商务与经济统计学 （英文版·第14版）课件 -ch10 Categorical Data Analysis

Chapter10CategoricalDataAnalysisContents1. CategoricalDataandtheMultinomialExperiment2. TestingCategoryProbabilities:One-WayTable3. TestingCategoryProbabilities:Two-WayContingencyTable4. AWordofCautionaboutChi-SquareTestsWhereWe’reGoingDiscussqualitative(i.e.,categorical)datawithmorethantwooutcomesPresentachi-squarehypothesistestforcomparingthecategoryproportionsassociatedwithasinglequalitativevariable–calledaone-wayanalysisPresentachi-squarehypothesistestforrelatingtwoqualitativevariables–calledatwo-wayanalysisCautionaboutthemisuseofchi-squaretests10.1CategoricalDataand

MultinomialExperimentQualitativeDataQualitativerandomvariablesyieldresponsesthatcanbeclassifiedExample:gender(male,female)QualitativedatathatfallinmorethantwocategoriesoftenresultfromamultinomialexperimentPropertiesofthe

MultinomialExperimentTheexperimentconsistsofnidenticaltrials.Therearekpossibleoutcomestoeachtrial.Theseoutcomesarecalledclasses,categories,orcells.Theprobabilitiesofthekoutcomes,denotedbyp1,p2,…,pk,remainthesamefromtrialtotrial,wherep1+p2+…+pk=1.Propertiesofthe

MultinomialExperiment(cont)Thetrialsareindependent.Therandomvariablesofinterestarethecellcounts,n1,n2,…,nk,ofthenumberofobservationsthatfallineachofthekclasses.10.2TestingCategoryProbabilities:One-WayTableMultinomialExperimentInthissection,weconsideramultinomialexperimentwithkoutcomesthatcorrespondtocategoriesofasinglequalitativevariable.Theresultsofsuchanexperimentaresummarizedinaone-waytable.Thetermone-wayisusedbecauseonlyonevariableisclassified.Typically,wewanttomakeinferencesaboutthetrueproportionsthatoccurinthekcategoriesbasedonthesampleinformationintheone-waytable.Chi-Square(

2)Test

forkProportionsTestsequality(=)ofproportionsonlyExample:p1=0.2,p2=0.3,p3=0.5OnevariablewithseverallevelsUsesone-waycontingencytableOne-Way

ContingencyTableShowsnumberofobservationsinkindependentgroups(outcomesorvariablelevels)Outcomes(k=3)NumberofresponsesCandidateTomBillMaryTotal352045100ATestofaHypothesisaboutMultinomialProbabilities:One-WayTable H0:p1=p1,0,p2=p2,0,…,pk=pk,0wherep1,0,p2,0,…,pk,0representthehypothesizedvaluesofthemultinomialprobabilities. Ha:Atleastoneofthemultinomialprobabilitiesdoesnotequalitshypothesizedvalue.ATestofaHypothesisaboutMultinomialProbabilities:One-WayTable(cont)whereEi=npi,0istheexpectedcell

count,thatis,theexpectednumberofoutcomesoftypeiassumingthatH0istrue.Thetotalsamplesizeisn.wherehas(k–1)dfandisthevalueoftheteststatistic.ConditionsRequiredforaValidTest:One-wayTableAmultinomialexperimenthasbeenconducted.Thisisgenerallysatisfiedbytakingarandomsamplefromthepopulationofinterest.Thesamplesizenislarge.Thisissatisfiedifforeverycell,theexpectedcellcountEi

willbeequalto5ormore.

2TestBasicIdeaComparesobservedcounttoexpectedcountassumingnullhypothesisistrueCloserobservedcountistoexpectedcount,themorelikelytheH0istrueMeasuredbysquareddifferencerelativetoexpectedcount-rejectlargevaluesFindingCriticalValueExampleWhatisthecritical

2

valueifk=3,and

=0.05?c20UpperTailAreaDF0.995…0.95…0.051...…0.004…3.84120.010…0.103…5.991

2Table(Portion)Ifni=E(ni),

2=0.DonotrejectH0df =k-1=25.991RejectH0

=0.05Example:

2Testfork

ProportionsApersonneldirectorwantstotesttheperceptionoffairnessofthreemethodsofperformanceevaluation.Of180employees,63ratedMethod1asfair,45ratedMethod2asfair,72ratedMethod3asfair.Atthe0.05

levelofsignificance,isthereadifferenceinperceptions?Example:

2Testfork

Proportions(cont)H0:Ha:

=n1=

n2=

n3=CriticalValue(s):p1=p2=p3=1/3Atleast1isdifferent0.0563

45

72

=0.05c20RejectH05.991Example:

2TestforkProportions(cont)Example:

2Testfork

Proportions(cont)H0:Ha:

=n1=

n2=

n3=CriticalValue(s):p1=p2=p3=1/3Atleast1isdifferent0.0563

45

72

=0.05c20RejectH05.991TestStatistic:Decision:Conclusion:

2=6.3Reject

at

=0.05Thereisevidenceofadifferenceinproportions10.3TestingCategoryProbabilities:Two-Way(Contingency)Table

