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1、Chapter 7 ANOVA First things first!Stage1Total variabilityBetween-treatmentvariabilityStage 2 FactorBvariabilityFactorAvariabilityWithin-treatmentvariabilityInteractionvariabilityAnalysis of Variance-ANOVANo-interaction effect means that the factor is independent with no impact on each other.Interac
2、tion effects represent the combined effects of factors on the dependent measure.Analysis of Variance-ANOVAANOVAOne-way ANOVATwo-way ANOVANo Interaction Effect in Two-way ANOVAInteraction Effect in Two-way ANOVAAnalysis of Variance-ANOVAFactor A:level of r as A1,A2,.,Ai,Ar;Factor B:level of s as B1,B
3、2,.,Bj,BS:(Ai,Bj):ij(i=1,2,3,r;j=1,2,3,s)ij means the (Ai,Bj)Factor B1B2BjBsA111121j1sA221222j2s.Aii1i2ijisArr1r2rjrsNo interaction:each varibale affects the outcome directly.Table x Two-Way ANOVA experiments with no interaction effectTwo-way ANOVA-No Interaction Two-way ANOVA ApproachClaim identifi
4、ed:The hypotheses of interest in two-way ANOVA without interaction effect as follows:Significance specified:rii10sjj10 as factor A ith effectij as factor B jth effectH01H02),.2,1,.2,1(,)2,0(,sjriNxijijjiijTwo-way ANOVA-No Interaction Two-way ANOVA ApproachSignificance specified:df identified:Critica
5、l value set:F(A)value:F(dfA,dfe)F(B)value:F(dfB,dfe)F calculated:dfdfA A=r r 1 1 dfdfB B =s s 1 1dfdfe e =(r(r 1)(s1)(s 1)1)=The hypotheses of interest in two-way ANOVA without interaction effect are as follows:=Two-way ANOVA-No Interaction SST=SSA+SSB+SSe dfdfA A=r r 1 1 dfdfB B=s s 1 1 dfdfe e=(=(
6、r r 1)(1)(s s 1)1)dfdfT T=n=n 1=1=rsrs 1 1=risjijTxxSS112)(21211riisjriiAxxsxxSS21211sjjrisjjBxxrxxSSAAAdfSSMS BBBdfSSMS eeedfSSMS 211risjjiijxxxxSSeTwo-way ANOVA-No Interaction Sum of squaresDegree of freedomMean squaresFCritical valuep-valueASSAr-1*BSSBs-1*ErrorSSE(r 1)(s-1)Total SSTrs-1FA F(r-1,(
7、r-1)(s-1),reject H01,significant difference in means(0.05:*or 0.01:*)FB F(s-1,(r-1)(s-1),reject H02,significant difference in means(0.05:*or 0.01:*)Otherwise,accept H0.1rSSMSAA1sSSMSBB)1)(1(srSSeMSeeAMSMS eBMSMS)1)(1(,1(srrFA)1)(1(,1(srsFBTable x.Two-way ANOVA with no interaction effectTwo-way ANOVA
8、-No Interaction Example:Two-way ANOVA-No Interaction ),.,2,1,.,2,1(,sjrixijjiijFactor BFactor A1234133.3525.4528.2528.95232.6026.0028.4027.70331.8525.2028.5028.95ijx62.284/)95.2850.2820.2585.31(68.284/)70.2740.2800.2660.32(00.294/)95.2825.2845.2535.33(321xxxmeans A53.283/)95.2870.2795.28(38.283/)50.
9、2840.2825.28(55.253/)20.2500.2645.25(60.323/)85.3160.3235.33(4321xxxxmeans B77.284/)53.2838.2855.2560.32(3/)62.2868.2800.29(xAssumption:Two-way ANOVA-No Interaction Example:),.,2,1,.,2,1(,sjrixijjiij0141.0767.28625.28092.0767.28675.28233.0767.280.29321332211xxxxxxxxii0233.0767.2853.28383.0767.2838.2
10、8217.3767.2855.25833.3767.2860.32432144332211xxxxxxxxxxjjTwo-way ANOVA-No Interaction)4,3,2,1,3,2,1(,jixjiijEstimated cell means:Factor BFactor A1234132.83325.78328.61728.767232.50825.45828.29228.442332.45825.40828.24228.392Factor BFactor A12341-0.515 0.3350.368-0.1822-0.09-0.54-0.1070.74330.610.21-
11、0.257-0.557 The residuals/errors:SSe=2.20)4,3,2,1,3,2,1(ji,xxijij2ijsijx Two-way ANOVA-No Interaction Formulas for sums of squares33.0)141.0()092.0(23.0(42221221riiriiAsxxsSS73.75)233.0()383.0()217.3(833.3(322221221riisjjBrxxrSS20.2SeS26.7877.28112112SSeSSSSxxxSSBArisjijrisjijTTwo-way ANOVA-No Inter
12、action Example:SourcedfSSMSFF crit.p valueA20.330.1660.4525.143 FINV(0.05,2,6).656 FDIST(0.45,2,6)B375.7325.2468.794.757 FINV(0.05,3,6).001 FDIST(25.24,3,6)Error62.200.367Total1178.26Factor A(Irrigation),F 0.05,no significant differFactor B(Land type),F F crit.,p value=.0010.01,extremely significant differANOVA TableTwo-way ANOVA-No Interaction