Design and Analysis of Experiments

1. Design and Analysis of Experiments Dr. Tai-Yue Wang Department of Industrial and Information Management National Cheng Kung University Tainan, TAIWAN, ROC

2. Blocking and Confounding in Two-Level Factorial Designs Dr. Tai-Yue Wang Department of Industrial and Information Management National Cheng Kung University Tainan, TAIWAN, ROC

3. Outline • Introduction • Blocking Replicated 2k factorial Design • Confounding in 2k factorial Design • Confounding the 2k factorial Design in Two Blocks • Why Blocking is Important • Confounding the 2k factorial Design in Four Blocks • Confounding the 2k factorial Design in 2pBlocks • Partial Confounding

4. Introduction • Sometimes it is impossible to perform all of runs in one batch of material • Or to ensure the robustness, one might deliberately vary the experimental conditions to ensure the treatment are equally effective. • Blocking is a technique for dealing with controllable nuisance variables

5. Introduction • Two cases are considered • Replicated designs • Unreplicated designs

6. Blocking a Replicated 2k Factorial Design • A 2k design has been replicated n times. • Each set of nonhomogeneous conditions defines a block • Each replicate is run in one of the block • The runs in each block would be made in random order.

7. Blocking a Replicated 2k Factorial Design -- example • Only four experiment trials can be made from a single batch. Three batch of raw material are required.

8. Blocking a Replicated 2k Factorial Design -- example • Sum ofSquares in Block • ANOVA

9. Confounding in The 2k Factorial Design • Problem: Impossible to perform a complete replicate of a factorial design in one block • Confounding is a design technique for arranging a complete factorial design in blocks, where block size is smaller than the number of treatment combinations in one replicate.

10. Confounding in The 2k Factorial Design • Short comings: Cause information about certain treatment effects (usually high order interactions ) to be indistinguishable from, or confounded with, blocks. • If the case is to analyze a 2k factorial design in 2p incomplete blocks, where p<k, one can use runs in two blocks (p=1), four blocks (p=2), and so on.

11. Confounding the 2k Factorial Design in Two Blocks • Suppose we want to run a single replicate of the 22 design. Each of the 22=4 treatment combinations requires a quantity of raw material, for example, and each batch of raw material is only large enough for two treatment combinations to be tested. • Two batches are required.

12. Confounding the 2k Factorial Design in Two Blocks • One can treat batches as blocks • One needs assign two of the four treatment combinations to each blocks

13. Confounding the 2k Factorial Design in Two Blocks • The order of the treatment combinations are run within one block is randomly selected. • For the effects, A and B: A=1/2[ab+a-b-(1)] B=1/2[ab-a+b-(1)] Are unaffected

14. Confounding the 2k Factorial Design in Two Blocks • For the effects, AB: AB=1/2[ab-a-b+(1)] is identical to block effect  AB is confounded with blocks

15. Confounding the 2k Factorial Design in Two Blocks • We could assign the block effects to confounded with A or B. • However we usually want to confound with higher order interaction effects.

16. Confounding the 2k Factorial Design in Two Blocks • We could confound any 2k design in two blocks. • Three factors example

17. Confounding the 2k Factorial Design in Two Blocks • ABC is confounded with blocks • It is a random order within one block.

18. Confounding the 2k Factorial Design in Two Blocks • Multiple replicates are required to obtain the estimate error when k is small. • For example, 23 design with four replicate in two blocks

19. Confounding the 2k Factorial Design in Two Blocks • ANOVA • 32 observations

20. Confounding the 2k Factorial Design in Two Blocks --example • Same as example 6.2 • Four factors: Temperature, pressure, concentration, and stirring rate. • Response variable: filtration rate. • Each batch of material is nough for 8 treatment combinations only. • This is a 24 design n two blocks.

21. Confounding the 2k Factorial Design in Two Blocks --example

22. Confounding the 2k Factorial Design in Two Blocks --example Factorial Fit: Filtration versus Block, Temperature, Pressure, ... Estimated Effects and Coefficients for Filtration (coded units) Term Effect Coef Constant 60.063 Block -9.313 Temperature 21.625 10.812 Pressure 3.125 1.563 Conc. 9.875 4.938 Stir rate 14.625 7.313 Temperature*Pressure 0.125 0.063 Temperature*Conc. -18.125 -9.063 Temperature*Stir rate 16.625 8.313 Pressure*Conc. 2.375 1.188 Pressure*Stir rate -0.375 -0.188 Conc.*Stir rate -1.125 -0.562 Temperature*Pressure*Conc. 1.875 0.938 Temperature*Pressure*Stir rate 4.125 2.063 Temperature*Conc.*Stir rate -1.625 -0.812 Pressure*Conc.*Stir rate -2.625 -1.312 S = * PRESS = *

23. Confounding the 2k Factorial Design in Two Blocks --example Factorial Fit: Filtration versus Block, Temperature, Pressure, ... Analysis of Variance for Filtration (coded units) Source DF Seq SS Adj SS Adj MS F P Blocks 1 1387.6 1387.6 1387.56 * * Main Effects 4 3155.2 3155.2 788.81 * * 2-Way Interactions 6 2447.9 2447.9 407.98 * * 3-Way Interactions 4 120.2 120.2 30.06 * * Residual Error 0 * * * Total 15 7110.9

24. Confounding the 2k Factorial Design in Two Blocks --example

25. Confounding the 2k Factorial Design in Two Blocks --example

26. Confounding the 2k Factorial Design in Two Blocks –example(Adj) ABCD Factorial Fit: Filtration versus Block, Temperature, Conc., Stir rate Estimated Effects and Coefficients for Filtration (coded units) Term Effect Coef SE Coef T P Constant 60.063 1.141 52.63 0.000 Block -9.313 1.141 -8.16 0.000 Temperature 21.625 10.812 1.141 9.47 0.000 Conc. 9.875 4.938 1.141 4.33 0.002 Stir rate 14.625 7.313 1.141 6.41 0.000 Temperature*Conc. -18.125 -9.062 1.141 -7.94 0.000 Temperature*Stir rate 16.625 8.312 1.141 7.28 0.000 S = 4.56512 PRESS = 592.790 R-Sq = 97.36% R-Sq(pred) = 91.66% R-Sq(adj) = 95.60% Analysis of Variance for Filtration (coded units) Source DF Seq SS Adj SS Adj MS F P Blocks 1 1387.6 1387.6 1387.56 66.58 0.000 Main Effects 3 3116.2 3116.2 1038.73 49.84 0.000 2-Way Interactions 2 2419.6 2419.6 1209.81 58.05 0.000 Residual Error 9 187.6 187.6 20.84 Total 15 7110.9

27. Another Illustration • Assuming we don’t have blocking in previous example, we will not be able to notice the effect AD. Now the first eight runs (in run order) have filtration rate reduced by 20 units

28. Another Illustration

29. Confounding the 2k design in four blocks • 2k factorial design confounded in four blocks of 2k-2 observations. • Useful if k ≧ 4 and block sizes are relatively small. • Example 25 design in four blocks, each block with eight runs. • Select two factors to be confound with, say ADE and BCE.

30. Confounding the 2k design in four blocks • L1=x1+x4+x5 • L2=x2+x3+x5 • Pairs of L1 and L2 group into four blocks

31. Confounding the 2k design in four blocks • Example: L1=1, L2=1  block 4 • abcde: L1=x1+x4+x5=1+1+1=3(mod 2)=1 L2=x2+x3+x5=1+1+1=3(mod 2)=1

32. Confounding the 2k design in 2pblocks • 2k factorial design confounded in 2p blocks of 2k-p observations.

33. Confounding the 2k design in 2pblocks

34. Partial Confounding • In Figure 7.3, it is a completely confounded case • ABC s confounded with blocks in each replicate.

35. Partial Confounding • Consider the case below, it is partial confounding. • ABC is confounded in replicate I and so on.

36. Partial Confounding • As a result, information on ABC can be obtained from data in replicate II, II, IV, and so on. • We say ¾ of information can be obtained on the interactions because they are unconfounded in only three replicates. • ¾ is the relative information for the confounded effects

37. Partial Confounding • ANOVA

38. Partial Confounding-- example • From Example 6.1 • Response variable: etch rate • Factors: A=gap, B=gas flow, C=RF power. • Only four treatment combinations can be tested during a shift. • There is shift-to-shift difference in etch performance. The experimenter decide to use shift as a blocking factor.

39. Partial Confounding-- example • Each replicate of the 23 design must be run in two blocks. Two replicates are run. • ABC is confounded in replicate I and AB is confounded in replicate II.

40. Partial Confounding-- example • How to create partial confounding in Minitab?

41. Partial Confounding-- example • Replicate I is confounded with ABC • STAT>DOE>Factorial >Create Factorial Design

42. Partial Confounding-- example • Design >Full Factorial • Number of blocks  2  OK

43. Partial Confounding-- example • Factors > Fill in appropriate information  OK  OK

44. Partial Confounding-- example • Result of Replicate I (default is to confound with ABC)

45. Partial Confounding-- example • Replicate II is confounded with AB • STAT>DOE>Factorial >Create Factorial Design • 2 level factorial (specify generators)

46. Partial Confounding-- example • Design >Full Factorial

47. Partial Confounding-- example • Generators …> Define blocks by listing …  AB • OK

48. Partial Confounding-- example • Result of Replicate II

49. Partial Confounding-- example • To combine the two design in one worksheet • Change block number 3 -> 1, 2 -> 4 in Replicate II • Copy columns of CenterPt, Gap, …RF Power from Replicate II to below the corresponding columns in Replicate I.

50. Partial Confounding-- example