信息与通信第六章英文简化版方差分析参考ch10DesignofExperimentsandAnalysis课件

1、Chapter 10Design of Experiments and Analysis of Variance第1页，共73页。One-Way ANOVA F-Test第2页，共73页。Types of Regression ModelsExperimentalDesignsOne-Way AnovaCompletely RandomizedRandomized BlockTwo-Way AnovaFactorial 第3页，共73页。One-Way ANOVA F-Test1.Tests the Equality of 2 or More (p) Population Means2.Var

2、iablesOne Nominal Scaled Independent Variable2 or More (p) Treatment Levels or ClassificationsOne Interval or Ratio Scaled Dependent Variable3.Used to Analyze Completely Randomized Experimental Designs 第4页，共73页。One-Way ANOVA F-Test Assumptions1.Randomness & Independence of ErrorsIndependent Random S

3、amples are Drawn for each condition2.NormalityPopulations (for each condition) are Normally Distributed3.Homogeneity of VariancePopulations (for each condition) have Equal Variances第5页，共73页。One-Way ANOVA F-Test HypothesesH0: 1 = 2 = 3 = . = pAll Population Means are EqualNo Treatment EffectHa: Not A

4、ll j Are EqualAt Least 1 Pop. Mean is DifferentTreatment EffectNOT 1 2 . p第6页，共73页。One-Way ANOVA F-Test HypothesesH0: 1 = 2 = 3 = . = pAll Population Means are EqualNo Treatment EffectHa: Not All j Are EqualAt Least 1 Pop. Mean is DifferentTreatment EffectNOT 1 2 . pXf(X)1 = 2 = 3Xf(X)1 = 23第7页，共73页

5、。Why Variances?Observe one sample from each treatment groupTheir means may be slightly differentHow different is enough to conclude population means are different?Depends on variability within each populationHigher variance in population higher variance in meansStatistical tests are conducted by com

6、paring variability between means to variability within each sample第8页，共73页。Two PossibleExperiment OutcomesSame treatment variationDifferent random variationACant reject equality of means!Reject equality of means!第9页，共73页。Two More PossibleExperiment OutcomesSame treatment variationDifferent random va

7、riationABDifferent treatment variationSame random variationCant reject equality of means!RejectReject第10页，共73页。1.Compares 2 Types of Variation to Test Equality of Means2.Comparison Basis Is Ratio of Variances 3.If Treatment Variation Is Significantly Greater Than Random Variation then Means Are Not

8、Equal4.Variation Measures Are Obtained by Partitioning Total VariationOne-Way ANOVA Basic Idea第11页，共73页。One-Way ANOVA Partitions Total Variation第12页，共73页。One-Way ANOVA Partitions Total VariationTotal variation第13页，共73页。One-Way ANOVA Partitions Total VariationVariation due to treatmentTotal variation

9、第14页，共73页。One-Way ANOVA Partitions Total VariationVariation due to treatmentVariation due to random samplingTotal variation第15页，共73页。One-Way ANOVA Partitions Total VariationVariation due to treatmentVariation due to random samplingTotal variationSum of Squares AmongSum of Squares BetweenSum of Squar

10、es TreatmentAmong Groups Variation第16页，共73页。One-Way ANOVA Partitions Total VariationVariation due to treatmentVariation due to random samplingTotal variationSum of Squares WithinSum of Squares Error (SSE)Within Groups VariationSum of Squares AmongSum of Squares BetweenSum of Squares Treatment (SST)A

11、mong Groups Variation第17页，共73页。Total VariationXGroup 1Group 2Group 3Response, X第18页，共73页。Treatment VariationXX3X2X1Group 1Group 2Group 3Response, X第19页，共73页。Random (Error) VariationX2X1X3Group 1Group 2Group 3Response, X第20页，共73页。SS=SSE+SST第21页，共73页。But第22页，共73页。Thus, SS=SSE+SST第23页，共73页。One-Way ANOV

12、A F-Test Test Statistic1.Test StatisticF = MST / MSEMST Is Mean Square for TreatmentMSE Is Mean Square for Error2.Degrees of Freedom1 = p -12 = n - pp = # Populations, Groups, or Levelsn = Total Sample Size第24页，共73页。One-Way ANOVA Summary TableSource ofVariationDegreesofFreedomSum ofSquaresMeanSquare

13、(Variance)FTreatmentp - 1SSTMST =SST/(p - 1)MSTMSEErrorn - pSSEMSE =SSE/(n - p)Totaln - 1SS(Total) =SST+SSE第25页，共73页。The F distributionTwo parametersincreasing either one decreases F-alpha (except for v2 F-Between groups 6109.7141 2 3054.85705 7.69 0.0007 Within groups 54045.8255 136 397.395776- Tot

