22雙因素析因設(shè)計(jì)雙因素析因設(shè)計(jì)實(shí)例分析某面包公司為城市里的許多超市提供包裝好的意大利薄片面包設(shè)計(jì)一個(gè)試驗(yàn)用來(lái)研究貨架的高度（底部、中部、上部）和貨架的寬度（常規(guī)、寬型）對(duì)該種面包銷售量的影響第六章析因設(shè)計(jì)分析雙因素析因設(shè)計(jì)多因素析因設(shè)計(jì)144雙因雙因素析因設(shè)計(jì)方差分析模型：=+i+j++i1,2,,a;j1,2,,b;k1,2,,試驗(yàn)單元:12個(gè)具有相似銷售量和顧客群的因素貨架的高度，三個(gè)水平(底部、中部、上部貨架的寬度，兩個(gè)水平(常規(guī)、寬型6個(gè)處理(33因素B因素AAB12AB1jAB1bAB21AB22AB2jAB2b┇┇┇┇┇┇┇ABi1ABi2ABijABib┇┇┇┇┇┇ABa1ABa2ABajABab66總平總平方和及自由度的分解(Y) 2i ) 2.j)2 ij .i . j)2 Y SSTO=SSASSBSSABdfabr1(a1)(b1)(a1)(b1)ab(r df588固定模=+i+j+iji1,2,,a;j1,2,,b;k1,2,,~N(+i+j+ij,2~N(0,2ijijijiij兩因素析因設(shè)計(jì)的方差分析表異自由度 E(MS)固定 E(MS)混合 隨機(jī) a- 2+br 2+r2br22+r2+br a1 a1 b- 2+ar 2+r2ar22+ar b1 (a-1) 22+r 2+r (b- (a1)(b1) rorab(r-1) 7DataDataInputHeight$Width$Sales@@;A1B147A1B2A1B143A1B2A2B162A2B2A2B168A2B2A3B141A3B2A3B139A3B2;PROCClassHeightMODELSales=HeightWidthHeight*Width;MeansHeight/LSD;MeansHeight/Tukey;MeansWidth/LSD;MeansWidth/Tukey;MeansHeight*Width/LSD;變異來(lái)源自由 均 期望均方固A因 a- MSA2 br a1 B因 b- MSB2 ar b1 互 (a-1)(b-1)MSAB2 (a1)(b1) 誤 ab(r― MSEFH0i0H1i=F*MSA/MSE~F(a-1ab(r-1如果H0是H0所有j0H1不是jF*MSB/MSE~F(b-1ab(r-1如果H0H0ij=0;H1:不是所有ij=F*=MSAB/MSE~F((a-1)(b-1),ab(r-1))如果H0是 DependentVariable:Sum MeanSquareFValuePr>F Height*Width tNA4B4B4A6A6Height Std222222實(shí)例分析水稻品種和栽培措施的試驗(yàn)因素A:(A1A29(3試驗(yàn)單元27依變量 Sum MeanSquareF ANOVAWithoutBlockDataInputVariety$Stand$BlockYield@@;A1B118A1B128A1B13A1B217A1B227A1B23A1B314A1B325A1B33A2B119A2B129A2B13A2B217A2B229A2B23A2B316A2B325A2B33A3B119A3B127A3B13A3B218A3B227A3B23A3B318A3B328A3B33;PROCClassVarietyStandMODELYield=VarietyStandVariety*StandMeansVariety 基于最小二乘的均值估計(jì)；MeansVariety/Tukey; 打印出t檢驗(yàn)(H0:LSM=0);PDIFF:MeansStand/LSD; MeansStand/Tukey; 度;TDIFF:打印出t值及顯著程度LSMeansVariety*Stand/STDERRPDIFF;22428224如如果AB互作是顯著的，就沒(méi)有必要AorB主效是否顯我們應(yīng)該檢驗(yàn)A因素在特定B水平中的表現(xiàn)或B因素在特定的A水平中的表現(xiàn)TukeyGrou Mean LeastLeastSquaresMeansforeffectVariety*StandPr>|t|forH0:LSMean(i)=LSMean(j)112437425568738699123456789123456789DataDataInputVariety$Stand$BlockYield@@;A1B118A1B128A1B13A1B217A1B227A1B23A1B314A1B325A1B33A2B119A2B129A2B13A2B217A2B229A2B23A2B316A2B325A2B33A3B119A3B127A3B13A3B218A3B227A3B23A3B318A3B328A3B33;PROCClassVarietyStandMODELYield=Variety|StandBlock;Contrast'Linear'Stand-101;Contrast'Quadratic'Stand1-21;MeansVariety/LSD;數(shù)量型因素的分析因素A是質(zhì)量型(A1A2B是數(shù)量型SAS命令ClassVarietyStandMODELYield=Variety|StandBlock;Contrast'Linear'Stand-101;Contrast'Quadratic'Stand1-2 SumSquares MeanSquareFValuePr>F 因Linear和Qadratic的線性組合的系數(shù)正交，所以Stand的平方和分解成了歸因于Linear和Quadratc的平方和比較X比較X是線性的還是二次Xici效應(yīng)的正交分解需根據(jù)效應(yīng)的水平數(shù)查相應(yīng)的正交分解系數(shù)表cc2iContrastTest(df=1)LinearContrastXL022QuadraticContrast62/66ContrastTest(df=LinearContrast-QuadraticContrast-DataInputVariety$BlockXLXQYield@@;A11-0.7071070.4082488A12-0.7071070.4082488A13-0.7071070.408248A110.000000-0.8164977A120.000000-0.8164977A130.000000-0.816497A110.7071070.4082484A120.7071070.4082485A130.7071070.408248A21-0.7071070.4082489A22-0.7071070.4082489A23-0.7071070.408248A210.000000-0.8164977A220.000000-0.8164979A230.000000-0.816497A210.7071070.4082486A220.7071070.4082485A230.7071070.408248A31-0.7071070.4082489A32-0.7071070.4082487A33-0.7071070.408248A310.000000-0.8164978A320.000000-0.8164977A330.000000-0.816497A310.7071070.4082488A320.7071070.4082488A330.7071070.408248;PROCClassVarietyMODELYield=VarietyXLXQMeansVariety/LSD;SumMeanFPr>211222SumMeanF221142確定B確定AB互作因素的取值X1×X3,確定B確定AB互作因素的取值X1×X3,X2×X3forA×X1×X4,X2×X4forA×二因素析因設(shè)計(jì)的回歸分析方法Yb0b1X1b2X2bpXp確定A因素的取 (df=2) - -ContrastContrast(df=c2i20226666DataDataInputX1X2X3X4BlockYield@@;A1L=X1*X3;A2L=X2*10-0.7071070.4082481810-0.7071070.4082482810-0.7071070.4082483100.000000-0.81649717100.000000-0.81649727100.000000-0.8164973100.7071070.40824814100.7071070.40824825100.7071070.408248301-0.7071070.4082481901-0.7071070.4082482901-0.7071070.4082483010.000000-0.81649717010.000000-0.81649729010.000000-0.8164973010.7071070.40824816010.7071070.40824825010.7071070.4082483-1-1-0.7071070.40824819-1-1-0.7071070.40824827-1-1-0.7071070.4082483-1-10.000000-0.81649718-1-10.000000-0.81649727-1-10.000000-0.8164973-1-10.7071070.40824818-1-10.7071070.40824828-1-10.7071070.4082483;PROCMODELYield=X1X2X3X4A1LA2L;YYY10-810-810-810-710-710-810410510401-901-901-801-701-901-6016015014---9---7---8---8---7---8--8--8--9SumSumMeanFPr>6111111tPr>------SumMeanFPr>41111tPr>--------B因素有三個(gè)水平：108cm2,163cm2,m163,55,11,C1播種密度為 (S ) 產(chǎn)量預(yù)測(cè)?