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Sampling Techniques Oswaldo Cruz Foundation
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Copyright 1977 by John Wiley Sons Inc, All nghts reserved Published simultaneously in Canada. Reproducuon or tran lation of any pan of this work beyond that. permined by Sections 107 or 108 of the 1976 United Slates Copy. right Act wuhout the perm1 sion of the copyright owner 1s unlaw. ful Requests for permission or funher mformat1on should be. addressed to the Pcrm1ss1ons Depanmcnt John Wiley Sons Inc. Library of Congress Cataloging In Puhllcatlon Data. Cochran William Gemmell 1909, Sampling techniques, Wiley series in probability and mathematical statistic. Includes bibliographical references and index, 1 Sampling Statistics I Title. QA276 6 C6 1977 001 4 222 77 728, ISBN 0 471 16240 X. Printed in the United States of America, 40 39 38 37 36.
As did the previous editions this textbook presents a comprehensive account of. sampling theory as it has been developed for use in sample surveys It cantains. illustrations to show how the theory is applied in practice and exercises to be. worked by the student The book will be useful both as a text for a course on. sample surveys in which the major emphasis is on theory and for individual. reading by the student, The minimum mathematical equipment necessary to follow the great bulk of. the material is a familiarity with algebra especially relatively complicated algeb. raic expressions plus a knowledge of probability for finite sample spaces includ. ing combinatorial probabilities The book presupposes an introductory statistics. course that covers means and standard deviations the normal binomial. hypergeometric and multinomial distributions the central limit theorem linear. regression and the simpler types of analyses of variance Since much of classical. sample survey theory deals with the distributions of estimators over the set of. randomizations provided by the sampling plan some knowledge of nonparamet. ric methods is helpful, The topics in this edition are presented in essentially the same order as in earlier. editions New sections have been included or sections rewritten primarily for one. of three reasons 1 to present introductions to topics sampling plans or methods. of estimation relatively new in the field 2 to cover further work done during the. last 15 years on older methods intended either to improve them or to learn more. about the performance of rival methods and 3 to shorten clarify or simplify. proofs given in previous editions, New topics in this edition include the approximate methods developed for the. difficult problem of attaching standard errors or confidence limits to nonlinear. estimates made from the results of surveys with complex plans These methods. will be more and more needed as statistical analyses e g regressions are. performed on the results For surveys containing sensitive questions that some. respondents are unlikely to be willing to answer truthfully a new device is to. present the respondent with either the sensitive question or an innocuous ques. tion the specific choice made by randomization is unknown to the interviewer. In some sampling problems it may seem economically attractive or essential in. countries without full sampling resources to use two overlapping lists or frames. as they are called to cover the complete population The method of double. sampling has been extended to cases where the objective is to compare the means. Vlll PREFACE, of a number of subgroups within the population There has been interesting. work on the attractive properties that the ratio and regression estimators have if it. can be assumed that the finite population is itself a random sample from an infinite. superpopulation in which a mathematical model appropriate to the ratio or. regression estimator holds This kind of assumption is not new I noticed. recently that Laplace used it around 1800 in a sampling problem but it clarifies. the relation between sample survey theory and standard statistical theory. An example of further work on topics included in previous editions is Chapter. 9A which has been written partly from materia 1 previously in Chapter 9 this was. done mainly to give a more adequate account of what seem to me the principal. methods produced for sampling with unequal probabilities without replacement. These include the similar methods given independently by Brewer J N K lh o. and Durbin Murthy s method the Rao Hartley Cochran method and Madow s. method related to systematic sampling with comparisons of the performances of. the methods on natural populations New studies have been done of the sizes of. components of errors of measurement in surveys by repeat measurements by. different interviewers by interpenetrating subsamples and by a combination of. the two approaches For the ratio estimator data from natural populations have. been used to appraise the small sample biases in the standard large sample. formulas for the variance and the estimated variance Attempts have also been. made to create less biased variants of the ratio estimator itself and of the formula. for estimating its sampling variance In stratified sampling there has been addi. tional work on allocating sample sizes to strata when more than one item is of. importance and on estimating sample errors when only one unit is to be selected. per stratum Some new systematic sampling methods for handling populations. having linear trends are also of interest, Alva L Finkner and Emil H Jebe prepared a large part of the lecture notes.