14、al 60155.5396 138 435.909707Bartletts test for equal variances: chi2(2) = 7.1931 Probchi2 = 0.027第33页，共73页。One-Way ANOVA F-Test Thinking ChallengeYoure a trainer for Microsoft Corp. Is there a difference in mean learning times of 12 people using 4 different training methods ( =.05)?M1M2M3M4101113189

15、1682359925Use the following table. 1984-1994 T/Maker Co.第34页，共73页。Summary Table(Partially Completed)Source ofVariationDegrees ofFreedomSum ofSquaresMeanSquare(Variance)FTreatment(Methods)348Error80Total第35页，共73页。F04.07One-Way ANOVA F-Test Solution*H0: 1 = 2 = 3 = 4Ha: Not All Equal = .051 = 3 2 = 8

16、Critical Value(s):Test Statistic: Decision:Conclusion:Reject at = .05There Is Evidence Pop. Means Are Different = .05FMSTMSE11610116.第36页，共73页。Summary Table Solution*Source ofVariationDegrees ofFreedomSum ofSquaresMeanSquare(Variance)FTreatment(Methods)4 - 1 = 334811611.6Error12 - 4 = 88010Total12 -

17、 1 = 11428第37页，共73页。10.26: condition 1 vs. 2Two-sample t test with equal variances- Group | Obs Mean Std. Err. Std. Dev. 95% Conf. Interval-+- 1 | 50 30.64 2.833439 20.03544 24.94599 36.33401 2 | 42 26.21429 3.65729 23.70195 18.82824 33.60033-+-combined | 92 28.61957 2.270253 21.7755 24.10999 33.129

18、14-+- diff | 4.425714 4.559216 -4.631963 13.48339-Degrees of freedom: 90 Ho: mean(1) - mean(2) = diff = 0 Ha: diff 0 t = 0.9707 t = 0.9707 t = 0.9707 P |t| = 0.3343 P t = 0.1671第38页，共73页。10.26 condition 2 vs. 3Two-sample t test with equal variances- Group | Obs Mean Std. Err. Std. Dev. 95% Conf. Int

19、erval-+- 2 | 42 26.21429 3.65729 23.70195 18.82824 33.60033 3 | 47 15.12766 2.290554 15.70325 10.51701 19.73831-+-combined | 89 20.35955 2.176533 20.53337 16.03415 24.68495-+- diff | 11.08663 4.220767 2.697394 19.47586-Degrees of freedom: 87 Ho: mean(2) - mean(3) = diff = 0 Ha: diff 0 t = 2.6267 t =

20、 2.6267 t = 2.6267 P |t| = 0.0102 P t = 0.0051第39页，共73页。Multiple Comparisons ProblemPAt least one of p intervals fails to contain the true difference= 1 PAll c intervals contain the true differences= 1 (1-alpha)c alphaIf comparing many pairs, need greater confidence for any one of them than you woul

21、d for rejecting equality of any one pair第40页，共73页。Multiple Comparisons Procedure1.Tells Which Population Means Are Significantly Different Example: 1 = 2 32.Post Hoc ProcedureDone After Rejection of Equal Means in ANOVAOutput From Many Statistical computer Programs various versions (Tukey, Bonferron

22、i, etc.)第41页，共73页。10.26 Multiple Comparisons (Bonferroni)Row Mean-|Col Mean | 1 2-+- 2 | -4.42571 | 0.872 | 3 | -15.5123 -11.0866 | 0.001 0.029第42页，共73页。Randomized Block Design第43页，共73页。Types of Regression ModelsExperimentalDesignsOne-Way AnovaCompletely RandomizedRandomized BlockTwo-Way AnovaFactor

23、ial 第44页，共73页。Randomized Block Design1.Experimental Units (Subjects) Are Assigned Randomly to BlocksBlocks are Assumed Homogeneous2.One Factor or Independent Variable of Interest2 or More Treatment Levels or Classifications3. One Blocking Factor第45页，共73页。Randomized Block DesignFactor Levels: (Treatm

24、ents)A, B, C, DExperimental UnitsTreatments are randomly assigned within blocksBlock 1ACDBBlock 2CDBABlock 3BADC . . .Block bDCAB第46页，共73页。Randomized Block F-Test1.Tests the Equality of 2 or More (p) Population Means2.VariablesOne Nominal Scaled Independent Variable2 or More (p) Treatment Levels or

25、ClassificationsOne Nominal Scaled Blocking VariableOne Interval or Ratio Scaled Dependent Variable3.Used with Randomized Block Designs 第47页，共73页。Randomized Block F-Test Assumptions1.NormalityProbability Distribution of each Block-Treatment combination is Normal2.Homogeneity of VarianceProbability Di

