1. Design and Analysis of Experiments
Dr. Tai-Yue Wang
Department of Industrial and Information Management
National Cheng Kung University
Tainan, TAIWAN, ROC
This is a basic course blah, blah, blah…

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
This is a basic course blah, blah, blah…

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 2p Blocks
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 of Squares 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
Block
Temperature
Pressure
Conc
Stir rate
Temperature*Pressure
Temperature*Conc
Temperature*Stir rate
Pressure*Conc
Pressure*Stir rate
Conc.*Stir rate
Temperature*Pressure*Conc
Temperature*Pressure*Stir rate
Temperature*Conc.*Stir rate
Pressure*Conc.*Stir rate
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 * *
Main Effects * *
2-Way Interactions * *
3-Way Interactions * *
Residual Error * * *
Total

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
Block
Temperature
Conc
Stir rate
Temperature*Conc
Temperature*Stir rate
S = PRESS =
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
Main Effects
2-Way Interactions
Residual Error
Total

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 2p blocks
2k factorial design confounded in 2p blocks of 2k-p observations.

33. Confounding the 2k design in 2p blocks

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

51. Partial Confounding-- example
STAT> DOE> Factorial> Define Custom Factorial Design
Factors  Gap, Gas Flow, RF Power

52. Partial Confounding-- example
Low/High > OK
Designs >Blocks>Specify by column  Blocks OK

53. Partial Confounding-- example
Now you can fill in collected data.

54. Partial Confounding-- example
ANOVA
Factorial Fit: Etch Rate versus Block, Gap, Gas Flow, RF
Estimated Effects and Coefficients for Etch Rate (coded units)
Term Effect Coef SE Coef T P
Constant
Block
Block
Block
Gap
Gas Flow RF
Gap*Gas Flow
Gap*RF
Gas Flow*RF
Gap*Gas Flow*RF
S = PRESS =
R-Sq = 97.60% R-Sq(pred) = 75.42% R-Sq(adj) = 92.80%

55. Partial Confounding-- example
ANOVA
Factorial Fit: Etch Rate versus Block, Gap, Gas Flow, RF
Analysis of Variance for Etch Rate (coded units)
Source DF Seq SS Adj SS Adj MS F P
Blocks
Main Effects
2-Way Interactions
3-Way Interactions
Residual Error
Total
* NOTE * There is partial confounding, no alias table was printed.

56. Partial Confounding-- example
ANOVA