Model Assisted Survey Sampling

Carl-Erik Sarndal Jan Wretman

Bengt Swensson

Model Assisted Survey Sampling

Springer

Contents

Preface

v

PARTI Principles of Estimation for Finite Populations and Important Sampling Designs CHAPTER 1 Survey Sampling in Theory and Practice 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10

Surveys in Society Skeleton Outline of a Survey Probability Sampling Sampling Frame Area Frames and Similar Devices Target Population and Frame Population Survey Operations and Associated Sources of Error Planning a Survey and the Need for Total Survey Design Total Survey Design The Role of Statistical Theory in Survey Sampling Exercises

3 3 4 8 9 12 13 14 17 19 20 22

CHAPTER 2 Basic Ideas in Estimation from Probability Samples

24

2.1 2.2 2.3 2.4 2.5 2.6 2.7

24 24 27 30 33 36 38

Introduction Population, Sample, and Sample Selection Sampling Design Inclusion Probabilities The Notion of a Statistic The Sample Membership Indicators Estimators and Their Basic Statistical Properties

ix

x 2.8 2.9 2.10 2.11

Contents The n Estimator and Its Properties With-Replacement Sampling The Design Effect Confidence Intervals Exercises

CHAPTER 3 Unbiased Estimation for Element Sampling Designs 3.1 Introduction 3.2 Bernoulli Sampling 3.3 Simple Random Sampling 3.3.1 Simple Random Sampling without Replacement 3.3.2 Simple Random Sampling with Replacement 3.4 Systematic Sampling 3.4.1 Definitions and Main Result 3.4.2 Controlling the Sample Size 3.4.3 The Efficiency of Systematic Sampling 3.4.4 Estimating the Variance 3.5 Poisson Sampling 3.6 Probability Proportional-to-Size Sampling 3.6.1 Introduction 3.6.2 nps Sampling 3.6.3 pps Sampling 3.6.4 Selection from Randomly Formed Groups 3.7 Stratified Sampling 3.7.1 Introduction 3.7.2 Notation, Definitions, and Estimation 3.7.3 Optimum Sample Allocation 3.7.4 Alternative Allocations under STSI Sampling 3.8 Sampling without Replacement versus Sampling with Replacement 3.8.1 Alternative Estimators for Simple Random Sampling with Replacement 3.8.2 The Design Effect of Simple Random Sampling with Replacement Exercises CHAPTER 4 Unbiased Estimation for Cluster Sampling and Sampling in Two ' o r More Stages 4.1 Introduction 4.2 Single-Stage Cluster Sampling 4.2.1 Introduction 4.2.2 Simple Random Cluster Sampling 4.3 Two-Stage Sampling 4.3.1 Introduction 4.3.2 Two-Stage Element Sampling 4.4 Multistage Sampling 4.4.1 Introduction and a General Result 4.4.2 Three-Stage Element Sampling 4.5 With-Replacement Sampling of PSUs

42 48 53 55 58

61 61 62 66 66 72 73 73 76 78 83 85 87 87 90 97 99 100 100 101 104 106 110 110 112 114

124 124 126 126 129 133 133 135 144 144 146 150

Contents

xi

4.6 Comparing Simplified Variance Estimators in Multistage Sampling Exercises

153 154

CHAPTER 5

Introduction to More Complex Estimation Problems

162

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10

162 163 166 169 172 176 181 184 186 190 190 192 197 205 207

Introduction The Effect of Bias on Confidence Statements Consistency and Asymptotic Unbiasedness n Estimators for Several Variables of Study The Taylor Linearization Technique for Variance Estimation Estimation of a Ratio Estimation of a Population Mean Estimation of a Domain Mean Estimation of Variances and Covariances in a Finite Population Estimation of Regression Coefficients 5.10.1 The Parameters of Interest 5.10.2 Estimation of the Regression Coefficients 5.11 Estimation of a Population Median 5.12 Demonstration of Result 5.10.1 ^ Exercises

.