=7.1850.722X11.650X30.943X1X3ABY101010101010101010101010隨機(jī)模隨機(jī)模~N(,2 22=+i+j+iji1, ,a;j1,,b;k1, ,ijk~N(0,2),~N(0,2j~N(0,2),~N(0,2i密 品種 品種 品種 76品種品種品 實(shí)例分析實(shí)例分析因素區(qū)組(3次重復(fù)A:(A1A2 密度9個(gè)處理(3試驗(yàn)單元:27個(gè)小區(qū)依變量: E(MS)RandomFactorA 2+r2br2 Factor b- 2+r2ar Interaction(a-1)(b-1)MSAB2+r FH0:20;H1:2 F*=MSA/MSAB~F(a-1,(a-1)(b-1))如果H0是H0:20;H1:2 F*=MSB/MSAB~F(b-1,(a-1)(b-1))如果H0是H0: 0;H1:2 F*=MSAB/MSE~F((a-1)(b-1),ab(r-1))如果H0是NotNotSelectingforErrorSourceDFTypeIIISSMeanSquareFValuePr>DataInputVariety$Stand$BlockYield@@; Random:要求GLM程序?qū)1B118A1B128A1B13A1B217A1B227A1B23 RandomA1B314A1B325A1B33 適當(dāng)?shù)腇A2B119A2B129A2B13 A2B217A2B229A2B23A2B316A2B325A2B33 確定作FA3B119A3B127A3B13A3B218A3B227A3B23A3B318A3B328A3B33 確定作F;PROCGLMdata=Rice;ClassVarietyStandBlock;MODELYield=VarietyStandVariety*StandBlock;RandomVarietyStandVariety*StandBlock/test;TestH=VarietyStandE=Variety*Stand; ClassVarietyStandBlock;MODELYield=VarietyStandVariety*StandBlock; TypeIMeanFPr>2242TypeIMeanFPr>224混合混合模~N(+i,2 2=+i+j+iji1, ,a;j1,,b;k1, ,iijk~N(0,2),ij~N(0,2),~N(0,20REML000000000實(shí)例分析實(shí)例分析因素區(qū)組(3次重復(fù)A:(A1A2 密度9個(gè)處理(3試驗(yàn)單元:27個(gè)小區(qū)依變量: E(MS)MixedFactor a- MSA2+r2+br a iFactor b- MSB2+ar Interaction(a-1)(b-1)MSAB2+r MSEFH0:20;H1:2 F*MSA/MSAB~F(a-1a-1)(b-1))H0H0:20;H1:2 F*MSB/MSE~F(b-1ab(r-1))如果H0H0:20;H1:2 F*MSAB/MSE~F((a-1b-1ab(r-1))H0是真PROCGLMdata=Rice;ClassPROCGLMdata=Rice;ClassVarietyStandBlock;MODELYield=VarietyStandVariety*StandBlock;RandomStandVariety*StandBlock/test;TestH=VarietyE=LSMeansVariety/STDERRPDIFFE=Variety*Stand;Adjust:requestamultiplecomparisonadjustmentforp-valueandconfidencelimitsforthedifferenceofLs-MeanAdjust=…選項(xiàng)改變了TDIFF和PDIFF的結(jié)果PROCMIXEDdata=Ricemethod=type3;ClassVarietyStandBlock;MODELYield=RandomStandVariety*StandBlock/SOLUTION;LSMeansVariety/ADJUST=TUKEY;DataInputVariety$Stand$BlockYield@@;A1B118A1B128A1B13A1B217A1B227A1B23A1B314A1B325A1B33A2B119A2B129A2B13A2B217A2B229A2B23A2B316A2B325A2B33A3B119A3B127A3B13A3B218A3B227A3B23A3B318