# Sampling, 3rd Edition

# Sampling, 3rd Edition

ISBN: 978-1-118-16293-4

Feb 2012

472 pages

## Description

**Praise for the Second Edition**

"This book has never had a competitor. It is the only book that takes a broad approach to sampling . . . any good personal statistics library should include a copy of this book."

—*Technometrics*

"Well-written . . . an excellent book on an important subject. Highly recommended."

—*Choice*

"An ideal reference for scientific researchers and other professionals who use sampling."

—*Zentralblatt Math*

**Features new developments in the field combined with all aspects of obtaining, interpreting, and using sample data**

*Sampling* provides an up-to-date treatment of both classical and modern sampling design and estimation methods, along with sampling methods for rare, clustered, and hard-to-detect populations. This Third Edition retains the general organization of the two previous editions, but incorporates extensive new material—sections, exercises, and examples—throughout. Inside, readers will find all-new approaches to explain the various techniques in the book; new figures to assist in better visualizing and comprehending underlying concepts such as the different sampling strategies; computing notes for sample selection, calculation of estimates, and simulations; and more.

Organized into six sections, the book covers basic sampling, from simple random to unequal probability sampling; the use of auxiliary data with ratio and regression estimation; sufficient data, model, and design in practical sampling; useful designs such as stratified, cluster and systematic, multistage, double and network sampling; detectability methods for elusive populations; spatial sampling; and adaptive sampling designs.

Featuring a broad range of topics, *Sampling*, Third Edition serves as a valuable reference on useful sampling and estimation methods for researchers in various fields of study, including biostatistics, ecology, and the health sciences. The book is also ideal for courses on statistical sampling at the upper-undergraduate and graduate levels.

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**Preface xv**

**Preface to the Second Edition xvii**

**Preface to the First Edition xix**

**1 Introduction 1**

1.1 Basic Ideas of Sampling and Estimation, 2

1.2 Sampling Units, 4

1.3 Sampling and Nonsampling Errors, 5

1.4 Models in Sampling, 5

1.5 Adaptive and Nonadaptive Designs, 6

1.6 Some Sampling History, 7

**PART I BASIC SAMPLING 9**

**2 Simple Random Sampling 11**

2.1 Selecting a Simple Random Sample, 11

2.2 Estimating the Population Mean, 13

2.3 Estimating the Population Total, 16

2.4 Some Underlying Ideas, 17

2.5 Random Sampling with Replacement, 19

2.6 Derivations for Random Sampling, 20

2.7 Model-Based Approach to Sampling, 22

2.8 Computing Notes, 26

Entering Data in R, 26

Sample Estimates, 27

Simulation, 28

Further Comments on the Use of Simulation, 32

Exercises, 35

**3 Confidence Intervals 39**

3.1 Confidence Interval for the Population Mean or Total, 39

3.2 Finite-Population Central Limit Theorem, 41

3.3 Sampling Distributions, 43

3.4 Computing Notes, 44

Confidence Interval Computation, 44

Simulations Illustrating the Approximate Normality of a Sampling Distribution with Small *n* and *N*, 45