from which the first edition of this book was written Some investigations that. provided background material were supported by the Office of Naval Research. Navy Department From discussions of recent developments in sampling or. suggestions about this edition I have been greatly helped by Tore Dalenius. David J Finney Daniel G Horvitz Leslie Kish P S R Sambasiva Rao Martin. Sandelius Joseph Sedransk Amode R Sen and especially Jon N K Rao whose. painstaking reading of the new and revised sections of this edition resulted in. many constructive suggestions about gaps weaknesses obscurities and selection. of topics For typing and other work involved in production of a typescript I am. indebted to Rowena Foss Holly Grano and Edith Klotz My thanks to all. William G Cochran, South Orleans Massachusetts, February 1977. CHAPTER PAGE, 1 INTRODUCTION 1, 1 1 Advantages of the Sampling Method 1. 1 2 Some Uses of Sample Surveys 2, 1 3 The Principal Steps in a Sample Survey 4. 1 4 The Role of Sampling Theory 8, 1 5 Probability Sampling 9. 1 6 Alternatives to Probability Sampling 10, 1 7 UseoftheNormalDistribution 11.
1 8 Bias and Its Effects 12, 1 9 The Mean Square Error 15. Exercises 16, 2 SIMPLE RANDOM SAMPLING 18, 2 1 Simple Random Sampling 18. 2 2 Selection of a Simple Random Sample 19, 2 3 Definitions and Notation 20. 2 4 Properties of the Estimates 21, 2 5 Variances of the Estimates 23. 2 6 The Finite Population Correction 24, 2 7 Estimation of the Standard Error from a Sample 25.
2 8 Confidence Limits 27, 2 9 An Alternative Method of Proof 28. 2 10 Random Sampling with Replacement 29, 2 11 Estimation of a Ratio 30. 2 12 Estimates of Means Over Subpopuiations 34, 2 13 Estimates of Totals Over Subpopulations 35. 2 14 Comparisons Between Domain Means 39, 2 15 Validity of the Normal Approximation 39. 2 16 Linear Estimators of the Population Mean 44, Exercises 45.
x CONTENTS, 3 SAMPLING PROPORTIONS, AND PERCENTAGES 50. 3 1 Qualitative Characteristics 50, 3 2 VariancesoftheSampleEstimates 50. 3 3 The Effect of Pon the Standard Errors 53, 3 4 The Binomial Distribution 55. 3 5 The Hypergeometric Distribution 55, 3 6 Confidence Limits 57. 3 7 Classification into More than Two Classes 60, 3 8 Confidence Limits with More than Two Classes 60.
3 9 TheConditionalDistributionofp 61, 3 10 Proportions and Totals Over Subpopulations 63. 3 11 Comparisons Between Different D mains 64, 3 12 Estimation of Proportions in Cluster Sampling 64. Exercises 68, 4 THE ESTIMATION OF SAMPLE SIZE 72, 4 1 A Hypothetical Example 72. 4 2 Analysis of the Problem 73, 4 3 The Specification of Precision 74. 4 4 The Formula for n in Sampling for Proportions 75. 4 5 Rare Items Inverse Sampling 76, 4 6 The Formula for n with Continuous Data 77.
4 7 Advance Estimates of Population Variances 78, 4 8 Sample Size with More than One Item 81. 4 9 Sample Size when Estimates Are Wanted for Subdivisions of the. Population 82, 4 10 Sample Size in Decision Problems 83. 4 11 The Design Effect Deff 85, Exercises 86, 5 STRATIFIED RANDOM SAMPLING 89. 5 1 Description 89, 5 2 Notation 90, 5 3 Properties of the Estimates 91. 5 4 The Estimated Variance and Confidence Limits 95. 5 5 Optimum Allocation 96, CONTENTS Xi, 5 6 Relative Precision of Stratified Random and Simple Random.
Sampling 99, 5 7 When Does Stratification Produce Large Gains in Precision 101. 5 8 Allocation Requiring More than 100 Per Cent Sampling 104. 5 9 Estimation of Sample Size with Continuous Data 105. 5 10 Stratified Sampling for Proportions 107, 5 11 Gains in Precision in Stratified Sampling for Proportions 109. 5 12 Estimation of Sample Size with Proportions 110. Exercises 111, SA FURTHER ASPECTS OF, STRATIFIED SAMPLING 115. 5A 1 Effects of Deviations from the Optimum Allocation 115. 5A 2 Effects of Errors in the Stratum Sizes 117, 5A 3 The Problem of Allocation with More than One Item 119. 5A 4 Other Methods of Allocation with More than One Item 121. 5A 5 Two Way Stratification with Small Samples 124. 5A 6 Controlled Selection 126, 5A 7 The Construction of Strata 127.
