot estimates as

similar as possible. To do this, one includes as much of the variability as possible within the plot,

thus increasing efficiency. However, long rectangular or large square plots have a large perimeter

that increases the number of decisions required on whether trees on the boundary are “in” or “out.”

Long plots are advantageous for remote sensing, especially low-level aerial photography and

videography. Numerous tree and stand variables, e.g., stocking (trees/ha) and mortality can be

measured with a high degree of reliability using remotely sensed imagery. However, sampling

subplots on the ground is desirable at this time to verify the remote sensed measurements and

adjust them by regression estimation if necessary.

Characteristics of plot types used in the USA are summarized in Table 3.

The following is an overview of the advantages of different subplot sizes and shapes (Schreuder

and Geissler 1999):

• Long rectangular plots are advantageous for low altitude photography measurements and plant

biodiversity estimates.

Table 3. Characteristics of plot types.

Plot/subplot FIA CVS 40 x 250 m 25 x 400 m 20 x 500 m

Plot

Area(ha) 1.000 1.000 1.000 1.000 1.000

Radius/dimensions (m) 56.42 56.42 m 40 x 250 m 25 x 400 m 20 x 500 m

Perimeter (m) 354.5 354.5 580 850 1040

Large subplot

Area (ha) 0.1012 0.0763 0.1000 0.1000 0.1000

Radius/dimensions (m) 17.95 15.58 25 x 40 25 x 40 20 x 50

Perimeter (m) 112.8 97.89 130 130 140

Medium subplot

Area (ha) 0.0168 0.020 0.020 0.020 0.020

Radius/dimensions (m) 7.32 8.02 10 x 20 10 x 20 10 x 20

Perimeter (m) 46.0 50.4 60 60 60

Small subplot

Area 0.001 0.004 0.020 0.020 0.001

Radius/dimensions (m) 1.78 3.57 10 x 20 10 x 20 2 x 5

Perimeter (m) 11.2 22.4 60 60 14
46 USDA Forest Service RMRS-GTR-126. 2004.

• Rectangular plots are easier to fly and interpret, and a 1-ha plot is a convenient size to fly and

photointerpret.

• Long narrow plot or transects are desirable to assess plant biodiversity (species richness and

identification of species) because one wishes to cover as many habitat conditions and as large

an area as possible to find rarer species. Boundary issues are relatively less important because

one only has to check to see if the occasional species not found in the subplots is in or out of

the plot.

• Circular subplots are advantageous for VRP sampling and for measuring other variables where

boundary issues are important, as in regeneration subplots.

• Transects are advantageous for traversing a large area to measure scattered or rare objects such

as woody debris on the ground.

• A series of small area samples such as soil cores are best for certain destructive and expensive

measurements such as cores for assessing soil quality and soil series measurements.

• Plot designs for animals are more general than for plants. The series of articles ending with

Schwarz and Seber (1999) hint at the possibility that with increasing technological improvement, animal populations may be sampled some day with the same ease as plant populations.

Radio tagging, recording devices, and traps can simplify animal sampling and are often needed.

Birds and large mammals cover large areas because of their mobility so sampling for them

requires large plots. Slow moving animals such as worms, snails, ants, and many insects can

be sampled on microplots similar to those used for plants described above but are often hard

to observe and traps may be required to find them. Birds are particularly difficult to sample

because they migrate so their populations are also reflected by conditions elsewhere. Counts

of birds are also influenced by season, the time of the day, and weather.

Problem: You are charged with developing a sampling strategy for the states of Chiapas in Mexico

and Colorado in the USA to estimate timber volumes in those states. What kind of ground plot(s) would

you recommend in the field?

Answer: Chiapas has considerable tropical forest with difficult travel conditions. It is likely that long

narrow plots, say 5 m x 100 m, might be best there. In Colorado travel in the forests would be easier

and VRP plots may be the best way so that trees are selected proportional to basal area.

Edge Effects When Sampling at Stand Boundaries

Randomly selected plots may fall close to a stand boundary, and part of such plots may cross

over into a different stand. These boundary plots have been dealt with in many ways, even to the

point of moving the plots away from the boundary or entirely eliminating them. Some practices can

seriously bias stand estimates, particularly in long skinny stands or fragmented landscapes where

there is a lot of edge. Trees along the edge may grow very differently in diameter and form, for

example where the bordering area is open, so ignoring boundary conditions is clearly wrong. Irregular shaped boundaries can introduce additional problems. For a complete technical treatment

of the issues, see Schreuder and others (1993), sec 7.11.3, and Iles (2003), chapter 14.

In a practical application, probably the most commonly used and accepted method to deal with

boundary plots is the mirage plot (Avery and Burkhart 1983, p. 221). To use the mirage technique,

place the plot without bias where it would fall, and if part of the plot falls outside the stand boundary, install a mirage plot. From the original plot center, tally all of the trees on the plot that are also

within the stand boundary. Measure the distance from the plot center to the boundary and install the

mirage plot the same distance on the other side of the boundary. Tally all of the trees on the mirage

plot that are within the stand boundary. In effect, the area of the plot that exists outside the stand

boundary is mirrored back inside the stand boundary, resulting in counting some trees twice from

points that are orthogonal projections of 1 2 ( , ) s s l l across the stand boundaries that truncate the area

of inclusion i a . Formally the mirage method works as follows:
USDA Forest Service RMRS-GTR-126. 2004. 47

 A sampling location 1 2 ( , ) s s l l is randomly located within an area A. If ir is the distance between

this location and tree i and Ri is the limiting distance for being included in the sample,

then

2

i

i

dbh R

F = or Ri = R. Depending on whether VRP or fixed area circular plots are used, unit ui is

included in the sample if i i r R ≤ . The inclusion area ai is a circular area concentric with ui but

truncated by the area boundary if it is within Ri of the tree. The weight attached to i y is an integer

multiple of / (0) A ai where the multiplier depends on whether ui can also be tallied.

The mirage method has problems with irregular boundaries and with inaccessibility, for example cliffs, swamps, water, or trespass. For such areas, a method called walkthrough (Ducey and

others 2004) has been introduced to address these shortcomings. For trees between the plot center

and the boundary, measure the distance from the plot center to the tree center. Following along the

same line, measure that same distance from the tree center to the boundary. If you are outside the

boundary, the tree is counted twice; otherwise, only once. The advantage is that you never need to

cross the boundary or worry about irregular shaped boundaries. A disadvantage may be that defining the boundary for each tree can be even more subjective often than for plots.

Design Issues

The following design issues are critical:

• Collect data on explanatory/stress variables such as rainfall deficiency, low soil moisture, exposure to pollution, etc. This type of data cannot usually be collected on plots but are essential

in building reliable models.

• Simplicity in design. This provides flexibility over time and ease in analysis.

• Consistency of design over time. This simplifies change estimation and identifying possible

cause-effect hypotheses.

• Flexibility to address new environmental or management issues while maintaining design

consistency.

• Flexibility to incorporate new measurement technologies while maintaining design consistency.

• Ability to treat each sample unit as a population. This is important for example in classifying

each sample to estimate acreage in forest types. This means, for example, no missing data for

a sample unit because of the design used. Of course this is frequently not feasible.

• Use interpenetrating sampling or similar methods so sampling intensity can be readily increased in time and space if needed. This is a nice feature of annualized inventories if

handled properly.

• Provide flexibility to accommodate replacement of plots to deal with damage caused by the

measurement process (for example, trampling or destructive sampling) or denial of access to

plots by private landowners—for example, sampling with partial replacement.

• Ability to handle missing data such as plots being inaccessible or landowners denying access

(as noted by C. Kleinn, inaccessibility may also be caused by land mines or wildlife such as

elephants and lions). Inaccessibility is best handled by setting aside a separate stratum for

such plots and clearly stating the estimated size of that stratum and how estimates if any are

generated for it.

• Implement a strong quality assurance program so that true changes in sample plots over time

will not be confounded with changes in measurement error or subtle details in measurement

protocol.

• Consider use of several plot designs at the same sample locations. Although this complicates

data collection, it may well be required when a large suite of parameters is of interest. For

example, for number of trees and total basal area of trees, very different plot designs are

efficient: fixed area and VRP plots, respectively.
48 USDA Forest Service RMRS-GTR-126. 2004.

Instrumentation

Measurement techniques are covered in great detail in Schreuder and others (1993), Chapter 7.

This section will serve as a supplemental update to that chapter. Although the instruments used

today by the forest practitioner are very different than in the past, the underlying principles remain

the same. In general, measurements are taken for the easily measured lengths and angles, and basic

trigonometric relationships are used to calculate the harder to measure elements. Technological

advances in electronics allow these measurements to be made easily, quickly, and accurately. In

addition, rugged handheld computers allow not only capturing these measurements, but also auditing and processing them.

New diameter measurement tools—The tool of choice for most remains the d-tape or caliper.

Two new tools however provide for convenience: the electronic caliper from Haglof and a new

electronic diameter measurement device, functionally equivalent to the Relaskop, from Laser Technology. The caliper looks like the traditional beam caliper, but it also has a digital readout of the

diameter as well as a data recorder; after a day’s field work, the data is downloaded to a computer

for processing. A promising new instrument, although not yet available commercially, is the electronic diameter measurement device. A lighted bar is superimposed on the tree, and the width of the

bar is manipulated with the controls to coincide with the diameter of the tree. A distance is entered

either manually or captured from a connected laser distance device. The distance to the tree, together with the width of the bar, allows the diameter to be calculated internally. With this instrument’s

2X magnification and vertical angle encoder, it can also be used for upper stem diameters.

New height measurement tools—The key to determining height is an accurate measurement of

horizontal distance to the tree. Laser distance measurement devices have proven themselves to be

very effective over the past few years. Laser Technology, Newcon Optik, LaserAce, Handlaser,

Opti-Logic, and others offer laser distance measurement. As with any new technology that is continually changing, search the World Wide Web for the latest information. Some have built in vertical angle encoders, and along with the internal logic they can display the height. An optional, addon flux-gate compass is available for some models.

Another recent addition to the practitioner’s toolbox is the Haglof Vertex Hypsometer, an ultrasonic distance measuring device. This system has two parts, a transponder and the hypsometer; the

transponder can be placed at the plot center or hung on a tree, and then the hypsometer is used to

determine the distance to the transponder, and optionally a vertical angle. Distance and height are

displayed on the screen. The problem of boundary trees, that is, those that occur at or near the

boundary of a plot, always arises when establishing sample units in the field. Measurement error

associated with such trees can be a source of considerable error in deriving plot estimates in forest

inventory. Ultrasonic distance measuring devices should make it easier to implement the miraging

or walkthrough methods described earlier for sampling boundary areas.

New data recording—Source point data collection on a handheld portable data recorder (PDR)

has many advantages over handwritten forms, particularly in light of the ease of data communication between the handheld and other electronic measurement devices. Direct capture of instrument

output by the PDR avoids the common input errors often encountered. Even with mechanical measurement processes, keying the data into the PDR avoids the possibility of transcription errors. In

addition, the PDR can be programmed to look for missing or illogical data entry values.

As the Microsoft Windows CE platform matures, many hardware and software solutions for

forestry are available as replacements for the DOS and other proprietary operating systems. There

are many choices of software for cruising, scaling, and sampling. Commercial software is available

through most hardware vendors, and is also available through public entities.

The ready availability of inexpensive personal data assistants (PDA) has made automated field

data collection much more affordable. With the addition of a hardshell case, the PDA has become

a very serviceable field unit. For production field use, however, the truly rugged units with integrated keypads are preferable.
USDA Forest Service RMRS-GTR-126. 2004. 49

Sampling for Coarse Woody Debris (CWD)

In CWD inventories, one may be interested in both standing and fallen woody material. Since

assessment of standing live and dead trees is usually done as part of a traditional timber inventory,

only the sampling of fallen woody debris is discussed here. The discussion draws heavily on the

review of Stahl and others (2001). We assume interest is in total volume and number of pieces. As

noted by Stahl and others, there is no obvious best way of sampling CWD. But in line with the

emphasis on simplicity in this book, strip or line sampling are favored. Strip sampling is the same

as the other fixed area sampling techniques discussed elsewhere and hence does not need further

elaboration here except that one needs to clearly decide when a log is in or out of the sample.

Usually it is best to call the log in if the center of the butt end is in the strip for both volume and

number of logs estimation. One could count a log in for volume if part of the log is in but the butt

center is not, but this can lead to such complications as possibly having volume with a zero estimate of number of logs. The advantage of this technique is that it is simple to implement since such

plots are easily laid out generally and material on the ground is readily accessible for measurements. There are also no problems with logs not lying horizontally or how crooked the stems and

branches are (the latter have to be considered for estimating number of CWD units).

In line intercept, also called line intersect sampling, all units intersected by an inventory line are

sampled. Usually the lines are laid out in segments with a specific spacing and orientation. Assuming the lines are laid out in a fixed direction, the inclusion probability of selection for a unit requires measurement of the projection of the length of the unit perpendicular to the orientation of

the survey line. Then the estimators for variable y, either total volume or number of units, is:

1

ˆ

sin

m

i

i i i

y Y L

= l w = ∑ (61)

where L is the spacing between survey lines laid out systematically across the entire population, m

is the number of lines, i l is the length of the unit, and wi the acute angle between the unit and the

survey line. If m lines of sizes i s are used, then the following ratio estimator should generally be

more efficient:

1

1

sin ˆ

m

i

i i i

m

i

i

y

l w

Y A

s

=

=

      =

∑

∑ (62)

with A the area being sampled. A complication of this sampling design can be sample logs parallel

to the direction of sampling. Such logs have a probability of selection of close to zero and as

indicated earlier with the Horvitz-Thompson estimator this can create seriously inflated estimates

if such logs are counted in even if they are a valid part of the sample. If they are counted out when

they should have been counted in, this clearly causes a bias in estimation. See Williams and Gove

(2003) for more details about the potential bias. This method has the considerable advantage in

that establishing and walking a line in the field is easy but suffers from the problems of having to

measure angles, having to compensate for logs not lying horizontally or for crooked stems and

branches, and deciding whether logs parallel to the line of sampling are in or out. A comprehensive

discussion of the theory and history behind line intersect sampling is given in Chapter 13 of DeVries

(1986).

Problem: Consider strip sampling where a log is counted in for volume but not for number of logs. If

part of the log is in the strip but the butt center is not, is it possible to

a. Have volume estimates but a zero count of number of logs?

b. Have a positive estimate of number of logs but with zero volume?

Answers: a. Yes b. No
50 USDA Forest Service RMRS-GTR-126. 2004.

Wildlife Sampling

Much of the theory of sampling finite plant populations is not applicable to sampling many

wildlife populations (Schreuder and others 1993, p. 326). Many animal species are mobile and

hide, making detection or measurement difficult and so sampling may affect their location. There is

usually no sampling frame and probabilities of selection have to be estimated usually after the

sample is drawn. The existence of a specific selection probability for an individual in the population is often mainly conceptual. As a result, sampling animal populations is usually more expensive

than sampling plant populations and more statistical assumptions have to be made to make estimation possible, so errors are more likely (Burnham 1980).

The primary parameters of interest in wildlife sampling are usually population size and rates of

birth, immigration, emigration, and mortality. Populations are classified often as either closed or

open. A closed population is assumed to have a constant size with the same members except for

known removals during a study. In an open population, births, immigrations, emigrations, and

deaths can occur.

Traditionally, only a single visit is made to a primary sample unit (psu). However, it is difficult

to obtain repeatable animal observations within one visit, because counts are influenced by weather,

time of day, and other factors. Leaving recording equipment in the field for a few weeks would

enable samples to be taken at all times, day and night, and under varying weather conditions,

making the observations much more repeatable. An important advantage of automatic recorders is

that nocturnal and shy animals can be observed.

As noted, sampling strategies for animals are considerably more complex than for vegetation.

Such devices as radio tags, classification of DNA samples from hairs and pellets encountered on

sample locations, and high-detail remote sensing should make animal sampling easier in the future.

Detailed procedures for sampling animal populations are given in Schwartz and Seber (1999) and

Thompson and others (1998).
USDA Forest Service RMRS-GTR-126. 2004. 51

V. Sampling Methods for Discrete Variables

Simple Random Sampling (SRS) for Classification Data

Assume that for a population of a given rare tree species it is important to determine the proportion of male and female trees, and the sex of a tree can only be obtained easily in the fall. From a

random sample of 50 trees, the number of females is 39. Then the estimate, p , of the proportion

that is female is:

p = Number having the specified attribute/Number observed (63)

= 50

39

= 0.78.

Standard error of estimate—The estimated standard error of p is

1

1

1 p

p p

n

s

n N

    −

   = −   −  

(64)

where: n = number of units observed.

In this example N is extremely large relative to n, and the finite-population correction (1-n/N)

can be ignored, so that

( )( )

( ) _

0.78 1 0.78

= 0.05918 p 50 1

s − = − .

Confidence limits—For certain sample sizes, confidence limits can be obtained from Appendix 3,

Table 3. In this example we found that in a sample of n = 50 trees, 39 were female. The estimated

proportion of females was 0.78 and, as shown in Table 3, the 95-percent confidence limits would be

0.64 and 0.88. For samples of 100 and larger the table does not show the confidence limits for

proportions higher than 0.50. These can easily be obtained, however, by working with the proportion of units not having the specified attribute. Thus suppose that, in a sample of n=1,000, the 95-

percent confidence interval for an observed fraction of 0.22 is 0.19 to 0.25. If the true population

proportion of males is within the limits of 0.19 and 0.25, the population proportion of females must

be within the limits of 0.75 and 0.81.

Confidence intervals for large sample—For large samples, the 95-percent confidence interval

can be computed as

1 2

2 p p s

n

  ± +    . (65)

Assume that a sample of n = 250 units has been selected and that 70 of these units are found to

have some specified attribute. Then,

70 0.280

250 p = = .

And, ignoring the finite-population correction,

( )( ) _

0.28 0.72

p 249

s = = 0.02845.
52 USDA Forest Service RMRS-GTR-126. 2004.

Then, the 95-percent confidence interval is:

0.280 ( ) ( )



 



 ± +

2 250

1 2 0.02845 = 0.280 ± 0.059 = 0.221 to 0.339.

Thus, unless a 1-in-20 chance has occurred, the true proportion is between the limits 0.22 and

0.34. For a 99-percent confidence interval we multiply p s by 2.6 instead of 2. (For samples of

n = 250 or 1,000, the confidence interval could be obtained from Appendix 3, Table 3. For this

example the table gives 0.22 to 0.34 as the limits.)

The above equation gives the normal approximation to the confidence limits. This approximation can be used for large samples. What qualifies as a large sample depends on the proportion of

items having the specified attribute. As a rough guide, the normal approximation will be good if the

common (base 10) logarithm of the sample size (n) is equal to or greater than

1.5 + 3 (|P – 0.5|)

where: P = our best estimate of the true proportion of the population having the specified attribute

and |P – 0.5| = the absolute value (i.e., algebraic sign ignored) of the departure of P from 0.5.

Thus, if our estimate of P is 0.2 then |P –0.5| is equal to 0.3. To use the normal approximation,

the log of our sample size should be greater than

1.5 + 3(0.3) = 2.4

so that n must be 251 (2.4 = log 251).

Sample size—Appendix 3, Table 3 may also be used as a guide to the number of units that should

be observed in a SRS to estimate a proportion with a specified precision. Suppose that we are

sampling a population in which about 40 percent of the units have a certain characteristic and we

wish to estimate this proportion to within ± 0.15 (at the 95-percent level). The table shows that for

a sample of size 30 with p = 0.40, the confidence limits would be 0.23 and 0.60. Since the upper

limit is not within 0.15 of p = 0.40, a sample of size 30 would not give the necessary precision. A

sample of n = 50 gives limits of 0.27 and 0.55. As each of these is within 0.15 of p = 0.40, we

conclude that a sample of size 50 would be adequate.

If the table suggests that a sample of over 100 will be needed, the size can be estimated by

( )( )( )

2

1

1

4 1

n

E

P PN

=

+ − for 95-percent confidence, and

( )( )( )

2

1

1

6.76 1

n

E

P PN

=

+ − for 99-percent confidence

where:

E = the precision with which P is to be estimated and

N = total number of units in the population.

The table indicates that to estimate a P of about 0.4 to within E = ± 0.05 (at the 95-percent

confidence level) would require somewhere between 250 and 1,000 observations. Using the first of

the above formulae (and assuming N = 5,000) we find,

( )

( )( )( )

2

1 357

0.05 1

4 0.4 0.6 5,000

n = =

+ .

If we have no idea of the value of P, we will have to make a guess at it in order to estimate the

sample size. The safest course is to guess a P as close to 0.5 as it might reasonably occur.
USDA Forest Service RMRS-GTR-126. 2004. 53

The following problem shows how dangerous it can be to sample for attributes without realizing

the implications exactly.

Problem. Industry and an environmental group are arguing about how much old growth there is in a

certain large forest. They agree upon the following definition of old growth: A hectare of forest is

considered old growth if it contains at least one tree with a diameter breast height of 100 cm. A

consultancy group is selected to make an inventory of the forest and decides to select 100 1-ha plots

randomly from the forest. Because it is expensive to measure all trees on the sample plots they propose randomly selecting 4 subplots of 0.1 ha each and then classify each hectare as to whether it is

old growth or not. Both industry and the environmental group want an unbiased estimate of old growth

for the forest. Would they get it with this approach?

