Two_Way_ANOVA

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1、3.3:TWO-WAYANALYSISOFVARIANCEØTheanalysisofvariancetechniqueshownpreviouslyiscalledaone-wayanalysisofvariancesincethereisonlyoneindependentvariable.ØThetwo-wayanalysisofvarianceisanextensionoftheone-wayanalysisofvariance;itinvolvestwoindependentvariables.ØTheind

2、ependentvariablesarealsocalledfactors.ØUsingthetwo-wayanalysisofvariance,theresearcherisabletotesttheeffectsoftwoindependentvariablesorfactorsononedependentvariable.ØItisananalysismethodforrandomizedblockdesign.ØThebenefitofconsideringotherfactorsisthatwecanredu

3、cetheerrorvariance.ØAsecondtreatmentvariablethanwhenincludedintheANOVAanalysiswillhavetheeffectofreducingtheSSEterm,isreferredtoasablockingvariable.ØThepurposeofdesigningarandomizedblockexperimentistoreducethewithintreatmentvariationtomoreeasilydetectdifferences

4、betweenthetreatmentmeans.28ConditionsorAssumptionsfortheTwo-WayANOVA1.Thepopulationsfromwhichthesampleswereobtainedmustbenormallyorapproximatelynormallydistributed.2.Thesamplesmustbeindependent.3.Thevariancesofthepopulationsfromwhichthesampleswereselectedmustbee

5、qual.4.Thegroupsmustbeequalinsamplesize.TheSumofSquaresThetotalvariationcanbepartitionintothreesourcesofvariation:SST=SSA+SSB+SSEThetreatmentssumofsquares,SSA:Theblockssumofsquares,SSB:28Thetotalsumofsquares,SST:Theerrorsumofsquares,SSE:SSE=SST-SSA–SSBMeanSquare

6、sforTreatment,MSA:MeanSquaresforBlock,MSB:MeanSquaresforError,MSE:28TestStatistic:1)2)HypothesisStatementTotestthetreatmentmeans:·Atwo-wayANOVAhasseveralnullhypotheses.·Thereisoneforeachindependentvariable.H0:m1=m2=m3=…=mk@H0:Thereisnodifferentbetweentreatmentme

7、ansH1:ThereisadifferentbetweentreatmentmeansTotesttheblockmeans:H0:ThereisnodifferentbetweenblockmeansH1:Thereisadifferentbetweenblockmeans.28AnalysisofVarianceSummaryTableSourceSumofdfMeanFSquaresSquaresTreatmentsSSAa-1MSABlocksSSBb-1MSBErrorSSE(a-1)(b-1)MSE___

8、______________________________________TotalSSTn-1Example:Arandomizedblockexperimentproducedthefollowingstatistics:a=5b=12SSA=1500SSB==1000SST=3500Testtodeterminewheth