Daily Precipitation Data, 46

Exercises, 50

**4 Sample Size 53**

4.1 Sample Size for Estimating a Population Mean, 54

4.2 Sample Size for Estimating a Population Total, 54

4.3 Sample Size for Relative Precision, 55

Exercises, 56

**5 Estimating Proportions, Ratios, and Subpopulation Means 57**

5.1 Estimating a Population Proportion, 58

5.2 Confidence Interval for a Proportion, 58

5.3 Sample Size for Estimating a Proportion, 59

5.4 Sample Size for Estimating Several Proportions Simultaneously, 60

5.5 Estimating a Ratio, 62

5.6 Estimating a Mean, Total, or Proportion of a Subpopulation, 62

Estimating a Subpopulation Mean, 63

Estimating a Proportion for a Subpopulation, 64

Estimating a Subpopulation Total, 64

Exercises, 65

**6 Unequal Probability Sampling 67**

6.1 Sampling with Replacement: The Hansen–Hurwitz Estimator, 67

6.2 Any Design: The Horvitz–Thompson Estimator, 69

6.3 Generalized Unequal-Probability Estimator, 72

6.4 Small Population Example, 73

6.5 Derivations and Comments, 75

6.6 Computing Notes, 78

Writing an R Function to Simulate a Sampling Strategy, 82

Comparing Sampling Strategies, 84

Exercises, 88

**PART II MAKING THE BEST USE OF SURVEY DATA 91**

**7 Auxiliary Data and Ratio Estimation 93**

7.1 Ratio Estimator, 94

7.2 Small Population Illustrating Bias, 97

7.3 Derivations and Approximations for the Ratio Estimator, 99

7.4 Finite-Population Central Limit Theorem for the Ratio Estimator, 101

7.5 Ratio Estimation with Unequal Probability Designs, 102

7.6 Models in Ratio Estimation, 105

Types of Estimators for a Ratio, 109

7.7 Design Implications of Ratio Models, 109

7.8 Computing Notes, 110

Exercises, 112

**8 Regression Estimation 115**

8.1 Linear Regression Estimator, 116

8.2 Regression Estimation with Unequal Probability Designs, 118

8.3 Regression Model, 119

8.4 Multiple Regression Models, 120

8.5 Design Implications of Regression Models, 123

Exercises, 124

**9 The Sufficient Statistic in Sampling 125**

9.1 The Set of Distinct, Labeled Observations, 125

9.2 Estimation in Random Sampling with Replacement, 126

9.3 Estimation in Probability-Proportional-to-Size Sampling, 127

9.4 Comments on the Improved Estimates, 128

**10 Design and Model 131**

10.1 Uses of Design and Model in Sampling, 131

10.2 Connections between the Design and Model Approaches, 132

10.3 Some Comments, 134

10.4 Likelihood Function in Sampling, 135

**PART III SOME USEFUL DESIGNS 139**

**11 Stratified Sampling 141**

11.1 Estimating the Population Total, 142

With Any Stratified Design, 142

With Stratified Random Sampling, 143

11.2 Estimating the Population Mean, 144

With Any Stratified Design, 144

With Stratified Random Sampling, 144

11.3 Confidence Intervals, 145

11.4 The Stratification Principle, 146

11.5 Allocation in Stratified Random Sampling, 146

11.6 Poststratification, 148

11.7 Population Model for a Stratified Population, 149

11.8 Derivations for Stratified Sampling, 149

Optimum Allocation, 149

Poststratification Variance, 150

11.9 Computing Notes, 151

Exercises, 155

**12 Cluster and Systematic Sampling 157**

12.1 Primary Units Selected by Simple Random Sampling, 159

Unbiased Estimator, 159

Ratio Estimator, 160

12.2 Primary Units Selected with Probabilities Proportional to Size, 161

Hansen–Hurwitz (PPS) Estimator, 161

Horvitz–Thompson Estimator, 161

12.3 The Basic Principle, 162

12.4 Single Systematic Sample, 162

12.5 Variance and Cost in Cluster and Systematic Sampling, 163

12.6 Computing Notes, 166

Exercises, 169

**13 Multistage Designs 171**

13.1 Simple Random Sampling at Each Stage, 173

Unbiased Estimator, 173

Ratio Estimator, 175

13.2 Primary Units Selected with Probability Proportional to Size, 176

13.3 Any Multistage Design with Replacement, 177

13.4 Cost and Sample Sizes, 177

13.5 Derivations for Multistage Designs, 179

Unbiased Estimator, 179

Ratio Estimator, 181

Probability-Proportional-to-Size Sampling, 181

More Than Two Stages, 181

Exercises, 182

**14 Double or Two-Phase Sampling 183**

14.1 Ratio Estimation with Double Sampling, 184

14.2 Allocation in Double Sampling for Ratio Estimation, 186

14.3 Double Sampling for Stratification, 186

14.4 Derivations for Double Sampling, 188

Approximate Mean and Variance: Ratio Estimation, 188

Optimum Allocation for Ratio Estimation, 189

Expected Value and Variance: Stratification, 189

14.5 Nonsampling Errors and Double Sampling, 190

Nonresponse, Selection Bias, or Volunteer Bias, 191

Double Sampling to Adjust for Nonresponse: Callbacks, 192

Response Modeling and Nonresponse Adjustments, 193

14.6 Computing Notes, 195

Exercises, 197

**PART IV METHODS FOR ELUSIVE AND HARD-TO-DETECT POPULATIONS 199**

**15 Network Sampling and Link-Tracing Designs 201**

15.1 Estimation of the Population Total or Mean, 202

