Sampling抽样技术统计学专业课课件.pptVIP

Sampling Design Analysis SharonL LohrArizonaStateUniversity ContentsCHAPTER1Introduction1 1ASampleControversy1 2RequirementsofaGoodSample1 3SelectionBias1 4MeasurementBias1 5QuestionnaireDesign1 6SamplingandNonsamplingErrorsCHAPTER2SimpleProbabilitySamples2 1TypesofProbabilitySamples 2 2FrameworkforProbabilitySampling2 3SimpleRandomSampling2 4ConfidenceIntervals2 5SampleSizeEstimation2 6SystematicSampling2 7RandomizationTheoryResultsforSimpleRandomSampling 2 8AModelforSimpleRandomSampling 2 9WhenShouldaSimpleRandom SampleBeUsed CHAPTER3RatioandRegressionEstimation3 1RatioEstimation3 2RegressionEstimation3 3EstimationinDomains3 4ModelsforRatioandRegressionEstimation 3 5Comparison CHAPTER4StratifiedSampling4 1WhatIsStratifiedSampling 4 2TheoryofStratifiedSampling4 3SamplingWeights4 4AllocatingObservationstoStrata4 5DefiningStrata4 6AModelforStratifiedSampling 4 7Post stratification4 8QuotaSampling CHAPTER5ClusterSamplingwithEqualProbabilities5 1NotationforClusterSampling5 2One StageClusterSampling5 3Two StageClusterSampling5 4UsingWeightsinClusterSamples5 5DesigningaClusterSample5 6SystematicSampling5 7ModelsforClusterSampling CHAPTER1IntroductionWhenstatisticsarenotbasedonstrictlyaccuratecalculations theymisleadinsteadofguide Themindeasilyletsitselfbetakeninbythefalseappearanceofexactitudewhichstatisticsretainintheirmistakes andconfidentlyadeptserrorsclothedintheformofmathematicaltruth AlexisdeTocqueville DEMOCRACYINAMERICA 1 1ASamplingControversyShereHite sbook WomenandLove ACulturalRevolutioninprogress 1987 84 ofwomenare notsatisfiedemotionallywiththeirrelationships p804 70 ofallwomen marriedfiveormoreyearsarehavingsexoutsideoftheirmarriages p856 95 ofwomen reportformsofemotionalandpsychologicalharassmentfrommenwithwhomtheyareinloverelationships p810 84 ofwomenreportformsofcondescensionfromthemenintheirloverelationships p809 Harassment toannoypersistentlysexualharassment uninvitedandunwelcomeverbalorphysicalbehaviorofasexualnatureespeciallybyapersoninauthoritytowardasubordinate asanemployeeorstudent Condescension 1 voluntarydescentfromone srankordignityinrelationswithaninferior 2 Theactofcondescendingoraninstanceofit 3 Patronizinglysuperiorbehaviororattitude Vignette Adecorativedesignplacedatthebeginningorendofabookorchapterofabookoralongtheborderofapage ThefollowingcharacteristicsofthesurveymakeHite sresultunsuitableforgenerali zation Thesamplewasself selected Thequestionnairesweremailedtospecificgroups Thequestionsinthesurveyaretoocomplicated Manyofthequestionsarevague usingwordssuchaslove Manyofthequestionsareleading 1 2RequirementsofaGoodSampleAperfectsampleshould 1 beascaled downversionofthepopulation 2 canmirrorcharacteristicsofthewholepopulationSomedefinitionstomakethenotionofagoodsamplemoreprecise ObservationunitAnobjectonwhichameasurementistaken TargetpopulationThecompletecollectionofobservationswewanttostudy SampleAsubsetofapopulation SampledpopulationThecollectionofallpossibleobservationunitsthatmighthavebeenchoseninasample Thepopulationfromwhichthesamplewastaken SamplingunitTheunitweactuallysample SamplingframeThelistofsamplingunits TargetpopulationSamplingframepopulation Sampledpopulation Notreachable Refusetorespond Notcapabletorespond Noteligibleforsurvey Inanidealsurvey thesampledpopulationwillbeidenticaltothetargetpopulation butthisidealisrarelymetexactly IntheHitestudyTargetpopulation alladultwomenintheUnitedStatesSampledpopulation womenbelongingtowomen sorganizationswhowouldreturnthequestionnaire IntheNationalCrimeVictimizationSurvey Targetpopulation allhouseholdsintheUnitedStatesSampledpopulation