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1 HomeworkThreeofBiostatisticsandExperimentalDesign1．Thefollowingdataareaportionoftheresponsescollectedduringaninter-laboratorystudy.Eachoftheseverallaboratorieswassentanumberofmaterialsthatwerecarefullychosentohavedifferentmeasurementvaluesonthecharacteristicofinterest.Thelaboratorieswererequiredtoperformthreeseparateanalysesofthetestmaterial.LaboratoryMaterialReplicate A12.212.312.21B15.515.015.3 C18.118.118.2 A12.612.312.72B15.015.515.2 C18.518.318.6 A12.712.812.73B15.315.215.2 C18.018.217.91)What’stheexperimentaldesign?Two-‐‑way nested designs(random model) 2)Writeoutthefactorsandtheirlevelsofthistrial;Factor A: Laboratory, Levels: 1 2 3; Factor B: Material, Levels: A B C. 3)SetupanANOVAmodelforthistrial,defineeachofvariablesinthemodel;FactorAFactorBinFactorAA1B1(1)B2(1)B3(1)A2B1(2)B2(2)B3(2)A3B1(3)B1(3)B1(3)Yijk=µ+αi+βj(i)+εijki=1,2,3;j=1,2,3;k=1,2,3.Yijkisobservationvalue;µispopulationmean;αiistheeffectundertheithlaboratoryfactor,αi~N(0,σ2α);i=1,2,3;βj(i)istheeffectofthejthmaterialfactorunderithlaboratoryfactor,βj(i)~N(0,σ2β);i=1,2,3;εijkisresidualerror,εijk~N(0,σ2ε),k=1,24)Ifthemodelisarandommodel(eacheffectisrandomeffectinmodel),writeoutthestatisticfortestingsignificanceofeacheffect.(不要求采用上表数据进行实际计算)CauseofVariationDegreesoffreedomMSE(MS)RandomA2MSAσ2ε+rσ2β+brσ2αB(A)6MSB(A)σ2ε+rσ2βError18MSEσ2εH0:σα2=0;H1:σα20F*=MSA/MSB(A)~F(2,6)ifH0istrueH0:σβ2=0;H1:σβ20F*=MSB(A)/MSE~F(6,18)ifH0istrue2 Machines 1 2 3 4 Operators123 123 123 123Mentalparts1799446 928676 885346 364062Mentalparts2627457 997968 755657 535647 2．Tostudythesurfaceroughnes（s表面光洁度）ofmetalparts(金属零件)producedby4differentmachines. Oneverymachine,3operatorswereassignedtoprocess2metalparts.Sincethese4machinesareindifferentplaces,total12operatorswererandomlyassignedtoeachmachine.Thefollowingdatawerecollected: 1)Whatexperimentaldesignwasemployedinthisstudy？Two-waynesteddesigns.(mixedmodel) 2)Pleasewriteoutthelinearregressionmodelforanalyzingthesedata,andtrytodefineeveryeffectsinthisregressionmodel.Yijk=μ+αi+βj(i)+εijk~N(μ+αi,σ2ε+σ2β)Yijkisobservationvalue; μ is population mean; αiistheeffectoftheithmachinefactor,Σαi=0,i=1,2,3,4; βj(i)istheeffectofthejthoperatorsfactorunderithmachinefactor,βj(i)~N(0,σ2β);i=1,2,3;εijkisresidualerror,εijk~N(0,σ2ε),k=1,2。3)Totestthesignificaneofeacheffectandcomparethedifferenceinsurfaceroughnessofmetalpartsbetweendifferentmachines,pleasewriteouttheSASprogramforthisanalysis(Note:ONLYSASprogramisneeded.).DataMachine;InputMachine$Operators$Result@@;Datalines;……………………;PROCGLM;ClassMachineOperators;MODELResult=MachineOperators(Machine);RandomOperators(Machine)/Test;MeansMachine/TukeyE=Operators(Machine);RUN;PROCMIXED;MODELResult=Machine;RandomOperators(Machine)/SOLUTION;LSMeansMachine/ADJUST=TUKEY;RUN;3 123456789Accumulatedprecipitation(X) 34.5 35.8 38.2 33.1 43.2 29.2 30.7 39.3 30.7Onsettimeofinsectdisease(Y) 12 7 9 16 -1 3 9 2 13 34.51235.8738.2933.11643.2-129.2330.7;939.3230.713 3．Inordertostudytherelationshipbetweentheonsettimeofoneinsectdiseaseofriceandaccumulatedprecipitationofplant,thesampledataislistedinfollowingtable TwodifferentmodelareemployedinanalysisbySAS,theoutputofSASisasfollows:Theresultsforthefirstlinearregressionmodel: Theresultsforthesecondlinearregressionmodel: 1)AccordingtoaboveoutputofSAS,writeouttheSASprogramforcorrespondinganalysis;Datazx;inputXY@@;Datalines; 4 Procreg;ModelY=X;Run;Procreg;ModelY=X/Noint;Run; 2)Writeoutthemodelsusedinanalysis,whichoneisbest?Why?model1:Yi=b0+b1Xi+ei;model2:Yi=b1Xi+ei.model2isbetter.Becausetheadjustedcoefficientofdeterminationofmodel2ishigherthanmodel1,anditsP-valueisalsolower.3)Forabove-mentionedmodels,makeconclusiononthelinearrelationship;Model1:P-value=0.13470.05doesn’thavesignificantlinearrelationshipsModel2:P-value=0.00720.05havesignificantlinearrelationships. 4)Writeoutthefinalregressionequationoftheonsettimeofinsectdiseaseontheaccumulatedprecipitation,what’sthedeterminationcoefficientofthemodel?What’sthestatisticalmeaning?Takemodel2:ŷ=0.20907X,R2=0.6152;ItexplainsthatthelinearrelationshipsbetweenOnsettimeofinsectdisease(Y)andAccumulatedprecipitation(X)canaccountfor61.52%ofthetotalvariation.5 HomeworkFourofBiostatisticsandExperimentalDesigns(1)Accordingtothedesignofexperiment,writeoutthedistributionofobservation；Yhij～N(μ+Gh,σ2D+σ2L+σ2GD+σ2GL+σ2DL+σ2ε)(2)Writeouttheformulaofcalculatingsumofsquaresforeacheffect；SSTO=SSG+SSD+SSL+SSGD+SSGL+SSDL+SSESSG=dl(Yh..−Y...)h∑2SSD=gl(Y.i.−Y...)i∑2SSL=gd(Y..j−Y...)j∑2SSGD=l(Yhi.−Yh..−Y.i.+Y...)i∑2h∑SSGL=d(Yh.j−Yh..−Y..j+Y...)j∑2h∑SSDL=g(Y.ij−Y.i.−Y..j+Y...)j∑2i∑SSE=(Yhij−Yhi.)j∑2i∑h∑(3)WriteoutthemeanexpectationofeachsumsquaresforeacheffectE(MSG)=σε2+lσGD2+dσGL2+dlg−1Gh2h∑E(MSD)=σε2+glσD2+gσDL2E(MSL)=σε2+gdσL2+gσDL2E(MSGD)=σε2+lσGD2E(MSGL)=σε2+dσGL26 E(MSDL)=σε2+gσDL2E(MSE)=σε2(4)Howtotestthesignificanceduetovarietyeffect？H0:Gh2h∑=0;H1:Gh2h∑0If H0 is true: