浙大生物统计作业3、4答案

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: