1. 10-1 of 29 ANOVA Experimental Design and Analysis of Variance McGraw-Hill/Irwin Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved.

2. 10-2 of 29 Experimental Design and Analysis of Variance 10.1Basic Concepts of Experimental Design 10.2One-Way Analysis of Variance 10.3The Randomized Block Design 10.4Two-Way Analysis of Variance

3. 10-3 of 29 Objective: To compare and estimate the effect of different treatments on the response variable. 10.1 Basic Concepts of Experimental Design Example 10.1 The Gasoline Mileage Case Does gasoline mileage vary with gasoline type? Type AType BType C x A1 =34.0x B1 =35.3x C1 =33.3 x A2 =35.0x B2 =36.5x C2 =34.0 x A3 =34.3x B3 =36.4x C3 =34.7 x A4 =35.5x B4 =37.0x C4 =33.0 x A5 =35.8x B5 =37.6x C5 =34.9 Response Variable:Gasoline mileage (in mpg) Treatments:Gasoline types – A, B, C

4. 10-4 of 29 10.2 One-Way Analysis of Variance Are there differences in the mean response  ,    …,  p associated with the p treatments? H 0 :   =    … =  p H a : At least two of the  ,  ,…,  p differ Or, is the between- treatment variability large compared to the within-treatment variability?

5. 10-5 of 29 Partitioning the Total Variability in the Response   p i= n j= iij p i= ii p n j= ij ii )x(x)xx(n)x 11 2 1 2 11 2  Squares of Squares of Squares of SumError SumTreatment Sum Total  yVariabilit y Treatment yVariabilit Within Between Total  SSE SST = SSTO

6. 10-6 of 29 F Test for Difference Between Treatment Means H 0 :   =    …=  p (no treatment effect) H a : At least two of the  ,    …,  p differ Test Statistic: Reject H 0 if F > F   or p-value <  F  is based on p-1 numerator and n-p denominator degrees of freedom.

7. 10-7 of 29 The One-Way Analysis of Variance Table DegreesSum of MeanF Sourceof FreedomSquaresSquaresStatistic Treatmentsp-1SSTMST = SSTF = MST p-1 MSE Errorn-pSSEMSE = SSE n-p Totaln-1SSTO Example 10.5 The Gasoline Mileage Case (Excel Output)

8. 10-8 of 29 Pairwise Comparisons, Individual Intervals Individual 100(1 -  )% confidence interval for  i -  h t  is based on n-p degrees of freedom. Example 10.6 The Gasoline Mileage Case (A vs B,  = 0.05)

9. 10-9 of 29 Pairwise Comparisons, Simultaneous Intervals Tukey simultaneous 100(1 -  )% confidence interval for  i -  h q  is the upper  percentage point of the studentized range for p and (n-p) from Table A.9. m denotes common sample size. Example 10.6 The Gasoline Mileage Case (A vs B,  = 0.05)

10. 10-10 of 29 Estimation of Individual Treatment Means Individual 100(1 -  )% confidence interval for  i t  is based on n-p degrees of freedom. Example 10.6 The Gasoline Mileage Case (Type B,  = 0.05)

11. 10-11 of 29 A randomized block design compares p treatments (for example, production methods) on each of b blocks (or experimental units; for example, machine operators.) A generalization of the paired difference design, this design controls for variability in experimental units by comparing each treatment on the same (not independent) experimental units. Blocks 1 2 3 … b Treatments 12...p12...p x ij = response from treatment i and block j 10.3 The Randomized Block Design

12. 10-12 of 29 Example: Randomized Block Design Example 10.7 The Defective Cardboard Box Case

13. 10-13 of 29 The ANOVA Table, Randomized Blocks DegreesSum of MeanF Sourceof FreedomSquaresSquaresStatistic Treatmentsp-1SSTMST = SSTF(trt) = MST p-1 MSE Blocksb-1SSBMSB = SSBF(blk) = MSB b-1 MSE Error(p-1)(b-1)SSEMSE = SSE (p-1)(b-1) Totalpb-1SSTO

14. 10-14 of 29 F Test for Treatment Effects H 0 : No difference between treatment effects H a : At least two treatment effects differ Test Statistic: Reject H 0 if F > F   or p-value <  F  is based on p-1 numerator and (p-1)(b-1) denominator degrees of freedom.

15. 10-15 of 29 F Test for Block Effects H 0 : No difference between block effects H a : At least two block effects differ Test Statistic: Reject H 0 if F > F   or p-value <  F  is based on b-1 numerator and (p-1)(b-1) denominator degrees of freedom.