26、stributions of all Block-Treatment combinations have Equal Variances第48页，共73页。Randomized Block F-Test HypothesesH0: 1 = 2 = 3 = . = pAll Population Means are EqualNo Treatment EffectHa: Not All j Are EqualAt Least 1 Pop. Mean is DifferentTreatment Effect1 2 . p Is Wrong 第49页，共73页。Randomized Block F-

27、Test HypothesesH0: 1 = 2 = 3 = . = pAll Population Means are EqualNo Treatment EffectHa: Not All j Are EqualAt Least 1 Pop. Mean is DifferentTreatment Effect1 2 . p Is Wrong Xf(X)1 = 2 = 3Xf(X)1 = 23第50页，共73页。The F Ratio for Randomized Block DesignsSS=SSE+SSB+SST第51页，共73页。Randomized Block F-Test Tes

28、t Statistic1.Test StatisticF = MST / MSEMST Is Mean Square for TreatmentMSE Is Mean Square for Error2.Degrees of Freedom1 = p -12 = n b p +1p = # Treatments, b = # Blocks, n = Total Sample Size第52页，共73页。Randomized Block F-Test Critical ValueIf means are equal, F = MST / MSE 1. Only reject large F!Al

29、ways One-Tail!Fapnp(,)10Reject H0Do NotReject H0F 1984-1994 T/Maker Co.第53页，共73页。Randomized Block F-Test ExampleYou wish to determine which of four brands of tires has the longest tread life. You randomly assign one of each brand (A, B, C, and D) to a tire location on each of 5 cars. At the .05 leve

30、l, is there a difference in mean tread life?Tire LocationBlockLeft FrontRight FrontLeft RearRight RearCar 1A: 42,000C: 58,000B: 38,000D: 44,000Car 2B: 40,000D: 48,000A: 39,000C: 50,000Car 3C: 48,000D: 39,000B: 36,000A: 39,000Car 4A: 41,000B: 38,000D: 42,000C: 43,000Car 5D: 51,000A: 44,000C: 52,000B:

31、 35,000第54页，共73页。F03.49Randomized Block F-Test SolutionH0: 1 = 2 = 3= 4Ha: Not All Equal = .051 = 3 2 = 12 Critical Value(s):Test Statistic: Decision:Conclusion:Reject at = .05There Is Evidence Pop. Means Are Different = .05F = 11.9933第55页，共73页。Exercise 10.47 What is the purpose of blocking on weeks

32、 in this study?c. Are the mean number of walkers different among the prompting conditions?d. Which pairwise means are significantly different?e. What assumptions are required for the analysis in c and d? 第56页，共73页。Factorial Experiments第57页，共73页。Types of Regression ModelsExperimentalDesignsOne-Way An

33、ovaCompletely RandomizedRandomized BlockTwo-Way AnovaFactorial 第58页，共73页。Factorial Design1.Experimental Units (Subjects) Are Assigned Randomly to TreatmentsSubjects are Assumed Homogeneous2.Two or More Factors or Independent VariablesEach Has 2 or More Treatments (Levels)3.Analyzed by Two-Way ANOVA第

34、59页，共73页。Factorial Design ExampleTreatmentFactor 2 (Training Method)FactorLevelsLevel 1Level 2Level 3Level 119 hr.20 hr.22 hr.Factor 1(High)11 hr.17 hr.31 hr.(Motivation)Level 227 hr.25 hr.31 hr.(Low)29 hr.30 hr.49 hr.第60页，共73页。Advantages of Factorial Designs1.Saves Time & Efforte.g., Could Use Sepa

35、rate Completely Randomized Designs for Each Variable2.Controls Confounding Effects by Putting Other Variables into Model3.Can Explore Interaction Between Variables第61页，共73页。Two-Way ANOVA第62页，共73页。Types of Regression ModelsExperimentalDesignsOne-Way AnovaCompletely RandomizedRandomized BlockTwo-Way A

36、novaFactorial 第63页，共73页。Two-Way ANOVA1.Tests the Equality of 2 or More Population Means When Several Independent Variables Are Used2.Same Results as Separate One-Way ANOVA on Each VariableBut Interaction Can Be Tested3.Used to Analyze Factorial Designs 第64页，共73页。Two-Way ANOVA Assumptions1.NormalityP

37、opulations are Normally Distributed2.Homogeneity of VariancePopulations have Equal Variances3.Independence of ErrorsIndependent Random Samples are Drawn第65页，共73页。Two-Way ANOVA Data TableXijkLevel i Factor ALevel j Factor BObservation kFactorFactor BA12.b1X111X121.X1b1X112X122.X1b22X211X221.X2b1X212X222.X2b2:aXa11Xa21.Xab1Xa12Xa22.Xab2第66页，共73页。Two-Way ANOVA Null Hypotheses1.No Difference in Means Due to Factor AH0: 1. = 2. =. = a.2.No Difference in Means Due to Factor BH0: .1. =