householdsinthesamplingframethatare athome andagreetoanswerquestions IntheNationalPesticideSurvey Targetpopulation allcommunitywatersystemsandruraldomesticwellsintheUnitedStates Sampledpopulation allcommunitywatersystemsandallidentifiabledomesticwellsoutsideofgovernmentreservationsthatbelongedtohouseholdswillingtocooperatewiththesurvey Inpublicopinionpolls Targetpopulation personswhowillvoteinthenextelectionSampledpopulation personswhocanbereachedbytelephoneandsaytheyarelikelytovoteinthenextelection 1 3SelectionBiasThefollowingexamplesindicatesomewaysinwhichselectionbiascanoccur Useasample selectionprocedurethat unknowntotheinvestigators dependsonsomecharacteristicassociatedwiththepropertiesofinterest Deliberatelyorpurposefullyselecta representative sample forinstance ajudgmentsample Misspecifythetargetpopulation Failtoincludeallthetargetpopulationinthesamplingframe calledunder coverage Substituteaconvenientmemberofapopulationforadesignatedmemberwhoisnotreadilyavailable Failtoobtainresponsesfromtheentirechosensample Thisiscallednon responseAllowthesampletoconsistentirelyofvolunteers CASESTUDYLiteraryDigestAneververyfamousmagazineinUSAwhobegantakingpollstoforecasttheoutcomeoftheUSApresidentialelectionin1912 theirpollsattainedareputationforaccuracybecausetheyforecastthecorrectwinnerineveryelectionbetween1912and1932 In1932 forexample thepollpredictedthatRooseveltwouldreceive56 ofthepopularvoteand474votesintheelectoralcollege intheactualelection Rooseveltreceived58 ofthepopularvoteand472votesintheelectoralcollege Electoralcollege intheU S abodyofpeoplerepresentingthestatesoftheU S whoformallycastvotesfortheelectionofthepresidentandvicepresident OnOctober31 1936 thepollpredictedTheoutcomeis RepublicanAlfLandon 55 PresidentRoosevelt 41 RepublicanAlfLandon 37 PresidentRoosevelt 61 Tworeasonsthataccountedfortheoutcome Oneproblemmayhavebeenundercoverageinthesamplingframe whichreliedheavilyontelephonedirectoriesandautomobileregistrationlist Thelowresponserate lessthan25 tothesurveywaslikelythesourceofmuchoftheerror OnelessontobelearntfromtheLiteraryDigestpollisthatthesheersizeofasampleisnoguaranteeofitaccuracy 1 4MeasurementBiasInfollowingcases itismostlikelytohappen Peoplesometimesdonottellthetruth PeopledonotunderstandthequestionsPeopleforgetPeoplegivedifferentanswerstodifferentinterviewersPeoplecatertotheinterviewersTheinterviewermayhavehisowninclinationtothesurveyCertainwordsmayhavevaguemeaningThequestionnairedoesn twordwellorisnotarrangedinagoodorder 1 5QuestionnaireDesignDecidewhatyouwanttofindoutAlwaystestyourquestionsbeforetakingthesurveyKeepitsimpleandclearUsespecificquestionsinsteadofgeneralonesRelateyourquestionstotheconceptofinterest Decidewhethertouseopenorclosedquestions openquestions therespondentsisnotpromptedwithcategoriesforresponse closedones multiplechoices ReporttheactualquestionaskedAvoidquestionsthatpromptormotivatetherespondenttosaywhatyouwouldliketohearUseforced choice ratherthanagreeordisagreequestionsAskonlyoneconceptineachquestionPayattentiontoquestion ordereffects 1 6samplingandnonsamplingerrorssamplingerrorsTheerrorthatresultsfromtakingonesampleinsteadofexaminingthewholepopulationnonsamplingerrorsTheerrorthatcannotbeattributedtothesample to samplevariability causedchieflybyfollowingcauses SelectionbiasIncorrectanswers IncompletevalueNonresponseSelectionbiasandmeasurementbiasareexamplesofnonsamplingerrorsInalotofcases nonsamplingerrorsmayhavemuchworseeffectonaccuracyofthesamplethansamplingones Whydoweusesampling Samplingcanprovidereliableinformationatfarlesscostthanacensus Datacanbecollectedmorequickly soestimatescanbepublishedinatimelyfashion Finally andlesswellknown estimatesbasedonsamplesurveysareoftenmoreaccuratethanthosebasedonacensusbecauseinvestigatorscanbemorecarefulwhencollectingdata CHAPTER2SimpleProbabilitySamplesProbabilitySampling inaprobabilitysample eachunitinthepopulationhasaknown butnotcertainlyequal probabilityofselection andachancemethodsuchasusingnumbersfromarandomnumbertableisusedtochoosethespecificunitstobeincludedinthesample 2 1TypesofProbabilitySamples1 Simplerandomsample2 Stratifiedsample3 Clustersample Asimplerandomsample SRS isthesimplestformofprobabilitysample AnSRSofsizenistakenwheneverypossiblesubsetofnunitsinthepopulationhasthesamechanceofbeingthesample Inastratifiedrandomsample thepopulationisdividedintosubgroupscalledstrata ThenanSRSisselectedfromeachstratum andtheSRSsinthestrataareselectedindependently Inaclustersample observationunitsinthepopulationareaggregatedintolargersamplingunits calledclusters ThenanSRSisdrawnundertheconditionthateachclusterisviewedasaunit 2 2FrameworkforProbabilitySamplingAspecialcaseforitisN 4 whichresultsin Itspossiblesamples n 2 are Example2 1 Example2 2 i 1 2 3 4 5 6 7 8 1 2 4 4 7 7 7 8 Theexpectedvalueof isthemeanofthesamplingdistributionof k 222830323436384042444648505258 1623746126473261 Thevarianceofthesamplingdistributionof i e is TheMeanSquaredError MSE ratherthanvariancetomeasuretheaccuracyofanestimatoris Anestimatorisunbiasedif Anestimatorispreciseifthefollowingissmall Anestimatorisaccurateifthefollowingissmall Someindicatorsforthepopulation Thepopulationtotalis Themeanofthepopulationis Thevarianceofthepopulationvaluesaboutthemeanis Thestandarddeviationofthepopulationvaluesaboutthemeanis Thecoefficientofvariation CV is Proportionisaspecialcaseofmean Thedistinctionbetweenmeanandproportionis inthecaseofmean thevariablecantakemorethantwovalues whereasinproportioncase itcantakeandcanonlytaketwovalues Wherethevariableis 2 3SimpleRandomSamplingTherearetwotypesofSimpleRandomSample 1SimpleRandomSamplewithreplacement SRSWR Inthiscase therearepossiblesamplesandwemaygetduplicates 2SimpleRandomSamplewithreplacement SRS Inthiscase therearepossiblesamplesandwemaynotgetduplicates ForestimatingthepopulationmeaninanSRS weusethesamplemean Theisanunbiasedestimatorofthepopulationmean andthevarianceofis Inwhichiscalledthefinitepopulationcorrection fpc Forestimatingthepopulationvariance weusethesamplevariance Anunbiasedestimatorofisasfollow Buttheestimatedvarianceofisusuallyreportedbyitsstandarderror SE Theestimatedcoefficientofvariationofanestimateis Allthisresultsapplytotheestimationofapopulationtotal t since Theunbiasedestimatoroftis Itsvarianceis Buttheunbiasedestimatorofthisvarianceis since Asforproportionvariable theparametersare thus Theestimatorsare Where 2 4ConfidenceIntervalsUsedtoindicatehowaccurateourestimatesare Usuallyappearsinthisway Ifwetakeasanexample thenwehave Thedistinctionbetweendistributionandsamplingdistributionfromit A distribution referstotheoriginaldistributionofavariabley whereasa samplingdistribution referstothedistributiongeneratedfromtheoriginalone likethedistributionofand Example2 1 11 4 14 4 15 4 16 4 17 4 29 4 Withauniformdistribution only8cases Withanon uniformdistribution upto15cases Thedistinctionbetweenlawoflargenumbersandcentrallimittheorem lawoflargenumbers saysthatthereisalmostnodifferencebetweensampleandpopulationmeanifnissufficientlylarge bothdependentlyorindependently withthesameordifferentdistribution whereas centrallimittheorem saysthatthedistributionofanysamplemeanconvergestonormaldistributionifnissufficientlylarge withthesameordifferentdistribution Bernoulli