Answer: No. With this approach one can only err in one way. A hectare can be classified as not being

old growth when in fact it is but it can never be classified as being old growth when it is not. Serious

bias can result in such an estimate of old growth. See Williams and others (2001) for an extensive

treatment of the issue involved. To obtain an unbiased estimate, all 100 1-ha plots would have to be

censused.

How to select a tree or a seed at random—If we try to estimate the proportion of trees in a stand

having a certain disease, we could do it by binomial sampling but this requires visiting every tree in

the population and at that time determining whether it is a sample tree or not. This is SRS but is

time consuming and results in a random sample size. Selecting trees completely at random then is

difficult to do in a practical manner, which explains why systematic sampling with a random start

is popular in such situations as a practical alternative.

In some populations, the individuals themselves are randomly located or can easily be made so.

A batch of seed is such a population. By thoroughly mixing the seed prior to sampling, it is possible

to select a number of individuals from one position in the batch and assume that this is equivalent

to a completely random sample. Those who have sampled seed warn against mixing in such a

manner that the light empty seeds tend to work towards the top of the pile. As a precaution, most

samplers select samples from several places in the pile with a scoop, combine them, and treat that

sample as a SRS.

Cluster Sampling for Attributes

In attribute sampling the cost of selecting and locating a unit is often very high relative to the

cost of determining whether or not the unit has a certain attribute. In such situations, cluster sampling is usually preferred over SRS. In cluster sampling, a group becomes the unit of observation,

and the unit value is the proportion in the group having the specified attribute.

In estimating the survival percentage of trees in a plantation, it is possible to choose individual

trees for observation by randomly selecting pairs of numbers and letting the first number stand for

a row and the second number designate the tree within that row. But it is inefficient to ignore all of

the trees that one walks by to get to the one selected. Instead, survival counts are made in a number

of randomly selected rows and averaged to estimate the survival percent if the same number of

trees occur in each row. This is a form of cluster sampling, the clusters being rows of planted trees.

The germination percent of a batch of seed can also be estimated by cluster sampling. Here the

advantage of clusters comes not in the selection of units for observation but from avoiding some

hazards of germination tests. Such tests are commonly made in small covered dishes. If all the

seeds are in a single dish, any mishaps (e.g., excess watering or fungus attack) may affect the entire

test. To avoid this hazard, it is common to place a fixed number of seeds (one or two hundred) in

each of several dishes. The individual dish then becomes the unit of observation and the unit value

is the germination percent for the dish.

When clusters are fairly large and all of the same size, the procedures for computing estimates of

means and standard errors are much the same as those described for measurement data. To illustrate, assume that 8 samples of 100 seeds each have been selected from a thoroughly mixed batch.
54 USDA Forest Service RMRS-GTR-126. 2004.

The 100-seed samples are placed in eight separate germination dishes. After 30 days, the following

germination percentages are recorded:

Dish number 1 2 3 4 5 6 7 8 Sum of percentages

Germination (percent) 84 88 86 76 81 80 85 84 664

If pi is the germination percent in the th i dish, the mean germination percent would be estimated

by

1 664 83.0

8

n

i

i

p

p

n

= = ==

∑

The variance of p would be computed by

( ) ( )

2

2

2 1 22 2

2 1

664

84 88 ... 84

8 = 14.5714

1 7

n

n i

i

i

i

p

p

p

n

s

n

=

=

      − + ++ −

= = −

∑

∑

and the standard error of p can be obtained as

2

1 p

p

s n

s

n N

  = −    

2

p

p

s

s

n = = 14.5714 1.35

8 = , if the finite-population correction is ignored.

Here n stands for the number of clusters sampled and N is the number of possible clusters in the

population. As in simple random sampling of measurement data, a confidence interval for the

estimated percentage can be computed by Students t 95-percent confidence interval: _

p p ±ts

with t = the value of Student’s t at the 0.05 level with n – 1 degrees of freedom. Thus, in this

example, t has 7 degrees of freedom and .05 t is 2.365. The 95-percent confidence interval is:

83.0 ± (2.365) (1.35) = 83.0 ± 3.19 = 79.8 to 86.2.

Transformation of percentages–If clusters are small (less than 100 units per cluster) or if some

of the observed percentages are greater than 80 or less than 20, it may be desirable to transform the

percentages before computing means and confidence intervals. This is done to approximate the

normal distribution better so that the confidence intervals should be more reliable. The common

transformation is arcsin percent . Appendix 3, Table 4 gives the transformed values for the observed percentages. For the data in the previous example, the transformed values are

Dish No. Percent Arcsin percent

1 84 66.4

2 88 69.7

3 86 68.0

4 76 60.7

5 81 64.2

6 80 63.4

7 85 67.2

8 84 66.4

Total 526.0
USDA Forest Service RMRS-GTR-126. 2004. 55

The mean of the transformed values is 65.75

8

526.0 = .

The estimated variance of these values is:

( ) ( )

8.1486

7

8

526 66.4 ... 66.4

2

2 2

2 =

+ + −

s =

and the standard error of the mean transformed value is

8.1486 1.0186 1.009

8 y s = ==

ignoring the finite population correction.

So the 95-percent confidence limits would be (using .05 t for 7 df’s = 2.365)

CI = 65.75 ± ( )( ) 2.365 1.009 = 65.75 ± 2.39 = 63.36 to 68.14.

Referring to the table again, we see that the mean of 65.75 from the acrsin transformation corresponds to a percentage of 83.1. The confidence limits correspond to percentages of 79.9 and 86.1.

In this case the transformation made little difference in the mean or the confidence limits, but in

general it is safer to use the transformed values even though some extra work is involved.

Other cluster-sampling designs—If we regard the observed or transformed percentages as equivalent to measurements, it is easy to see that any of the designs described for continuous variables can

also be used for cluster sampling of attributes. In place of individuals, the clusters become the units

of which the population is composed.

Stratified random sampling might be applied when we wish to estimate the mean germination

percent of a seed lot made up of seed from several sources. The sources become the strata, each of

which is sampled by two or more randomly selected clusters of 100 or 200 seeds. Similarly we

might stratify a plantation into sections (strata), ones with high expected mortality and ones with

lower expected mortality in order to assess survival percentage of trees by section. Two or more

rows would be randomly selected in each section. In both cases not only might this be more efficient in estimating overall germination or survival percentages but we also can generate estimates

for the strata, which might be of interest in their own right.

With seed stored in a number of canisters of 100 kg each, we might use two-stage sampling, the

canisters being primary sample units and clusters of 100 seeds being the secondary sample units. If

the canisters differed in volume (or the different sections in the plantation were of different importance), they (or the sections) could be sampled at different intensities, a form of unequal probability sampling.

Cluster Sampling for Attributes With Unequal-Sized Clusters

Frequently when sampling for attributes, it is convenient to let a plot be the sample unit. On each

plot we count the total number of individuals and the number having the specified attributes. Even

though the plots are of equal area, the total number of individuals may vary from plot to plot; thus,

the clusters will be of unequal size. In estimating the proportion of individuals having the attribute,

we definitely do not want to average the proportions for all plots because that would give the same

weight to plots with few individuals as those with many.

In such situations, we might use the ratio-of-means estimator. Suppose that a pesticide has been

sprayed on an area of small scrub oaks and we wish to determine the percentage of trees killed. To

make this estimate, the total number of trees ( )i x and the number of dead trees ( )i y is determined

on 20 plots, each 0.04-ha in size.
56 USDA Forest Service RMRS-GTR-126. 2004.

Plot No. trees (xi) No. dead trees (yi)

1 15 11

2 42 32

3 128 98

4 86 42

5 97 62

68 6

7 28 22

8 65 51

9 71 48

10 110 66

11 63 58

12 48 32

13 26 16

14 160 126

15 103 80

16 80 58

17 32 25

18 56 44

19 49 24

20 84 59

Total 1135 960

Mean 67.55 48.0

The ratio-of-means estimate of the proportion of trees killed is

48.0 0.7106

67.55

y p

x == = .

The estimated standard error of p is

2 22

2

1 2

1 y x yx

p

s p s ps n

s

xn N

  + −   = −       

where:

2

y s = variance of individual y values,

2

x s = variance of individual x values,

yx s = covariance of y and x, and

n = number of plots observed.

In this example

( )

892.6316

19

20

960 11 32 ... 59

2

2 2 2

2 =

+ + + −

s y =

( ) 2

22 2

2

1351 15 42 ... 84

20 1,542.4711

19 x s

+ ++ −

= =

( )( ) ( )( ) ( )( ) ( )( ) 960 1351

11 15 32 42 ... 59 84

20

19 yx s

+ ++ −

= =1,132.6316.
USDA Forest Service RMRS-GTR-126. 2004. 57

With these values (but ignoring the fpc),

( )

( ) ( ) ( )( ) 2

2

1 892.6316 0.7106 1542.4711 2 0.7106 1132.6316

= 0.026

67.55 20 p s

  + − =  

  .

As in any use of the ratio-of-means estimator, the results may be biased if the proportion of units

in a cluster having a specified attribute is related to the size of the cluster. For large samples, the

bias will often be trivial.

Sampling of Count Variables

Statistical complications often arise in handling data such as number of weevils in a cone, number of seedlings on a 0.0004-ha plot, and similar count variables having no fixed upper limit. Small

counts and those with numerous zeroes are especially troublesome. They tend to follow distributions (Poisson, Negative Binomial, etc.) with which it is difficult to work. If count variables cannot

be avoided, the sampler’s best course may be to define the sample units so that most of the counts

are large and to take samples of 30 units or more. It may then be possible to apply the procedures

given for continuous variables.

In order to estimate the number of larvae of a certain insect in the litter of a forest tract, 30 2

cm

samples were taken at 600 randomly selected points (Freese 1962). The litter was carefully examined and the number of larvae recorded for each sample. The counts varied from 0 to 6 larvae per

plot. The number of plots on which the various counts were observed gave the following results:

Count 0 1 2 3456Total

Number of plots 256 244 92 21 4 1 2 600

The counts are close to following a Poisson distribution (see Appendix 2). To permit the application of normal distribution methods, the units were redefined. The new units consist of 15 of the

original units selected at random from the 600. There are a total of 40 of the new units, and unit

values are the total larvae count for the 15 selected observations. The values for the 40 redefined

units are

14 13 16 13 13 14 15 12

16 18 11 7 9 10 11 10

12 14 13 14 14 13 9 17

15 8 12 5 13 15 13 10

12 12 20 10 9 14 15 13

Total = 504

By the procedures for simple random sampling of a continuous variable, the estimated mean y

per unit is

12.6

40

504 _

y = = .

The variance 2

y s is

( ) ( )2

2 2 2

2

504

14 16 13

40

39

s y

+ ++

=



= 8.8615.

Ignoring the finite population correction, the standard error of the mean is

40

8.8615 _ =

y

s = 0.47.
58 USDA Forest Service RMRS-GTR-126. 2004.

The new units have a total area of 1.35 2

m ; hence to estimate the mean number of larvae per ha

the mean per unit must be multiplied by 10000/1.35 = 666.67

Thus, the mean per ha is (666.67) (12.6) = 8400.04 and the standard error of the mean per ha is

(666.67) (0.47) = 313.33.

As an approximation we can say that unless a 1-in-20 chance has occurred in sampling, the mean

count per ha is within the limits 8400.04 ± 2(313.33) or 7773.34 to 9026.66.
USDA Forest Service RMRS-GTR-126. 2004. 59

VI. Remote Sensing and Other Ancillary Information

Remote Sensing and Photography

Remote sensing can be defined as the science and art of obtaining information about objects,

areas, and phenomena under investigation through analysis of data acquired by some device not in

contact with these objects, areas, or phenomena (Lillesand and Kiefer 1987). Remote sensing has a

number of significant advantages not attainable by ground sampling from an inventory and monitoring point of view. It provides a synoptic view of the study area, can be collected quickly over a

large area, provides information about land cover in visible and nonvisible portions of the electromagnetic spectrum, is increasingly acquired and processed digitally, and provides a permanent

record of the situation at the time.

Remote sensing sensors are either passive or active. Passive ones receive signals from the target

itself, and active ones transmit a known signal. Passive remote sensing technologies useful in

natural resource applications today include photographic and electro-optical imaging systems such

as satellite borne sensors and airborne scanners. Their sensing capabilities extend from ultraviolet

to well into the microwave. Active sensors including RADAR and LIDAR are just beginning to

prove useful for selected applications.

Brief descriptions of the three types of sensors follow:

1. Photographic systems include camera, film, and a platform (usually an aircraft) to carry

them. These systems now are often integrated with geographic positioning systems (GPS) and

other electronics to help identify and record the location and position of the camera over the target

to be photographed. Image resolution is primarily a function of camera lens resolution, film resolution, degradation due to image motion (forward motion, pitch and roll, and vibration), and film

processing. If all goes well, a negative or reversal film positive will resolve 50 line pairs per millimeter or more. Sixty or even 70 line pairs per millimeter resolved on the film are not uncommon.

Paper prints still are limited to about 25-30 line pairs per millimeter. The size of ground features

resolved is a function of the above, plus the camera lens focal length and the flying height above

the terrain. This results in photographs of a given scale, which, together with the image resolution

resident in the system, determines what will be possible to see and interpret from the final photographs. Improvements in cameras and camera mounts, and integration of GPS with the aircraft and

camera system, make pinpoint plot photography at very large scale (1:3000 to 1:1000) operationally feasible. Use of computer equipment and geographic information systems (GIS) make flight

planning relatively easy and reliable. Recorded information on the photo center files makes it

possible to plot out a map of a photo flight shortly after the mission is completed. Because of these

acquisition and display technology improvements, very low altitude photography may have real

possibilities as a principal source of information for hard-to-sample areas such as wilderness or it

can at least decrease the amount of ground sampling needed. An old but still important reference on

the use of photography for inventory is Aldrich (1979).

2. Electro-optical imaging sensors collect data as arrays of pixels. A pixel is defined as the

smallest unit or cell of a raster image. It is usually assumed to be square in shape and consists of a

digital number that represents the brightness value recorded for that pixel within a single spectral

band. The ground resolution of the pixel is usually understood to be the distance that one side of the

pixel represents on the ground. The key issue is to extract useful information from the spectral band

data using image analysis (Holmgren and Thuresson 1998).

Such sensors include multispectral scanners, the main one of which currently is the thematic

mapper (TM). The spatial resolution for TM is 30 m for six of the seven bands carried by Landsats

4 and 5. Landsat 7 has eight bands, one of which is a black and white band with 8 m resolution.

Band 6, the thermal IR band, has a spatial resolution of 120 m. Advanced Very High Resolution

Radiometer (AVHRR) is used regularly by the US National Oceanic and Atmospheric Administration. This was designed for daily high spatial-resolution images of regional cloud patterns for
60 USDA Forest Service RMRS-GTR-126. 2004.

weather forecasting. The bands were designed to discriminate between clouds, water, ice, snow,

and land. One band was subsequently modified on the operational sensor prior to NOAA-6 to allow

also for observations suitable for vegetation studies. AVHRR is very coarse in its coverage. This is

advantageous for getting a smaller number of observations for a very large area but is quite limited

in its spatial resolution (from 1.1 km to a maximum of 3.5 km at nadir) for the same reason. The

French system SPOT (System probatoire d’observation de la terre) is a commercial alternative to

TM. It has higher resolution (10 m) but is much more expensive to acquire. Newer satellite systems

with much higher resolution are now available, generally from commercial sources.

Czaplewski (1999) reviews remote sensing sources available and soon to be available for inventory purposes. He distinguishes the following categories:

• Low-resolution satellite data include AVHRR, MODIS, Orb View-2, ERS-2, and SPOT 4. Such

data are inexpensive and have a 1,000-2,900 km (600-1,800 mile) swath width. Because of

this wide swath, spatial resolution is poor with a pixel representing 64-128 ha (158-316 acres)

in size. Such data have been useful for very large-scale maps of forested landscapes for global

change models, and to detect hot spots of serious deforestation in heavily forested landscapes.

But they are too coarse to reliably measure and monitor most forest conditions.

• Medium-resolution satellite data include Landsat 5 and 7, Radarsat, SPOT 2 and 4, IRS C and

D, P 2 and 5, Spin 2, EOS AM-1m, and CBERS 1 and 2 with pixel sizes of 10-30 m (33 to 98

feet) wide. They are more expensive with a 50-160 km (30 to 100 mile) swath width. Such

systems can separate forest from non-forest, and can identify some forest types and density

classes. Landsat can generally identify clearcuts but not most partial cuts. Advanced regeneration after land clearing, urban centers, and size, shape, and connectivity of forest patches

can also be measured. High quality data without clouds are generally available every 1-2

years except in humid tropical areas and many boreal forests.

• High-resolution satellites include Ikonos-2, OrbView 3 and 4, EROS B1 and B2, SPOT 5, and

Quickbird 1 and 2 with 3.2- 9.6 km (2-6 mile) swath width and pixel size of from 1-3 m (3-10

feet) wide. These sensors have capabilities, limitations, and costs similar to high altitude

56.25 cm square (9-inch square) 1:40000 small scale aerial photography—as available from

the USA Geological Service (USGS) national aerial photography program (NAPP), which

covers an area about 8 km (5 miles) wide. Such satellite and photo data can be used to reliably

distinguish some forest types, several stages of stand development, clearcuts and many partial cut areas, regeneration after land clearing, and concentrated tree mortality. Forest stands,

land use, distance to adjacent roads, water bodies, forest fragmentation, and various types of

urbanization can be photo interpreted.

• Large scale aerial photography with scales from 1:2500 to 1:12000 is routinely acquired by

aerial survey companies for small sites. Each photo covers an area 0.16-3.2 km (0.1 to 2

miles) wide. Interpreters can reliably identify many forest cover conditions such as 10 broad

forest types, 5 stages of stand development, 3 stand density classes, clearcut and partial cut

areas, regeneration success rates, natural or artificial stand origin, 3-5 severity levels of tree

mortality, most indicators of urbanization and fine-scale forest fragmentation, and stand size,

shape, and edge measurements.

Aldrich (1979) notes that forest diseases are less easily detected and evaluated than insect damage with aerial photography because it takes a long time for visible symptoms of disease to show

up. The symptoms are often not uniform over the forest and are more subtle than insect damage.

Dwarf mistletoe, Dutch elm disease, oak wilt, basal canker of white pine, ash dieback, Fomes

annosus, sulfur dioxide damage, and ozone damage are detectable with some degree of success.

Large-scale color and color infrared (CIR) film (1:1584) are needed to ascertain the degree of

damage, while 1:8000-1:16000 scales of CIR photography can be used to define and delineate the

boundaries of the disease. 70 mm color and CIR photography can be used as part of a sampling

strategy within susceptible forest types for damage assessment. Of course disturbances to the
USDA Forest Service RMRS-GTR-126. 2004. 61

vegetation caused by windstorm, flood, fire, or human activities are relatively easy to detect on

aerial photography as is change in these characteristics if the photography is repeated.

Aerial color video system imaging may have real utility especially for annualized inventories to

find out more about changes observed in certain areas of special interest. Videocameras can be

easily mounted on a variety of aircraft for either vertical or oblique sensing. Images can be digitized easily for computer-aided interpretation and can be used immediately since development is

not needed. Video equipment is portable, versatile, easy to use, can tolerate different light conditions, and is cheaper to operate than photographic systems. Also, the operator can view the imagery

on a monitor in the plane at the time of acquisition, can adjust the exposure settings interactively,

and can record comments in flight. Also, the high rate of picture acquisition (30 frames/sec) provides extra data. Its disadvantages are its low spatial resolution relative to film, the difficulty of

obtaining hard copy images from the data, practical limits on field-of-view because of the small

tape format, difficulty in calibrating the cameras because of the automatic exposure control, and

vignetting problems with the many near-IR video sensors since the camera optics are not designed

for this wave length band. The value of common color video systems for natural resources and

agricultural applications is limited because of the difficulty of extracting discrete spectral data

from a composite video signal and the lack of spectral bands outside the visible resolution.

3. Microwave sensors are generating considerable interest at this time but the applications in

forestry are still limited (see for example Lefsky and others 2002). The main data sets of nonphotographic images currently comprise those collected by Landsat 1-5 and 7, SPOT, and AVHRR.

These are available on computer-compatible media and in electronically reconstituted photographs.

Both computer-aided media and conventional photointerpretation methods are used to interpret

such data. Key references regarding inventory and monitoring using remote sensing are: Holmgren

and Thuresson (1998), USDA Forest Service (1998), and Lefsky and others (2002).

Accuracy of Remotely Sensed Information

Management of lands by agencies, such as those of two federal agencies in the USA, the

National Forest System of the USDA Forest Service, and the Bureau of Land Management of the

U.S. Department of the Interior (USDI), requires reliable maps of variables such as percent forest

cover, stand structure, and vegetation types. Such maps also require frequent updating and generating them is expensive. It is natural that remote sensing sources such as TM are used for this purpose

since it is facilitated by the frequent, large-scale, digital acquisition. Considerable work has gone

into making such maps. However, although TM contains useful information, the amount is limited.