16. 10-16 of 29 Example: Randomized Block ANOVA Example 10.7 The Defective Cardboard Box Case Minitab Output Analysis of Variance for Defects Source DF SS MS F P Method 3 90.917 30.306 47.43 0.000 Operator 2 18.167 9.083 14.22 0.005 Error 6 3.833 0.639 Total 11 112.917 Data Summary

17. 10-17 of 29 Estimation of Treatment Differences Under Randomized Blocks, Individual Intervals Individual 100(1 -  )% confidence interval for  i  -  h  t  is based on (p-1)(b-1) degrees of freedom. Example 10.8 The Defective Cardboard Box Case (4 vs 1) t  with (3-1)(4-1) = 6 degrees of freedom.

18. 10-18 of 29 Tukey simultaneous 100(1 -  )% confidence interval for  i  -  h  q  is the upper  percentage point of the studentized range for p and (p-1)(b-1) from Table A.9. Example 10.8 The Defective Cardboard Box Case (4 vs 1) q  for 4 and 6. Estimation of Treatment Differences Under Randomized Blocks, Simultaneous Intervals

19. 10-19 of 29 A two factor factorial design compares the mean response for a levels of factor 1 (for example, display height) and each of b levels of factor 2 ( for example, display width.) A treatment is a combination of a level of factor 1 and a level of factor 2. 10.4 Two-Way Analysis of Variance Factor 1 12...a12...a Factor 2 1 2 3 … b x ijk =response for the k th experimental unit (k=1,…,m) assigned to the i th level of Factor 1 and the j th level of Factor 2

20. 10-20 of 29 Example: Two-Way Analysis of Variance Example 10.9 The Shelf Display Case

21. 10-21 of 29 Example: Graphical Analysis of Bakery Demand

22. 10-22 of 29 Possible Treatment Effects in Two-Way ANOVA

23. 10-23 of 29 Two-Way ANOVA Table DegreesSum of MeanF Sourceof FreedomSquaresSquaresStatistic Factor 1a-1SS(1)MS(1) = SS(1)F(1) = MS(1) a-1 MSE Factor 1b-1SS(2)MS(2) = SS(2)F(2) = MS(2) b-1 MSE Interaction(a-1)(b-1)SS(int)MS(int) = SS(int) F(int) = MS(int) (a-1)(b-1) MSE Errorab(m-1)SSEMSE = SSE ab(m-1) Totalabm-1SSTO

24. 10-24 of 29 Example: Two-Way ANOVA Example 10.9 The Shelf Display Case Minitab Output Analysis of Variance for Demand Source DF SS MS F P Height 2 2273.88 1136.94 185.62 0.000 Width 1 8.82 8.82 1.44 0.253 Interaction 2 10.08 5.04 0.82 0.462 Error 12 73.50 6.12 Total 17 2366.28 Data Summary

25. 10-25 of 29 F Tests for Treatment Effects H 0 : No difference between treatment effects H a : At least two treatment effects differ Test Statistics: Reject H 0 if F > F   or p-value <  F  is based on a-1 and ab(m-1) degrees of freedom. F  is based on b-1 and ab(m-1) degrees of freedom. F  is based on (a-1)(b-1) and ab(m-1) degrees of freedom. Main Effects Interaction

26. 10-26 of 29 Estimation of Treatment Differences Under Two-Way ANOVA, Factor 1 Individual 100(1 -  )% confidence interval for  i  -  i’  t  is based on ab(m-1) degrees of freedom. Tukey simultaneous 100(1 -  )% confidence interval for  i  -  i’  q  is the upper  percentage point of the studentized range for a and ab(m-1) from Table A.9. Example 10.10 The Shelf Display Case (M vs B)

27. 10-27 of 29 Estimation of Treatment Differences Under Two-Way ANOVA, Factor 2 Individual 100(1 -  )% confidence interval for   j -   j’ t  is based on ab(m-1) degrees of freedom. Tukey simultaneous 100(1 -  )% confidence interval for   j -   j’ q  is the upper  percentage point of the studentized range for b and ab(m-1) from Table A.9.

28. 10-28 of 29 Experimental Design and Analysis of Variance 10.1Basic Concepts of Experimental Design 10.2One-Way Analysis of Variance 10.3The Randomized Block Design 10.4Two-Way Analysis of Variance Summary: