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1、Sampling:Design&AnalysisSharon L.LohrArizona State UniversityContentsCHAPTER 1 Introduction 1.1 A Sample Controversy 1.2 Requirements of a Good Sample 1.3 Selection Bias 1.4 Measurement Bias 1.5 Questionnaire Design 1.6 Sampling and Nonsampling Errors CHAPTER 2 Simple Probability Samples 2.1 Types o
2、f Probability Samples 2.2 Framework for Probability Sampling 2.3 Simple Random Sampling 2.4 Confidence Intervals 2.5 Sample Size Estimation 2.6 Systematic Sampling 2.7 Randomization Theory Results for Simple Random Sampling*2.8 A Model for Simple Random Sampling*2.9 When Should a Simple Random Sampl
3、e Be Used?CHAPTER 3 Ratio and Regression Estimation 3.1 Ratio Estimation 3.2 Regression Estimation 3.3 Estimation in Domains 3.4 Models for Ratio and Regression Estimation*3.5 ComparisonCHAPTER 4 Stratified Sampling 4.1 What Is Stratified Sampling?4.2 Theory of Stratified Sampling 4.3 Sampling Weigh
4、ts 4.4 Allocating Observations to Strata 4.5 Defining Strata 4.6 A Model for Stratified Sampling*4.7 Post-stratification 4.8 Quota Sampling CHAPTER 5 Cluster Sampling with Equal Probabilities 5.1 Notation for Cluster Sampling 5.2 One-Stage Cluster Sampling 5.3 Two-Stage Cluster Sampling 5.4 Using We
5、ights in Cluster Samples 5.5 Designing a Cluster Sample 5.6 Systematic Sampling 5.7 Models for Cluster Sampling*CHAPTER 1 IntroductionWhen statistics are not based on strictly accurate calculations,they mislead instead of guide.The mind easily lets itself be taken in by the false appearance of exact
6、itude which statistics retain in their mistakes,and confidently adepts errors clothed in the form of mathematical truth.-Alexis de Tocqueville,DEMOCRACY IN AMERICA1.1 A Sampling ControversyShere Hites book“Women and Love:A Cultural Revolution in progress”(1987):84%of women are not satisfied emotiona
7、lly with their relationships(p804).70%of all women married five or more years are having sex outside of their marriages(p856).95%of women report forms of emotional and psychological harassment from men with whom they are in love relationships(p810).84%of women report forms of condescension from the
8、men in their love relationships(p809).-*Harassment:to annoy persistentlysexual harassment:uninvited and unwelcome verbal or physical behavior of a sexual nature especially by a person in authority toward a subordinate(as an employee or student)*Condescension:1.voluntary descent from ones rank or dig
9、nity in relations with an inferior;2.The act of condescending or an instance of it.3.Patronizingly superior behavior or attitude.*Vignette:A decorative design placed at the beginning or end of a book or chapter of a book or along the border of a page.The following characteristics of the survey make
10、Hites result unsuitable for generali-zation.The sample was self-selected.The questionnaires were mailed to specific groups.The questions in the survey are too complicated.Many of the questions are vague,using words such as love.Many of the questions are leading.1.2 Requirements of a Good SampleA per
11、fect sample should:1.be a scaled-down version of the population;2.can mirror characteristics of the whole populationSome definitions to make the notion of a good sample more precise:Observation unit An object on which a measurement is taken.Target population The complete collection of observations w
12、e want to study.Sample A subset of a population.Sampled population The collection of all possible observation units that might have been chosen in a sample.The population from which the sample was taken.Sampling unit The unit we actually sample.Sampling frame The list of sampling units.Target popula
13、tion Sampling frame population Sampled population NotreachableRefuse torespondNot capable to respondNot eligible for surveyIn an ideal survey,the sampled population will be identical to the target population,but this ideal is rarely met exactly.In the Hite studyTarget population:all adult women in t
14、he United StatesSampled population:women belonging to womens organizations who would return the questionnaire.In the National Crime Victimization Survey:Target population:all households in the United StatesSampled population:households in the sampling frame that are at home and agree to answer quest
15、ions.In the National Pesticide Survey:Target population:all community water systems and rural domestic wells in the United States.Sampled population:all community water systems and all identifiable domestic wells outside of government reservations that belonged to households willing to cooperate wit
16、h the survey.In public opinion polls:Target population:persons who will vote in the next electionSampled population:persons who can be reached by telephone and say they are likely to vote in the next election 1.3 Selection Bias The following examples indicate some ways in which selection bias can oc
17、cur:Use a sample-selection procedure that,unknown to the investigators,depends on some characteristic associated with the properties of interest.Deliberately or purposefully select a representative sample.for instance:”a judgment sample”Misspecify the target population.Fail to include all the target