slawoflargenumbersis Linderbergandlevy scentrallimittheoremis 2 5SampleSizeEstimationAninvestigatoroftenmeasuresseveralvariablesandhasanumberofgoalsforasurvey AnyonedesigninganSRSmustdecidewhatamountofsamplingerrorintheestimatesistolerableandmustbalancetheprecisionoftheestimateswiththecostofthesurvey Followthesestepstoestimatethesamplesize Askquestionsas A Whatisexpectedofthesample B HowmuchprecisiondoIneed C Whataretheconsequencesofthesampleresults D Howmucherroristolerable Findanequationrelatingthesamplesizenandourexpectationsofthesample Estimateanyunknownquantitiesandsolveforn Ifthesamplesizeyoucalculatedinlaststepismuchlargerthanyoucanafford Gobackandadjustsomeoftheexpec tationsforthesurveyandtryagain Specifythetolerableerror FindanequationSolvingforn wehave Inrelativeprecisioncase wehave 2 7RandomizationTheoryResultsforSimpleRandomSampling ToverifyandDefine thenwehave Asaconsequenceofequation 2 18 inordertocalculatethevarianceof notethat Nowletusprove Proof CHAPTER3RatioandRegressionEstimationExampleincensusbyLaplace 3 1RatioEstimationEstimationmethodusingauxiliaryvariableDefine andThen Ratioandregressionestimationbothtakeadvantageofthecorrelationcoefficientofxandyinthepopulation thehigherthecorrelation thebettertheywork Definethepopulationcorrelationcoefficientofxandytobe 3 1 1Whydoweuseratioestimation Sometimeswesimplywanttoestimatearatio Averageyieldperacre Percapitaincome Theratioofliabilitiestoassets Theratioofthefishcaughttothehoursspentfishing Sometimeswewanttoestimateapopulationtotal butthepopulationsizeNisunknown Ratioestimationisusedtoincreasetheprecisionofestimatedmeansandtotals Ratioestimationisusedtoadjustestimatesfromthesamplesothattheyreflectdemographicaltotals Population 4000Sample 400 2700females1300males 240females160males tobecomeateacherSimpleestimation Ratioestimation 240females160males 84females40males Ratioestimationisusedtoadjustnonresponse 3 1 2BiasandMeanSquaredErrorofRatioEstimatorsRatioestimationisbiased unlikeSRSestimation butwithareducedvarianceasacompensatorforthepresenceofbias Since weget Thenwehave Consequently asshownbyHartleyandRoss 1954 Anotherwaytoshowthisis Thebiasofissmallif 1 thesamplesizenislarge 2 thesamplingfractionn Nislarge 3 islarge4 issmall 5 thecorrelationRiscloseto1 Theproofforitis IntheworksofHartley Ross anunbiasedestimatorfortheparameterBisgivenafterthefollowingisproved Theunbiasedestimatoris TheapproximatedMSEofis TheapproximatedMSEofwillbesmallif 1 thesamplesizenislarge 2 thesamplingfractionn Nislarge 3 thedeviationaboutthelineislarge 4 islarge 5 thecorrelationRiscloseto 1 Theproofforitis And And Thereforeweget Inpractice Bisunknown thenletItfollowsfromabove wewillhave TheCIscanbeconstructedas Let sjuststudytwoexamplesinthetextbook 3 1 2 1AccuracyoftheMSEApproximationFor 3 6 tobeagoodapproximationofMSE wewantalargesamplesizen 30 and3 1 2 2Advantageofratioestimation Then Sototheaccuracyoftheapproximation holds ifandonlyif 3 2RegressionEstimationRatioestimationworksbestifthedataarewellfitbyastraightlinethroughtheorigin Butsometimes dataappeartobeevenlyscatteredaboutastraightlinethatdoesnotgothroughtheorigin thatis thedatalookasthoughtheusualstraight lineregressionmodelwouldprovideagoodfit where Hereristhesamplecorrelationcoefficientofxandy Liketheratioestimator theregressionestimatorisbiased Thebiasis LetThen Thatis TheapproximationMSEissmallif 1 thesamplesizenislarge 2 thesamplingfractionn Nislarge 3 issmall 4 thecorrelationRiscloseto 1or1 Let thentheStandardErroris CHAPTER4StratifiedSampling4 1WhatIsStratifiedSampling stratum 1 ahorizontallayerofmaterial