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. Two-Level Fractional 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 The One-Half Fraction of the 2k factorial Design
The One-Quarter Fraction of the 2k factorial Design
The General 2k-p Fractional Factorial Design
Alias Structures in Fractional Factorials and Other Designs
Resolution III Designs
Resolution IV and V Designs
Supersaturated Designs

4. Alias Structures I Fractional Factorials and other Designs
Assuming that we use the following regression equation to fit the experimental results:
where y is an n x 1 vector of the response
X1 is an n x p1 matrix
β1 is a p1 x 1 vector
Thus the estimated of β1 via LSE is

5. Alias Structures I Fractional Factorials and other Designs
Suppose that the true model is
where X2 is an n x p2 matrix with additional variables
β2 is a p2 x 1 vector

6. Alias Structures Fractional Factorials and other Designs
Thus the expected parameters
The matrix is called alias matrix
The elements of A operating on β2 identify the alias relationships for the parameters in the vector β1

7. Alias Structures Fractional Factorials and other Designs
Example: 23-1 design with I=ABC

8. Alias Structures Fractional Factorials and other Designs
Regression model
So, for the four runs

9. Alias Structures Fractional Factorials and other Designs
Suppose the true model is
and

10. Alias Structures Fractional Factorials and other Designs
Now try to find A

11. Alias Structures Fractional Factorials and other Designs

12. Alias Structures Fractional Factorials and other Designs
Comparison
[A] A+BC
[B] B+AC
[C] C+AB

13. Resolution III Designs -- Constructing
Resolution III designs are useful for screening (5 – 7 variables in 8 runs, variables in 16 runs, for example)
A saturated design has k = N – 1 variables
Examples of saturated design:

14. Resolution III Designs -- Constructing
Case of

15. Resolution III Designs -- Constructing
Can be used to generate factors fewer than 7
For example,

16. Resolution III Designs – Fold over
By combining fractional factorial designs that certain signs are switched , one can systematically isolate effects of the potential interest
This type of sequential experiments is called a fold over of the original design

17. Resolution III Designs – Fold over
For the case of
Reversing the sign in factor D 

18. Resolution III Designs – Fold over
Reversed effects
[A]’  A-BD+CE+FG
[B]’  B-AD+CF+EG
[C]’  C+AE+BF+DG
[D]’  D-AB-CG-EF
[-D]’  -D+AB+CG+EF
[E]’  E+AC+BG-DF
[F]’  F+BC+AG-DE
[G]’  G-CD+BE+AF
Original effects

19. Resolution III Designs – Fold over
Assuming the three-factor and higher interactions are insignificant, one can combine the two fractions
For effect of the factor D
½ [D]+1/2[D]’ D
For effects
½ [D]-1/2[D]’ AB+C+EF

20. Resolution III Designs – Fold over
In general, if we add to a fractional design of resolution III or higher a further fraction with signs of a single factor reversed, the combined design will provide the estimates of the man effect of that factor and its two-factor interactions
This is a single-factor fold over

21. Resolution III Designs – Fold over
If we add to a fractional design of resolution III a second fraction with signs of all the factors are reversed, the combined design break the alias link between all main effects and their two-factor interaction.
This is a full fold over

22. Resolution III Designs – example (7—1/9)
Eye focus, Response= time
7 factors
Screening experiment

23. Resolution III Designs – example (7—2/9)
STAT>DOE>Create Factorial Design
2 level fractional (default)
Number of factor  7
Choose 1/8 fractional

24. Resolution III Designs – example (7—3/9)
STAT>DOE>Analyze Factorial Design
Only A, B, D are significant

25. Resolution III Designs – example (7—4/9)
Examining the alias structure
We are not sure if A or BD, B or AD, D or AB are significant!!!!!
Alias Structure (up to order 3)
I + A*B*D + A*C*E + A*F*G + B*C*F + B*E*G + C*D*G + D*E*F
A + B*D + C*E + F*G + B*C*G + B*E*F + C*D*F + D*E*G
B + A*D + C*F + E*G + A*C*G + A*E*F + C*D*E + D*F*G
C + A*E + B*F + D*G + A*B*G + A*D*F + B*D*E + E*F*G
D + A*B + C*G + E*F + A*C*F + A*E*G + B*C*E + B*F*G
E + A*C + B*G + D*F + A*B*F + A*D*G + B*C*D + C*F*G
F + A*G + B*C + D*E + A*B*E + A*C*D + B*D*G + C*E*G
G + A*F + B*E + C*D + A*B*C + A*D*E + B*D*F + C*E*F

