observationslet美人3

CDAChapter3partIITwo-waytableswithorderedclasscationsLetu1≤u2≤.
.
.
≤uIdenotescoresfortherowvariableX,andletν1≤ν2≤.
.
.
≤νJdenotecolumnYscores.
ConsiderthehypothesisH0:XandYareindependentVSH1:YisalinearfunctionofXMantel-Haenszel(MH)statisticM2=(n1)r2whereristhePearsoncorrelationcoecientbetweenXandYbasedontheabovescores.
Forlargesample,M2isapproximatelychi-squaredwithdf=1.
Example:IshappinessassociatedwithpoliticalideologyWithscores(1,2,3)foreachvariable,thecorrelationisr=0.
135.
ThelineartrendteststatisticM2=(3211)(0.
135)2=5.
85Thisshowsstrongevidenceofassociation(P=0.
016)SensitivitytochoiceofscoresCochran(1954)notedthat".
.
.
Ifthesetofscoresispoor,inthatitbadlydistortsanumericalscalethatreallydoesunderlietheorderedclassication,thetestwillnotsensitive"Formostdatasets,dierentchoicesofmonotonescoresgivesimilarresultsScoresthatarelineartransformsofeachother,suchas(1,2,3,4)and(0,2,4,6),havethesameabsolutecorrelationandhencethesameM2.
Resultsmaydependonthescores,whendataarehighlyunbalanced,withsomecategorieshavingmanymoreobservationsthanothers.
Itisusuallybettertoselectscoresthatreectperceiveddistancesbetweencategories.
SensitivitytochoiceofscoresEquallyspacedscoresoftenprovideareasonablecompromisewhencategoricallabelsdonotsuggestobviouschoicesWhenunsure,dosensitivityanalysisandcheckwhetherresultsaresimilarIfyouchooseasetofscores,andgetasignicantresult,thenthissuggeststhatrowandcolumnarenotindependent.
However,youshouldn'tkeepchoosingalotofdierentscoresuntilyougetasignicantresult.
Example:InfantbirthdefectsbymaternalalcoholconsumptionWithYscore{1,2,3,4,5},M2=1.
83andP=0.
18WithYscoreasmidranks{(1+17,114)/2,24,365.
5,32,013,32,473,32,555.
5},M2=0.
35andP=0.
55WithYscore{0,0.
5,1.
5,4.
0,7.
0},M2=6.
57andP=0.
01MonotonetrendalternativestoindependenceConsiderthehypothesis:H0:XandYareindependentVSH1:YisamonotonefunctionofXGammateststatistic:z=γ/SEwhereSEisthestandarderrorofγdrivedbasedonthedeltamethod.
UnderH0,zfollowsastandardnormaldistribution.
Fortheexampleonhappinessandpoliticalideology,γ=0.
185,z=0.
185/0.
078=2.
37andthetwosidedP=0.
018.
Anapproximate95%CIforγis0.
185±1.
96(0.
078)=(0.
032,0.
338)Thetrueassociationseemstoberelativelyweak.
ExtrapowerwithordinaltestsConsiderthesameexampleofhappinessandpoliticalideology,Ignoringtheorderingofthecategories,Pearsonchi-squaredstatisticsfortestingindependenceisX2=7.
07withdf=4,P=0.
13MHtestisM2=5.
85withdf=1,P=0.
016Gammatestz=2.
37,P=0.
018Thelattertwoareordinaltestswhichhavemorepowerinthisexamplebecausetheyaredesignedtodetectlinearormonotonepatterns,whereastheX2andG2refertothemostgeneralalternative,wherebycellprobabilitiesexhibitanytypeofstatisticaldependence.
Trendtestsfor2XJtablesUsingscoresfortheYvariable,contructtheMHtest,whichdetectdierencesbetweenthetworowmeansofthescoresonYWithmidrankscoresforY,theMHtestisalsocalledWilcoxonorMann-Whitneytest.
Itistwo-samplet-testwithranks.
TheMHtestisalsoequivalenttothetestbasedonz=CDSE0whereCandDarenumbersofconcordantanddiscordantpairsrespectively.
FindscoreCIforthemeasure=P(Y1>Y2)P(Y2>Y1)NominalXordinaltablesandIX2tablesExtensiontoanominalrowvariablewithmorethantwocategories,theMann-WhitneytestextendstoKruskal–WallistestwhichisANOVAtestonranksoftheYvalues.
InIX2tables,Yisbinary.
Thelineartrendstatisticthenreferstoalineartrendintheprobabilityofeitherresponsecategory,suchastheprobabilityofmalformationasafunctionofalcohoconsumption.
Thetestinthiscase,oftencalledCochran-Armitagetrendtest,whichisalsorelatedtologisticregression.
Small-sampleinferenceforcontingencytablesSmall-sampleinferenceforcontingencytablesConditionalontherowandcolumnmargins,theprobabilityofallpossibletables.
Letθbetheoddsratio.
ConsidertheonesidedhypothesisH0:θ=1vsH1:θ>1TheP-valueequalsthesumoftheprobabilitiesofthetablesthathavelargen11,thereforePvalue=0.
0238.
Fisher'sexacttestfor2x2tablesConditioningonbothsetsofmarginaltotals,Fisher'sexactteststatisticisthetableprobabilityFisher'sexacttestfor2x2tablesFortheone-sidedalternative,thesameP-valueresultsusingtablesorderedaccordingtolargern11,largeroddsratio,orlargerdierenceofproportions.
Foratwo-sidedalternative,themostcommonapproachsumsprobabilitiesofequallyorlesslikelytables,thatis,Pvalue=P(p(n11)≤p(to))fortheobservedvaluetoInthepreviousexample,thetwosidedP-valueis0.
0238+0.
0238=0.
0476