1. Chapter 22 Design of Experiments

2. Experiments
Experiment: A study in which certain independent variables are manipulated, their effect on one or more dependent variables is determined, and the levels (values) of these independent variables are assigned at random to the experimental units in study.
Trial and Error Experiments
Lack direction and focus
Waste valuable resources
May never reach optimal solution

3. Design of Experiments (DOE)
Design of Experiment: A method of experimenting with the complex interactions among parameters in a process or product.
It provides a layout of the different factors and the values at which those factors are to be tested.
It is useful to identify the effects of hidden variables

4. Design of Experiments (DOE) Terminology
Factor: A factor is an input variable that can be manipulated during the experiment (e.g., temperature)
Level: A level is one of the settings available for a factor (e.g., level 1 = 120F, level 2 = 150F)
Qualitative factors: levels vary by category
Quantitative factors: levels vary by numerical degree
Fixed levels: levels are set at specified values
Random levels: levels are randomly selected from all possible values

5. Design of Experiments (DOE) Terminology
Response variable: The variable(s) of interest used to describe the reaction of a process to variations in control variables (factors)
Effect: The effect is the change exhibited by he response variable when the factor level is changed
Main Effect
Interactions: Two or more factors that together produce a result different than what the result of their separate effects would be

6. Design of Experiments (DOE) Terminology
Experimental Unit: The entity to which a specific treatment combination is applied (e.g. PC board, silicon wafer)
Blocking: A schedule for conducting treatment combinations in an experimental study such that any effects on the experimental results due to a known change (e.g. raw materials, operators, machines, etc.) become concentrated in the levels of the blocking variable (e.g. shift)

7. Design of Experiments (DOE) Terminology
Treatment: The specific combination of levels for each factor used for a particular run
Run: An experimental trail, the application of one treatment
Replicate: Repeat of the treatment condition
Repetition: Multiple runs of a particular treatment condition

8. Design of Experiments (DOE) Terminology
Noise Factor: An uncontrollable, but measurable, source of variation in the function characteristics of a process or a product
Confounding factor: A factor’s effect on the dependent variable cannot be distinguished from the effect of the independent variable. X and Y are confounded when there is a third variable Z that influences both X and Y; such a variable is then called a confounder of X and Y.
Alias: When the estimate of an effect also includes the influence of one or more other effects, the effects are said to be aliased. For example, if the estimate of effect D in a four factor experiment actually estimates (D + ABC), then the main effect D is aliased with the 3-way interaction ABC.

9. Types of Experiments
Trial-error methods -- Introduce a change and see what happens
Running special lots -- Produced under controlled conditions
Pilot runs – Set up to produce a desired effect
One-factor at a time experiments – Vary 1 factor and keep all other factors constant
Planned comparisons of two methods – Background variables considered in plan
Experiments with 2~4 factors – Study separate effects and interactions
Experiments with 5~20 factors – Screening studies
Comprehensive experimental plan with many phases – Modeling, multiple factor levels, optimization

10. How does DOE help to improve a process?
Screening Designs -- to identify the “vital few” process factors, large number of factors at two levels
Characterization Design – to study effects of a small number of factors, full factorial models
Optimization Design -- to study complicated effects and interactions of one or two number of factors, full factorial models

11. Steps in DOE Define Design Statement of Problem
Choice of Response (dependent variable).
Selection of factors (independent variables) to be varied.
Choices of levels of these factors
Qualitative or Quantitative
Fixed or random
Design
Number of observation to be taken
Order of experimentation.
Method of randomization to be used.
Mathematical model to describe the experiment.
Hypothesis to be test.

12. Steps in DOE Analysis Data collection and processing.
Computation of test statistics and preparation of graphics.
Interpretation of results.

13. Treatment Combination (1,1) Treatment Combination (1,2)
Complete Factorial Experiment Design
Factor B
Factor A
Level 1
Level 2
Level 3
Treatment Combination (1,1)
Obs 1
Obs 2
Obs 3
Treatment Combination (1,2)

14. Nested Experiment Design
Factor A
A-level 1
A-level 2
A-level 3
Factor B
B-level 1
B-level 2
B-level 3
B-level 4
B-level 5
B-level 6

15. Completely Randomized Single-Factor Experiments
An observation, Yij, for the ith observation and jth treatment may contain three parts
Yij = m + tj + eij
m: a common effect for the whole experiment, a fixed parameter
tj: the effect of the jth treatment, tj = mj - m
eij: random error present in the ith observation and jth treatment. NID (0, 2) where 2 is the common variance within all treatments

16. Completely Randomized Single-Factor Experiments
Fixed Model
t1, t2, .... , tj, , tk are considered to be fixed parameters
H0: t1 = t2 = = tk = 0
H1: At least one of the treatment effect  0

17. Completely Randomized Single-Factor Experiments
Random Model
t1, t2, .... , tj, , tk are considered to be random variables
tj’s are NID(0, 2)
H0: 2= 0
H1: At least one of the treatment effect  0

18. Completely Randomized Single-Factor Experiments
Example 1
We are interested in knowing if there is a difference of the resistance to abrasion of 4 different fabrics, A, B, C, D. A single factor randomized experiment is run. The four treatments (levels) are A, B, C, and D. (Fixed Effect)

19. Example 1
H0: t1 = t2 = = tk = 0
H1: At least one of the treatment effect  0
Treatment
Obs. A B C D 1
1.93
2.55
2.40
2.33 2
2.38
2.72
2.68 3
2.20
2.75
2.31
2.28 4
2.25
2.70

20. Example 1 One-way ANOVA Source DF SS MS F P Treatment 3 .5201 .1734
8.53
0.0026
Residual Error 12
.2438
.0203
Total 15
.7639
S = R-Sq = 68.09% R-Sq(adj) = 60.11%
Individual 95% CIs For Mean Based on Pooled StDev
Level N Mean StDev
A (------*-----)
B (-----*-----)
C (------*-----)
D (------*-----)
Pooled StDev =

21. Example 2
H0: t1 = t2 = = tk = 0
H1: At least one of the treatment effect  0
Output
Obs. A B C 1 4 2 -3 8 3 5 -2 7 -1 6

22. Example 2 Source DF SS MS F P Machine 2 124.13 62.07 25.86 0.000
Error
Total
S = R-Sq = 81.17% R-Sq(adj) = 78.03%
Individual 95% CIs For Mean Based on Pooled StDev
Level N Mean StDev
A (----*----)
B (----*----)
C (----*----)
Pooled StDev = 1.549

23. ANOVA Rationale Factor Treatment (level #1) Treatment (level #2)
Factor
Treatment (level #1)
Treatment (level #2)
Treatment (level #j)
Treatment (level #k)
Obs. #1
Y11
Y12
Y1j
Y1k
Obs. #2
Y21
Y22
Y2j
Y2k
Obs. #3
Y31
Y32
Y3j
Y3k
Obs. #i
Yi1
Yi2
Yij
Yik m1
m22 mj mk

24. ANOVA Rationale Yij = m + tj + eij Yij = m + (mj – m) + (Yij – mj)
Yij = m + (mj – m) + (Yij – mj)
Yij - m = (mj – m) + (Yij – mj)

25. SStotal = SStreatment +SSerror
ANOVA Rationale
SStotal = SSbetween+SSwithin
SStotal = SStreatment +SSerror

26. ANOVA Rationale

27. Single Factor ANOVA Test
Source df SS MS F
p-value
Treatmt.
k-1
SStreatment
MStreatment
P(F(1, 2)  f)
Error
N-k
SSerror
MSerror
Totals
N-1
SStotal

28. After ANOVA, there is a difference. What do we do next?
Identify the treatment(s) which make the difference.
Compare the means

29. Two-way ANOVA High Pressure Med Pressure Low Pressure High Temp 39 3218 30 31 20 35 28 21 43 25 29 26
Med Temp 38 10 22 15 36
Low Temp 24 37 27

30. Two-way ANOVA Two-way ANOVA: Photoresist versus Pressure, Temp
Source DF SS MS F P
Pressure
Temp
Interaction
Error
Total
S = R-Sq = 56.44% R-Sq(adj) = 46.76%

31. Two-way ANOVA Individual 95% CIs For Mean Based on Pooled StDev
Pressure Mean
H (-----*----)
L (----*----)
M (----*-----)
Temp Mean
H ( * )
L ( * )
M ( * )

32. Factorial Experiments
Full Factorial design consists of all possible combinations of all selected levels of the factors to be investigated

33. Example
Interested to find the effect of both Temperature and Altitude on the Current Flow in IC.
Two temperature settings:
25oC and 55oC.
Two altitude settings: 0K ft. and 3K ft. A

34. One-factor-at-a-time Experiment
Holding one variable constant while changing the other.
At 0K ft. (Altitude)
Temperature
25oC
55oC
Current
210 mA
240 mA
Observed: Temperature  , Current  by 30 mA
By chance?
True increase?
How do we make sure?

35. One-factor-at-a-time Experiment (Cont.)
Repeat the experiment again!
At 0K ft. (Altitude)
Temperature
25oC
55oC
Current (I)
210 mA
240 mA
Current (II)
205 mA
230 mA
Temperature  , Current  by (30+25)/2 = 27.5 mA
How about the current under altitude?

36. One-factor-at-a-time Experiment (Cont.)
At 25oC (Temperature)
Altitude
0K ft
3K ft
Current (I)
210 mA
180 mA
Current (II)
205 mA
185 mA
Altitude  , Current  by (30+20)/2 = 25 mA
At 55oC, would current increase /decrease as Altitude increase?
Run two more experiments under 55oC.
8 experiments total!!