18、 population in the sampling frame,called under-coverage.Substitute a convenient member of a population for a designated member who is not readily available.Fail to obtain responses from the entire chosen sample.This is called non-responseAllow the sample to consist entirely of volunteers CASE STUDY
19、Literary Digest An ever very famous magazine in USA who began taking polls to forecast the outcome of the USA presidential election in 1912.their polls attained a reputation for accuracy because they forecast the correct winner in every election between 1912 and 1932.In 1932,for example,the poll pre
20、dicted that Roosevelt would receive 56%of the popular vote and 474 votes in the electoral college;in the actual election.Roosevelt received 58%of the popular vote and 472 votes in the electoral college.Electoral college:(in the U.S.)a body of people representing the states of the U.S.,who formally c
21、ast votes for the election of the president and vice president.On October 31,1936,the poll predictedThe outcome is:Republican Alf Landon:55%President Roosevelt:41%Republican Alf Landon:37%President Roosevelt:61%Two reasons that accounted for the outcome:One problem may have been undercoverage in the
22、 sampling frame,which relied heavily on telephone directories and automobile registration list;The low response rate(less than 25%)to the survey was likely the source of much of the error.One lesson to be learnt from the Literary Digest poll is that the sheer size of a sample is no guarantee of it a
23、ccuracy1.4 Measurement BiasIn following cases,it is most likely to happen:People sometimes do not tell the truth.People do not understand the questionsPeople forgetPeople give different answers to different interviewersPeople cater to the interviewersThe interviewer may have his own inclination to t
24、he surveyCertain words may have vague meaningThe questionnaire doesnt word well or is not arranged in a good order1.5 Questionnaire DesignDecide what you want to find outAlways test your questions before taking the surveyKeep it simple and clearUse specific questions instead of general onesRelate yo
25、ur questions to the concept of interest.Decide whether to use open or closed questions(open questions:the respondents is not prompted with categories for response;closed ones:multiple choices)Report the actual question askedAvoid questions that prompt or motivate the respondent to say what you would
26、 like to hearUse forced-choice,rather than agree or disagree questionsAsk only one concept in each questionPay attention to question-order effects1.6 sampling and nonsampling errorssampling errorsThe error that results from taking one sample instead of examining the whole populationnonsampling error
27、sThe error that can not be attributed to the sample-to-sample variability,caused chiefly by following causes:Selection biasIncorrect answersIncomplete valueNonresponseSelection bias and measurement bias are examples of nonsampling errors In a lot of cases,nonsampling errors may have much worse effec
28、t on accuracy of the sample than sampling onesWhy do we use sampling?Sampling can provide reliable information at far less cost than a census.Data can be collected more quickly,so estimates can be published in a timely fashion.Finally,and less well known,estimates based on sample surveys are often m
29、ore accurate than those based on a census because investigators can be more careful when collecting dataCHAPTER 2 Simple Probability Samples Probability Sampling:in a probability sample,each unit in the population has a known(but not certainly equal)probability of selection,and a chance method such
30、as using numbers from a random number table is used to choose the specific units to be included in the sample.2.1 Types of Probability Samples1,Simple random sample2,Stratified sample3,Cluster sampleA simple random sample(SRS)is the simplest form of probability sample.An SRS of size n is taken when
31、every possible subset of n units in the population has the same chance of being the sample.In a stratified random sample,the population is divided into subgroups called strata.Then an SRS is selected from each stratum,and the SRSs in the strata are selected independently.In a cluster sample,observat
32、ion units in the population are aggregated into larger sampling units,called clusters.Then an SRS is drawn under the condition that each cluster is viewed as a unit.2.2 Framework for Probability Sampling A special case for it is N=4,which results in:Its possible samples(n=2)are:Example 2.1:Example 2
33、.2:i1,2,3,4,5,6,7,81,2,4,4,7,7,7,8The expected value of ,is the mean of the sampling distribution of :k22 28 30 32 34 36 38 40 42 44 46 48 50 52 581 6 2 3 7 4 6 12 6 4 7 3 2 6 1The variance of the sampling distribution of ,i.e ,is:The Mean Squared Error(MSE)rather than variance to measure the accura
