第六章(英文-简化版)-方差分析参考资料-ch10-Design of Experiments and Analysis.ppt

Click to edit Master title,Click to edit Master text styles,Second Level,Third Level,Fourth Level,Fifth Level,10-,*,2003 Pearson Prentice Hall,Chapter 10,Design of Experiments and,Analysis of Variance,One-Way ANOVA F-Test,Types of Regression Models,Experimental,Designs,One-Way,Anova,Completely Randomized,Randomized Block,Two-Way,Anova,Factorial,One-Way ANOVA F-Test,1.Tests the Equality of 2 or More(,p,)Population Means,2.Variables,One Nominal Scaled Independent Variable,2 or More(,p,)Treatment Levels or Classifications,One Interval or Ratio Scaled Dependent Variable,3.Used to Analyze Completely Randomized Experimental Designs,One-Way ANOVA F-Test Assumptions,1.Randomness&Independence of Errors,Independent Random Samples are Drawn for each condition,2.Normality,Populations(for each condition)are Normally Distributed,3.Homogeneity of Variance,Populations(for each condition)have Equal Variances,One-Way ANOVA F-Test Hypotheses,H,0,:,1,=,2,=,3,=.=,p,All Population Means are Equal,No Treatment Effect,H,a,:Not All,j,Are Equal,At Least 1 Pop.Mean is Different,Treatment Effect,NOT,1,2,.,p,One-Way ANOVA F-Test Hypotheses,H,0,:,1,=,2,=,3,=.=,p,All Population Means are Equal,No Treatment Effect,H,a,:Not All,j,Are Equal,At Least 1 Pop.Mean is Different,Treatment Effect,NOT,1,2,.,p,X,f(X),1,=,2,=,3,X,f(X),1,=,2,3,Why Variances?,Observe one sample from each treatment group,Their means may be slightly different,How different is enough to conclude population means are different?,Depends on variability within each population,Higher variance in population,higher variance in means,Statistical tests are conducted by comparing variability between means to variability within each sample,Two PossibleExperiment Outcomes,Same treatment variation,Different random variation,A,Cant reject equality of means!,Reject equality of means!,Two More PossibleExperiment Outcomes,Same treatment variation,Different random variation,A,B,Different treatment variation,Same random variation,Cant reject equality of means!,Reject,Reject,1.Compares 2 Types of Variation to Test Equality of Means,2.Comparison Basis Is Ratio of Variances,3.If Treatment Variation Is Significantly Greater Than Random Variation then Means Are,Not,Equal,4.Variation Measures Are Obtained by Partitioning Total Variation,One-Way ANOVA,Basic Idea,One-Way ANOVA,Partitions Total Variation,One-Way ANOVA,Partitions Total Variation,Total variation,One-Way ANOVA,Partitions Total Variation,Variation due to treatment,Total variation,One-Way ANOVA,Partitions Total Variation,Variation due to treatment,Variation due to random sampling,Total variation,One-Way ANOVA,Partitions Total Variation,Variation due to treatment,Variation due to random sampling,Total variation,Sum of Squares Among,Sum of Squares Between,Sum of Squares Treatment,Among Groups Variation,One-Way ANOVA,Partitions Total Variation,Variation due to treatment,Variation due to random sampling,Total variation,Sum of Squares Within,Sum of Squares Error(SSE),Within Groups Variation,Sum of Squares Among,Sum of Squares Between,Sum of Squares Treatment(SST),Among Groups Variation,Total Variation,X,Group 1,Group 2,Group 3,Response,X,Treatment Variation,X,X,3,X,2,X,1,Group 1,Group 2,Group 3,Response,X,Random(Error)Variation,X,2,X,1,X,3,Group 1,Group 2,Group 3,Response,X,SS=SSE+SST,But,Thus,SS=SSE+SST,One-Way ANOVA F-Test Test Statistic,1.Test Statistic,F,=,MST,/,MSE,MST,Is Mean Square for Treatment,MSE,Is Mean Square for Error,2.Degrees of Freedom,1,=,p,-1,2,=,n,-,p,p,=#Populations,Groups,or Levels,n,=Total Sample Size,One-Way ANOVA Summary Table,Source of,VariationDegrees,of,Freedom,Sum of,Squares,Mean,Square,(Variance),F,Treatment,p-1,SST,MST=,SST/(p-1),MST,MSE,Error,n-p,SSE,MSE=,SSE/(n-p),Total,n-1,SS(Total)=,SST+SSE,The F distribution,Two parameters,increasing either one decreases F-alpha(except for v2 