Multiplicity Estimator, 202

Horvitz–Thompson Estimator, 204

15.2 Derivations and Comments, 207

15.3 Stratification in Network Sampling, 208

15.4 Other Link-Tracing Designs, 210

15.5 Computing Notes, 212

Exercises, 213

**16 Detectability and Sampling 215**

16.1 Constant Detectability over a Region, 215

16.2 Estimating Detectability, 217

16.3 Effect of Estimated Detectability, 218

16.4 Detectability with Simple Random Sampling, 219

16.5 Estimated Detectability and Simple Random Sampling, 220

16.6 Sampling with Replacement, 222

16.7 Derivations, 222

16.8 Unequal Probability Sampling of Groups with Unequal Detection Probabilities, 224

16.9 Derivations, 225

Exercises, 227

**17 Line and Point Transects 229**

17.1 Density Estimation Methods for Line Transects, 230

17.2 Narrow-Strip Method, 230

17.3 Smooth-by-Eye Method, 233

17.4 Parametric Methods, 234

17.5 Nonparametric Methods, 237

Estimating *f* (0) by the Kernel Method, 237

Fourier Series Method, 239

17.6 Designs for Selecting Transects, 240

17.7 Random Sample of Transects, 240

Unbiased Estimator, 241

Ratio Estimator, 243

17.8 Systematic Selection of Transects, 244

17.9 Selection with Probability Proportional to Length, 244

17.10 Note on Estimation of Variance for the Kernel Method, 246

17.11 Some Underlying Ideas about Line Transects, 247

Line Transects and Detectability Functions, 247

Single Transect, 249

Average Detectability, 249

Random Transect, 250

Average Detectability and Effective Area, 251

Effect of Estimating Detectability, 252

Probability Density Function of an Observed Distance, 253

17.12 Detectability Imperfect on the Line or Dependent on Size, 255

17.13 Estimation Using Individual Detectabilities, 255

Estimation of Individual Detectabilities, 256

17.14 Detectability Functions other than Line Transects, 257

17.15 Variable Circular Plots or Point Transects, 259

Exercise, 260

**18 Capture–Recapture Sampling 263**

18.1 Single Recapture, 264

18.2 Models for Simple Capture–Recapture, 266

18.3 Sampling Design in Capture–Recapture: Ratio Variance Estimator, 267

Random Sampling with Replacement of Detectability Units, 269

Random Sampling without Replacement, 270

18.4 Estimating Detectability with Capture–Recapture Methods, 271

18.5 Multiple Releases, 272

18.6 More Elaborate Models, 273

Exercise, 273

**19 Line-Intercept Sampling 275**

19.1 Random Sample of Lines: Fixed Direction, 275

19.2 Lines of Random Position and Direction, 280

Exercises, 282

**PART V SPATIAL SAMPLING 283**

**20 Spatial Prediction or Kriging 285**

20.1 Spatial Covariance Function, 286

20.2 Linear Prediction (Kriging), 286

20.3 Variogram, 289

20.4 Predicting the Value over a Region, 291

20.5 Derivations and Comments, 292

20.6 Computing Notes, 296

Exercise, 299

**21 Spatial Designs 301**

21.1 Design for Local Prediction, 302

21.2 Design for Prediction of Mean of Region, 302

**22 Plot Shapes and Observational Methods 305**

22.1 Observations from Plots, 305

22.2 Observations from Detectability Units, 307

22.3 Comparisons of Plot Shapes and Detectability Methods, 308

**PART VI ADAPTIVE SAMPLING 313**

**23 Adaptive Sampling Designs 315**

23.1 Adaptive and Conventional Designs and Estimators, 315

23.2 Brief Survey of Adaptive Sampling, 316

**24 Adaptive Cluster Sampling 319**

24.1 Designs, 321

Initial Simple Random Sample without Replacement, 322

Initial Random Sample with Replacement, 323

24.2 Estimators, 323

Initial Sample Mean, 323

Estimation Using Draw-by-Draw Intersections, 323

Estimation Using Initial Intersection Probabilities, 325

24.3 When Adaptive Cluster Sampling Is Better than Simple Random Sampling, 327

24.4 Expected Sample Size, Cost, and Yield, 328

24.5 Comparative Efficiencies of Adaptive and Conventional

Sampling, 328

24.6 Further Improvement of Estimators, 330

24.7 Derivations, 333

24.8 Data for Examples and Figures, 336

Exercises, 337

**25 Systematic and Strip Adaptive Cluster Sampling 339**

25.1 Designs, 341

25.2 Estimators, 343

Initial Sample Mean, 343

Estimator Based on Partial Selection Probabilities, 344

Estimator Based on Partial Inclusion Probabilities, 345

25.3 Calculations for Adaptive Cluster Sampling Strategies, 347

25.4 Comparisons with Conventional Systematic and Cluster Sampling, 349

25.5 Derivations, 350

25.6 Example Data, 352

Exercises, 352

**26 Stratified Adaptive Cluster Sampling 353**

26.1 Designs, 353

26.2 Estimators, 356

Estimators Using Expected Numbers of Initial Intersections, 357

Estimator Using Initial Intersection Probabilities, 359

26.3 Comparisons with Conventional Stratified Sampling, 362

26.4 Further Improvement of Estimators, 364

26.5 Example Data, 367

Exercises, 367

**Answers to Selected Exercises 369**

**References 375**

**Author Index 395**

**Subject Index 399**