For example, it is unlikely to be useful for stand structure, a difficult to measure variable even on

the ground. Similarly, vegetation types are difficult to define and interpretation may vary from one

user to another. Ideally, TM information should be combined with geo-referenced field inventory

and other mapped data to provide the necessary information for management decisions.

Remote sensing researchers desire a single coefficient to represent the accuracy of a thematic

map and of each category displayed (Rosenfield and Fitzpatrick-Lins 1986). Usually the results of

an accuracy assessment of a map are displayed in a matrix called a contingency table (called error

matrix in remote sensing) where the columns indicate the classes defined by the standard of comparison and rows indicate the mapped ones. The elements in the contingency table are the counts in

the row/column classes with the number in the last row the total count in that row class and the

numbers in the last column the total count in that column class. An obvious first estimator of

overall accuracy is the ratio of the sum of all correct counts over the total number of counts in the

contingency table. But ideally we also want estimators of the errors of commission (the proportions

of diagonal values to column sums = user’s accuracy) and of omission (the proportions of diagonal

values to row sums = producer’s accuracy). A widely accepted coefficient of agreement is the

Kappa statistic (K) estimated by:

ˆ 0

1

c

c

p p K

p

− = − (66)
62 USDA Forest Service RMRS-GTR-126. 2004.

where:

0

,

k

ij ij

i j

p pw = ∑ = weighted proportion of units that agree,

. .

,

k

c ij i j

i j

p wpp = ∑ = weighted proportion of units with expected chance agreement, and

. .

1 1

,

k k

i ij j ij

j i

p pp p = == = ∑ ∑

where wij is the assigned weight of importance of agreement for (i,j) with 1 wij = for all i,j for the

simple unweighted Kappa statistic and 0 1 ≤ ≤ wij for the weighted Kappa. Unequal weights can

be assigned if the accuracy of some classes is more important than for others, with the disadvantage

inherent in this that such weights would be subjective. Here Kˆ = 0 indicates that obtained agreement equals chance agreement, ˆK > 0 indicates greater than chance agreement, ˆK < 0 less than

chance agreement, and Kˆ = 1 is perfect agreement.

Then user’s accuracy = 1–(number correctly classified to be correct in the diagonal/number in

that row) and producer’s accuracy = 1–(number classified to be correct in the diagonal/number in

that column).

To assess accuracy, we need a probabilistically selected sample of size n on which both truth and

the map values to be assessed are available (Schreuder and others 2003). For specificity we only

discuss the situations discussed in that paper here. We assume plots are used. Truth should be

defined exactly for each variable and measured accordingly. It should not be defined as the best

readily available information as is done frequently in remote sensing. For percent tree cover 1 ( ) y ,

very low altitude photography may best be used; for species composition 2 ( ) y such photography

should be combined with ground sampling; for stand structure 3 ( ) y more emphasis is likely required on ground sampling with the photography providing some utility; for other variables such as

understory vegetation especially in dense stands, reliance may have to be placed completely on

ground sampling. Call the values of the variable of interest on plot i, ,( 1, 2,3) i y i = and the corresponding value of the variable on the map to be assessed for accuracy ,( 1, 2,3) i x i = . For a simple

random sample of n plots:

• We have n plots with i y from truth coverage.

• For many variables, it is likely that some plots will contain more than one class.

• Truth may be obtained from photography alone or a combination of photo and ground information, depending on the variable of interest. However, if only photo information is used for

percent cover, the truth is obtained error-free for the whole plot whereas if ground sampling is

involved, the plot information may have sampling error.

Location error for the mapped information is assumed negligible since we do not know what it

is. If present it will likely lead to an underestimate of the actual accuracy. Unless we have detailed

information about errors in plot locations, we cannot correct for them. For a certain number of the

n plots, all the information falls within one or more categories for the variable of interest for both

the truth and the mapped information. The following treats the case of both x and y labeling only

the same two “truth” classes occurring on a truth plot. The extension to more than two classes is

straightforward.

For a given plot, assume that the part x labeled ij x is part of or covers the part ij y called that by

the truth plot. If the truth plot and mapped plot could be overlaid completely, this assumption is not

needed. But the truth plot may only provide estimated areas of the plot area in the classes of interest

for 2 y and 3 y ; this assumption is required since we won’t know what part of the plot belongs to the

category estimated. That is the situation we currently have to live with. Generally violation of this

assumption will result in higher estimates of accuracy than actually obtained.
USDA Forest Service RMRS-GTR-126. 2004. 63

Continuous variables will have to be put into classes in order to determine whether mapped

correctly or not. This can be done objectively; for example, for percent tree cover use the 10 classes

0 to 10%, 10+ to 20%, ....90+ to 100%.

We then have the following determination for each of the n plots k (k = 1,...,n):

If truth calls it ij y and ij y ′ with plot area weights ,

y y w w ij ij′ such that 1 yy y www ij ij iother ++ = ′ and x

calls it the same with plot area weights ,

x x w w ij ij′ such that 1 xx x www ij ij iother ++ = ′ , then if

, xy x y ww ww ij ij ij i ≤ ≤′ ′j , correct classification for the plot gets a value of ( ) / x x w w ij ij + ′ n for p0 .

, y x w w ijother ijother are the percent (weights) of plot areas for which y or x or both define a condition

on the plot not recognized by the other. The weight given to all partially or totally correctly classified plots is 1 2 ( ) / z z w w ij ij + ′ n for ij p where z1 and z2 are the smaller of ,

y x w w ij ij and ,y x w w ij i ′ ′j , respectively. The response variable for each plot k is then 1 2 ( )/ z z

ij ij ij p ww = + ′ n where 0 1/ ij ≤ ≤ p n .

Plots classified correctly or 0.80 correct are counted as 1/n and 0.80/n, respectively.

Calculate 0

,

k

ij ij

i j

p pw = ∑ and then compute the Kappa statistic in equation (66). By repeatedly

taking n plots with replacement from the n sample plots B (say) 2,000 times and applying the above

computation of the ij p to each sample, we generate a series of B estimates for each cell of our

contingency table as well as for producer and user accuracy and a Kappa statistic for each. With

this bootstrapping we then can construct confidence limits around all the cells in the table as well

as for the Kappa statistic by treating the B samples as independent estimates of the same quantities.

An example of a contingency table with user, producer, overall accuracies, and the Kappa statistic

based on results in Table 4 for a sample of n = 200 plots are:

User’s accuracy for class 1 is 60.3/101 = 0.60,

 for class 2 is 44.2/53 = 0.83, and

 for class 3 is 30.8/46 = 0.70.

Producer’s accuracy for class 1 is 60.3/70 = 0.86,

 for class 2 is 44.2/73 = 0.61, and

 for class 3 is 30.8/57 = 0.54,

with overall accuracy 135.3/200 = 0.68.

Then 0 p = 0.68 and c p = 0.33 and 

0.68 0.33 0.35 ˆ 0.52

1 0.33 0.67

K − = == − .

Then by repeatedly taking say 2,000 with replacement samples from the 200 sample plots, we

compute for each sample the accuracies and the Kappa statistics again and construct confidence

limits around the above producer’s, user’s, and overall accuracy as well as around the Kappa statistic.

The contingency tables are the basic product from the accuracy assessment. Users should study

those tables in order to attempt to explain the causes of misclassifications. Some are obvious while

others need investigating. Misclassifications may result from problems with the technology used,

user errors, registration errors, errors in the final preparation of map products, or in calculations in

the accuracy assessments. Studying the results is essential in that it may explain or uncover errors

that can be corrected.

Table 4. A numerical example of a contingency table for forest cover class.

Cover class 1 Cover class 2 Cover class 3 Row totals

Map class 1 60.3 20.7 20.0 101

Map class 2 2.6 44.2 6.2 53

Map class 3 7.1 8.1 30.8 46

Column totals 70 73 57 200
64 USDA Forest Service RMRS-GTR-126. 2004.

It is also desirable for a manager to know how serious a misapplication of a treatment to an area

may be expected to be if the area is thought to belong in one category when in fact it belongs to

another one. There would be different consequences in applying a treatment to a category close to

the desired one than to a very different one.

Summary of what is needed for accuracy assessments:

• Define truth for the variables of interest and where and how to measure it. Minimize, or if

possible, eliminate measurement errors in truth by observers.

• Decide on using either pixel accuracy assessments or polygon accuracy assessments.

• Ensure an adequate sample size in each of the categories of interest for the variables of interest.

• Define different types of accuracy or give some of these a different label than accuracy.

• Determine the implications of achieving a stated accuracy in terms of making correct or incorrect management decisions.

• Combine/integrate the accuracy assessments for the variables of interest and use the information also to improve the maps developed.

Global Positioning System for Spatial Location Needs

Spatial location is critical for success in forest inventory and monitoring because present needs

require mapping this information.

Traditional methods of determining geographic location are still used, but more often than not,

these former methods now supplement the Global Positioning System (GPS) when used in natural

resources analysis. The GPS uses satellites to locate ground positions, usually within 150 meters,

and often to less than 10 meters. With this system the location of aircraft and plots can be established rapidly with reasonably accuracy. GPS does not require the use of known geodetic markers

for autonomous observation. Also, measurements can be made any time in any weather, with the

exception of possibly large solar storms. However, because the measurements require a clear line

of sight to the satellites, establishing location is difficult in a forest with a very dense canopy, deep

valleys or gorges, or similar situations. Additionally, a number of errors do happen if the user does

not use the GPS receiver correctly such as using the incorrect datum. Moreover, atmospheric and

satellite-signal-path-errors occur naturally and need to be recognized and adjusted or reported inside the accuracy statements by the user. GPS has been used successfully as the basis of a sophisticated navigation and flight recording system controlling the acquisition of large-scale aerial photographs in West Australia (Biggs and others 1989, Biggs and Spencer 1990). Remote sensing has

been oversold in the USA but will one day fulfill its promise partially because of reliable GPS

systems. Accurate GPS will be even more reliable and precise in the future. For example, algorithms are being developed that correct satellite signals and make positioning more accurate. It

behooves potential users to keep abreast of the technology because it is increasingly critical for

inventory and monitoring.

Geographic Information System (GIS)

A Geographic Information System provides for entering, storing, manipulating, analyzing, and

displaying spatial data. Data can be represented by points, lines, or polygons with the associated

variables. Such data can be represented by raster or grid data on the one hand and by vector data on

the other.

The raster system stores the data in a grid or pixel format with bounded geodetic values such as

latitude and longitude, while the vector system uses a series of x, y coordinates to define the limits

of the attribute of interest. Grid data are computationally easier to manipulate but usually require

large amounts of storage space. Vector data require less storage and usually represent discrete data

more accurately. Though the vector system may retain the shape of a discrete feature more correctly (has better resolution), it is computationally more time-consuming and difficult to render
USDA Forest Service RMRS-GTR-126. 2004. 65

and analyze. Satellite imagery, digital pictures, and digital elevation models are examples of grid

data; property boundaries, structure outlines, utility poles, and utility lines are examples of vector

data. A GIS is a computerized system that can play a critical role in inventory, in manipulating and

processing data, and in assessing land use and land cover. It has emerged as an effective tool in

defining and focusing discussion relative to the merits of alternative land use allocations. For example, it gives the analysts the ability to simulate the effects of changes in management (Green

1993).

The GIS should have the capabilities to:

• Input many forms of data such as: analog and digital maps, textual or tabular information, and

images.

• Store and maintain information with the necessary spatial relationships.

• Manipulate data, as in search and retrieval, and to do computations efficiently on the data.

• Provide levels of modeling taking into account data interrelationships and possible cause-andeffect responses of the relevant factors.

• Present tabulations, video displays, and computer generated maps of existing or derived information.

A good GIS depends primarily on good data. In addition high-speed computers, a variety of

peripheral input-output devices, and powerful software are required. Articles by Congalton and

Green (1992), Green (1993), and Bolstad and Smith (1992) give a good overview of GIS and

Lachowski and others (1992) present a useful example of integrating remote sensing and GIS.

Small Area Estimation

There is considerable interest in management agencies to have reliable spatial information. In

the past foresters and other land managers cruised or sketch mapped an area usually to decide what

is where. Managers avoided statistical sampling because it might give reliable data on how much

was there but not where. Frequent legal challenges changed this in the USA. Now interest is in

obtaining reliable (defensible) mapped and statistical data together. One such area of current research is referred to as small area estimation, basically a model-building approach using statistical

data in combination with ancillary data such as TM, GIS, topographic maps, and other related

information.

Small area estimation techniques represent a substantial improvement in terms of quality of

data, especially in defensibility of data-based management decisions relative to what used to be

done when managers relied on subjective information. Small area estimates have been claimed to

have standard errors similar to those for classical sampling. The trouble is the comparison is made

for the entire population of interest whereas managers are also interested in predictions for much

smaller areas such as polygons used as a basis for management. Standard errors for individual

predictions can be large, as one would expect, given the variability encountered on the ground in

forests.

For successful small area estimation, two conditions need to be met. First of all there should

ideally be a good correlation between sampled and non-sampled areas either nearby or from other

ancillary sources such as remote sensing. This usually requires a much more intensive grid than the

5,000-m grid now used by FIA. Secondly the spatial locations for both the sampled areas and the

ancillary data need to be accurate. Given these conditions it should be possible to develop good

prediction models.

Considerable work in small area estimation of forest resources is now being done. For example,

there is innovative work being done in Finland where the more homogeneous conditions relative to

other countries may make small area methods more useful. Multiple imputation methods (including regression models) and k-nearest neighbor techniques have been proposed for continuous variables. In these techniques, field sample information is extrapolated to the entire population where

information on sample locations is input to non-sampled locations by some criteria such as similar
66 USDA Forest Service RMRS-GTR-126. 2004.

TM readings for the sampled and non-sampled locations. In multiple imputations for each unit

without sample data, a series of say 100 predictions are made using randomly selected data and an

underlying model and database. Then the data sets are analyzed separately and pooled into a final

result, usually an average of the results.

Franco-Lopez (1999) reviews methods for projecting and propagating forest plot and stand information. As he notes, considerable effort has been expanded in Nordic countries combining forest monitoring information, remote sensing, and geographic information systems (GIS) to develop

maps for forest variables such as cover type, stand density, and timber volume with emphasis on

the k-nearest neighbor technique. He confides that while his results are poor for Minnesota, they

are representative of those obtained by other methods in this region.

Lin (2003) presents a semi-parametric bootstrap method for estimating the precision of estimates. In general: 1. Fit the best fitting model say i i i y x =+ + αβ ε resulting in the estimated model:

ˆ ˆ i i y = + α β x . 2. Compute the residuals ˆ

i ε and calculate the scaled residuals

1

ˆ ˆ / , 1,...,

n

ii i

i

εε ε n i n =

=− = ∑ . 3. Bootstrap the residuals , 1,..., i εi = n , i.e. take a sample of n residuals

with replacement from the n residuals. Do this say 1,000 times, each sample constituting a bootstrap sample. 4. For each bootstrap sample compute ˆ ˆ ˆ i i i y x =+ + αβ ε. 5. Refit the model to each of

the bootstrap samples using the sample points ˆ ,i i y x and predict for each of the samples at the

desired locations; the variability between these estimates is then used for the bootstrap variance for

that location. Lin’s results for predicting mortality, total basal area, and number of live trees on the

Siuslaw National Forest in Oregon showed errors averaging nearly 100 percent for plots on a

1.36-km (0.85-mile) grid base using data from sampled 1-ha plots on a 2.72-km (1.7-mile) grid.

To obtain reliable predictions today, additional information is required, such as that available

from improved remote sensors or large-scale photos and various maps combined with expertise

from local ecologists. Also, at present it is still necessary to correct for location errors with models.

Making such corrections requires considerable information on the extent and location of the errors.

Hopefully, improvements in GPS-type sensors will reduce location errors in the future and improve

results from small area estimation techniques.
USDA Forest Service RMRS-GTR-126. 2004. 67

VII. Sampling for Rare Events

Sampling rare populations is an order of magnitude more difficult than sampling common ones.

Yet, assessing such populations can be critical. For example, the world is losing many plant and

animal species and people may want to preserve at least some of these species. Knowing how many

of a species exist and where and why in specific areas is critical for their preservation. The costs of

locating rare populations are considerable and often exceed the costs of measurement. The fundamental problem is that several attempts may be needed to identify sample units with the rare trait in

the overall population such as rare mushroom or tree species. Also, identifying the actual species or

other attributes may require very specialized knowledge that only a few people have. Possible

approaches are:

1. Screening. A large sample from the total population is examined to identify members of the

rare population or at least areas where it is more likely to occur. If the latter is possible then

such areas are sampled with much higher intensity than other areas for frequency of occurrence.

2. Multiplicity sampling and adaptive sampling. Basically, these are techniques that rely on locating some of the units with the rare attribute and then obtaining additional information about

them, which is then used to locate others, thus reducing the cost of the survey.

a. In multiplicity sampling a selected sample unit yields information about itself as well as

about other units. Obviously this is more applicable to human surveys than vegetation

surveys.

b. In adaptive sampling a sample of units is selected probabilistically and units in the neighborhood of a sampled unit are added if the attributes of interest for those units satisfy a

given criterion. The cleverness of the approach is that all units in the population are put in

non-overlapping clusters and all units in the sample clusters are measured. Clusters can

vary greatly in size and shape. Adaptive sampling is a probabilistic procedure but is hard to

implement, and analysis of the results is difficult.

3. In multiple frame sampling, a sample is taken from an existing partial list and an additional one

from the total population to screen for units with the characteristics of interest. The weakness

of this approach is the overlap of frames (for which Kalton and Anderson, 1977 give some

solutions) and the expense of screening and sampling the screened part of the population.

4. In snowball sampling, a necessary condition is that units contain information about each other.

Then a frame of units is created in the rare population by sampling a few units and through

them identifying others. Clearly again, this is more likely to be fruitful with human populations than with vegetation. Once a frame has been developed, a probabilistic sample is drawn.

The weakness of this approach is the degree of completeness of the frame. An advantage is

that rare units are identified more quickly than with other methods.

5. Sequential sampling. Select an initial probabilistic sample of sufficient size to give the desired

sample size (n) of members of the rare population based on the rate of incidence observed.

This will yield 1 n members of the rare population and an estimate of incidence. If 1 n n < , a

second sample is selected to produce the remaining 1 n n − members of the rare population

based on the incidence obtained in the first sample. This procedure is generally expensive and

hence is not practical in most vegetation surveys.

Several of these techniques may become more useful in vegetation and animal population surveys as rapid, simplified DNA identification methods are developed.
68 USDA Forest Service RMRS-GTR-126. 2004.

VIII. Multiple Level Sampling

Previously, we discussed single-level surveys where either the variables of interest or these

variables plus covariates on the same sample units were measured. When covariates were measured, we assumed that the covariate values for all units in the population or, at least the population

total, were known. Often the covariates are useful for estimation and are cheaper to collect but

unknown. It often pays to collect covariate information on a large sample and the variable of interest on a subsample.

This approach is referred to as multilevel sampling. For example, in a timber sale we might

obtain ocular estimates of diameter at breast height D (and hence 2 D ) for a large sample of trees

and measure actual volume of some of these trees. Or, in estimating recreational use of an area, we

may use traffic counters recording counts of vehicles at the entrance to the park on a large number

of days and actually count the number of users on a subset of these days.

Multilevel sampling can be separated into multiphase and multistage sampling.

Multistage Sampling

Refers to sampling designs where the ultimate sample units, called elements, are selected in

stages. Samples at each stage are taken from the sample units comprising clusters of units selected

in the previous stage. Interest is in estimation of attribute totals or means per element, such as

biomass per tree rather than per ha. The population is first divided into a number of primary sample

units (PSU), some of which are selected as the first stage sample. These selected PSUs are then

subdivided into a series of secondary sample units (SSU), some of which are randomly selected as

the second stage sample. This process can be repeated of course with additional stages if necessary.

The procedure has the advantage of concentrating work on a relatively small number of PSUs after

which much less effort is usually needed to obtain the second and later samples.

The main reasons for selecting a multistage sample are:

1. Drawing a set of units from a population such as trees in a large forest or recreation users of a

park over a full season is expensive. It is difficult to obtain a list of all the trees and even more

difficult to determine all users of a park.

2. Even if a list of population units was available, efficiency might dictate that groups of units

(clusters) rather than single units be chosen and that only some units in each cluster are

measured. For example, it is usually cheaper to sample 20 randomly located clusters of 30

trees in a forest than 600 randomly located trees and we may want to only sample 10 out of the

30 trees in each cluster because of the homogeneity in the cluster or the expense of measuring

all 30 trees. In sampling recreation users, it is clearly easier to select and subsample random

days on which to interview all users rather than attempt to randomly sample individual users

or days, respectively.

Generally, though as indicated earlier, there is a definite tradeoff in efficiency between cluster

sampling and random sampling of units because units close together are often more similar than

those further apart and it often pays to measure only some of them in each selected cluster.

Sampling can be in a large number of stages. We illustrate how this works with the simple and

often practically useful situation of 2-stage sampling with SRS at each stage. Assume N groups or

clusters with Mi units (i = 1, ..., N) in the i

th cluster. Our total of interest can now be written as:

.

11 1

N N Mi

ij i

ij i

Y y Y == == = ∑∑ ∑ (67)

In 2-stage sampling a random sample of n is selected out of the N clusters but instead of measuring all units in the cluster, a random sample of mi units is chosen in each. Thus, the cluster total

Yi. is first estimated by
USDA Forest Service RMRS-GTR-126. 2004. 69

.