F,-,Between groups 6109.7141 2 3054.85705 7.69 0.0007Within groups 54045.8255 136 397.395776,-,Total 60155.5396 138 435.909707,Bartletts test for equal variances:chi2(2)=7.1931,Prob,chi2=0.027,One-Way ANOVA F-Test Thinking Challenge,Youre a trainer for Microsoft Corp.Is there a difference in,mean,learning times of 12 people using 4 different training methods(,=.05,)?,M1,M2,M3,M4,1011131891682359925,Use the following table.,1984-1994 T/Maker Co.,Summary Table(Partially Completed),Source of,Variation,Degrees of,Freedom,Sum of,Squares,Mean,Square,(Variance),F,Treatment,(Methods),348,Error,80,Total,F,0,4.07,One-Way ANOVA F-Test Solution*,H,0,:,1,=,2,=,3,=,4,H,a,:,Not All Equal,=,.05,1,=,3,2,=,8,Critical Value(s):,Test Statistic:,Decision:,Conclusion:,Reject at,=.05,There Is Evidence Pop.Means Are Different,=.05,F,MST,MSE,116,10,11,6,.,Summary Table Solution*,Source of,Variation,Degrees of,Freedom,Sum of,Squares,Mean,Square,(Variance),F,Treatment,(Methods),4-1=3,348,116,11.6,Error,12-4=8,80,10,Total,12-1=11,428,10.26:condition 1 vs.2,Two-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 33.12914,-+-,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=0.1671,10.26 condition 2 vs.3,Two-sample t test with equal variances,-,Group|,Obs,Mean Std.Err.Std.Dev.95%Conf.Interval,-+-,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=2.6267 t2.6267,P|t|=0.0102 P t=0.0051,Multiple Comparisons Problem,PAt least one of p intervals fails to contain the true difference,=1 PAll c intervals contain the true differences,=1 (1-alpha),c,alpha,If comparing many pairs,need greater confidence for any one of them than you would for rejecting equality of any one pair,Multiple Comparisons Procedure,1.Tells Which Population Means Are Significantly Different,Example:,1,=,2,3,2.,Post Hoc,Procedure,Done After Rejection of Equal Means in ANOVA,Output From Many Statistical computer Programs various versions(,Tukey,Bonferroni,etc.),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,Randomized Block Design,Types of Regression Models,Experimental,Designs,One-Way,Anova,Completely Randomized,Randomized Block,Two-Way,Anova,Factorial,Randomized Block Design,1.Experimental Units(Subjects)Are Assigned Randomly to Blocks,Blocks are Assumed Homogeneous,2.One Factor or Independent Variable of Interest,2 or More Treatment Levels or Classifications,3.One Blocking Factor,Randomized Block Design,Factor Levels:,(Treatments),A,B,C,D,Experimental Units,Treatments are randomly assigned within blocks,Block 1,A,C,D,B,Block 2,C,D,B,A,Block 3,B,A,D,C,.,.,.,.,.,.,.,.,.,.,.,.,.,.,.,Block b,D,C,A,B,Randomized Block F-Test,1.Tests the Equality of 2 or More(,p,)Population Means,2.Variables,One Nominal Scaled Independent Variable,2 or More(,p,)Treatment Levels or Classifications,One Nominal Scaled Blocking Variable,One Interval or Ratio Scaled Dependent Variable,3.Used with Randomized Block Designs,Randomized Block F-Test Assumptions,1.Normality,Probability Distribution of each Block-Treatment combination is Normal,2.Homogeneity of Variance,Probability Distributions of all Block-Treatment combinations have Equal Variances,Randomized Block F-Test Hypotheses,H,0,:,1,=,2,=,3,=.=,p,All Population Means are Equal,No Treatment Effect,H,a,:Not All,j,Are Equal,At Least 1 Pop.Mean is Different,Treatment Effect,1,2,.,p,Is,Wrong,Randomized Block F-Test Hypotheses,H,0,:,1,=,2,=,3,=.