34、cy of an estimator is:An estimator is unbiased if:An estimator is precise if the following is small:An estimator is accurate if the following is small:Some indicators for the population:The population total is:The mean of the population is:The variance of the population values about the mean is:The
35、standard deviation of the population values about the mean is:The coefficient of variation(CV)is:Proportion is a special case of mean:The distinction between mean and proportion is:in the case of mean,the variable can take more than two values;whereas in proportion case,it can take and can only take
36、 two values.Where the variable is:2.3 Simple Random SamplingThere are two types of Simple Random Sample:1 Simple Random Sample with replacement(SRSWR).In this case,there are possible samples and we may get duplicates;2 Simple Random Sample with replacement(SRS).In this case,there are possible sample
37、s and we may not get duplicates.For estimating the population mean in an SRS,we use the sample mean:The is an unbiased estimator of the population mean ,and the variance of is:In which is called the finite population correction(fpc):For estimating the population variance ,we use the sample variance:
38、An unbiased estimator of is as follow:But the estimated variance of is usually reported by its standard error(SE):The estimated coefficient of variation of an estimate is:All this results apply to the estimation of a population total,t,since:The unbiased estimator of t is:Its variance is:But the unb
39、iased estimator of this variance is:sinceAs for proportion variable,the parameters are:thusThe estimators are:Where:2.4 Confidence Intervals Used to indicate how accurate our estimates are.Usually appears in this way:If we take as an example,then we have:The distinction between distribution and samp
40、ling distribution from it:A“distribution”refers to the original distribution of a variable y,whereas a“sampling distribution”refers to the distribution generated from the original one,like the distribution of and .Example 2.1 11/4,14/4,15/4,16/4,17/4,29/4 With a uniform distribution,only 8 cases.Wit
41、h a non-uniform distribution,up to 15 cases.The distinction between law of large numbers and central limit theorem:“law of large numberslaw of large numbers”says that there is almost no difference between sample and population mean if n is sufficiently large,both dependently or independently,with th
42、e same or different distribution;whereas“central central limit theoremlimit theorem”says that the distribution of any sample mean converges to normal distribution if n is sufficiently large,with the same or different distribution.Bernoullis law of large numbers is:Linderberg and levys central limit
43、theorem is:2.5 Sample Size EstimationAn investigator often measures several variables and has a number of goals for a survey.Anyone designing an SRS must decide what amount of sampling error in the estimates is tolerable and must balance the precision of the estimates with the cost of the survey.Fol
44、low these steps to estimate the sample size:Ask questions as:A:What is expected of the sample?B:How much precision do I need?C:What are the consequences of the sample results?D:How much error is tolerable?Find an equation relating the sample size n and our expectations of the sample Estimate any unk
45、nown quantities and solve for n.If the sample size you calculated in last step is much larger than you can afford.Go back and adjust some of the expec-tations for the survey and try again.Specify the tolerable error Find an equationSolving for n,we have:In relative precision case,we have 2.7 Randomi
46、zation Theory Results for Simple Random Sampling*To verify and Define ,then we have As a consequence of equation(2.18)in order to calculate the variance of ,note that:Now let us prove:Proof:CHAPTER 3 Ratio and Regression Estimation Example in census by Laplace 3.1 Ratio Estimation Estimation method
47、using auxiliary variable Define:and Then,Ratio and regression estimation both take advantage of the correlation coefficient of x and y in the population;the higher the correlation,the better they work.Define the population correlation coefficient of x and y to be:3.1.1 Why do we use ratio estimation
48、?Sometimes we simply want to estimate a ratio;Average yield per acrePer capita incomeThe ratio of liabilities to assetsThe ratio of the fish caught to the hours spent fishingSometimes we want to estimate a population total,but the population size N is unknown;Ratio estimation is used to increase the
49、 precision of estimated means and totals.Ratio estimation is used to adjust estimates from the sample so that they reflect demographical totals.Population:4000Sample:400 2700 females1300 males240 females160 males to become a teacherSimple estimation:Ratio estimation:240 females160 males84 females40
50、malesRatio estimation is used to adjust nonresponse.3.1.2 Bias and Mean Squared Error of Ratio Estimators Ratio estimation is biased,unlike SRS estimation,but with a reduced variance as a compensator for the presence of bias.Since:we get:Then we have:Consequently,as shown by Hartley and Ross(1954).A