1

ˆ mi

ij

i i

j i

y Y M = m = ∑ (68)

for each of the n clusters sampled. Our estimated total is:

.

11 1

ˆ ˆ n n mi

ij

i i

ij i i

N N y Y M Y

n mn == =

  = =     ∑∑ ∑ (69)

with variance

22 2 2 2

1

(1 ) (1 ) ˆ ( )

N

a b i iwi

i i

NM f M f N V Y

n nm

σ σ

=

− − = + ∑ (70)

where:

2

2 1

( 1)

N

i

i

i a

b

M Y Y

M

N

σ =

    −

  = −

∑

, the between-cluster variance, and

( )2

.

2 1

1

Mi

ij i

j

wi

i

y Y

M

σ =

−

= −

∑

, the within-cluster variance.

Here . .

.

1 1

, , , , and ,

N N

ii i i

ii a

i i i i

n mY Y M ff Y Y M NM M N N = =

== = = = ∑ ∑ the average number of SSUs

per PSU. Similarly, an unbiased variance estimator ˆ v(Y)is:

2 22

2 2

1

(1 ) ˆ ( ) (1 )

n

b i iwi

a

i i

s M fs N

v Y NM f

nn m =

− = −+ ∑ (71)

where:

( )2

2 1

.

1

ˆ , , 1

i

n

i i m

ij i

b i

j i a

My y y Y

s y y n m NM

=

=

−

= == −

∑

∑ and 

2

2 .

1

( )

1

mi

ij i

wi

j i

y y

s = m

− = − ∑ .

There is a considerable literature on multistage sampling, but this subject is still best discussed

in the book by Murthy (1967).

Multiphase Sampling

In multiphase sampling the same size of sample units are retained at each level (phase) but with

fewer sample units selected at each consecutive one. In the last phase the variable of interest is

measured and is combined with covariate information from the early phases either in design (stratification or pps sampling) or estimation (regression or ratio estimation). In multiphase sampling a

complete frame of units is required since a sample of units is selected at each phase. The main

reason for using multiphase sampling is to reduce the cost of sampling by collecting a large amount
70 USDA Forest Service RMRS-GTR-126. 2004.

of relatively cheap information on covariates that are correlated with the variables of interest and

then measuring the variables of interest on a smaller sample. Stratified double sampling and double

sampling for regression or ratio estimation are two examples. Specifically:

1. For stratified double sampling, the large (first phase) sample information is used to construct

strata from which the second phase samples are selected. Typically this is done if interest is in

specific subpopulations (strata) or the strata are more homogeneous than the overall population so that efficiency is gained by stratification. For example, in traditional large-scale timber

surveys we might have a large sample of say ' n 1-ha plots from remote sensing or photos

classified into primarily large timber, pole timber, and regeneration. Clearly, if interest is in

volume, those three strata would be of interest in their own right and are likely to be much

more homogeneous (if sufficiently well done by remote sampling) than the overall population. A subsample of those 1-ha plots would then be sampled on the ground for volume by

stratum. Similarly in sampling a large park for recreation use, we might take a large sample of

photos on sample days to count users, use that information to divide the park into strata of

heavy, moderate, and low use days, and then sample these three strata on a subset of those

same sample days. The estimator of the total in both cases is:

1

1 1

ˆ ˆ

nh

K K hi

i

dst h h h st

h h h

y

Y N w N y Ny

n

=

= =

= ==

∑

∑ ∑ (72)

where K = number of strata, 

'

'

ˆ h

h

Nn N

n = is the estimated number of sample units in stratum h, '

'

h

h

n

w

n = is the estimated weight for stratum h for the first phase sample with '

h n and n′ the first

phase sample sizes for stratum h and overall respectively, h n is the second phase sample in stratum

h, and st y is the estimated mean for stratum h for the sample of h n units in that stratum. The variance of this estimator is:

( ) 2

22 2

1

11 1 ˆ 1

K

h h

dst

h h

W S V Y NS N

nN n v =

    = −+ −       ′ ′   ∑ (73)

with 2 S the population variance of y, 2

h S the variance of y in stratum h, / h h v = n n 1 and h y and

st y the sample mean for stratum h and overall sample mean for stratified sampling respectively.

An almost unbiased sample estimator of ˆ ( ) V Ydst , if both 1/N and 

1

n′

 are negligible, is:

( ) 2 2

2 22 1 2

11 1

ˆ ( )

KK K

h h

dst h h h h st

hh h h

w s g v Y N N ws N w y y == = n n

= −+ − ∑∑ ∑′ (74)

where 1 1

N n

g N

− ′ = − .

Strata may be of different degrees of interest and vary in homogeneity, so varying sampling rates

may be desirable. This requires knowledge of or an estimate of the variability within the strata in

order to allocate n. If such knowledge is available, one can then optimally allocate the sample to the

strata. Assume that there is information available or easily collectable on a variable x correlated

with y. Then applying the simple cost function:

1

K

h h

h

C Cn n C n =

= + ′′ ′∑
USDA Forest Service RMRS-GTR-126. 2004. 71

where C′ is the cost of classifying a unit for the first phase and Ch is the cost of measuring a unit in

stratum h, the expected cost E(C) is:

1

( )

K

h h h

h

E C Cn n CvW =

= + ′′ ′∑ . (75)

Then the optimal n' can be computed from by substituting

2 2

0

1

ˆ K

h yh y h yh

h h

C

v s s ws

C =

′   = −     ∑

for h v where 2

y s and 2yh s are the estimated variance for variable y in the population and stratum h

respectively and w0h is the estimated stratum weight for stratum h based on the preliminary information. More complex cost functions are discussed in the literature, especially Hansen and others

(1953), but usually insufficient information is available to assume a better cost function, so it

makes sample size computations more difficult and sample size determination seems fairly insensitive to improved cost functions.

2. For double sampling with ratio or regression estimators, a linear relationship is assumed between the covariates and the variables of interest as shown in the general linear model in (39).

For instance in the timber example above, one may have confidence that the information on the

1-ha remotely sensed or photo plots is linearly related to the same information as measured on the

ground. Or, similarly, the photo counts of recreational users might be linearly related to the actual

counts on the ground. Clearly, whether such a linear relationship exists as a useful approximation

or there is a useful but unknown relationship between the remote sensing and the ground information in both cases determines whether stratified double sampling or double sampling with ratio/

regression estimation is more efficient and reliable. The regression estimator of the overall total is:

1 1

11 1

1 ˆˆˆ nn n

i i

gr gr gr

ii i ii i

y x Y aN bX == = ππ π

   =+ − + −       ∑∑ ∑ (76)

where:

1 1

1

1

n n

i i

gr

i i ii ii

gr n

i i i

y x b

v v

a

v

π π

π

= =

=

−

=

∑ ∑

∑

11 1 1

2 2

11 1

1

1

nn n n

ii i i

ii i i ii ii i i i i

gr nn n

i i

ii i ii i i i i

xy y x

vvvv b

x x

vv v

ππ ππ

ππ π

== = =

== =

−

=

  −   

∑∑ ∑∑

∑∑ ∑

with 1

12 1 2

1 11 1 1

111 ˆˆ ˆˆ ,, , ,,

n

i

n nn n n

i i ii i

s s ja j ii i i i i i ja i s

x

x x v N NN X Xx

v N

π

π ππ π π

′ ′

=

= == = =

= = = = ==

∑

∑ ∑∑ ∑ ∑   
72 USDA Forest Service RMRS-GTR-126. 2004.

where π ja is the probability of selecting unit j in the sample of n ' units and π i the probability of

selecting unit i in the sample of n units and 1

n

i

i i i

s

s

y

v

y N

= π =

∑   . The π i may not always be computable.

Deriving a classical variance estimator for this estimator is difficult and this is an example where

bootstrap variance estimation would be the method of choice.

Illustration: A large sample of n’ plots is measured for plot basal areas on aerial photos. These could

be stratified into K strata, selecting either a subsample of n plots in the K strata or a SRS of n out of the

n’ plots which are then measured on the ground. Using BATi and VTi to denote basal area on plot i

as measured on the photo plots and volume as measured on the ground plots,

we then have:

n’ plots with , 1,..., BAT i i = n′

n plots with , 1,..., , 1,..., BAT h K i hi h = = n and , 1,..., , 1,..., VT h Ki hi h = = n for stratified double

sampling or , 1,..., BAT i i = n and , 1,..., VT i i = n for double sampling with regression.

Whether stratified double sampling or double sampling with a regression estimator would be used,

depends on the relationship expected between BATi and VTi . If there is expected to be a linear

relationship, regression estimation would be used, otherwise double sampling for stratification is indicated.

For stratified double sampling one would use (72) and (74) to estimate total volume and its variance.

For double sampling with regression one would use (76) with a bootstrap variance estimator.

If one expects the relationships between the covariates and the variables of interest to go through

the origin approximately, a double sampling with a ratio of means estimator can be used:

1

1 1

1

ˆ ˆ ˆ ˆ ˆ

n

i

i i HT

drm n

i HT

i i

y

Y Y X X

x X

π

π

=

=

      = =            

∑

∑ . (77)

In this case too it is best to use bootstrapping to estimate the variance of ˆYdrm . Here too the π i

may not always be computable.

Multilevel sampling methods in forestry are common especially for large scale surveys. For

example:

1. Double sampling for stratification is used in large-scale surveys such as FIA. Areas are stratified usually into forested vs. non-forested areas by either photography or more commonly

now by data collected from remote sensing sources such as the Landsat Thematic Mapper

Satellite (TM) and then ground plots are measured in those strata. In the past, with primary

interest in timber, prestratification was used. Now post-stratification is used because plots are

grid-based. Newer remote sensing sources will define small features on the ground better and

locations of both the ground and remote sensing information can be pinpointed more accurately with improved GPS units. It is likely that more detailed stratification and regression

estimation will improve estimation in the future.

2. VRP sampling with subsequent selection of trees by either Poisson sampling proportional to

estimated tree heights or another subsampling scheme were frequently used in timber sales.

Clearly combinations of multiphase and multistage sampling can be desirable too. For instance,

in example 1 above we might select a random sample of trees on the selected ground plots. This

design then would be double sampling for stratification with random subsampling.
USDA Forest Service RMRS-GTR-126. 2004. 73

IX. Monitoring Over Time

Managers of biological resources are always interested in change over time in timber volume,

mortality, wildlife habitat, degree of urbanization, change from forest land to agricultural land, etc.

There are three major sampling options to consider in sampling over time:

1. Complete replacement sampling (CRP)

2. Complete remeasurement sampling (CR)

3. Sampling with partial replacement (SPR), a combination of a and b.

In CRP sampling, a completely new set of sample plots is used each time. Such a design is

simple and cheap to implement since plot locations do not have to be monumented for future use

and one does not have to worry about the plots being treated different from other parts of the

population or changes in the underlying population. CRP sampling is efficient for estimating current attributes but not efficient for estimating change relative to CR and SPR.

In CR sampling all sample plots are remeasured periodically. This requires that they remain

representative of the population over time, so that the plots should not be visited excessively, and

should be treated no differently from other parts of the population. CR sampling is the most efficient of the methods available for change estimation.

In SPR a random subset of the permanent plots is remeasured as well as a new set of plots, i.e.,

it is a combination of CRP and CR sampling. Regression estimation between the remeasured and

new plots is used to “update” the plots that were not remeasured. SPR can be efficient when trying

to balance precision between current and change estimation.

Duncan and Kalton (1987) summarized the properties of the three options nicely. They also list

another method that is a combination of the other three (Table 5).

Both CR and CRP sampling are special cases of SPR sampling from an estimation point of view

so we only present SPR sampling for two occasions here:

If n sample units are selected from N units at both occasions with m units common to both, then

u = n–m units are not shared.

Let ˆ ˆ , Yim i Yu , and ˆYin equal the estimates of Yi , the population total on the i

th occasion (i = 1,2),

based on the m, u, and n units respectively, β

ˆ = the regression coefficient estimator based on the

m common units, 2 σ1 and 2 σ 2 the variances of y at times 1 and 2, σ12 the covariance of y between

times 1 and 2 and ρ the correlation between measurements at times 1 and 2. Then an unbiased

estimator of Y2 based on the u new units at time 2 is:

2

1

2

ˆ

u

i

i

u

y

Y N

u

= =

∑ , (78)

with variance: 2 2

2 2

ˆ () / V Y N u = σ u , (79)

and variance estimator: 2

2 2

ˆ ( ) (1) / u v Y s = u (80)

with 2

2 s (1)the within-sample variance of the u 2i y measurements.

Equivalently, a regression-based estimator of Y2 using the m common units at times 1 and 2 to

update the total from time 1 is:

2 2

ˆ ˆ ˆˆ ˆ( ) Y mr m in im =+ − Y YY β (81)
74 USDA Forest Service RMRS-GTR-126. 2004.

with variance:

2 2

2 2

2

ˆ () 1 mr

u V Y N

m n

   σ ρ = −      . (82)

Combining estimates 2

ˆY u and 2ˆY mr to obtain an improved estimate of Y2 is usually done by weighing them inversely proportional to their sample variances, so we obtain:

( ) 12 22

2

ˆ ˆ ˆ ˆ ˆ ˆ

wY wY u mr

Y

w

+ = (83)

with approximate variance 2

ˆ V( ) 1/ Y  w

Table 5. Objectives and properties of four remeasurement designs (adapted from Duncan and Kalton 1987).

Complete replace- Complete remeasurement sampling ment sampling Sampling with partial Combination of CR with

Sampling objective (CRP) (CR) replacement (SPR) CRP or SPR

a) Estimate population Automatically takes Needs mechanism for Needs mechanism for Remeasurement component

parameters at distinct into account taking population taking into account needs mechanism for taking

times. population changes. changes into account. population changes population changes into

b) Estimate average during life of replace- account.

values of population ment group. Composite

parameters. estimates can be used

to produce efficient

estimates.

c) Estimate net change. Estimates combined Needs mechanism for Needs mechanism for Remeasurement component

effect of changing taking population taking into account needs mechanism for taking

values and changing changes into account. population changes population changes into

population. Variance of change during life of account. Variance of change in

reduced by positive remeasurement. remeasurement component

correlation of values Composite estimation reducedby positive correlation

between surveys. can be used to produce of values between surveys.

efficient estimates.

d) Estimate components Not possible. Well-suited for these Can be used for change Complete remeasurement

of change over time. populations. estimation or aggregate component is well-suited for

e) Aggregate data for information over time these purposes. Not possible

 individuals over time. periods shorter than the for complete replacement

time a sample is to be surveys component.

replaced in sample. Only

the sample to be

replaced can be used.

f) Collect data on events Not possible. Can construct long- Can construct long-term Can construct limited longoccurring in specific time term history of events history of events but on term history of events.

periods. by combining data a more limited

from several surveys. basis than complete

remeasurement surveys.

g) Cumulate samples Excellent for static Not useful for static Of some use for static Complete remeasurement

over time. characteristics and characteristics, but characteristic and survey component is

for new events. useful for new useful for new events. excellent. Complete

events. remeasurement survey

component is useful for

new events but not for

static characteristics.

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○

○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○
USDA Forest Service RMRS-GTR-126. 2004. 75

with 2 2

12 1 2 w =+= + w w 1/ 1/ σ σ

and variance estimator:

2

2 2

1

4 ˆˆ ˆ ( ) ˆ () 1 ˆ ˆ

i i

i i

ww w

v Y w

w d =

  − = +     ∑ (84)

with 1 2 12 1 2

2 2

1 1 ˆ ˆ ˆˆ ˆ , , , 1, 1 ˆ ˆ () ( ) u mr

w w ww w d m d u

vY vY

= = = + = − =− , and w is estimated by wˆ .

Two estimators of change Y are possible, the most obvious one being:

1 2 1 Y Y ˆ ˆ = −Yˆ . (85)

A desirable property of such an estimate is that it is consistent with the estimates at the two

occasions.

A more efficient estimator than 1 Yˆ in general takes advantage of the regression based on the m

common units as in 2 Yˆ above. This estimator is:

11 2 2

2

ˆ ˆ ˆ ˆ ˆ

ˆ

wY wY Y

w

+ = . (86)

Here 1 Yˆ and 2 Yˆ are the estimators of change from the m remeasured and u unmatched plots

respectively, where

22 22

1 2 12 1 2

1 2 ˆ ˆ 2 , ss s ss

w w

mm m u u

=+− =+

are unbiased estimators respectively of:

22 22

1 2 12 1 2

1 2 w 2 , w

mm m u u

σσ σ σσ

=+− =+ ,

1 2 ww w ˆ ˆ = + ˆ estimates ww w = +1 2 , and

2

ˆ V( ) 1/ Y = w (87)

with approximate variance estimator:

where: ( )

2

2 2

1

4 ˆˆ ˆ ( ) ˆ 1 ˆ ˆ

i i

i i

ww w

v Y w

w d =

  − = +     ∑ (88)

1 2 d = − =− m du 1, 1.

Problem: Show how CR and CPR sampling are special cases of SPR sampling.

Answer: Set ρ = 0 in (82) to get the variance for CR sampling and ρ =1 to get the variance for

CPR sampling.

The matching proportion in SPR sampling depends on the correlation ρ between the measurements at the two times. It should not exceed 0.5 for optimizing the estimator 2 Yˆ . SPR sampling

quickly becomes much more complicated in estimation when more than two occasions are measured (Schreuder and others 1993). But a serious disadvantage of SPR sampling, that of variance
76 USDA Forest Service RMRS-GTR-126. 2004.

estimation, has been eliminated. With bootstrapping it should be simple to generate variance estimates for any number of remeasurements and estimation schemes.

All of the FIA units in the USA now use complete remeasurement sampling, although one formerly used sampling with partial replacement. In general, SPR sampling is probably most efficient

but becomes quite complex from an analysis point of view, which makes it hard to deal with the

numerous special requests of estimates for specific subpopulations of the survey population covered.
USDA Forest Service RMRS-GTR-126. 2004. 77

X. Building Models and Cause-Effect

The statement by Box and Draper (1987) that “all models are wrong; the practical question is

how wrong do they have to be to not be useful” is generally accepted in the statistical world, and

can be paraphrased as “all models are wrong and some are useful.” The utility of models is often

assessed by the degree of correlation between the variables of interest and covariates, but note that

correlation does not prove causation (Kish 1967).

Much if not all of research revolves around model building, and the potential misuse of models

has been greatly facilitated by the ready availability of computers and easy use of regression programs. Ideally, a researcher observes the real world or carefully studies substantive scientific theories. Models are then developed on the basis of the insights accorded, recognizing the fact that

besides the explanatory variables, there are other sources of variation to be considered. Kish (1967)

separates all sources of variation into four classes:

1. The explanatory or experimental variables that are the objectives of the research in explaining

or establishing a relationship between both the dependent (often called the response variables

in this context) and the independent (often called predictor variables in this context) variables.

2. Extraneous variables that can be controlled either in sample selection or estimation.

3. Extraneous (unmeasured, often unmeasurable) variables that may be confounded with the

variables in class 1 above.

4. Extraneous, difficult to control or uncontrollable variables that have to be treated as randomized errors. In ideal experiments, they can be randomized, whereas in surveys they can only

be assumed to be randomized.

In all research one wants to place as many extraneous variables as possible in class 2. Since this

usually cannot be done, we have experiments and surveys. Experiments, the conduct of a systematic, controlled test or investigation, tries to control the variables in class 3 as much as possible by

trying to place all of the third class of variation into the fourth through randomization. In an ideal

experiment, there are no variables in the third class. In an ideal survey, all variables in class 3 are

separated from those in class 1 through regression adjustments, matching of units, and standardization.

If there was complete command over the research situation, one could introduce the desired

effects and measurements into controlled and randomized portions of the target population with

firm experimental controls and build a “true” model (Kish 1967). Such situations are rare so that

we have experiments that are strong on control through randomization but weak on representing

actual populations of interest and frequently weak on the natural setting of the model being built.

Surveys often are feasible when experiments cannot be done, the most obvious being that we do not

experiment on humans. Surveys are strong on representation but are often weak on control. The

latter is a ready explanation of why so many studies are published claiming this or that chemical is

bad for you and subsequent research does not support such claims.

Often survey data are used to build models, to lead to a better understanding of what is going on.

Many models appear to have poor predictive ability; for example, this is true for the staple of forest

research: building growth and mortality models. A missing ingredient is key data that would help in

identifying cause-effect relationships such as daily rainfall, atmospheric deposition, soil moisture

content, etc. Such data cannot be collected yet in a practical manner in conjunction with natural

resources surveys, but development of new instrumentation should make that feasible some day.

Until this information is routinely available for the plots, prediction models for growth, mortality,

erosion, and other key variables are unlikely to be reliable.

Large-scale surveys such as FIA and the Natural Resources Inventory (NRI) of the National

Resources Conservation Service (NRCS) can establish trends in change for large areas, can be used

to suggest and identify potential cause-effect relationships, and can suggest useful hypotheses to
78 USDA Forest Service RMRS-GTR-126. 2004.

document relationships (Olsen and Schreuder 1997). Inferences about possible cause-effect relationships have to be interpreted cautiously because screening of data makes it difficult to define the

population of inference (see for example Schreuder and Thomas 1991). It is unfortunate in the USA

that there are two natural resources surveys, Forest Inventory and Analysis of the USFS and the

NRI of the NRCS where both agencies are in the USDA. Narrowly focused surveys seem to be the

rule in most other countries too as evidenced from descriptions in European Commission (1997).

Complementariness of the data collected would make it more likely to identify promising causeeffect relationships for a wider range of resource variables. For example, it would be desirable to

have the reliable information on soils collected by the NRI also available on the FIA plots to

develop better growth and yield models.

Mosteller and Tukey (1977) identify three criteria, of which two have to be satisfied to infer

cause-effect relationships: consistency, responsiveness, and mechanism. Consistency implies the

presence and magnitude of the effect y, associated with a minimal level of the suspected causal

agent x. Responsiveness is established by an experimental exposure of the suspected causal agent

and reproducing the symptoms. Mechanism demonstrates the biological or ecological process causing

the observed effect. To establish all three criteria is difficult. For example, the consistency between

smoking and cancer was established in the 1950s. The responsiveness was well documented then

and in the 1960s, but the actual causal mechanism was not established until the 1990s (Peifer

1997). And this linkage was relatively easy to establish because the effect is dramatic; see for

example Taubes (1995).

Feinstein (1988) advocated the following scientific principles for establishing cause-effect: stipulate a hypothesis prior to analysis, study a well-defined cohort having a statistical factor in common, collect high-quality data, study possible explanations, and avoid detection bias.

Hill (1965) in the epidemiological literature suggests a weight of evidence approach consisting

of nine criteria for inferring causality: strength, consistency, specificity, temporality, biological

gradient, plausibility, coherence, experimental evidence, and analogy. Strength refers to having a

high magnitude of an effect associated with exposure to the stressor; consistency to repeatedly

observing the association of the observed effect and stressor under different conditions; specificity

to the degree of the effect being more likely to be diagnostic of the stressor and the ease of associating it with an effect; temporality to the fact that the stressor always precedes the effect in time;

biological gradient to the change in effect with corresponding changes in the stressor; plausibility

that the association between effect and stressor is consistent with physical, chemical, and biological principles; experimental evidence that changes in effect are documented after experimental

manipulation or through recovery of the population following relief of the stressor; and analogy is

having similar stressors associated with similar effects. The more of these criteria that are satisfied,

the more weight can be given to the evidence that there is probable cause.

Survey data can only provide information for identifying possible cause-effect relationships.

Establishing that there is a correlation between possible cause and effect variables is a useful first

step in this indentification For readers who want more information on how to maximize the possibilities of such identification, we refer them to Olsen and Schreuder (1997) and Gadbury and

Schreuder (2003).
USDA Forest Service RMRS-GTR-126. 2004. 79

XI. Forest Sampling Situations

Pitfalls

Now that you have studied sampling intensively you may think you know what to do. This

section covers some major errors committed by serious samplers over time in the USA.

1. Moving subplots. FIA used to select 0.4 ha (1 acre) primary sample units subsampled by a

series of 5 or 10 VRP subplots. As is logical, in some cases some of the subplots might fall in

a different condition than the center subplot. For example, subplots 1-3 might be sampling a

pine plantation and subplots 4-5 in a hardwood stand. The decision was made as early as the

1930s and continued into the 1990s by several FIA units to keep all subplots in the same

forest type. For example if subplot 1 (the center one) was in a pine plantation, all subplots not

falling in the pine plantation would be moved by some systematic rule into the pine plantation. This procedure biases the results (Williams and others 1996).

2. Averaging conditions. Related to the above, another unit did not move the subplots in the

above situation but made the equally undesirable decision to average forest types, i.e., they

did not keep track of what type was being sampled. They would call the plot described in (1)

above a mixed pine-hardwood stand.

Situations 1 and 2 led to the interesting situation where two states in the USA that are quite

similar showed huge differences in the area in pine/hardwood stands as a percentage of the

total in forest.

3. VRP sampling to get 6-8 trees per point. A reasonable recommendation was made in a forest

mensuration textbook that in VRP sampling one should select a prism factor yielding on

average 6-8 trees/point. This recommendation was followed up incorrectly in several places

in the Western USA. Field crews would take various prisms to the field with them and then

select one at a location that would yield them between 6 and 8 trees. This biased approach

surprisingly was supported by several prominent biometricians. In several “experiments” of

this method, little or no bias resulted but one author got involved in a situation in California

where such serious bias was noted (see Wensel and others 1980 and Schreuder and others

1981).

4. Misuse of model predictions. A FIA unit developed growth and mortality models based on

growth and yield studies and used those models to update the information on plots they could

not remeasure from a cost point of view. The predicted plot values were then used as real plot

data for generating state-wide estimates.

5. Dropping subplots to meet production targets. FIA program managers put heavy emphasis on

meeting production targets. This is why one unit approved of the elimination of subplot 4 to

meet the production of 8 plots per 2 weeks for crews if they felt they could not meet their

production targets. This biases the results, especially if crews decide to judiciously drop subplots 4 such as in difficult sampling conditions.

6. Forgetting probabilities of selection. A government agency selected a timber cruising sample

using stratified sampling to obtain volume estimates for different strata. Ten years later they

decided they wanted to revisit the locations for other purposes but had not kept track of the

probabilities of selection. They wanted to treat the existing sample as a SRS for remeasurement

purposes. Schreuder and Alegria (1995) illustrate how this may seriously bias results.

7. Treating subplots as plots because their information is considered uncorrelated. Clearly the

subplots are not independent observations and hence should not be treated as such.

8. Different results by different agencies. Two agencies in the same department estimated

vastly different areas of forest in several states. It turned out that this was due to differences

in interpreting a common definition of forest, definition of what is a tree, and standards in

measurement and estimation techniques. Several of these differences are also tied into
80 USDA Forest Service RMRS-GTR-126. 2004.

considering forest as a use class (Goebel and others 1998). A forester may prefer to see as

much forest in a state as possible, while a range manager may see the same land as range.

Problem: Using a tree growth model developed from growth and yield study data, how would the

predictions for such a model compare to the actual growth of trees of the same species on inventory

plots?

Answer: Growth and yield studies typically use plots with 100 percent stocking levels with a much

more favorable environment than that of inventory plots which are more likely impacted by insects and

diseases, by human activities, etc. It is therefore more likely that the predictions will yield overestimates of the actual growth of the inventory trees.

Problem: An unmotivated crew using the situation described in #5 decides to always drop subplot 4

when they know it is difficult to measure. What are the consequences?

Answer: Clearly this will bias estimates for the area inventoried because it changes the probabilities

of selection of the subplots and hence the plot that they are part of. It is really not possible to answer

which way the bias will go. Some of the subplots may be in areas difficult to measure because they are

in a swamp while others may be in highly productive areas where vines and underbrush make access

to the subplots difficult and others may be on a very unproductive, steep cliff.

Problem: A well-intentioned crew using the situation described in #5 decides to measure subplot 4

only when they know it contains nice “timber” trees. What are the consequences?

Answer: This should bias certain estimates upward such as those for timber volume. It might not have

much effect on estimates for variables not much correlated with niceness of trees such as number of

trees or mortality.

Problem: A crew used multiple prisms selecting the one giving 6-8 trees at each point in VRP sampling. Generating estimates for the 100 VRP plots they took generates an estimate of 100,000 3

m for

the area. They sell the timber based on that amount of volume. The company that buys the timber finds

only 60,000 3 m . They are not happy and file a lawsuit against the company that did the inventory. Both

sides approach you, a reknown inventory expert, to testify on their behalf. Since truth is more important to you than money, you can pick either side. Which one would you pick?

Answer: It would be more sensible to take the side of the company that bought the timber. Certainly

the inventory method used was faulty as indicated in #3 above.

Suggestions

Our experience leads us to believe that one can do better than what is available now in regards to

surveys. It is hard to change existing surveys. Hence:

1. Be more flexible, less hidebound.

2. Do research on what is available and what can be done better.

3. Document well what you are doing.

4. Observe the “Keep it simple” (KIS) principle in design, less so in estimation.

5. Use creative, competent analysts.
USDA Forest Service RMRS-GTR-126. 2004. 81

6. Focus on the objectives. For example, there may be need to resolve potential conflicts between

what is wanted for timber surveys for which there is often strong political support vs. ecology

parameters where such support may be less powerful. Anticipate what may be needed in the

future too.

7. Keep up on the world literature and contribute to it.

8. Define measurable variables (see Schreuder and others 1993, p. 292, specifically the warning

by Innes for example).

Over time, we have learned the following lessons:

1. The objectives of a successful survey will change over time and will become more encompassing.

2. Don’t lock yourself into existing approaches. Allow for change. An example is plot design

where in the USA we have gone from rectangular plot sampling to variable radius plot or VRP

sampling, to circular plot sampling and more than likely at a future date should move to

sampling using different plots for different variables including a long rectangular or square

plot closely tied to remotely sensed information. Over time, more information can be collected by remote sensing. Large-scale surveys are getting away from a pure timber orientation

to being more ecologically based so that we are interested also in linear features such as

ripararian areas, understory vegetation, and rare and endangered plant species. Because plots

can be more accurately colocated on both the remote sensing and the actual ground plots by

the use of geographic positioning systems (GPS) over time, and because more and more

detail can be discerned with newer remote sensing platforms, more efficient estimation will

be possible by combining ground and remote sensing information in statistical regression

estimation models.

3. The estimates/analyses can be and should be as defensible as possible. A fundamental principle

in FIA is to keep things simple: KIS (keep it simple). Our recommendation is to keep the

design simple but allow for more complexity in the analyses, since different people want to

use the data in different ways. We are likely to have much controversy in analyzing the annualized data sets before agreement is reached.
82 USDA Forest Service RMRS-GTR-126. 2004.

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USDA Forest Service RMRS-GTR-126. 2004. 85

XIII. Glossary

Accuracy. Freedom from error or the closeness of a measurement or estimate to the true value.

More broadly, it is the degree to which a statement or quantitative result approaches the truth.

Note that 2 2 Accuracy Precision Bias = + using these statistical definitions. Thus, if bias is eliminated, Accuracy = Precision.

Asymptotically unbiased. Estimation bias goes to 0 as sample size approaches population size. It

is the same as consistency as used by Cochran (1977).

Attribute. Units classified as having or not having some specific quality.

Basal area (per site). The cross-sectional area at breast height of all trees on the site.

Basal area (per tree). The cross-sectional area of a tree at breast height.

Bias. A systematic error introduced into sampling, measurement, or estimation by selecting or

favoring, possibly unintentionally, one outcome or answer over others.

Breast height. The point on a tree stem at 1.4 m (4'6") in the USA, New Zealand, Burma, India,

Malaysia, South Africa, and some other countries and 1.3 m (4'3") above ground in continental

Europe, Great Britain, Australia, Canada, and Mexico.

Consistency. The same as asymptotically unbiased as defined above.

Continuous variable. A variable expressed in a numerical scale of measurement, where any interval of it can be subdivided into an infinite number of values.

Correlation coefficient. A measure of the degree of linear association between two variables that

is unaffected by the sizes or scales of the variables.

Covariance. A variance or measure of association between paired measurements of two variables.

Covariate. A quantitative, often explanatory variable in a model such as a regression model.

Covariates are often important in improving estimation.

dbh. The diameter at breast height of a tree.

Discrete variable. Qualitative variables or those represented by integral values or ratios of integral

values.

Double sampling. Two levels of sampling where the first level provides information on covariates

and the second on the variable of interest to estimate parameter(s).

Efficient estimator. An estimator that predicts a parameter more reliably than competing estimators where reliability is usually measured by the ratio of the mean square errors of the estimators.

Estimate. The numerical value calculated from an estimator for a sample.

Estimator. A function of the values in a sample or a formula used for estimating a parameter based

on a sample.

Estimator of population mean. The formula used in estimating the population mean from a sample.

Estimator of population variance. The formula used in estimating the population variance from

a sample.

Experiment. The conduct of a systematic, controlled test or investigation.

Global positioning system (GPS). A system using satellites to locate ground positions.

Inference. The drawing of conclusions based on data or observations.

Mean. The average value of a variable for all units in a population or sample.

Median. The value of a variable so that half of the values are larger and half are smaller than this

value in a population or sample.

Mode. The value of a variable that occurs most frequently in a population or sample.
86 USDA Forest Service RMRS-GTR-126. 2004.

Multilevel sampling. A sampling design where more than one phase or stage of sampling is used.

The first levels are used to collect information on covariates useful for more efficient estimation

of the ultimate parameter(s) of interest for which information is usually collected at the last

phase or stage.

Parameter. A characteristic or function of the values of the units in a population, i.e., the population characteristic of interest, such as average volume per ha or total volume of trees in a forest.

Population. An aggregate of items each with a common characteristic or common set of characteristics. In the statistical sense, a population is an assembly of individual units formed in order to

describe the population quantitatively. For example, it might be all the trees in a particular forest

stand or all the users of a recreation area.

pps sampling. A sampling design where sample units are selected with a probability proportional

to a measure of size, usually a covariate such as dbh or basal area in the case of tree volume.

Precision. Relative freedom from random variation. In sampling it is expressed as the standard

error of the estimate and relates to the degree of clustering of sample values about their own

average or the reproducibility of an estimate in repeated sampling. It is also used to indicate the

resolving power of a measuring device.

Probabilistic sampling. Procedures in which samples are selected such that all units and each pair

of units in the population have a positive probability of selection.

Randomization. A deliberately haphazard arrangement of observations to simulate selection by

chance.

Sample surveys. The design and execution of surveys to provide estimates of characteristics (parameters) of well-defined finite populations.

Sample unit. A unit from a population, i.e., a tree or all trees located within a plot (i.e., fixed-area,

strip or point sample).

Sample. A subset of a population used to obtain estimates of one or more of its parameters. In this

book we focus on probabilistic samples. For example, a sample can be the diameters (dbh) of all

trees on a sample of plots or the amount of time spent picnicking by users of a recreation area on

given days.

Sampling design. A formalized method of selecting a sample from the population, for example

simple random sampling.

Sampling frame. A list of all sample units used to represent a population.

Sampling strategy. Comprises both the sampling design and the estimator(s) used, for example

simple random sampling with the estimator of the population mean, say the sample mean.

Single-level sampling. A sampling design where units are selected directly from the sampling

frame of the population.

Standard deviation. The square root of the variance defined below.

Statistical inference. Expressing the connection between the unknown “state of nature” and observed information in probabilistic terms.

Statistical survey. Involves the design and execution of surveys to provide estimates of characteristics of well-defined finite populations.

Unequal probability sampling. Sampling designs where units are selected with different probabilities. These probabilities need to be known for unbiased estimation.

Unit. The basic sample unit used; e.g., that used in the last stage of multistage sampling.

Variable. A characteristic that varies from unit to unit; for example, the age of a tree.

Variance. The average of the deviations squared between the values of the variables and the overall mean in the case of a population or between the values of the variables and the sample mean

in the case of a sample; in the first case it is a population parameter, in the second a sample

statistic.
USDA Forest Service RMRS-GTR-126. 2004. 87

Appendix 1. Inference

Inductive logic, the drawing of conclusions from analysis of observed data about unobserved

parameters or underlying laws, is one of the most controversial issues in philosophy (Gregoire

1998, Schreuder and Gregoire 2001). Although inference, the drawing of conclusions based on

data or observations, is not limited to the narrow field of scientific and statistical inference, the

latter is important in this contentious world, and a proper understanding of it is crucial to discussing the role of sampling in the inferential process. Scientific inference becomes statistical inference when the connection between the unknown “state of nature” and the observed information is

expressed in probabilistic terms (Dawid 1983).

Statistical inference comprises the whole field of statistics, its focus being what is logically

implied by the information available (Fraser 1983). Cramer (1946) summarizes the role of statistical inference as having three functions: description, analysis, and prediction. Description is the

reduction of data sets to as small a set of numbers as possible, such as the mean, variance, skewness

of a distribution, etc. This enables one to describe a population as concisely and briefly as possible

and can allow for comparison between populations. Analysis is the summarization of data for a

particular purpose or objective. Examples are: What are the estimates of certain population characteristics? Did the sample arise from a given distribution? Or given two samples, did they arise from

the same population or not? Statistics provides methods of how to do such analyses. Statistical

methods are used to predict and explain phenomena, often a very challenging task.

Ideally, statistical inference would always be based on Bayes theorem, which combines prior information with information from surveys or experiments and would be acceptable to many statisticians if

the prior belief is objective. The problem is that usually prior information is subjective, where subjective

indicates that the information available varies from person to person. Objective prior information indicates that people would normally agree on it. As an example of subjective prior information, a forest

industry person could believe that there is plenty of old growth distributed nicely over the forest for

habitat for endangered species whereas an environmentalist could equally strongly believe that the old

growth in the forest is limited and badly distributed. People willing to accept prior subjective information are called Bayesians and rely on Bayes theorem for inference. Non-Bayesians or frequentists, a

majority, use classical inference procedures relying only on objective data often based on normality

assumptions and large sample theory based on the central limit theorem and related statistical properties.

It is our belief that Bayesian procedures should be used when immediate logically defensible decisions

need to be made, and classical ones when building a body of scientific knowledge. A forest manager who

has to make decisions about whether or not to cut old growth and where for management purposes, may

well choose to use all his prior information to construct a (subjective) prior distribution to combine with

actual sample data in order to use Bayes theorem to make such decisions. Such decisions can be defended at least on the basis of a systematic approach. Scientific databases can be used by different users

applying different priors to make their decisions.

Statistical inference from sample surveys can be either model-based or design-based. In modelbased sampling, inference relies on a statistical model to describe how the probability structure of

the observed data depends on uncontrollable chance variables and, frequently, on other unknown

nuisance variables. Such models can be based on a theoretical understanding of the process by

which the data were generated, experimental techniques used, or past experience with similar processes. For inference in design-based sampling, reliance is placed on probabilistic sampling. It is

the most widely accepted approach now. The following is a brief summary of both approaches.

In non-probabilistic or model-based sampling, inference is made by specifying an underlying

superpopulation model ξ for the values of the variable in the actual population being sampled. The

actual values are considered to be random variables from this superpopulation. It is then assumed

that the actual population or a sample from it is a sample from this superpopulation of interest.

Then using ξ , for estimator Yˆ of the quantity of interest Y, the distribution of Y-Yˆ can be derived

for the specific sample and the model-based mean square error of Y-Yˆ can be obtained and estimated, leading to a model-based prediction interval for Y. Inference extends to parameters of the
88 USDA Forest Service RMRS-GTR-126. 2004.

superpopulation model, so that the inference space is broader than for design-based inference.

Sample units do not have to be chosen at random or with known probability as long as they are not

selected based on their values of interest i y , i =1,...,N .

Conclusions and inferences rely heavily on the model assumed, which can be a serious liability

if the model is not specified correctly. But if a model is correctly specified, an increase in precision

can be expected over the design-based approach. Our experience is that very few models are reliable. The statement by Box and Draper (1987) that “all models are wrong; the practical question is

how wrong do they have to be to not be useful” deserves consideration. Nonetheless, models are

useful in building a body of knowledge in every subject and may have to be relied on when a quick

decision needs to be made. The Bayesian approach to inference with subjective priors fits well into

the model-based inference approach, although many advocates of model-based sampling would

not consider themselves Bayesians at all. As noted by Koch and Gillings (1983), model-based

inference encompasses Bayesian and superpopulation inference since the validity of the claimed

generality is model dependent, i.e., is sensitive to model misspecification.

The design-based approach to inference relies heavily on probabilistic sampling, in which each sample

unit of the population has a positive probability of being selected and the probability of each sample can

be calculated. The statistical behavior of estimators of a population attribute is based on these probabilities

and the probability-weighted distribution of all possible sample estimates. The distribution of the variable,

probabilistic or otherwise, is not considered here. An obvious weakness of this approach is that samples

that were not drawn are considered heavily in evaluating the properties of the inference procedure, yet

should not inference about a population parameter be based solely on the actual sample drawn?

Nevertheless, the approach is objective and the only assumption made is that observed units are selected

at random so the validity of the inference only requires that the targeted and sampled populations are the

same. And, careful attention to sample selection within the framework of probabilistic sampling will

eliminate some undesirable samples from consideration and give others a low probability of selection.

The whole idea behind probabilistic sampling is to make the sample representative of the population

being sampled. However, as illustrated especially well in Kruskal and Mosteller (1979), “representative”

is subject to a wide array of interpretations.

Smith (1994, p.17), formerly a strong advocate of model-based inference, states: “My view is

that there is no single right method of inference. All inferences are the product of man’s imagination

and there can be no absolutely correct method of inductive reasoning. Different types of inference

are relevant for different problems and frequently the approach recommended reflects the

statistician’s background such as science, industry, social sciences or government...I now find the

case for hard-line randomization inference based on the unconditional distribution to be

acceptable...Complete reconciliation is neither possible nor desirable. Vive la difference.”

A crucial difference between design-based and model-based approaches is that for the former

inference is made about the finite, usually large population sampled, whereas model-based sampling makes inferences about superpopulations by the compulsory use of models. The inference

then is about the actual population that is represented in some sense by the existing population,

assuming that the models used underlie the real population.

Deming (1975) recommends that a distinction should be made between enumerative (or descriptive)

and analytical (or comparative) surveys. In enumerative surveys interest is in a finite, identifiable, and

unchanging population from which samples are drawn. Action is taken on the population of units studied

(e.g., all forests in the state of Montana) at the time of sampling to decide how much timber to harvest.

This is the type of survey conducted by Forest Inventory and Analysis (FIA) program of the USDA FS.

Here design-based inference is indicated. In contrast, analytical surveys focus on populations where

action is to be taken on the process or cause system, the purpose being to improve conditions in the

future. For the Montana forests, we are still talking about the same forests but different conditions exist

when we apply treatments to improve conditions. For example, land management agencies such as the

National Forest System (NFS) of the US Forest Service may be interested in collecting information on

managing rare and endangered wildlife species so as to create or modify existing vegetation conditions

to increase the number of such animals in the forests over time. Although we still want to take a designbased sample here from the existing population, the inference clearly is for populations of the future so
USDA Forest Service RMRS-GTR-126. 2004. 89

Objectives

Define the target

or process about which

to draw inference

Type of survey needed (1)

Enumerative

survey

Define

sampling

frame

Is frame identical to

target population

Assess relevance

of frame (2)

Define sampling procedure

Select

SRS (3)

Other

Probabilistic

Sample (4)

Select

nonrandom

sample (5)

Analytical

Survey

Determine assessment process

Define sampling procedure

Determine relevance of sampled

Process (6)

Select desired

sample

no

yes

is model-based in the sense that we are extrapolating from the existing to future populations. This is

clarified further by the following.

Deming (1975) suggests that in enumerative surveys a 100 percent sample of the population

provides the complete answer to the questions posed, whereas in analytical surveys the answer is

still inconclusive. Hahn and Meeker (1993) make the further distinction that analytical studies

require the assumption, usually not verifiable, that the process about which one wants to make

inferences is statistically identical to that from which the sample was selected. Figure A-1 from

Hahn and Meeker (1993) illustrates the differences between analytical and enumerative surveys. A

problem with this distinction is that often in surveys conducted by land management agencies,

there is interest in both types of inference. For example: one aim may be to determine how much

timber to harvest and from where or what areas to delineate for weed or erosion control (requiring

enumerative surveys) and another to assess what needs to be done to improve habitat for wildlife or

diminish it for noxious weeds (requiring analytical surveys).

Figure A-1. A Comparison of Enumerative and Analytical Surveys. Reprinted with permission from The

American Statistician. Copyright 1993 by the American Statistical Association. All rights reserved. The

numbers refer to the following comments:

(1) Is the purpose of the study to draw conclusions about an existing finite population (enumerative

study), or is it to act on and/or predict the performance of a (frequently future) process (analytic

study)?

(2) Statistical intervals apply to the frame from which the sample is taken. When the frame does not

correspond to the target population, inferences about the target population could be biased, and a

statistical interval provides only a lower bound on the total uncertainty.

(3) Most statistical intervals assume a simple random sample from the frame.

(4) More complex statistical intervals than those for simple random samples apply; see Cochran (1977).

(5) Statistical intervals do not apply. If they are calculated, they generally provide only a lower bound on

the total uncertainty.

(6) Statistical intervals apply to the sampled process, and not necessarily to the process of interest. Thus,

any statistical interval generally provides only a lower bound on the total uncertainty with regard to

the process of interest.
90 USDA Forest Service RMRS-GTR-126. 2004.

Appendix 2. Distributions

Populations are discrete or finite ( N < ∞ ) but often are assumed to be infinitely large ( ) N = ∞

since continuous distributions are more likely to approximate the real population. Infinite populations have properties that are crucial to statistical inference. A few of the more important distributions are presented below as well as some key results from statistical theory based on continuous

distributions. The material is a condensation of material discussed in Schreuder and others (1993).

Distributions are often characterized by their moment generating function (mgf).

Definition. If Y is a random variable with probability density f(y), then the expected value, E, of ty e is called the mgf of Y if it exists for every value of t in some interval 2 2 − << h t h . This is

denoted by

() ( ) ( ) ty ty

y

m t Ee e f yd = = ∑ y

where for discrete distributions, f(y) is the probability mass function, and

() ( ) ( ) ty ty m t Ee e f ydy

∞

−∞

= = ∫ where

for continuous distributions, f(y) is the probability density function (Mood and others 1974). The

logarithm of the mgf, called the cumulant generating function, is often used. The moments of this

function are called the cumulants.

m(t) generates the moments of distributions. For example, in survey sampling we are often

interested in estimating the first two moments of the normal distribution. The first moment is the

mean and is obtained from the mgf by differentiating with respect to t once and setting t = 0.

Similarly the second moment, the variance, is obtained by differentiating the mgf with respect to t

twice and setting t = 0 and then subtracting the first moment squared at t = 0.

Continuous Distributions

Three important continuous distributions are the normal, gamma, and multivariate.

Normal Distribution

The cumulative distribution, usually simply called distribution, is defined as

2 ( ) { } (1/ 2 exp[ 1/ 2{( ) / } ] ( ; , )

y y

F y P Y y y dy f y d πσ μ σ μ σ y

−∞ −∞

= ≤= − − = ∫ ∫

with parameters μ and 2 σ ( 0) σ > , and −∞ < < ∞ Y . The parameters μ and 2 σ are called the

mean and variance of the distribution, respectively.

The mgf of the normal distribution is

 2 2 m( ) exp( / t tt = + μ σ 2)

with mean '

1 m'(0) = = μ μ and variance 2 2

2 m''(0) [ '(0)] − == m μ σ so in fact the mean and variance

are the two parameters of the distribution. Although there are numerous situations where the normal distribution approximates the distribution of units in a population (such as the heights of all

trees in a large plantation), it is most commonly used as a convenient approximation to other distributions. The normal distribution is important in probability theory because it is the limiting distribution of almost any standardized sums (or means) of random variables as the number of variables

in the sum increases.
USDA Forest Service RMRS-GTR-126. 2004. 91

If a statistic θ

∧

 is an unbiased estimator of a parameter θ , with estimated variance ˆ v(θ ), and

is approximately normally distributed, then the statistic ˆ ˆ t = − ( )/ ( θθ θ v ) follows Student’s

 t-distribution with density 2 ( 1)/ 2 ( ) {( 1) / 2)}(1/ )(1 / ) /{ ( / 2) (1/ 2)} v ft v v t v v τ ττ − + =+ + where

v is the number of degrees of freedom on which the estimate of the standard error is based

and /2 1

0

( / 2) v t τ v te dt

∞

− − = ∫ . The t-distribution is fundamental to constructing confidence intervals,

and tables for this distribution are widely available (see Appendix 3, Table 2).

Gamma Distribution

This distribution appears naturally as the distribution of the sum of squares of independent,

standard, normally distributed random variables, 1 2 , ,..., Z Z Zn . Then 2

1

n

i

i

Z

=

∑ has a 2 χ distribution

with parameter n where 2 χ is a special case of the gamma distribution and n is the number of

degrees of freedom. The gamma distribution is often used in survey sampling to enable comparisons of sampling strategies. The distribution is:

1

0 0

( ) [ ] exp( / ) /{ ( 1)} ( )

y y

F y P Y y y y dy f y dy α α β βτα − = ≤= − − = ∫ ∫

with parameters α and β > 0 , y>0, and ( 2)

0

( 1)

y

t t edt α τ α − − − = ∫ .

The gamma distribution has the mgf

m( ) (1 ) t t α β − = −

with mean μ =αβ and variance 2 μ αβ 2 = .

Multivariate Distributions

The multivariate normal distribution has been studied much more extensively than other multivariate distributions and is used more frequently for inference among multivariate continuous distributions than is the normal distribution among univariate continuous distributions. The bivariate

normal distribution is defined as 21 2 2 2

2 2

( , ) ( , ) (2 1 ) exp{[ 1/{2(1 )}]{( ) / 2 {( ) / }

{( ) / ( ) / }]

xy x x x x

yy y y

F x y P X xY y x x

y y

πσ σ ρ ρ μ σ ρ μ σ

μσ μ σ

− = ≤ ≤= − − − − − −

− +−

with −∞ < < ∞ −∞ < < ∞ x y , , where 2 2 ( ), ( ), ( ), ( ) μ σσ x yx y === = E X u EY V X VY ,

and 2 2 ( )( ) /( ) /( ),( 1 1) ρ μ μ σσ σ σσ ρ = − − = −< < EX Y x y x y xy x y is called the correlation between X

and Y,

and σ xy is the covariance between X and Y.

Discrete Distributions

Important examples of discrete distributions are the binomial, hypergeometric, Poisson, and

multinomial distributions.

Binomial Distribution

 If n independent trials are made (such as whether a load of logs should be sampled or not) and

each trial has the probability p of outcome i occurring, then the number of times in which i occurs
92 USDA Forest Service RMRS-GTR-126. 2004.

may be represented as a random variable Y following the binomial distribution with parameters n

and p. This distribution is defined as the distribution of a random variable Y (= number of occurrences of i) for which

[ ] [ !/{ !( )!}] (1 ) y n y PY y n y n y p p − == − − (y = 0,1,2,...,n).

The mgf of the binomial distribution is

( ) (1 )

t n m t p pe =−+ .

From the mgf, the mean and variance are derived as

' '2

22 1 μ μμ μ = =− = − np p p , ( ) (1 ) where ' 2

2 μ =+ − np n n( 1) p .

There are many approximations to the binomial distribution. Such approximations often involve

limiting distributions that arise when one or both parameters converge to a specific value. Limiting

distributions are the (discrete) Poisson distribution discussed below (n p →∞ → , 0) with np = a

constant θ , and the normal distribution, which is a special case of the standardized binomial variable (Y np np p − − ) / (1 ) as n → ∞ .

Hypergeometric Distribution

The classic situation in which it arises in forestry is as follows.

Suppose we have a population with N trees, M of which are dead, and N - M of which are alive.

If n trees are drawn at random without replacement, then the probability of selecting y dead trees is

PY y M y M y N M N M n y n y N n N n [ ] { !/[ !( )!]}{( )!/[( )!( )!]}/[ !/{ !( )! = = − − − −+ − − }]

for max(0, ) min( , n N M y nM − + ≤≤ )with parameters M, N, and n and n! = n(n-1)(n-2)...1 and 0! = 1.

The mgf for the hypergeometric distribution is

( ) [( )!( )!/ !] ( , ; 1; )

t m t N n N M N H n MN M n e = − − −− − −+

where 2 Hz z z (α β γ αβ γ α α β β γ γ , ; ; ) 1 ( / )( /1!) [ ( 1) ( 1)]/{ ( 1)} / 2! =+ + + + + +...

is a hypergeometric function which converges for absolute value of z<1.

The mean is μ = nM / N and variance 2 μ =− − − [( ) /( 1)] ( / )(1 / N n N nM N M N).

Several approximations to the hypergeometric distribution exist, a simple one being the binomial distribution [ ] [ !/{ !( )!}] / ) (1 / ) y n y PY y n y n y M N M N − == − − which is usually adequate when

n < 0.1N. Confidence intervals can be constructed as described under the binomial distribution

assuming either the binomial or normal approximation to the hypergeometric distribution.

Poisson Distribution

If the future lifetime of an item of equipment (say a chainsaw) is independent of its present age,

then the lifetime can be represented by a random variable Y with distribution

[ ] / !, 0,1, 2,...; 0 y PY y e y y θ θ θ − == = > where θ is the only parameter (= np in the binomial

distribution with n → ∞ as p → 0 ). A widely quoted application of this distribution concerns the

number of soldiers annually kicked to death by mules in an army at the middle of the 19th century.

(An analogous situation might be the number of loggers killed by falling trees in a forest.) The

probability of death was small and the number of soldiers exposed was large. It is doubtful that the

conditions of independence and constant probability (p) were satisfied but the data available were

satisfactorily fitted by this distribution.

The moment generating distribution is ( ) exp[ ( 1)] t m t e = − θ

with mean μ =θ and variance μ θ 2 = .
USDA Forest Service RMRS-GTR-126. 2004. 93

Multinomial Distribution

Multivariate discrete distributions are often closely related to univariate ones. For example, the

marginal distribution of individual variables is generally a simple binomial, Poisson, or 

hypergeometric distribution or a distribution obtained by modifying or compounding one of the

univariate distributions. For example, the joint distribution of the random variables

1 2 , ,..., k n n n representing the number of occurrences of events 1 2 , ,..., O O Ok in n trials is the multinomial distribution

12 1 ( , ,..., ) ! ( / !) k nj Pn n n n p n k j jj = Π =

with 0 ≤ nj for all j = 1, . . . , k and 1

k

j

j

n n =

∑ = . This distribution is a natural extension of the binomial

distribution, which is a special case if k = 2. The joint distribution of any subset s < k is also a

multinomial, hence the important subset where we have two classes, that is, “class 1” and “all

others,” is also a binomial.

The mgf of the multinomial is 1

1 )1 ( ,..., ( ... ) k t t n m k k t t pe p e = ++ where the moments of the

( 1,..., ) i n i k = are simply

'

1( ) , 1,..., i μ t pi = = k

and

2 ( ) (1 ) ii i μ tp p = − .

Note that the covariance of ni and j n and the correlation between them are respectively

(, ) Cov n n np ij i = − pj

and

( , ) /{(1 )(1 ) Cor n n p p p p i j ij i j =− − −

Confidence intervals for any pi or i i n p are constructed by treating the multinomial as a binomial with probability of selecting class i, pi , and of all other classes, 1− pi . Then the discussion

for constructing confidence intervals under the binomial is appropriate.

Multivariate Hypergeometric Distribution

A generalization of the hypergeometric distribution is the multivariate hypergeometric distribution defined as follows: If there is a population of N units, Ni of which are of type i (i = 1, . . . , k)

so that 1

k

i

i

N N =

∑ = and a sample of size n is taken without replacement from the N units, then

12 1 ( , ,..., ) { !/[( )! !]}/[ !/{( )! !}] k Pn n n N N n n N N n n k i i i ii =∏ − − =

is the multivariate hypergeometric distribution with

1

;0 , 1,...,

k

i ii

i

n n n Ni k =

∑ = ≤≤ =

The moments of the multivariate hypergeometric distribution are analogous to those for the

hypergeometric distribution and the correlation between ni and j n is

( , ) /{( )( )} C ij ij i j orr n n N N N N N N =− − − .

Laws of Large Numbers

In inductive inference we determine something about a population of interest, say its mean, by

examining a sample from the population. The following theorems assuming random sampling are

critical to survey sampling inference. A finite number of values of Y can be used to make reliable
94 USDA Forest Service RMRS-GTR-126. 2004.

inferences about E(Y), the average of an infinite (or very large finite) number of values of Y (i.e.,

1

/

N

i

i

y N =

∑ the average for the whole population). For simple random samples, the following three

theorems apply:

(1) Theorem 1(Tchebysheff’s Inequality)

For a distribution F(y) with mean μ and finite variance 2 σ , and if y is the mean of a random

sample of size n from this distribution and α any positive number, then

2 P ny n [ / / ] 1 1/ − ≤ − ≤ ≥− ασ μ ασ α .

(2) Theorem 2 (Weak Law of Large Numbers)

Let F(y) be a distribution with mean μ and finite variance 2 σ and let ε and δ be two specified

small numbers where ε > 0 and 0 1 < < δ . If n is any integer greater than 2 2 σ εδ /( ) and n y is the

mean of a random sample of size n from F(y), then

[ ]1 P yn −< − < ≥− ε με δ .

Thus this theorem states that for any two small numbers ε and δ , where ε > 0 and 0 1 < < δ ,

there is an integer n such that for a random sample of size n or larger from the distribution of the

population of y-values F(y), the probability that the mean of the sample of y-values n y is arbitrarily

close to the population mean μ can be made as close to 1 as desired. The weak law of large

numbers is an example of convergence in probability. The following theorem exemplifies the notion of convergence in distribution. It underlies the wide application of the t-distribution in constructing confidence intervals around estimates of parameters of interest and highlights the critical

importance of the normal distribution.

(3) Theorem 3 (Central Limit Theorem)

Let F(y) be a distribution with mean μ and finite variance 2 σ and let n y be the mean of a

random sample of size n from F(y). If ( )/ n n zy n = − μ σ , then the distribution of n z approaches

the standard normal distribution as n increases without bound.

This theorem states that the mean n y of a random sample from any distribution with finite mean

μ and variance 2 σ is approximately distributed as a normal variable with mean μ and finite

variance 2 σ / n . Since we usually deal with populations of size N with N considerably less than ∞ ,

n cannot increase without bound, so the applicability of this theorem to finite populations is arguable. In many cases, when n is not too small and the distribution of y is not too far from symmetry,

the distribution of n z will be approximately normal for inference.
USDA Forest Service RMRS-GTR-126. 2004. 95

Appendix 3. Tables

Tabulated values are frequently used in data analysis and hypothesis testing. Among the more

common tables (included here) are: Table of random numbers, Distribution of Student’s t, confidence intervals (95 percent) for the binomial distribution, ArcSine transformation, and two-tailed

significance levels of correlation coefficient r.

The tables presented here have been generated using the open source package R. The code used

to create these tables can be downloaded from http://www.r-project.org/ and can be modified and

used for more detailed tables to meet specific needs.

Table 1. Table of random numbers.

55862 47692 92962 37452 24651 87211 80143 24086 72731 65043

97923 60028 26117 73239 61498 11036 01350 60940 45478 60901

17929 18397 91162 93398 56598 01268 68729 94782 04323 25374

88428 55580 07083 85487 57816 64538 86549 25001 79018 20327

94726 02973 87423 03166 20079 63784 07889 05779 66550 07533

33594 96394 84905 60462 20100 83140 26129 16337 74811 65944

38484 42163 27173 84056 59407 91873 92328 39655 49040 90167

23393 82742 38862 26980 75368 02847 01053 75732 71076 88682

33066 92829 97349 50095 97538 55777 58994 22487 99974 03351

12453 15870 57421 79329 17487 03304 44107 25050 84297 75759

42973 12644 92911 56030 27487 73955 56921 33478 91622 26207

43397 07534 66071 65568 44354 19675 08492 81531 52961 95150

75824 23611 05961 23306 12030 99024 51409 09220 46436 25592

24270 90235 66887 91822 57484 20615 81048 06121 22790 30293

32155 55651 50165 40420 87805 94167 47014 03412 58232 01190

49128 68603 70371 73609 85851 27406 97846 10701 92339 75438

47881 21969 79326 14985 78919 70848 55693 15446 28657 08951

93028 12394 31791 66834 14037 59579 96851 03082 07339 70364

26557 21485 75834 65133 40810 62393 44524 88053 88774 22159

95069 64989 22449 98023 87914 58086 11783 21285 45201 65647

51473 69229 92738 22769 14238 94509 42403 66017 96637 19124

98957 80805 46441 33404 51741 28407 68943 08679 16198 15385

87592 01776 43166 78814 34823 73796 68837 75250 10046 10791

84100 16277 41706 72797 06683 16386 77954 53505 53812 78249

69462 24705 80114 02004 55594 08060 47933 63441 69719 09729

93608 75353 64552 45577 88199 58313 12097 63335 06115 16543

37241 51581 55513 98131 71564 23230 66610 06202 79080 36601

06855 76107 80948 61777 88260 44776 46862 15308 61721 61179

33864 01654 05023 13244 33936 93647 20956 30452 14649 74309

45167 23633 72357 55846 90477 73288 92447 79028 36627 20670

10376 76848 45729 44801 98129 64002 41643 11779 87581 70343

76057 26051 60638 94221 44527 99577 97790 44854 70527 18809

48157 59061 05958 68478 05364 39412 04714 51510 50286 81665

19216 87580 99632 26032 19660 21029 94943 14525 10052 81280

79799 94944 63412 38049 70926 37288 14214 59185 61717 97117

66450 25680 24907 14967 16427 12253 36493 25117 51827 24131

49643 44121 78339 23361 96049 32878 24339 84647 19902 21751

64688 52256 57958 17785 53250 65071 35052 10527 60814 66955

83199 86497 83727 31070 56191 20996 47452 64613 32828 50908

48957 96114 11718 10502 07780 43378 33125 27659 42041 17867

31520 90234 63855 30387 33228 22565 86551 90405 64928 35050

79921 55566 73325 84683 81471 77280 05845 56210 31429 92431

61925 86171 34231 66531 77294 69358 54647 96733 59454 22476

72272 85835 76108 32805 99814 70078 86787 06660 33271 06007

10468 53992 14394 29949 50095 84011 27467 89068 41882 89295
96 USDA Forest Service RMRS-GTR-126. 2004.

62553 31050 81817 50871 43793 11283 49570 55558 05747 46890

35447 95380 11951 12916 34031 76096 63437 93967 60777 42642

19103 27155 01880 79915 52008 63297 47031 29449 92174 31646

30337 62493 42159 33595 18578 17139 14653 05651 19936 74640

12368 37150 93911 77900 07710 44900 63036 09541 87084 70367

33570 88254 10300 95538 98565 40765 37423 39175 74793 16095

52449 30644 15956 64412 55242 70402 27976 08474 54064 17177

73618 42175 28226 68731 34494 83981 68435 13824 22642 85836

50313 70672 33804 69796 09370 97272 36512 37197 79443 68344

03759 11496 52998 41981 61478 56898 16524 99470 81524 78416

43816 09467 87381 69089 03858 55216 48997 44554 16474 62920

80437 07966 42383 76849 95261 64131 56586 93570 30307 65755

14066 48327 42374 32218 25867 36283 96019 91864 89841 50048

31114 11873 48718 83532 08240 96875 44598 90183 58624 51209

87913 07045 90226 53779 98807 84334 71407 75478 86113 25203

99132 12065 93418 08473 37561 73214 50471 47693 79254 09753

97629 31824 60163 55879 29385 79623 21296 64812 91567 82636

08978 72131 78267 03037 92041 89849 68399 91036 49158 98049

26668 13492 49538 04136 44762 29368 13949 77645 54521 62896

54846 41553 88909 84887 83911 17358 79433 98950 26714 33734

72087 63890 49539 35081 99740 97436 29133 23521 99344 35706

65497 79878 10520 11005 75748 94319 02478 72131 97000 56085

95803 73813 71494 87266 27061 52087 53429 04310 07814 50188

40865 42512 47260 19632 58445 89434 79864 73372 31074 65604

04433 29545 35821 07371 65764 91799 98243 52226 31903 79077

02057 08958 81921 22160 54557 91189 97243 52332 76120 41564

05041 35418 76418 43272 43076 34316 21812 98938 06714 09484

43891 14181 24985 29895 21869 01045 71527 20064 55337 97291

10374 41917 24948 25856 86498 46629 96251 41806 78913 18636

32879 79602 49067 60820 08714 92253 82848 73409 48597 34394

86673 82907 67765 73961 66363 61262 38162 31243 18387 11775

29063 82178 45025 54215 10231 28407 71873 58663 25027 97921

54587 40236 85404 66748 30574 95912 89247 74995 37696 73460

48607 87689 11871 98132 61211 32425 62083 99140 17050 57359

16094 97671 02064 35310 48094 23033 92444 40069 62889 29614

63697 17999 81953 97997 58143 55029 40358 52536 21476 68069

13117 55809 88704 52420 31357 59400 50199 33963 15282 12459

77399 52580 05822 09809 20640 47579 56527 83490 30383 51673

76618 27999 79590 15016 94053 10365 60327 50400 84668 75029

32875 25165 79676 05502 90404 32841 93419 72246 59709 65307

27458 54831 40982 14291 01684 19623 02560 37877 17419 23878

58754 72564 55632 06415 58533 69342 67019 16555 39796 07811

80475 59074 95621 92668 30545 48770 18343 64267 67114 85963

83370 87361 36193 46322 08986 50128 96736 52654 62464 84932

58667 32519 54144 10160 57730 78138 79983 91235 21796 61710

72694 94654 53848 76727 91635 81324 80402 89686 14023 66006

54230 03232 69368 30694 91077 07709 43411 54098 27967 06669

93447 01796 87049 02472 51265 20130 78615 59145 12773 61529

73087 46442 16168 64092 55380 39620 56090 28236 20743 46986

90745 38867 06363 80949 62878 76653 32971 27592 30049 12427

65291 78320 73014 50550 23378 95816 01401 81341 19325 16530

87733 37580 60372 32473 83102 66290 59967 32447 84792 54735

51120 84671 75765 89097 89408 43351 39652 19391 02850 72261

12974 05910 82732 72030 61112 91125 66991 20928 77852 05238

12779 27311 21722 01344 32040 15520 25040 86340 27990 33335

Table 1. Continued.
USDA Forest Service RMRS-GTR-126. 2004. 97

Table 2. Distribution of Student’s t.

Probability of a larger value of t, sign ignored

df 0.5 0.1 0.05 0.01 0.001

1 1.00 6.31 12.71 63.66 636.62

2 0.82 2.92 4.30 9.92 31.60

3 0.76 2.35 3.18 5.84 12.92

4 0.74 2.13 2.78 4.60 8.61

5 0.73 2.02 2.57 4.03 6.87

6 0.72 1.94 2.45 3.71 5.96

7 0.71 1.89 2.36 3.50 5.41

8 0.71 1.86 2.31 3.36 5.04

9 0.70 1.83 2.26 3.25 4.78

10 0.70 1.81 2.23 3.17 4.59

11 0.70 1.80 2.20 3.11 4.44

12 0.70 1.78 2.18 3.05 4.32

13 0.69 1.77 2.16 3.01 4.22

14 0.69 1.76 2.14 2.98 4.14

15 0.69 1.75 2.13 2.95 4.07

16 0.69 1.75 2.12 2.92 4.01

17 0.69 1.74 2.11 2.90 3.97

18 0.69 1.73 2.10 2.88 3.92

19 0.69 1.73 2.09 2.86 3.88

20 0.69 1.72 2.09 2.85 3.85

21 0.69 1.72 2.08 2.83 3.82

22 0.69 1.72 2.07 2.82 3.79

23 0.69 1.71 2.07 2.81 3.77

24 0.68 1.71 2.06 2.80 3.75

25 0.68 1.71 2.06 2.79 3.73

26 0.68 1.71 2.06 2.78 3.71

27 0.68 1.70 2.05 2.77 3.69

28 0.68 1.70 2.05 2.76 3.67

29 0.68 1.70 2.05 2.76 3.66

30 0.68 1.70 2.04 2.75 3.65

40 0.68 1.68 2.02 2.70 3.55

60 0.68 1.67 2.00 2.66 3.46

120 0.68 1.66 1.98 2.62 3.37

∞ 0.67 1.64 1.96 2.58 3.29
98 USDA Forest Service RMRS-GTR-126. 2004.

Table 3. Confidence intervals (95 percent) for the binomial distribution.

Observed n = 10 n = 15 n = 20 n = 30 n = 50 n = 100

0 0 31 0 22 0 17 0 12 0 7 0 4

1 0 45 0 32 0 25 0 17 0 11 0 5

2 3 56 2 40 1 32 1 22 0 14 0 7

3 7 65 4 48 3 38 2 27 1 17 1 9

4 12 74 8 55 6 44 4 31 2 19 1 10

5 19 81 12 62 9 49 6 35 3 22 2 11

6 26 88 16 68 12 54 8 39 5 24 2 13

7 35 93 21 73 15 59 10 42 6 27 3 14

8 44 97 27 79 19 64 12 46 7 29 4 15

9 55 100 32 84 23 68 15 49 9 31 4 16

10 69 100 38 88 27 73 17 53 10 34 5 18

11 45 92 32 77 20 56 12 36 6 19

12 52 96 36 81 23 59 13 38 6 20

13 60 98 41 85 25 63 15 40 7 21

14 68 100 46 88 28 66 16 42 8 22

15 78 100 51 91 31 69 18 45 9 24

16 56 94 34 72 20 47 9 25

17 62 97 37 75 21 49 10 26

18 68 99 41 77 23 51 11 27

19 75 100 44 80 25 53 12 28

20 83 100 47 83 26 55 13 29

21 51 85 28 57 13 30

22 54 88 30 59 14 31

23 58 90 32 61 15 32

24 61 92 34 63 16 34

25 65 94 36 64 17 35

26 69 96 37 66 18 36

27 73 98 39 68 19 37

28 78 99 41 70 19 38

29 83 100 43 72 20 39

30 88 100 45 74 21 40

31 47 75 22 41

32 49 77 23 42

33 51 79 24 43

34 53 80 25 44

35 55 82 26 45

36 58 84 27 46

37 60 85 28 47

38 62 87 28 48

39 64 88 29 49

40 66 90 30 50

41 69 91 31 51

42 71 93 32 52

43 73 94 33 53

44 76 95 34 54

45 78 97 35 55

46 81 98 36 56

47 83 99 37 57

48 86 100 38 58

49 89 100 39 59

50 93 100 40 60
USDA Forest Service RMRS-GTR-126. 2004. 99

Table 4. The ArcSine Percentage transformation. Transformation of binomial percentages in the

margins to angles equals information in degrees.

% 0 1 2 3 4 5 6 78 9

0 0.00 0.57 0.81 0.99 1.15 1.28 1.40 1.52 1.62 1.72

0.1 1.81 1.90 1.99 2.07 2.14 2.22 2.29 2.36 2.43 2.50

0.2 2.56 2.63 2.69 2.75 2.81 2.87 2.92 2.98 3.03 3.09

0.3 3.14 3.19 3.24 3.29 3.34 3.39 3.44 3.49 3.53 3.58

0.4 3.63 3.67 3.72 3.76 3.80 3.85 3.89 3.93 3.97 4.01

0.5 4.05 4.10 4.14 4.17 4.21 4.25 4.29 4.33 4.37 4.41

0.6 4.44 4.48 4.52 4.55 4.59 4.62 4.66 4.70 4.73 4.76

0.7 4.80 4.83 4.87 4.90 4.93 4.97 5.00 5.03 5.07 5.10

0.8 5.13 5.16 5.20 5.23 5.26 5.29 5.32 5.35 5.38 5.41

0.9 5.44 5.47 5.50 5.53 5.56 5.59 5.62 5.65 5.68 5.71

1 5.74 6.02 6.29 6.55 6.80 7.03 7.27 7.49 7.71 7.92

2 8.13 8.33 8.53 8.72 8.91 9.10 9.28 9.46 9.63 9.80

3 9.97 10.14 10.30 10.47 10.63 10.78 10.94 11.09 11.24 11.39

4 11.54 11.68 11.83 11.97 12.11 12.25 12.38 12.52 12.66 12.79

5 12.92 13.05 13.18 13.31 13.44 13.56 13.69 13.81 13.94 14.06

6 14.18 14.30 14.42 14.54 14.65 14.77 14.89 15.00 15.12 15.23

7 15.34 15.45 15.56 15.68 15.79 15.89 16.00 16.11 16.22 16.32

8 16.43 16.54 16.64 16.74 16.85 16.95 17.05 17.16 17.26 17.36

9 17.46 17.56 17.66 17.76 17.85 17.95 18.05 18.15 18.24 18.34

10 18.43 18.53 18.63 18.72 18.81 18.91 19.00 19.09 19.19 19.28

11 19.37 19.46 19.55 19.64 19.73 19.82 19.91 20.00 20.09 20.18

12 20.27 20.36 20.44 20.53 20.62 20.70 20.79 20.88 20.96 21.05

13 21.13 21.22 21.30 21.39 21.47 21.56 21.64 21.72 21.81 21.89

14 21.97 22.06 22.14 22.22 22.30 22.38 22.46 22.54 22.63 22.71

15 22.79 22.87 22.95 23.03 23.11 23.18 23.26 23.34 23.42 23.50

16 23.58 23.66 23.73 23.81 23.89 23.97 24.04 24.12 24.20 24.27

17 24.35 24.43 24.50 24.58 24.65 24.73 24.80 24.88 24.95 25.03

18 25.10 25.18 25.25 25.33 25.40 25.47 25.55 25.62 25.70 25.77

19 25.84 25.91 25.99 26.06 26.13 26.21 26.28 26.35 26.42 26.49

20 26.57 26.64 26.71 26.78 26.85 26.92 26.99 27.06 27.13 27.20

21 27.27 27.35 27.42 27.49 27.56 27.62 27.69 27.76 27.83 27.90

22 27.97 28.04 28.11 28.18 28.25 28.32 28.39 28.45 28.52 28.59

23 28.66 28.73 28.79 28.86 28.93 29.00 29.06 29.13 29.20 29.27

24 29.33 29.40 29.47 29.53 29.60 29.67 29.73 29.80 29.87 29.93

25 30.00 30.07 30.13 30.20 30.26 30.33 30.40 30.46 30.53 30.59

26 30.66 30.72 30.79 30.85 30.92 30.98 31.05 31.11 31.18 31.24

27 31.31 31.37 31.44 31.50 31.56 31.63 31.69 31.76 31.82 31.88

28 31.95 32.01 32.08 32.14 32.20 32.27 32.33 32.39 32.46 32.52

29 32.58 32.65 32.71 32.77 32.83 32.90 32.96 33.02 33.09 33.15

30 33.21 33.27 33.34 33.40 33.46 33.52 33.58 33.65 33.71 33.77

31 33.83 33.90 33.96 34.02 34.08 34.14 34.20 34.27 34.33 34.39

32 34.45 34.51 34.57 34.63 34.70 34.76 34.82 34.88 34.94 35.00

33 35.06 35.12 35.18 35.24 35.30 35.37 35.43 35.49 35.55 35.61

34 35.67 35.73 35.79 35.85 35.91 35.97 36.03 36.09 36.15 36.21

35 36.27 36.33 36.39 36.45 36.51 36.57 36.63 36.69 36.75 36.81

36 36.87 36.93 36.99 37.05 37.11 37.17 37.23 37.29 37.35 37.41

37 37.46 37.52 37.58 37.64 37.70 37.76 37.82 37.88 37.94 38.00

38 38.06 38.12 38.17 38.23 38.29 38.35 38.41 38.47 38.53 38.59

39 38.65 38.70 38.76 38.82 38.88 38.94 39.00 39.06 39.11 39.17

40 39.23 39.29 39.35 39.41 39.47 39.52 39.58 39.64 39.70 39.76

41 39.82 39.87 39.93 39.99 40.05 40.11 40.16 40.22 40.28 40.34

42 40.40 40.45 40.51 40.57 40.63 40.69 40.74 40.80 40.86 40.92

43 40.98 41.03 41.09 41.15 41.21 41.27 41.32 41.38 41.44 41.50

44 41.55 41.61 41.67 41.73 41.78 41.84 41.90 41.96 42.02 42.07

45 42.13 42.19 42.25 42.30 42.36 42.42 42.48 42.53 42.59 42.65

46 42.71 42.76 42.82 42.88 42.94 42.99 43.05 43.11 43.17 43.22

47 43.28 43.34 43.39 43.45 43.51 43.57 43.62 43.68 43.74 43.80

48 43.85 43.91 43.97 44.03 44.08 44.14 44.20 44.26 44.31 44.37

49 44.43 44.48 44.54 44.60 44.66 44.71 44.77 44.83 44.89 44.94

50 45.00 45.06 45.11 45.17 45.23 45.29 45.34 45.40 45.46 45.52

51 45.57 45.63 45.69 45.74 45.80 45.86 45.92 45.97 46.03 46.09

52 46.15 46.20 46.26 46.32 46.38 46.43 46.49 46.55 46.61 46.66

53 46.72 46.78 46.83 46.89 46.95 47.01 47.06 47.12 47.18 47.24
100 USDA Forest Service RMRS-GTR-126. 2004.

54 47.29 47.35 47.41 47.47 47.52 47.58 47.64 47.70 47.75 47.81

55 47.87 47.93 47.98 48.04 48.10 48.16 48.22 48.27 48.33 48.39

56 48.45 48.50 48.56 48.62 48.68 48.73 48.79 48.85 48.91 48.97

57 49.02 49.08 49.14 49.20 49.26 49.31 49.37 49.43 49.49 49.55

58 49.60 49.66 49.72 49.78 49.84 49.89 49.95 50.01 50.07 50.13

59 50.18 50.24 50.30 50.36 50.42 50.48 50.53 50.59 50.65 50.71

60 50.77 50.83 50.89 50.94 51.00 51.06 51.12 51.18 51.24 51.30

61 51.35 51.41 51.47 51.53 51.59 51.65 51.71 51.77 51.83 51.88

62 51.94 52.00 52.06 52.12 52.18 52.24 52.30 52.36 52.42 52.48

63 52.54 52.59 52.65 52.71 52.77 52.83 52.89 52.95 53.01 53.07

64 53.13 53.19 53.25 53.31 53.37 53.43 53.49 53.55 53.61 53.67

65 53.73 53.79 53.85 53.91 53.97 54.03 54.09 54.15 54.21 54.27

66 54.33 54.39 54.45 54.51 54.57 54.63 54.70 54.76 54.82 54.88

67 54.94 55.00 55.06 55.12 55.18 55.24 55.30 55.37 55.43 55.49

68 55.55 55.61 55.67 55.73 55.80 55.86 55.92 55.98 56.04 56.10

69 56.17 56.23 56.29 56.35 56.42 56.48 56.54 56.60 56.66 56.73

70 56.79 56.85 56.91 56.98 57.04 57.10 57.17 57.23 57.29 57.35

71 57.42 57.48 57.54 57.61 57.67 57.73 57.80 57.86 57.92 57.99

72 58.05 58.12 58.18 58.24 58.31 58.37 58.44 58.50 58.56 58.63

73 58.69 58.76 58.82 58.89 58.95 59.02 59.08 59.15 59.21 59.28

74 59.34 59.41 59.47 59.54 59.60 59.67 59.74 59.80 59.87 59.93

75 60.00 60.07 60.13 60.20 60.27 60.33 60.40 60.47 60.53 60.60

76 60.67 60.73 60.80 60.87 60.94 61.00 61.07 61.14 61.21 61.27

77 61.34 61.41 61.48 61.55 61.61 61.68 61.75 61.82 61.89 61.96

78 62.03 62.10 62.17 62.24 62.31 62.38 62.44 62.51 62.58 62.66

79 62.73 62.80 62.87 62.94 63.01 63.08 63.15 63.22 63.29 63.36

80 63.43 63.51 63.58 63.65 63.72 63.79 63.87 63.94 64.01 64.09

81 64.16 64.23 64.30 64.38 64.45 64.53 64.60 64.67 64.75 64.82

82 64.90 64.97 65.05 65.12 65.20 65.27 65.35 65.42 65.50 65.57

83 65.65 65.73 65.80 65.88 65.96 66.03 66.11 66.19 66.27 66.34

84 66.42 66.50 66.58 66.66 66.74 66.82 66.89 66.97 67.05 67.13

85 67.21 67.29 67.37 67.46 67.54 67.62 67.70 67.78 67.86 67.94

86 68.03 68.11 68.19 68.28 68.36 68.44 68.53 68.61 68.70 68.78

87 68.87 68.95 69.04 69.12 69.21 69.30 69.38 69.47 69.56 69.64

88 69.73 69.82 69.91 70.00 70.09 70.18 70.27 70.36 70.45 70.54

89 70.63 70.72 70.81 70.91 71.00 71.09 71.19 71.28 71.37 71.47

90 71.57 71.66 71.76 71.85 71.95 72.05 72.15 72.24 72.34 72.44

91 72.54 72.64 72.74 72.85 72.95 73.05 73.15 73.26 73.36 73.46

92 73.57 73.68 73.78 73.89 74.00 74.11 74.21 74.32 74.44 74.55

93 74.66 74.77 74.88 75.00 75.11 75.23 75.35 75.46 75.58 75.70

94 75.82 75.94 76.06 76.19 76.31 76.44 76.56 76.69 76.82 76.95

95 77.08 77.21 77.34 77.48 77.62 77.75 77.89 78.03 78.17 78.32

96 78.46 78.61 78.76 78.91 79.06 79.22 79.37 79.53 79.70 79.86

97 80.03 80.20 80.37 80.54 80.72 80.90 81.09 81.28 81.47 81.67

98 81.87 82.08 82.29 82.51 82.73 82.97 83.20 83.45 83.71 83.98

99 84.26 84.29 84.32 84.35 84.38 84.41 84.44 84.47 84.50 84.53

99.1 84.56 84.59 84.62 84.65 84.68 84.71 84.74 84.77 84.80 84.84

99.2 84.87 84.90 84.93 84.97 85.00 85.03 85.07 85.10 85.13 85.17

99.3 85.20 85.24 85.27 85.30 85.34 85.38 85.41 85.45 85.48 85.52

99.4 85.56 85.59 85.63 85.67 85.71 85.75 85.79 85.83 85.86 85.90

99.5 85.95 85.99 86.03 86.07 86.11 86.15 86.20 86.24 86.28 86.33

99.6 86.37 86.42 86.47 86.51 86.56 86.61 86.66 86.71 86.76 86.81

99.7 86.86 86.91 86.97 87.02 87.08 87.13 87.19 87.25 87.31 87.37

99.8 87.44 87.50 87.57 87.64 87.71 87.78 87.86 87.93 88.01 88.10

99.9 88.19 88.28 88.38 88.48 88.60 88.72 88.85 89.01 89.19 89.43

Table 4. Continued.

% 0 1 2 3 4 5 6 78 9
USDA Forest Service RMRS-GTR-126. 2004. 101

Table 5. Two-tailed significance levels of the correlation coefficient r.

Significance level

df 0.1 0.05 0.01 0.001

1 0.988 0.997 1.000 1.000

2 0.900 0.950 0.990 0.999

3 0.805 0.878 0.959 0.991

4 0.729 0.811 0.917 0.974

5 0.669 0.754 0.875 0.951

6 0.621 0.707 0.834 0.925

7 0.582 0.666 0.798 0.898

8 0.549 0.632 0.765 0.872

9 0.521 0.602 0.735 0.847

10 0.497 0.576 0.708 0.823

11 0.476 0.553 0.684 0.801

12 0.458 0.532 0.661 0.780

13 0.441 0.514 0.641 0.760

14 0.426 0.497 0.623 0.742

15 0.412 0.482 0.606 0.725

16 0.400 0.468 0.590 0.708

17 0.389 0.456 0.575 0.693

18 0.378 0.444 0.561 0.679

19 0.369 0.433 0.549 0.665

20 0.360 0.423 0.537 0.652

21 0.352 0.413 0.526 0.640

22 0.344 0.404 0.515 0.629

23 0.337 0.396 0.505 0.618

24 0.330 0.388 0.496 0.607

25 0.323 0.381 0.487 0.597

26 0.317 0.374 0.479 0.588

27 0.311 0.367 0.471 0.579

28 0.306 0.361 0.463 0.570

29 0.301 0.355 0.456 0.562

30 0.296 0.349 0.449 0.554

40 0.257 0.304 0.393 0.490

50 0.231 0.273 0.354 0.443

60 0.211 0.250 0.325 0.408

70 0.195 0.232 0.302 0.380

80 0.183 0.217 0.283 0.357

90 0.173 0.205 0.267 0.338

100 0.164 0.195 0.254 0.321

150 0.134 0.159 0.208 0.264

200 0.116 0.138 0.181 0.230

300 0.095 0.113 0.148 0.188

400 0.082 0.098 0.128 0.164

500 0.073 0.088 0.115 0.146
102 USDA Forest Service RMRS-GTR-126. 2004.