=,p,All Population Means are Equal,No Treatment Effect,H,a,:Not All,j,Are Equal,At Least 1 Pop.Mean is Different,Treatment Effect,1,2,.,p,Is,Wrong,X,f(X),1,=,2,=,3,X,f(X),1,=,2,3,The F Ratio for Randomized Block Designs,SS=SSE+SSB+SST,Randomized Block F-Test Test Statistic,1.Test Statistic,F,=,MST,/,MSE,MST,Is Mean Square for Treatment,MSE,Is Mean Square for Error,2.Degrees of Freedom,1,=,p,-1,2,=,n b p+1,p,=#Treatments,b,=#Blocks,n,=Total Sample Size,Randomized Block F-Test Critical Value,If means are equal,F,=,MST,/,MSE,1.Only reject large,F,!,Always One-Tail!,F,a,p,n,p,(,),1,0,Reject H,0,Do Not,Reject H,0,F,1984-1994 T/Maker Co.,Randomized Block F-Test Example,You 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,level,is there a difference in,mean,tread life?,Tire LocationBlock,Left Front,Right Front,Left Rear,Right Rear,Car 1,A:42,000,C:58,000,B:38,000,D:44,000,Car 2,B:40,000,D:48,000,A:39,000,C:50,000,Car 3,C:48,000,D:39,000,B:36,000,A:39,000,Car 4,A:41,000,B:38,000,D:42,000,C:43,000,Car 5,D:51,000,A:44,000,C:52,000,B:35,000,F,0,3.49,Randomized Block F-Test Solution,H,0,:,1,=,2,=,3,=,4,H,a,:,Not All Equal,=,.05,1,=,3,2,=,12,Critical Value(s):,Test Statistic:,Decision:,Conclusion:,Reject at,=.05,There Is Evidence Pop.Means Are Different,=.05,F=11.9933,Exercise 10.47,What is the purpose of blocking on weeks 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?,Factorial Experiments,Types of Regression Models,Experimental,Designs,One-Way,Anova,Completely Randomized,Randomized Block,Two-Way,Anova,Factorial,Factorial Design,1.Experimental Units(Subjects)Are Assigned Randomly to Treatments,Subjects are Assumed Homogeneous,2.Two or More,Factors,or Independent Variables,Each Has 2 or More Treatments(Levels),3.Analyzed by Two-Way ANOVA,Factorial Design Example,Treatment,Factor 2 (Training Method),Factor,Levels,Level 1,Level 2,Level 3,Level 1,19 hr.,20 hr.,22 hr.,Factor,1,(High),11 hr.,17 hr.,31 hr.,(Motivation),Level 2,27 hr.,25 hr.,31 hr.,(Low),29 hr.,30 hr.,49 hr.,Advantages of Factorial Designs,1.Saves Time&Effort,e.g.,Could Use Separate Completely Randomized Designs for Each Variable,2.Controls Confounding Effects by Putting Other Variables into Model,3.Can Explore Interaction Between Variables,Two-Way ANOVA,Types of Regression Models,Experimental,Designs,One-Way,Anova,Completely Randomized,Randomized Block,Two-Way,Anova,Factorial,Two-Way ANOVA,1.Tests the Equality of 2 or More Population Means When Several Independent Variables Are Used,2.Same Results as Separate One-Way ANOVA on Each Variable,But Interaction Can Be Tested,3.Used to Analyze Factorial Designs,Two-Way ANOVA Assumptions,1.Normality,Populations are Normally Distributed,2.Homogeneity of Variance,Populations have Equal Variances,3.Independence of Errors,Independent Random Samples are Drawn,Two-Way ANOVA Data Table,X,i,j,k,Level i Factor A,Level j Factor B,Observation k,Factor,Factor B,A,1,2,.,b,1,X,111,X,121,.,X,1b1,X,112,X,122,.,X,1b2,2,X,211,X,221,.,X,2b1,X,212,X,222,.,X,2b2,:,:,:,:,:,a,X,a11,X,a21,.,X,ab1,X,a12,X,a22X,ab2,Two-Way ANOVA Null Hypotheses,1.No Difference in Means Due to Factor A,H,0,:,1,.,=,2,.,=.=,a,.,2.No Difference in Means Due to Factor B,H,0,:,.,1,.,=,.,2,.,=.=,.,b,.,3.No Interaction of Factors A&B,H,0,:,AB,ij,=0,Total Variation,Two-Way ANOVA,Total Variation Partitioning,Variation Due to Treatment A,Variation Due to Random Sampling,Variation Due to Interaction,SSE,SSA,SS(AB),SS(Total),Variation Due to Treatment B,SSB,Source of,Variation,Degrees of,Freedom,Sum of,Squares,Mean,Square,F,A,(Row),a-1,SS(A),MS(A),MS(A),MSE,B,(Column),b-1,SS(B),MS(B),MS(B),MSE,AB,(Interaction),(a-1)(b-1),SS(AB),MS(AB),MS(AB),MSE,Error,n-,ab,SSE,MSE,Total,n-1,SS(Total),Two-Way ANOVA Summary Table,Same as Other Designs,Interaction,1.Occurs When Effects of One Factor Vary According to Levels of Other Factor,2.When Significant,Interpretation of Main Effects(A&B)Is Complicated,3.Can Be Detected,In Data Table,Pattern of Cell Means in One Row Differs From Another Row,In Graph of Cell Means,Lines Cross,Graphs of Interaction,Effects of Motivation(High or Low)&Training Method(A,B,C)on Mean Learning Time,Interaction,No Interaction,Average,Response,A,B,C,High,Low,Average,Response,A,B,C,High,Low,Exercise 10.63,Conclusion,1.Described Analysis of Variance(ANOVA),2.Explained the Rationale of ANOVA,3.Compared Experimental Designs,4.Tested the Equality of 2 or More Means,Completely Randomized Design,Factorial Design,