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1、NOV分析课件ANalysis Of VAriance(ANOVA)ANOVAANOVA is a statistical analysis technique in which the is a statistical analysis technique in which the total variation is divided into meaningful sources of total variation is divided into meaningful sources of variation.variation.Recall from Recall from Measu

2、rement System AnalysisMeasurement System Analysis,2 2Analysis of Variance(ANOVA)ANOVA is often used for studying the relationship ANOVA is often used for studying the relationship between a between a response variableresponse variable(Y)and one or more (Y)and one or more explanatory explanatory oror

3、 predictor variables predictor variables(Xs).The (Xs).The predictor variables are also called predictor variables are also called factorsfactors or or treatmentstreatments.While the response is While the response is quantitativequantitative,the predictors,the predictors may be either may be either q

4、uantitativequantitative or or qualitativequalitative.However,quantitative predictors are analyzed as if However,quantitative predictors are analyzed as if they are qualitative(or categorical).they are qualitative(or categorical).3 3Analysis of Variance(ANOVA)To study the effect of one or more factor

5、s on a To study the effect of one or more factors on a response,each factor having two or more levels.response,each factor having two or more levels.ANOVA can be used toANOVA can be used ton n determine the statistical significance of effects determine the statistical significance of effectsn n calc

6、ulate the components of variance calculate the components of variancen n estimate the contribution to variation by each estimate the contribution to variation by each identified sourceidentified sourcen n estimate the underlying estimate the underlying noisenoise within the process within the proces

7、s4 4ANOVA PrincipleN(1,1)N(2,2)N(3,3)N(4,4)5 5ANOVA-Terminologyy y is the overall mean of all observations across all factor levels.is the overall mean of all observations across all factor levels.y_ _8 8ANOVA-Terminology i i=y yi i y y is the is the effecteffect of the factor at its of the factor a

8、t its i ithth level.level.y_1234_ _9 9ANOVA-Terminology ij ij=y=yij ij y yi i is the is the residualresidual or or errorerror of the of the j jthth observation at observation at the the i ithth level of the factor.level of the factor.y_14243143_1010ANOVA-Terminology y_Total DeviationUnexplained Devi

9、ationExplained DeviationTotal Deviation =yij y_Unexplained Deviation =yij yi_Explained Deviation =yi y_Total Deviation =Explained Deviation +Unexplained Deviation1111ANOVA table1)1)DFDFTotalTotal=DF=DFFactorFactor +DF +DFErrorErrorDFDFErrorError=DF=DFTotalTotal DF DFFactorFactor=(=(N 1N 1)()(k 1k 1)

10、=N kN k2)2)SSSSTotalTotal=SS=SSFactorFactor +SS +SSErrorError1212ANOVA TableTotal Deviation =Explained Deviation+Unexplained DeviationTotal Deviation =Explained Deviation+Unexplained DeviationSSTotal=SSExplained+SSUnexplained =SSFactor +SSError1313Hypothesis TestingH H0 0:1 1=2 2=k kall group means

11、are equalall group means are equalH Ha a:i i j j for some for some i i j jat least one pair of group means is not equalat least one pair of group means is not equal ANOVA verifies the null hypothesis by comparing the variance ANOVA verifies the null hypothesis by comparing the variance between the g

12、roups against the variation within a group between the groups against the variation within a group mean:mean:The null hypothesis is rejected ifThe null hypothesis is rejected if .1414Hypothesis TestingIn general,As will be seen in following examples,the actual formulation of F*is dependent on the ty

13、pe of ANOVA model.1515ANOVA-Assumptionsn nThe observations are mutually independent.The observations are mutually independent.n nS Stat tat NNonparametrics onparametrics R Runs Testuns Testn nThe The k k groups exhibit homogeneity of variance.groups exhibit homogeneity of variance.i.e.i.e.1 1=2 2=k

14、k n nS Stat tat A ANOVA NOVA Test for Equal Test for Equal V Variancesariancesn nThe The residualsresiduals are normally distributed.are normally distributed.i.e.i.e.e NID(e NID(0,0,)n nS Stat tat B Basic Statistics asic Statistics NNormality Testormality Test1616ANOVA-Assumptions1717ANOVA-ModelsFix

15、ed Effects(ANOVA I)ModelFixed Effects(ANOVA I)Modeln nfactor levels in the experiment are specifically chosenfactor levels in the experiment are specifically chosenn nthe conclusion is relevant only for the chosen levels and may not be the conclusion is relevant only for the chosen levels and may no

16、t be extended to similar treatments that were not consideredextended to similar treatments that were not consideredRandom Effects(ANOVA II)ModelRandom Effects(ANOVA II)Modeln nfactor levels are randomly drawn from a large population of factor levels are randomly drawn from a large population of trea

17、tmentstreatmentsn nthe conclusion is valid for the entire population of treatmentsthe conclusion is valid for the entire population of treatmentsMixed Effects(ANOVA III)ModelMixed Effects(ANOVA III)Modeln none or more factors are one or more factors are fixed factorsfixed factors while others are wh

18、ile others are random random factorsfactors1818ANOVA-ModelsA A k kWay ANOVA model involves the study of Way ANOVA model involves the study of k k factors.factors.k kWay ANOVA I Way ANOVA I:k k factors;factors;all factors of fixed effectall factors of fixed effectk kWay ANOVA IIWay ANOVA II:k k facto

19、rs;factors;all factors of random effectall factors of random effect(Will not be discussed in GB course)(Will not be discussed in GB course)k kWay ANOVA IIIWay ANOVA III:k k factors;factors;some factors of fixed some factors of fixed effecteffect(Will not be discussed in GB course)(Will not be discus

20、sed in GB course)others of random effectothers of random effect1919One-Way ANOVA-I ModelwherewhereY Yij ijis the value of the response variable in the jis the value of the response variable in the jthth trial trial for the i for the ithth factor level or treatment factor level or treatment.is the ov

21、erall mean for all observationsis the overall mean for all observations i i=i i .is the effect of the i.is the effect of the ithth treatment treatment s.t.s.t.ij ijis the random error component is the random error component s.t.s.t.The hypothesis may now be rephrased:The hypothesis may now be rephra

22、sed:H H0 0:1 1 =2 2 =r r =0 =0H Ha a:not all not all i i equal 0 equal 02020One-Way ANOVA-I Model2121ExampleAn engineer want to evaluate the effect of the An engineer want to evaluate the effect of the percentage of the strengthener of the adhesive to percentage of the strengthener of the adhesive t

23、o the bonding force(breakage force of the assy)of a the bonding force(breakage force of the assy)of a bonded actuator.bonded actuator.The engineer decides to investigate four levels of The engineer decides to investigate four levels of percentage of the strengthener (ie,5%,10%percentage of the stren

24、gthener (ie,5%,10%15%&20%).Six samples were tested at each 15%&20%).Six samples were tested at each percentage level.percentage level.The data is stored in“4.ANOVA.MTW”The data is stored in“4.ANOVA.MTW”2222ExampleS Stat tat A ANOVA NOVA OOne-wayne-way2323Examplen nSession WindowsSession WindowsOne-w

25、ay ANOVA:Force versus StrengtherOne-way ANOVA:Force versus StrengtherAnalysis of Variance for Force Analysis of Variance for Force Source DF SS MS F PSource DF SS MS F PStrength 3 382.79 127.60 19.61 0.000Strength 3 382.79 127.60 19.61 0.000Error 20 130.17 6.51Error 20 130.17 6.51Total 23 512.96Tota

26、l 23 512.96 Individual 95%CIs For Mean Individual 95%CIs For Mean Based on Pooled StDev Based on Pooled StDevLevel N Mean StDev -+-+-+-+-Level N Mean StDev -+-+-+-+-5 6 10.000 2.828 (-*-)5 6 10.000 2.828 (-*-)10 6 15.667 2.805 (-*-)10 6 15.667 2.805 (-*-)15 6 17.000 1.789 (-*-)15 6 17.000 1.789 (-*-

27、)20 6 21.167 2.639 (-*-)20 6 21.167 2.639 (-*-)-+-+-+-+-+-+-+-+-Pooled StDev=2.551 10.0 15.0 20.0 25.0Pooled StDev=2.551 10.0 15.0 20.0 25.02424Example2525Example2626ExampleS Stat tat R Regression egression R Re esidual Plotssidual Plots2727Example2828One-Way ANOVA-II ModelwherewhereY Yij ijis the v

28、alue of the response variable in the jis the value of the response variable in the jthth trial trial for the i for the ithth factor level or treatment factor level or treatment.is the overall mean for all observationsis the overall mean for all observations i i=i i .is the effect of the iis the effe

29、ct of the ithth factor level factor levels.t.s.t.ij ijis the random error component is the random error component s.t.s.t.The hypothesis may now be rephrased:The hypothesis may now be rephrased:H H0 0:=0 =0H Ha a:0 02929One-Way ANOVA-II ModelThere is no difference in calculation of the F-Statistic b

30、etween a One-Way ANOVA-I Model and a One-Way ANOVA-II Model.The difference lies in the conclusion.3030ExampleAn engineer is interested in winding machine variability in An engineer is interested in winding machine variability in peel strength of a coil.The engineer selects four winding peel strength

31、 of a coil.The engineer selects four winding machines at random and determines the peel strength of machines at random and determines the peel strength of four samples chosen at random from each winding four samples chosen at random from each winding machine.machine.3131ExampleS Stat tat A ANOVA NOV

32、A G General Linear Modeleneral Linear Model3232Examplen nSession WindowsSession WindowsGeneral Linear Model:Peel Strength versus MachineGeneral Linear Model:Peel Strength versus MachineFactor Type Levels Values Factor Type Levels Values Machine random 4 1 2 3 4Machine random 4 1 2 3 4Analysis of Var

33、iance for Peel Str,using Adjusted SS for TestsAnalysis of Variance for Peel Str,using Adjusted SS for TestsSource DF Seq SS Adj SS Adj MS F PSource DF Seq SS Adj SS Adj MS F PMachine 3 89.187 89.187 29.729 15.68 0.000Machine 3 89.187 89.187 29.729 15.68 0.000Error 12 22.750 22.750 1.896Error 12 22.750 22.750 1.896Total 15 111.937 Total 15 111.937 3333Example3434End of PresentationRev 1 25 Jun 02Rev 1 25 Jun 02