26. Resolution III Designs – example (7—5/9)
Note that ABD is one of the word in defining relation, do not project into a full 23 factorial in ABD
It does project into two replicates of a 23-1 design.
23-1 is a resolution III design, too
Try fold over

27. Resolution III Designs – example (7—6/9)
2nd fraction:
STAT>DOE>Modify Design
Specify  fold all factor OK

28. Resolution III Designs – example (7—7/9)

29. Resolution III Designs – example (7—8/9)
Collecting data
STAT>DOE>Analyze Factorial Design

30. Resolution III Designs – example (7—9/9)
Though B, D, BD, and AF are significant, B and D are distinguishable
BD is aliased with CE and FG
AF is aliased with CD and BE.
A + B*C*G + B*E*F + C*D*F + D*E*G
B + A*C*G + A*E*F + C*D*E + D*F*G
C + A*B*G + A*D*F + B*D*E + E*F*G
D + A*C*F + A*E*G + B*C*E + B*F*G
E + A*B*F + A*D*G + B*C*D + C*F*G
F + A*B*E + A*C*D + B*D*G + C*E*G
G + A*B*C + A*D*E + B*D*F + C*E*F
A*B + C*G + E*F A*C + B*G + D*F
A*D + C*F + E*G A*E + B*F + D*G
A*F + B*E + C*D A*G + B*C + D*E
B*D + C*E + F*G

31. Resolution III Designs – Fold over
To find the defining relation for a combined design, one can assume that the first fraction has L words and the fold over fraction has U words.
Thus the combined design will have L+U-1 words used as a generators.

32. Resolution III Designs – Fold over
For example,
Generators for the first fraction:
I=ABD, I=ACE, I=BCF, I=ABCG
Generators for the second fraction:
I=-ABD, I=-ACE, I=-BCF, I=ABCG
We have switched the signs on the generators with an odd number of letters

33. Resolution III Designs – Fold over
The complete defining relations for the combined design are:
I=ABCG=BCDE=ACDF=ADEG=BDFG
=ABEF=CEFG

34. Resolution III Designs – Fold over
Usually the second fraction are different from the first fraction in day, time, shift, material, methods.
This leads to the blocking situation.

35. Resolution III Designs – Plackett-Burman Designs
For the case of k=N-1 variables in N runs, where N is a multiple of 4, one can use fold over if N is a power of 2.
However, N=12, 20, 24, 28 and 36, The Placket-Burman is of interest.
Because these design cannot be represented as cubes, called non-geometric designs.
Two ways to generate these designs, check example 8.

36. Resolution III Designs – Plackett-Burman Designs
Upper half: for N=12, 20, 24, and 36
Lower half: for N=28

37. Resolution III Designs – Plackett-Burman Designs
Example for Upper half: N=12 and k=11
Turn into the first column

38. Resolution III Designs – Plackett-Burman Designs
Shift down one row!
Add “-” sign

39. Resolution III Designs – Plackett-Burman Designs
Example for Lower Half: N=28 and k=27 Y Z X

40. Resolution III Designs – Plackett-Burman Designs
N=28 and k=27
Arrange the design into
X Y Z
Z X Y
Y Z X
Add “-” sign to the 28th row

41. Resolution III Designs – Plackett-Burman Designs
Alias structure
Messy and complicated
Main effects are partially aliased with every two-factor interaction not involving itself
Non-regular design
For the case of N=12
Projected into three replicates of a full 22 design in any two of the original 11 factors
Projected into a full 23 factorial plus a 23-2III fractional factorial

42. Resolution III Designs – Plackett-Burman Designs
The resolution II Placket-Burman design has Projectivity 3.
It will collapse into a full factorial in any subset of the three factors.

43. Resolution III Designs – example (8—1/7)
12 factors
If fractional is used, all 12 main effects are aliased with four two-factor interactions.
Additional experiments could be required
Use 20 run Placket-Burman design
Two kinds of designs, one is to follow the text and Minitab
The other is to follow Example 8 in the text.

44. Resolution III Designs – example (8—2/7)
Add “+” sign X1 X2 X3 X4 X5 X6 X7 X8 X9
X10
X11
X12
X13
X14
X15
X16
X17
X18
X19 1 +1 2 -1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Reverse “+” and “-” sign in text

45. Resolution III Designs – example (8—3/7)
Corrected Table 8.25

46. Resolution III Designs – example (8—4/7)
Alternate P-B design for N=20

47. Resolution III Designs – example (8—5/7)
No effect is significant according to traditional analysis.

48. Resolution III Designs – example (8—6/7)
Use stepwise regression
Stepwise Regression: y versus X1, X2, ...
Alpha-to-Enter: 0.1 Alpha-to-Remove: 0.15
Response is y on 19 predictors, with N = 20
Step
Constant X
T-Value
P-Value X
T-Value
P-Value
x1x
T-Value
P-Value
x1x
T-Value
P-Value X
T-Value
P-Value X
T-Value
P-Value S
R-Sq
R-Sq(adj)

49. Resolution III Designs – example (8—7/7)
Fitted model:

50. Resolution IV and V Designs -- Resolution IV Designs
A 2k-p fractional is of resolution IV if the main effects are clear of two-factor interactions and some two-factor interactions are aliased with each other.
Any 2k-pIV design must contain at least 2k runs. Resolution IV designs that contain 2k runs are called minimal designs.
Resolution IV designs maybe obtained from resolution III designs by the process of fold over.

51. Resolution IV and V Designs -- Resolution IV Designs

52. Resolution IV and V Designs -- Resolution IV Designs
How many runs are needed for different number of factors and resolution.

53. Resolution IV and V Designs -- Resolution IV Designs
Example: 18 runs for k=9

54. Resolution IV and V Designs -- Resolution IV Designs
Example: 12 runs for k=6

55. Resolution IV and V Designs -- Resolution IV Designs
These designs are non-regular designs
They are resolution IV with minimum runs
No quarantine on orthogonal
Useful alternative in screening the main effects
If two-factor interaction are proven important, step-wise regression is used to estimate
The price paid for reducing run number is the complicated alias table

56. Resolution IV and V Designs – Sequential Experimentation with Resolution IV Designs
Fold over in resolution III is to separate the main effect
We can’t use the full fold-over procedure given previously for Resolution III designs – it will result in replicating the runs in the original design.
That is, runs are in different order!!

57. Resolution IV and V Designs – Sequential Experimentation with Resolution IV Designs
Switching the signs in a single column allows all of the two-factor interactions involving that column to be separated.

58. Resolution IV and V Designs – Sequential Experimentation with Resolution IV Designs
Example: 6 factors

59. Resolution IV and V Designs – Sequential Experimentation with Resolution IV Designs
Alias

60. Resolution IV and V Designs – Sequential Experimentation with Resolution IV Designs
Half-Normal

61. Resolution IV and V Designs – Sequential Experimentation with Resolution IV Designs
Since AB is aliased with CE, we do not know whether this is AB or CE or both.
Fold over !!
Setting up a new fraction of 26-1IV and changing sign of factor A
STAT>DOE>Modify design  fold over
Specify  fold just one factorAOK

62. Resolution IV and V Designs – Sequential Experimentation with Resolution IV Designs

63. Resolution IV and V Designs – Sequential Experimentation with Resolution IV Designs

64. Resolution IV and V Designs – Sequential Experimentation with Resolution IV Designs
Sometimes we can use partial fold over to reduce the run number
In previous example, one can select half of the new fraction.
Here we choose the “-” half of the new fraction because the “-” part has better response in the original 16 runs

65. Resolution IV and V Designs – Sequential Experimentation with Resolution IV Designs

66. Resolution IV and V Designs – Sequential Experimentation with Resolution IV Designs

67. Resolution IV and V Designs –Resolution V Designs
In Resolution V designs, main effects and two-factor interactions do not have other main effect and two-factor interactions as their aliases.
For k=6, 26-1V is required
How about non-regular design?
How about k=8?
non-regular designs are not orthogonal!!
The precision of estimation is higher than the orthogonal one.

68. Resolution IV and V Designs –Resolution V Designs

69. Resolution IV and V Designs –Resolution V Designs