Appendix 4. Statistical Analysis Worked Examples

The best way to understand statistics is to work through many examples. Unfortunately, most

examples are computationally intensive. But with the computing power available on the desktop,

there are many good statistical computing packages available. In this Appendix, we show the commands as well as output for many of the examples presented throughout. We choose the R language

because it is freely available and because it provides an extensive array of statistical analysis procedures. The R package can be downloaded from the R project web site at http://www.r-project.org/.

The application of the methods discussed throughout this book is computationally intensive. We

present examples of these analyses in this section; these worked examples may be used as starting

points or templates for other analyses. Included in this Appendix are descriptions of the data sets as

well as a variety of sampling methods with results. The data sets as well as the programs can be

downloaded from RMRS http://www.fs.fed.us/rm/ftcol/index.shtm.

File Name Description

schreuderworkedexamples.xls Original large data set with description and summary of results

surinam.csv Text export of large data set that is read by R

macros.r Miscellaneous function definitions used by the R programs

schreuder.r Calculations used in the text body

schreudertables.r Development of tables used in appendices

workedexamples.r Worked examples using the large data set

Analysis Software

The choice of software running on Windows, Linux, or other platforms is very broad. The commercially available packages such as SAS, SPSS, or S/S-Plus run on the full range of platforms,

from PC to mainframe. Because of the uniqueness and selective availability of each of these packages, we do not attempt to work these examples in terms of these systems. Instead, we illustrate the

analyses with the readily available open source package, R. The R data handling package is extremely robust and powerful, and it offers a wide array of statistical analysis procedures; most

programs written for the widely available commercial packages S and S-Plus will run under the R

system. Links from the R home page will take you to the downloads for the package itself (the

complete installation contains the executables along with complete documentation), as well as

contributed packages and various electronic publications in English, Spanish, French, and German.

Data Sets

The first data set is a small data set of 10 trees that is presented in Table 1 in the text body.

Although contrived, it is an easy data set to analyze by hand.

The second data set consists of a 60 ha stem-mapped population of trees from a tropical forest in

Surinam. These data were used and described by Schreuder and others (1997). The tree heights and

volumes were added by using trees of the same size from FIA data for very different species by

necessity. This population of 6,806 trees has the relative spatial location of the trees and is used to

illustrate the efficiency of several sampling strategies. The population stem map is displayed in

Figure A-2 below.
USDA Forest Service RMRS-GTR-126. 2004. 103

The attributes recorded for each tree were:

Column names Description

Diameter_cm Diameter of tree measured in centimeters

Longitude X-offset of tree measured in 0.1m

Lattitude Y-offset of tree measured in 0.1m

Height_m Height of the tree in m

Volume_cum Volume of the tree in m3

Subplot Subplot identification based on grid labeled with letters for one dimension and

numbers for the other

DBHClass Diameter class

Diameter_in Diameter of tree measured in inches - hard conversion

Height_ft Height of the tree in feet - hard conversion

Volume_cuft Volume of the tree in cubic ft - hard conversion

CC Crown class of (D)ominant or (S)ubdominant derived from height

The tree locations are indicated with circles that are proportional to the diameter of the tree. Ten

trees were selected at random from the population to illustrate simple random sampling; these trees

are indicated with thick circles. Ten 30-m by 30-m plots were also randomly selected to illustrate

cluster sampling; these plots are indicated with thick squares. Stratifed sampling is illustrated by

categorizing the trees as either dominants or subdominants on the basis of height.

Results

The results from analyzing the small data set are tabulated in the main body text. The worked

example calculations can be created by running the R program in the file schreuder.r.

The Surinam data set can be used for realistic exercises in applying the methods discussed in

this book. One of the most useful steps in any analysis is to produce some descriptive statistics,

either tabular or graphical. Some useful graphics include the boxplot. Examples of the boxplot for

the volume (in m3

) for the entire population and the boxplot for the stratified population are shown

in Figure A-3.

5000 10000 15000

1000 2000 3000 4000 5000

Location oftrees w ith circles proportionalto diam eters

Sam ple plots as thick squares and sam ple trees as thick circles

Distance (m) along X

Distance (m) along Y

Figure A-2. Stem map for the Surinam population with sample locations for a SRS and a cluster sample.

Location of trees with circles proportional to diameters

Sample plots as thick squares and sample trees as thick circles
104 USDA Forest Service RMRS-GTR-126. 2004.

Clearly, the mean volume per tree is related to the crown position of the tree, and thus would be

useful in stratifying the population.

The sampling methods discussed in this paper all have strengths and weaknesses. As an exercise, the original population and five samples were used to demonstrate actual calculations performed in the R analysis program. The commands to duplicate these results are in the R command

file named workedexamples.r

Appendix 3, Table 6 summarizes some of the results.

The first row of the table contains the population parameters calculated from all 6,806 trees; this

is the benchmark against which all estimates are judged. The next two lines of the table are for two

contrived samples that picked three small trees and then three large trees. Even though this is a

woefully inadequate sample size, either could actually result from a random trial. Both result in

poor estimates of the population parameter. The nature of estimates resulting from a random draw

do not guarantee reliable estimates. The next trial was a SRS of size 10 selected from this population. The estimate, again sample based, is better, but still not reliable. A random sample of 10

clusters resulted in the measurement of 73 trees and yielded a reliable result. A stratified sample

measured 10 trees, but this time, five from the dominant and five from the subdominant classes.

This particular trial resulted in another reliable estimate of the population parameter.

Figure A-3. Sample boxplots.

Table 6. Summary results for population parameters of Surinam population and results for some

samples.

Method Size Mean Variance

Original population 6806 1.89 8.30

Sample of small trees 3 0.40 0.004

Sample of large trees 3 8.44 10.54

Simple random sample 10 4.15 14.94

Cluster sample 10 plots 73 trees 2.25 3.17

Stratified sample 10 2.35 7.29
USDA Forest Service RMRS-GTR-126. 2004. 105

The output resulting from running the large data set analysis file follow:

***********************************************************

Worked examples:

Surinam data set.

Summary information for the population.

The following variables are available:

 Diameter.cm Longitude Lattitude Height.m Volume.cum SubPolt x2 x3 DBHClass

Diameter.in Height.ft Volume.cuft CC

The basic pop statistics for these variables are:

Statistical summary:

 Diameter.cm Height.m Volume.cum

Mean 41.79327 24.28212 1.886360

Variance 326.25685 71.81564 8.295587

N 6806.00000 6806.00000 6806.000000

The statistics are saved to file: ..//Data/AllSurinamResults.csv

With the distribution summary:

 Diameter.cm Height.m Volume.cum

“Min. : 25.00 “ “Min. : 9.20 “ “Min. : 0.176 “

“1st Qu.: 29.00 “ “1st Qu.:19.50 “ “1st Qu.: 0.569 “

“Median : 36.00 “ “Median :22.60 “ “Median : 0.961 “

“Mean : 41.79 “ “Mean :24.28 “ “Mean : 1.886 “

“3rd Qu.: 48.00 “ “3rd Qu.:25.90 “ “3rd Qu.: 1.903 “

“Max. :165.00 “ “Max. :70.90 “ “Max. :35.978 “

 NA NA NA

***********************************************************

See plot (in another window) for spatial arrangement of trees.

In addition, type:

 identify(surinam$Longitude, surinam$Lattitude,surinam$Diameter.cm)

to interactively click on points to identify diameter.

NOTE: be sure to rt-click-Stop if you do this.

See plot (in another window) example of boxplot for diameter.

***********************************************************

Suppose we select a SRS of size three from the population, say observations:

 4 34 216

The basic sample statistics for this sample are:

Statistical summary:

 Diameter.cm Height.m Volume.cum

Mean 88 42.66667 8.444333

Variance 39 90.20333 10.535722

n 3 3.00000 3.000000

The statistics will be saved to file: ..//Data/SurinamSample1Results.csv

***********************************************************

Suppose we select another SRS of size three from the population, say observations:

 3 814 1278

The basic sample statistics for this sample are:

Statistical summary:

 Diameter.cm Height.m Volume.cum

Mean 28 18.10 0.402333333

Variance 12 50.89 0.004058333

n 3 3.00 3.000000000

The statistics will be saved to file://Data/SurinamSample2Results.csv

***********************************************************

Compare the estimates from these two samples with the actual population parameters.

***********************************************************

Now let us select a true random sample of size 10, say observations:

 654 1008 1040 3038 3587 4529 4564 5470 5628 6030

The basic sample statistics for this sample are:

Statistical summary:

 Diameter.cm Height.m Volume.cum
106 USDA Forest Service RMRS-GTR-126. 2004.

Mean 57.3000 32.9100 4.15060

Variance 683.1222 238.1877 14.93925

n 10.0000 10.0000 10.00000

***********************************************************

How did we do in estimating the population parameters?

***********************************************************

***********************************************************

We could also use plots or clusters to sample.

Suppose we select a true random sample of 10 plots

The plot samples selected are:

 Diameter.cm Height.m Volume.cum IsOnPlot

37 35 25.0 1.196 2

46 85 32.3 3.761 2

75 32 19.2 0.710 8

77 28 20.1 0.544 8

78 64 46.9 5.116 8

79 28 20.1 0.545 8

80 30 28.7 0.734 8

81 28 22.3 0.677 8

82 46 20.1 1.403 8

84 46 14.9 0.927 8

85 26 22.3 0.585 8

86 40 23.5 1.220 8

87 27 21.3 0.648 8

88 25 17.5 0.375 8

89 32 20.1 0.743 8

90 32 26.8 0.936 8

763 57 21.9 2.520 7

767 31 23.8 0.737 7

770 41 29.3 1.547 7

776 47 23.5 1.834 7

2765 42 25.0 1.358 9

2766 25 21.0 0.397 9

2768 57 18.6 1.731 9

2771 32 25.0 0.874 9

2775 35 19.8 0.942 9

2776 27 12.2 0.286 9

2778 39 26.5 1.311 9

2779 42 24.1 1.540 9

2780 42 22.6 1.281 9

2878 44 20.7 1.205 6

2879 29 19.5 0.515 6

2880 33 22.3 0.816 6

2881 36 23.2 1.033 6

2882 71 32.3 2.962 6

2883 82 47.2 7.719 6

2885 25 20.4 0.373 6

2886 29 22.3 0.584 6

2887 60 28.4 3.769 6

2888 26 22.6 0.555 6

2889 25 17.4 0.390 6

2890 33 21.0 0.780 6

2891 40 17.7 1.049 6

2892 72 42.5 4.457 6

3191 26 14.6 0.295 3

3192 65 44.2 5.306 3

3193 29 14.6 0.414 3

3196 32 19.1 0.658 3

4116 28 18.3 0.480 1

4120 89 40.3 7.985 1

4121 29 17.7 0.505 1

4123 52 19.9 1.676 1

4124 25 23.5 0.487 1

4127 26 15.8 0.407 1

4129 72 27.4 5.467 1

4386 43 25.0 1.359 4

4393 27 22.6 0.697 4
USDA Forest Service RMRS-GTR-126. 2004. 107

5863 30 18.9 0.541 5

5864 36 21.3 0.920 5

5866 33 24.7 0.721 5

5867 117 63.6 22.413 5

5868 65 48.9 6.584 5

5872 31 21.0 0.732 5

5874 120 67.6 25.699 5

5875 36 27.7 1.302 5

5876 31 21.4 0.717 5

5877 46 23.2 1.701 5

5878 29 19.5 0.571 5

5879 38 19.8 0.973 5

6118 28 22.3 0.617 10

6119 38 12.8 0.450 10

6120 26 21.6 0.485 10

6123 43 25.0 1.359 10

6125 96 46.3 9.713 10

Cluster Statistics for the volume (CuM):

 [1] [2] [3] [4] [5] [6] [7]

mi 7.000000 2.000000 4.000000 2.000000 12.00000 14.000000 4.0000000

ybari 2.429571 2.478500 1.668250 1.028000 5.23950 1.871929 1.6595000

vari 9.337635 3.289613 5.904263 0.219122 80.47928 4.574484 0.5448577

 [8] [9] [10]

mi 14.000000 9.0000000 5.00000

ybari 1.083071 1.0800000 2.52480

vari 1.422464 0.2456065 16.28365

Thus the estimates of the volume for the total population are:

 Mean: 2.245466 and variance: 3.167461

***********************************************************

***********************************************************

Assume we stratify the population by crown class.

Stratified Surinam data set.

Summary info for the stratified population.

The following variables are available:

 Diameter.cm Longitude Lattitude Height.m Volume.cum SubPolt x2 x3 DBHClass

Diameter.in Height.ft Volume.cuft CC IsSRS IsOnPlot IsStratifiedSample

The basic population statistics for the dominants are:

Statistical summary:

 Diameter.cm Height.m Volume.cum

Mean 45.86105 27.10766 2.400007

Variance 377.56259 71.87200 10.712216

N 4829.00000 4829.00000 4829.000000

The statistics will be saved to file://Data/DomSurinamResults.csv

While the statistics for the suppressed trees are:

Statistical summary:

 Diameter.cm Height.m Volume.cum

Mean 31.85736 17.380475 0.6317299

Variance 61.79857 4.544404 0.1742217

N 1977.00000 1977.000000 1977.0000000

The statistics will be saved to file://Data/SupSurinamResults.csv

***********************************************************

Compare the parameters from these two strata with the single population parameters.

***********************************************************

We can sample from these two strata with the results:

Stratified Stats for the volume (CuM):

 [1] [2]

IDi “D” “S”

nh “5” “5”

ybarh “3.053” “0.6374”

varh “10.1608805” “0.3070853”

Resulting in population estimates of:

 Mean: 2.351319 and variance: 7.29087

See plot (in another window) example of boxplot for diameter, by dominance.
108 USDA Forest Service RMRS-GTR-126. 2004.

Index

A

adaptive 67

aerial 4, 19, 44, 45, 60, 61, 64, 72, 82

arithmetic mean 8, 13

AVHRR 59, 60, 61

B

basal area 4, 5, 8, 18, 26, 29, 30, 33, 40, 41, 44, 46, 47, 66, 72, 84-86

Bayesian 87, 88

beta 83

binomial 10, 21, 53, 57, 91-93, 95

binomial sampling 42, 53

Bitterlich sampling 40

bivariate normal 91

bootstrapping 33, 34, 63, 72, 76

C

census 2, 3

CIR photography 60

cluster 6, 7, 22, 27-29, 31, 33, 44, 45, 53-55, 57, 68, 69, 103, 107

coefficient of variation 16

complete remeasurement 73, 76

complete remeasurement sampling 73, 76

confidence interval 34, 42, 43, 51, 52, 54

consistent 8, 33, 42, 43, 75, 78

continuous 9, 10, 55, 57, 63, 65, 85, 90, 91

correlation coefficient 17, 18, 85, 95

count method 82

D

descriptive 88, 103

design-based 83, 87, 88

discrete 9, 10, 51, 61, 64, 85, 90-93

distributions 10, 57, 90-93

DNA 50, 67

double sampling 36, 70, 71, 72, 85

E

ease of implementation 29

Edge Effect 46

effective 26, 29, 31, 33, 48, 65

enumerative 88, 89

estimation bias 8, 85

F

FIA 1. See also Forest Inventory and Analysis Program

finite 2, 9, 22, 50, 51, 54, 55, 57, 68, 82, 83, 85, 86, 88, 90, 93, 94

fixed 5, 9, 10, 25, 29, 40, 41, 44, 47, 49, 53, 57, 86

forest inventory and analysis 3, 78, 84, 88

Forest Inventory and Analysis Program 1
USDA Forest Service RMRS-GTR-126. 2004. 109

G

gamma 90, 91

geographic information system 59, 64, 66, 82

Global Positioning system 85

H

height 2, 7-10, 13, 14, 16, 18, 19, 40, 48, 53, 59, 68, 85, 103, 105-107

Horvitz-Thompson estimator 22, 25, 29, 34-36, 39, 49

hypergeometric 91-93

hypsometer 48

I

Inference 78, 82, 83, 85, 87

Instruments

Hypsometer 48

Relaskop 40

inventory 1, 3, 4, 48, 49, 53, 59-61, 64, 65, 80, 82-84

J

jackknife 36

L

Landsat 59, 60

line intercept 45, 49

M

mapping 64

mean 3, 5, 7-29, 32-39, 43, 53-58, 68, 70, 72, 85-87, 90-94, 104-107

mean-of-ratios 36

median 12, 13, 85, 105

methodology 2

Microwave 59, 61

mirage method 46, 47

missing data 5, 47

mode 12, 13, 32, 85

model-based 6, 83, 87-89

monitoring 1, 2, 4, 59, 61, 64, 66, 73, 82-84

multinomial 91, 93

multiphase 68, 69, 72

multiphase sampling 69

multiplicity sampling 67

multistage 68, 69, 72, 82, 86

multistage sampling 68, 69, 72, 86

multivariate normal 91

N

negative binomial 57

normal 2, 10, 21, 33, 43, 52, 54, 57, 90-92, 94

O

optimal 27, 33, 43, 71
110 USDA Forest Service RMRS-GTR-126. 2004.

P

parameter 7, 9, 12-14, 16, 18, 25, 39, 42, 43, 85, 86, 88, 91, 92, 104

permanent 4, 59, 73

Pitfalls 79

point 4, 30, 40, 44, 46, 48, 59, 73, 76, 79, 80, 84-86

Poisson 10, 21, 40-42, 57, 72, 84, 91-93

precision 8, 33, 34, 43, 45, 52, 66, 73, 85, 86, 88

prior 44, 53, 60, 78, 87

probabilistic 2, 4, 6, 22, 24, 25, 33, 67, 86-88

probabilistic sampling 1

R

random sample 6, 13, 24, 31, 39, 42, 44, 51, 53, 62, 68, 72, 83, 94, 104-106

randomization 6, 77, 86, 88

ratio-of-means 34, 36, 37, 39, 55-57

regression 16-18, 34-39, 45, 65, 69, 70-77, 81-85

regression estimators 36, 37, 71

Relaskop 48

relative 10, 16, 22, 31, 34, 41, 45, 51, 53, 61, 65, 73, 86, 102

remote sensing 4, 27, 45, 50, 59, 60-66, 70-72, 81-84

S

sample plots 4, 5, 18, 37, 47, 53, 63, 73

sample size 8, 10, 13, 14, 17, 22-25, 29, 39, 40-43, 51-53, 64, 67, 70, 71, 85, 104

sample survey 2, 7, 16, 20, 42, 83, 84, 86, 87

sampling 1-10, 14-19, 22-76, 79-93, 102-104

screening 67, 78

selection 6-8, 12, 22-34, 39, 40-42, 44, 49, 50, 53, 72, 77, 79, 80, 83, 84, 86, 88

sequential sampling 67

simple linear regression 39

simple random 6, 13, 16, 22, 25, 27, 30, 33-35, 42, 51, 54, 57, 62, 83, 86, 94, 103

simple random sampling (SRS) 22, 30, 51

size of 5, 16, 17, 24, 37, 39, 40, 44, 47, 57, 59, 60, 69

small area estimation 3, 65, 66, 83

snowball sampling 67

SPOT 60, 61

standard deviation 12-16, 19, 35, 36, 86

standard error 14, 15, 19, 35, 36, 37, 43, 51, 54, 55-58, 86, 91

statistical 1, 10, 21, 33, 42, 50, 65, 77, 78, 81-84, 87-90, 102

statistical inference 83, 86, 87, 90

stratification 26, 27, 31, 34, 39, 69, 70, 72, 84

stratified 7, 22, 26, 27, 29, 30, 31, 33, 35, 39, 70-72, 79, 103, 104, 107

strip 44, 49, 86

suggestions 1, 80, 83

survey sampling 2, 3, 82-84, 90, 91, 93

systematic 8, 22, 32, 33, 42, 44, 53, 77, 79, 85, 87

systematic sampling 31, 32

T

t-distribution 91, 94

transects 46
USDA Forest Service RMRS-GTR-126. 2004. 111

U

unequal probability 22, 25, 26, 31, 33, 34, 44, 55

unequal probability sampling 25, 86

V

variable probability 7

variable radius plots 5

variance 16, 19, 20, 86, 105-107

variance estimation 29, 72, 75

vertical 4, 48, 61

volume 28, 103, 105-107

VRP sampling 40, 41, 44, 46, 72, 79-81

W

walkthrough 47, 48, 82

weighted regression 35

wildlife sampling 50
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