2TestofIndependenceShowsifarelationshipexistsbetweentwoqualitativevariablesOnesampleisdrawnDoesnotshowcausalityUsestwo-waycontingencytable

2TestofIndependenceContingencyTableAtwo-waytable

,calledacontingencytable,

showsthemultinomialcountdataclassifiedontwoscales,ordimensions,ofclassification.Here,thedimensionarehousestyleandhouselocation.Levelsofvariable2Levelsofvariable1

2TestofIndependenceContingencyTableThesymbolsrepresentingthecellcountsforthemultinomialexperimentinprevioustableareshownbelowintheobservedcountstable.So,n11

representsthenumberofbuyerswhopreferasplit-levelhouseinanurbanenvironment.ObservedCountsforContingencyTable

2TestofIndependenceContingencyTableThemarginalprobabilitiesforeachrowandcolumnarecomputedfromtheobservedcountdata.Forexample,pr1=p11

+p12andpc1=p11+p21.Ifthetwo

classifications

are

independent,wemusthavep11

=pr1pc1,p21=pr2pc1

p12=pr1pc2,p22=pr2pc2ProbabilitiesforContingencyTableFindingExpectedCellCountsfor

aTwo-WayContingencyTableTheestimateoftheexpectednumberofobservationsfallingintothecellinrowiandcolumnjisgivenbywhereRi=totalforrowi,Cj=totalforcolumnj,andn=samplesize.GeneralFormofaTwo-Way(Contingency)TableAnalysis:

2-TestforIndependenceH0:Thetwoclassificationsareindependent.Ha:Thetwoclassificationsaredependent.whereRejectionregion:p-value:wherehas(r–1)(c–1)df.ConditionsRequiredforaValid

2-Test:ContingencyTableThenobservedcountsarearandomsamplefromthepopulationofinterest.Wemaythenconsiderthenconsiderthistobeamultinomialexperimentwithr

cpossibleoutcomes.Thesamplesize,n,willbelargeenoughsothat,foreverycell,theestimatedexpectedcountÊij

willbeequalto5ormore.112

160Marginalprobability==0.7Example:FindingExpectedCellCountsforTwo-WayContingencyTableLocation Urban Rural

HouseStyleObs. Obs. TotalSplit–Level 63 49 112Ranch 15 33 48Total 78 82 16078

160Marginalprobability==0.4875Example:FindingExpectedCellCountsforTwo-WayContingencyTable(cont)112

160Marginalprobability==0.7Location Urban Rural

HouseStyleObs. Obs. TotalSplit–Level 63 49 112Ranch

15 33 48Total 78 82 160Example:FindingExpectedCellCountsforTwo-WayContingencyTable(cont)78

160Marginalprobability==0.4875112

160Marginalprobability==0.7Jointprobability=112

16078

160Location Urban Rural

HouseStyleObs. Obs. TotalSplit–Level 63 49 112Ranch 15 33 48Total 78 82 160Expectedcount=160·112

16078

160=54.6Example:FindingExpectedCellCountsforTwo-WayContingencyTable(cont)

HouseLocation

Urban

Rural

HouseStyle

Obs.

Exp.

Obs.

Exp.

Total

Split-Level

63

112·78

16054.6

49

112·82

16057.4

112

Ranch

15

48·78

16023.4

33

48·82

16024.6

48

Total

78

78

82

82

160

Example:ConductingaTwo-WayAnalysisAsarealtoryouwanttodetermineifhousestyleandhouselocationarerelated.Atthe0.05levelofsignificance,isthereevidenceofarelationship?Example:ConductingaTwo-WayAnalysis(cont)H0:

Ha:

=df=

CriticalValue(s):NoRelationshipRelationship0.05(2–1)(2–1)=1c20RejectH03.841

=0.05Eij

5inallcells,asrequiredExample:ConductingaTwo-WayAnalysis(cont)112·82

16048·78

16048·82

160112·78

160Example:ConductingaTwo-WayAnalysis(cont)Example:ConductingaTwo-WayAnalysis(cont)H0:

Ha:

=df=

CriticalValue(s):NoRelationshipRelationship0.05(2–1)(2–1)=1c20RejectH03.841

=0.05TestStatistic:Decision:Conclusion:

2=8.41Rejectat

=0.05Thereisevidenceofarelationship10.4AWordofCautionabout

Chi-SquareTestsCautionaboutthe

2TestThe

2isoneofthemostwidelyappliedstatisticaltoolsandalsooneofthemostabusedstatisticaltool.Becertaintheexperimentsatisfiestheassumptions.Becertainthesampleisdrawnfromthecorrectpopulation.Avoidusingwhentheexpectedcountsareverysmall.Cautionaboutthe

2TestIfthe

2valuedoesnotexceedtheestablishedcriticalvalueof

2,donotacceptthehypothesisofindependence.YouriskaTypeIIerror.Avoidconcludingthattwoclassificationsareindependent,evenwhen

2issmall.Ifacontingencytable

2valuedoesexceedthecriticalvalue,wemustbecarefultoavoidinferringthatacausalrelationshipexistsbetweentheclassifications.Thee