PART II Estimation through Linear Modeling, Using Auxiliary Variables CHAPTER 6 The Regression Estimator 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8

Introduction Auxiliary Variables The Difference Estimator Introducing the Regression Estimator Alternative Expressions for the Regression Estimator The Variance of the Regression Estimator Comments on the Role of the Model Optimal Coefficients for the Difference Estimator Exercises

219 219 219 221 225 230 234 238 239 242

CHAPTER 7 Regression Estimators for Element Sampling Designs

245

7.1 Introduction 7.2 Preliminary Considerations 7.3 The Common Ratio Model and the Ratio Estimator 7.3.1 The Ratio Estimator under SI Sampling 7.3.2 The Ratio Estimator under Other Designs 7.3.3 Optimal Sampling Design for the n Weighted Ratio Estimator 7.3.4 Alternative Ratio Models 7.4 The Common Mean Model 7.5 Models Involving Population Groups 7.6 The Group Mean Model and the Poststratified Estimator 7.7 The Group Ratio Model and the Separate Ratio Estimator

245 245 247 249 252 253 255 258 260 264 269

Contents 7.8 7.9

7.10

7.11 7.12 7.13

Simple Regression Models and Simple Regression Estimators Estimators Based on Multiple Regression Models 7.9.1 Multiple Regression Models 7.9.2 Analysis of Variance Models Conditional Confidence Intervals 7.10.1 Conditional Analysis for BE Sampling 7.10.2 Conditional Analysis for the Poststratification Estimator Regression Estimators for Variable-Size Sampling Designs A Class of Regression Estimators Regression Estimation of a Ratio of Population Totals Exercises

272 275 276 281 283 284 287 289 291 294 297

CHAPTER 8

Regression Estimators for Cluster Sampling and Two-Stage Sampling 303 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12

Introduction ^ The Nature of the Auxiliary Information When Clusters of Elements Are Selected Comments on Variance and Variance Estimation in Two-Stage Sampling Regression Estimators Arising Out of Modeling at the Cluster Level The Common Ratio Model for Cluster Totals Estimation of the Population Mean When Clusters Are Sampled Design Effects for Single-Stage Cluster Sampling Stratified Clusters and Poststratified Clusters Regression Estimators Arising Out of Modeling at the Element Level Ratio Models for Elements The Group Ratio Model for Elements The Ratio Model Applied within a Single PSU Exercises

303 304 307 308 312 314 315 319 322 327 330 332 333

PART III Further Questions in Design and Analysis of Surveys CHAPTER 9 Two-Phase Sampling 9.1 ' 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9

Introduction Notation and Choice of Estimator The 7i* Estimator , Two-Phase Sampling for Stratification Auxiliary Variables for Selection in Two Phases Difference Estimators Regression Estimators for Two-Phase Sampling Stratified Bernoulli Sampling in Phase Two Sampling on Two Occasions 9.9.1 Estimating the Current Total 9.9.2 Estimating the Previous Total 9.9.3 Estimating the Absolute Change and the Sum of the Totals Exercises

343 343 345 347 350 354 356 359 366 368 370 376 377 379

Contents

xiii

CHAPTER 10 Estimation for D o m a i n s

386

10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9

Introduction The Background for Domain Estimation The Basic Estimation Methods for Domains Conditioning on the Domain Sample Size Regression Estimators for Domains A Ratio Model for Each Domain Group Models for Domains Problems Arising for Small Domains; Synthetic Estimation More on the Comparison of Two Domains Exercises

CHAPTER 11 Variance Estimation

^

386 387 390 396 397 403 405 408 412 413

418

11.1 Introduction 11.2 A Simplified Variance Estimator under Sampling without Replacement 11.3 The Random Groups Technique 11.3.1 Independent Random Groups 11.3.2 Dependent Random Groups 11.4 Balanced Half-Samples 11.5 The Jackknife Technique 11.6 The Bootstrap 11.7 Concluding Remarks Exercises

418 421 423 423 426 430 437 442 444 445

CHAPTER 12 Searching for Optimal Sampling Designs

447

12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8

Introduction Model-Based Optimal Design for the General Regression Estimator Model-Based Optimal Design for the Group Mean Model Model-Based Stratified Sampling Applications of Model-Based Stratification Other Approaches to Efficient Stratification Allocation Problems in Stratified Random Sampling Allocation Problems in Two-Stage Sampling 12.8.1 The n Estimator of the Population Total 12.8.2 Estimation of the Population Mean 12.9 Allocation in Two-Phase Sampling for Stratification 12.10 A Further Comment on Mathematical Programming 12.11 Sampling Design and Experimental Design Exercises

447 448 455 456 461 462 465 471 471 475 478 480 481 481

CHAPTER 13 Further Statistical Techniques for Survey D a t a

485

13.1 Introduction 13.2 Finite Population Parameters in Multivariate Regression and Correlation Analysis

485 486

xiv

Contents

13.3 The Effect of Sampling Design on a Statistical Analysis 13.4 Variances and Estimated Variances for Complex Analyses 13.5 Analysis of Categorical Data for Finite Populations 13.5.1 Test of Homogeneity for Two Populations 13.5.2 Testing Homogeneity for More than Two Finite Populations 13.5.3 Discussion of Categorical Data Tests for Finite Populations 13.6 Types of Inference When a Finite Population Is Sampled Exercises

491 494 500 500 507 510 513 520

PART IV

A Broader View of Errors in Surveys CHAPTER 14 Nonsampling Errors and Extensions of Probability Sampling Theory

525

14.1 Introduction 14.2 Historic Notes: The Evolution of the Probability Sampling Approach 14.3 Measurable Sampling Designs 14.4 Some Nonprobability Sampling Methods 14.5 Model-Based Inference from Survey Samples 14.6 Imperfections in the Survey Operations 14.6.1 Ideal Conditions for the Probability Sampling Approach 14.6.2 Extension of the Probability Sampling Approach 14.7 Sampling Frames 14.7.1 Frame Imperfections 14.7.2 Estimation in the Presence of Frame Imperfections 14.7.3 Multiple Frames 14.7.4 Frame Construction and Maintenance 14.8 Measurement and Data Collection 14.9 Data Processing 14.10 Nonresponse Exercises

525 525 527 529 533 537 537 538 540 540 543 545 545 546 548 551 553

CHAPTER 15 Nonresponse

556

15.1 Introduction 15.2 Characteristics of Nonresponse 15.2.1 Definition of Nonresponse 15.2.2 Response Sets 15.2.3 Lack of Unbiased Estimators 15.3 Measuring Nonresponse 15.4 Dealing with Nonresponse 15.4.1 Planning of the Survey 15.4.2 Callbacks and Follow-Ups 15.4.3 Subsampling of Nonrespondents 15.4.4 Randomized Response 15.5 Perspectives on Nonresponse 15.6 Estimation in the Presence of Unit Nonresponse 15.6.1 Response Modeling

556 556 556 557 558 559 563 564 564 566 570 573 575 575

•

Contents

xv

15.6.2 A Useful Response Model 15.6.3 Estimators That Use Weighting Only 15.6.4 Estimators That Use Weighting as Well as Auxiliary Variables 15.7 Imputation Exercises

577 580 583 589 595

CHAPTER 16 Measurement Errors

601

16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 16.10 16.11

Introduction On the Nature of Measurement Errors The Simple Measurement Model Decomposition of the Mean Square Error . ^ The Risk of Underestimating the Total Variance Repeated Measurements as a Tool in Variance Estimation Measurement Models Taking Interviewer Effects into Account Deterministic Assignment of Interviewers Random Assignment of Interviewers to Groups Interpenetrating Subsamples A Measurement Model with Sample-Dependent Moments Exercises

CHAPTER 17 Quality Declarations for Survey D a t a 17.1 Introduction 17.2 Policies Concerning Information on Data Quality 17.3 Statistics Canada's Policy on Informing Users of Data Quality and Methodology Exercise

601 602 605 608 612 614 617 618 622 627 630 634

637 637 638 641 648

APPENDIX A Principles of N o t a t i o n

649

APPENDIX B The MU284 Population

652

APPENDIX C The Clustered M U 2 8 4 Population

660

APPENDIX D

The CO 124 Population

662

References Answers to Selected Exercises Author Index

666 680 684

Subject Index

688