A3B328A3B33;SourceTypeSourceTypeIIIExpectedMeanSAS中給出的Stand的期望均方有錯(cuò)誤，不應(yīng)該包含上表紅色的那一項(xiàng)TypeIMeanFPr>2242TypeIMeanFPr>224ResultsofPROCNotSelectingforErrorSourceDFTypeIIISSMeanSquareFValuePr>224224ResultsofResultsofPROC對(duì)Stand效應(yīng)進(jìn)行顯著性檢123123StandardPr>123LeastSquaresMeansforeffectVarietyPr>|t|forH0:2Var(Res)+3Var(Var*Sta)2Var(Res)+3Var(Var*Sta)+942ErrorFPr>F44Cov預(yù)測(cè)值預(yù)測(cè)值SolutionforFixed tValuePr>|t| - - - - SolutionforRandom EstimateSE tValuePr>|t| 0.7181 0.7181 - 0.7181 - EstimateSE tValuePr>|t| 0.8577 0.8577 - 0.8577 - 0.8577 0.8577 - 0.8577 - - 0.8577 - - 0.8577 - 0.8577 - 16- -0.06548 16- <.0001YYijkm=+i+j++ij+ik+jk+ijk+i1,2,,a;j1,2,,b;k1,2,,c;m1,2,,6.225-35歲的人的、脂肪含量、吸煙程度對(duì)壓力試驗(yàn)中的忍受能力的影響因素8個(gè)處理(22平方和與自由度的分解 )2 +cr(Y...Y...Y...+Y....)2+br(Y...Y...Y... i +ar(Y...Y...Y... k+r )2 i.k i . ..k SSTO=SSASSBSSCSSABSSACSSBCSSABCdfTOdfAdfBdfCdfABdfACdfBCdfABCdfabr1(a1)(b1)(c1)(a1)(b1)(a1)(c1)(b1)(c1)(a1)(b1)(c1)abc(rANOVA表(固定模 E(MS) a- 2+bcr a1 b- 2+acr b1 c- 2+abr c1 (a-1)(b- (a1)(b1) (a-1)(c- (a1)(c1) (b-1)(c- (b1)(c1)j (a-1)(b- (a1)(b1)(c (c- jErrorAbc(r- HH0所有ij0H1不是所有ijF*=MSAB/MSE~F((a-1)(b-1),abc(r-1))如果H0是H0所有ik0H1不是所有ikF*=MSAC/MSE~F((a-1)(c-1),abc(r-1))如果H0是真H0:jk0H1不是所有jkF*=MSBC/MSE~F((b-1)(c-1),abc(r-1))如果H0H0:ijk0H1不是所有ijkF*=MSABC/MSE~F((a-1)(b-1)(c-1),abc(r-1))如H0FH0i0H1iF*=MSA/MSE~F(a-1,abc(r-1))H0是真H0j=0H1jF*=MSB/MSE~F(b-1,abc(r-1))如果H0是真H0k0H1不是所有kF*=MSC/MSE~F(c-1,abc(r-1))如果H0是真H0:2=H0:2= H1:2F*MSA/MSC1~F(a-1dfC1H0是MSC1=MSAB+MSAC(MSABMSACMSABC df df (MSAB (MSAC (MSABCH0:2=0;H1:2F*=MSB/MSC2~F(b-1,dfC2)H0MSC2=MSAB+MSBC(MSABMSBCMSABCH:2=0;H01C (MSAB (MSBC (MSABC2 22dfF*=MSC/MSC3~F(c-1,dfC3)如果H0MSC3=MSAC+MSBC(MSACMSBCMSABC Cdf(MSAC)2(MSBC)2(MSABCdf隨機(jī)模 E(MS)Random 2+r +cr2+br2bcr 2+r +cr2+ar2acr 2+r +br2+ar2abr 2+r +cr 2+r +br 2+r +ar 2+r 實(shí)實(shí)例分析25-35歲的人的、脂肪含量、吸煙程度對(duì)力試驗(yàn)中的忍受能力的影因素 8個(gè)處理(22H0:2=0;H1:2> F*MSAB/MSABC~F((a1)(b1a1)(b1)(c1))H0H:2=0;H:2> F*=MSAC/MSABC~F((a1)(c1),(a1)(b1)(c1))H0H:2=0;H: > F*=MSBC/MSABC~F((b1)(c1),(a1)(b1)(c1)H0H:2=0;H: > F*=MSABC/MSE~F((b1)(c1),abc(r1))如果H0固固定模PROCClassSexFatMODELScore=Sex|Fat|Smoking;MeansSex/Tukey;MeansFat/Tukey;MeansSmoking/Tukey;LSMeansSex*Fat/STDERRPDIFF;LSMeansSex*Smoking/STDERRPDIFF;LSMeansFat*Smoking/STDERRPDIFF;LSMeansSex*Fat*Smoking/STDERRDataInputSex$Fat$Somking$Score@@;A1B1C124A1B1C218A1B1C129A1B1C219A1B1C125A1B1C223A2B1C120A2B1C215A2B1C122A2B1C210A2B1C118A2B1C211A1B2C115A1B2C215A1B2C115A1B2C220A1B2C112A1B2C213A2B2C116A2B2C210A2B2C19 A2B2C214A2B2C111A2B2C2; Pr>F NABScorePr>123412341234SASSASPROCClassSexFatMODELScore=Sex|Fat|Smoking/P;RandomFatSmokingSex*FatTestH=FatSmokingE=Fat*Smoking;MeansSex/Tukey;殘EstimatesofSex*Fat*SmokingEstimatesofSex*Fat*Smoking MeanFPr>1111111TestsofHypothesesforMixed ysisofError:MS(S*F)+MS(S*M)-MS(S*F*M)+11E-Error:MS(S*F)+MS(F*M)-MS(S*F*M)+11E-Error:MS(S*F)+MS(F*M)-MS(S*F*M)+11E-Error:Cov0實(shí)例分析棉花品種、播期、密度試驗(yàn)因素(混合隨機(jī)區(qū)組(3個(gè)重復(fù)質(zhì)量型A:品種(A1質(zhì)量型B:播期(B1數(shù)量型度(3500株/畝,5000株/畝,12(22試驗(yàn)單元36依變量:產(chǎn)量DataDataInputVariety$Time$Stand$$YieldA1B1C1R112A1B1C1R214A1B1C1R313A1B1C2R112A1B1C2R211A1B1C2R311A1B1C3R110A1B1C3R29A1B1C3R39A1B2C1R110A1B2C1R29A1B2C1R39A1B2C2R19A1B2C2R29A1B2C2R38A1B2C3R16A1B2C3R26A1B2C3R37A2B1C1R13A2B1C1R22A2B1C1R34A2B1C2R14A2B1C2R23A2B1C2R34A2B1C3R17A2B1C3R26A2B1C3R37A2B2C1R12A2B2C1R22A2B2C1R33A2B2C2R13A2B2C2R24A2B2C2R35A2B2C3R15A2B2C3R27A2B2C3R37;PROCClassVarietyTimeStandMODELYield=Variety|Time|StandBlock;Contrast'Linear'Stand-101;Contrast'Quadratic'Stand1-21;MeansVariety/Tukey;MeansTimeLSMeansVariety*Time/STDERRPDIFF;LSMeansVariety*Stand/STDERRPDIFF;LSMeansTime*Stand/STDERRPDIFF;LSMeansVariety*Time*Stand/STDERRPROCGLM的分析結(jié) DFFPr> Variety*Time*Stand TukeyGrou TukeyGrou 86420 Pr> 12341234 6 20 VarTim Std_QYldVar Std_QYldVarTim Std_Q -1-0.7071070.408248 -1-0.7071070.408248 -1-0.7071070.408248 - -0.816497 - -0.816497 - -0.816497 -10.7071070.408248 -10.7071070.408248 -10.7071070.408248 1-0.7071070.408248 1-0.7071070.408248 1-0.7071070.408248 -0.816497 -0.816497 -0.816497 0.7071070.408248 0.7071070.408248 0.7071070.408248- -1-0.7071070.408248 - -1-0.7071070.408248 - -1-0.7071070.408248- - -0.816497 - - -0.816497 - - -0.816497- -10.7071070.408248 - -10.7071070.408248 - -10.7071070.408248- 1-0.7071070.408248 - 1-0.7071070.40