5A 8 Numberof Strata 132, 5A 9 Stratification After Selection of the Sample Poststratification 134. 5A 10 Quota Sampling 135, 5A 1 l Estimation from a Sample of the Gain Due to Stratification 136. 5A 12 Estimation of Variance with One Unit per Stratum 138. 5A 13 Strata as Domains of Study 140, 5A 14 Estimating Totals and Means Over Subpopulations 142. 5A 15 SamplingfromTwoFrames 144, Exercises 146, 6 RATIO ESTIMATORS 150. 6 1 Methods of Estimation 150, 6 2 The Ratio Estimate 150.
6 3 Approximate Variance of the Ratio Estimate 153. 6 4 Estimation of the Variance from a Sample 155, 6 5 Confidence Limits 156. 6 6 Comparison of the Ratio Estimate with Mean per Unit 157. 6 7 Conditions Under Which the Ratio Estimate Is a Best Linear. Unbiased Estimator 158, 6 8 Bias of the Ratio Estimate 160. xii CONTENTS, 6 9 Accuracy of the Formulas for the Variance and Estimated. Variance 162, 6 10 Ratio Estimates in Stratified Random Sampling 164. 6 11 The Combined Ratio Estimate 165, 6 12 Comparison of the Combined and Separate Estimates 167.
6 13 Short Cut Computation of the Estimated Variance 169. 6 14 Optimum Allocation with a Ratio Estimate 172, 6 15 UnbiasedRatio typeEstimates 174. 6 16 Comparison of the Methods 177, 6 17 Improved Estimation of Variance c 178. 6 18 ComparisonofTwoRatios 180, 6 19 Ratio of Two Ratios 183. 6 20 Multivariate Ratio Estimates 184, 6 21 Product Estimators 186. Exercises 186, 7 REGRESSION ESTIMATORS 189, 7 1 The Linear Regression Estimate 189.
7 2 Regression Estimates with Preassigned b 190, 7 3 Regression Estimates when b Is Computed from the Sample 193. 7 4 Sample Estimate of Variance 195, 7 5 Large Sample Comparison with the Ratio Estimate and the Mean. per Unit 195, 7 6 Accuracy of the Large Sample Formulas for V y 1 and v y 1 197. 7 7 Bias of the Linear Regression Estimate 198, 7 8 The Linear Regression Estimator Under a Linear Regression. 7 9 Regression Est mates in Stratified Sampling 200. 7 10 Regression Co1 ffi cients Estimated from the Sample 201. 7 11 Comparison of the Two Types of Regression Estimate 203. Exercises 203, 8 SYSTEMATIC SAMPLING 205, 8 1 Description 205.
8 2 Relation to Cluster Sampling 207, 8 3 Variance of the Estimated Mean 207. 8 4 Comparison of Systematic with Stratified Random Sampling 212. 8 5 Populations in Random Order 212, CONTENTS xiii. 8 6 Populations with Linear Trend 214, 8 7 Methods for Populations with Linear Trends 216. 8 8 Populations with Periodic Variation 217, 8 9 AutocorrelatedPopulations 219. 8 10 Natural Populations 221, 8 11 Estimation of the Variance from a Single Sample 223.
8 12 Stratified Systematic Sampling 226, 8 13 Systematic Sampling in Two Dimensions 227. 8 14 Summary 229, Exercises 231, 9 SINGLE STAGE CLUSTER SAMPLING. CLUSTERS OF EQUAL SIZES 233, 9 1 Reasons for Cluster Sampling 233. 9 2 ASimpleRule 234, 9 3 Comparisons of Precision Made from Survey Data 238. 9 4 Variance in Terms of Intracluster Correlation 240. 9 5 Variance Functions 243, 9 6 ACostFunction 244, 9 7 Cluster Sampling for Proportions 246.
Exercises 2 47, 9A SINGLE STAGE CLUSTER SAMPLING, CLUSTERS OF UNEQUAL SIZES 249. 9A 1 Cluster Units of Unequal Sizes 249, 9 A 2 Sampling with Probability Proportional to Size 250. 9A 3 Selection with Unequal Probabilities with Replacement 252. 9A 4 The Optimum Measure of Size 255, 9A 5 Relative Accuracies of Three Techniques 255. 9A 6 Sampling with Unequal Probabilities Without Replacement 258. 9A 7 The Horvitz Thompson Estimator 259, 9A 8 Brewer s Method 261. 9A 9 Murthy s Method 263, 9A 10 Methods Related to Systematic Sampling 265.
9A 11 The Rao Hartley Cochran Method 266, 9A 12 NumericalComparisons 267. methods produced for sampling with unequal probabilities without replacement These include the similar methods given independently by Brewer J N K lh o and Durbin Murthy s method the Rao Hartley Cochran method and Madow s method related to systematic sampling with comparisons of the performances of the methods on natural populations

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