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1、15CountDatainTablesTheanalysisofcountdatawithcategoricalexplanatoryvariablescomesundertheheadingofcontingencytables.Thegeneralmethodofanalysisforcontingencytablesinvolveslog-linearmodelling,butthesimplestcontingencytablesareoftenanalysedbyPearson’schi-squared,Fisher’sexacttestortestso
2、fbinomialproportions(seep.365).15.1Atwo-classtableofcountsYoucount47animalsand?ndthat29ofthemaremalesand18arefemales.Arethesedatasuf?cientlymale-biasedtorejectthenullhypothesisofanevensexratio?Withanevensexratiotheexpectednumberofmalesandfemalesis47/2=23.5.ThesimplesttestisPearson’sch
3、i-squaredinwhichwecalculate(observed?expected)2χ2=.expectedSubstitutingourobservedandexpectedvalues,weget(29?23.5)2+(18?23.5)2χ2==2.574468.23.5Thisislessthanthecriticalvalueforchi-squaredwith1degreeoffreedom(3.841),soweconcludethatthesexratioisnotsigni?cantlydifferentfrom50:50.Therei
4、sabuilt-infunctionforthis:observed<-c(29,18)chisq.test(observed)Chi-squaredtestforgivenprobabilitiesdata:observedX-squared=2.5745,df=1,p-value=0.1086TheRBook,SecondEdition.MichaelJ.Crawley.?2013JohnWiley&Sons,Ltd.Published2013byJohnWiley&Sons,Ltd.600THERBOOKwhichindicatesthatasexratio
5、ofthissizeormoreextremethanthiswouldarisebychancealoneabout10%ofthetime(p=0.1086).Alternatively,youcouldcarryoutabinomialtest:binom.test(observed)Exactbinomialtestdata:observednumberofsuccesses=29,numberoftrials=47,p-value=0.1439alternativehypothesis:trueprobabilityofsuccessisnotequal
6、to0.595percentconfidenceinterval:0.46379940.7549318sampleestimates:probabilityofsuccess0.6170213Youcanseethatthe95%con?denceintervalfortheproportionofmales(0.46,0.75)contains0.5,sothereisnoevidenceagainsta50:50sexratiointhesedata.Thepvalueisslightlydifferentthanitwasinthechi-squaredte
7、st,buttheinterpretationisexactlythesame.15.2SamplesizeforcountdataHowmanysamplesdoyouneedbeforeyouhaveanychanceofdetectingasigni?cantdeparturefromequality?Supposeyouarestudyingsexratiosinfamilies.Howmanyfemalechildrenwouldyouneedtodiscoverinafamilywithnomalesbeforeyoucouldconcludethat
8、afath