especiallyoneofseveralparallellayersarrangedoneontopofanother 2 alevelofsocietycomposedofpeoplewithsimilarsocial cultural oreconomicstatusstrata thepluralformofstratumstratify toform arrange ordepositinlayers WhydoweuseStratifiedSampling 1 tobeprotectedfromthepossibilityofobtainingvery bad sample Population Sample 2500Females 1500Males 150Males 250Females 2 tohavedataofknownprecisionforsubgroups Population Sample 3500Females 500Males 250Females 150Males 3 inordertoadministerasamplemoreconvenientlyortolowercostforthesurvey Mailsurvey Personalinterview Telephoneinterview 4 toproducemorepreciseestimatesforthewholepopulation ifdonecorrectly BloodPressure byage ConcentrationofPlants bytypeofterrain Youngsters theMiddle aged Woodland Marsh Dotages Dryland 4 2TheoryofStratifiedSamplingDefinition Instratifiedrandomsampling thesimplestformofstratifiedsampling WeindependentlytakeanSRSfromeachstratumsothatnhobservationsarerandomlyselectedfromthepopulationunitsinstratumh DefineShtobethesetofnhunitsintheSRSforstratumh NotationforStratificationThepopulationsizeis valueofjthunitinstratumh populationtotalinstratumh overallpopulationtotal populationmeaninstratumh overallpopulationmean populationvarianceinstratumh Thecorrespondingquantitiesforthesampleare Curlicue adecorativecurlortwistincalligraphyorinthedesignofanobject Thenwehavetheestimationforthepopulationas andThepropertiesoftheseestimatorsfollowdirectlyfromthepropertiesofSRSestimators Unbiasedness Varianceoftheestimators Varianceestimatorsforstratifiedsamples 4 3SamplingWeightsDenote samplingweight and samplingfraction Thenweget andthus 4 4AllocatingObservationstoStrata4 4 1ProportionalAllocationAnallocationmethodistoletthenumberofsampledunitineachstratumbeproportionaltothesizeofthecorrespondingstratum Theprobabilityofselection hjisthesamen Nforallstrata sinceAsforSRS wehave PopulationANOVAtable ThereforeIfonly Thefollowingwillhold Virtually witharelativelylargerstratumsize wehaveAndthisresultsin 4 4 2OptimalAllocationLetand Weget Wesampleheavilywithinastratumif Thestratumaccountsforalargepartofthepopulation Thevariancewithinthestratumislarge wesamplemoreheavilytocompensatefortheheterogeneity Samplinginthestratumisinexpensive 4 4 4DeterminethesamplesizeofthepopulationSince Ifthefpc sarenegligibleandifthenormalapproximationisvalid theCIcanbe Supposetheabsoluteerroris i e wethenhave Practically theinthevwillbesubstitutedwith CHAPTER5ClusterSamplingwithEqualProbabilitiesRelativeConcepts ClusterPrimarysamplingunits psu s Secondarysamplingunits ssu s Definition seetheexampleonthetextbookWhydoweuseClusterSampling 1 Constructingsamplingframelistofobservationunitsmaybedifficult expensive orimpossible2 Thepopulationmaybewidelydistributedgeographicallyandmayoccurinnaturalclusterssuchashouseholdsorschools SimilaritiesanddifferencesbetweenstratumandclusterSimilaritiesbothareagroupingofthemembersofthepopulation Differences 1 forstratasampling selectanSRSsampleunitswithineverystratum forstratasampling selectanSRSclustersfromthepopulation theunitsineachclusterconstitutingthewholesample 2 Instratifiedsamplingcase thesamplingmayincreasetheprecision whereasinclustersampling thesamplingmaydecreasetheprecision butmayresultinmoreinformationperdollarspent 5 1NotationforClusterSamplingyij measurementforjthelementinithpsupsulevel populationquantities N numberofpsu sinthepopulationMi numberofssu sinithpsu totalnumberofpsu sinthepopulation totalintheithpsu s populationtotal populationvarianceofthepsutotalssulevel populationquantities populationmean populationmeanintheithpsu populationvariance perssu populationvariancewithintheithpsuSampleQuantities n