37. Factorial Arrangement
25oC
55oC
0K ft
210 mA
240 mA
3K ft
180 mA
200 mA
No Interaction!

38. Factorial Arrangement
Factorial Fit: Current versus Altitude, Temp
Estimated Effects and Coefficients for Current (coded units)
Term Effect Coef
Constant
Altitude
Temp
Altitude*Temp
Analysis of Variance for Current (coded units)
Source DF Seq SS Adj SS Adj MS F P
Main Effects * *
2-Way Interactions * *
Residual Error * * *
Total

39. Factorial Arrangement
25oC
55oC
0K ft
210 mA
240 mA
3K ft
180 mA
160 mA
Interaction!

40. Factorial Arrangement
Analysis of Variance for Current (coded units)
Source DF Seq SS Adj SS Adj MS F P
Main Effects * *
2-Way Interactions * *
Residual Error * * *
Total
Estimated Coefficients for Current using data in uncoded units
Term Coef
Constant
Altitude
Temp
Altitude*Temp

41. Factorial Arrangement
More efficient: less experiments, more information.
Temperature effect
Altitude effect
Interaction
Better identify the optimal setting

42. One-factor-at-a-time Experiment

43. Factorial Arrangement

44. Factorial Experiment (An Example)
Subject: Vacuum Tube
Interested to find the effect of
Exhaust index ( in seconds)
Pump heater Voltage (in volts)
On the pressure inside a tube ( m-6 Hg)
Index: 60, 90, 150
Voltage: 127, 220

45. Factorial Experiment (An Example)
Pump Pressure
Exhaust
Index
Voltage 60 90
150
127 48 28 7 58 33 15
220 62 14 9 54 10 6

46. Factorial Experiment (An Example) One-way ANOVA
One-way ANOVA: Pressure versus Group
Source DF SS MS F P
Group
Error
Total
S = R-Sq = 97.29% R-Sq(adj) = 95.03%
Individual 95% CIs For Mean Based on Pooled StDev
Level N Mean StDev
(----*---)
(---*---)
(---*---)
(---*---)
(----*---)
(---*---)
Pooled StDev = 4.813

47. Factorial Experiment (An Example) Two-way ANOVA
Two-way ANOVA: Pressure versus Exhaust, Voltage
Source DF SS MS F P
Exhaust
Voltage
Interaction
Error
Total
S = R-Sq = 97.29% R-Sq(adj) = 95.03%
Individual 95% CIs For Mean Based on Pooled StDev
Exhaust Mean
(---*---)
(---*---)
(---*---)
Voltage Mean
( * )
( * )

48. Factorial Experiment (An Example)
General Linear Model: Pressure versus Exhaust, Voltage
Factor Type Levels Values
Exhaust fixed , 90, 150
Voltage fixed , 220
Analysis of Variance for Pressure, using Adjusted SS for Tests
Source DF Seq SS Adj SS Adj MS F P
Exhaust
Voltage
Exhaust*Voltage
Error
Total
S = R-Sq = 97.29% R-Sq(adj) = 95.03%

49. Factorial Experiment (An Example)
A significant interaction means that the effect of exhaust index on the vacuum tube pressure at one voltage is different from its effect at the other voltage.

50. Factorial Experiment (An Example) One-way ANOVA

51. Factorial Experiment (An Example) Two-way ANOVA

52. Factorial Experiments (2 Factors)
Yijk =  + Ai + Bj + ABij + k(ij)
H01: A1 = A2 = = Aa = 0
H02: B1 = B2 = = Bb = 0
H03: AB11 = AB12 = = ABab = 0

53. 2x2 Factorial Experiments (2 Factors, 2 Levels each)
Yijk =  + Ai + Bj + ABij + k(ij)
i=1,2
j=1,2
k=1, 2, …, n

54. Example
Interested to find the effect of both Temperature and Altitude on the Current Flow in IC.
Two temperature settings:
25oC and 55oC.
Two altitude settings:
0K ft. and 3K ft. A

55. Factorial Arrangement
25oC
55oC
0K ft
210 mA
240 mA
3K ft
180 mA
200 mA
Notations for Treatment Combinations
Factor A
Factor B
Low (0)
High (1)
A0B01
A1B0a
A0B1b
A1B1ab

56. Normal Order 22 Factorial Experiments: 1, a, b, ab.
1, a, b, ab, c, ac, bc, abc.
24 Factorial Experiments:
1, a, b, ab, c, ac, bc, abc,
d, ad, bd, abd, cd, acd, bcd, abcd.

57. Effect of a Factor Factor A Factor B Low (0) High (1) T1 T2 T3 T4
EA ={(T2- T1)+ (T4- T3)}/2 =(1/2){ -T1+T2-T3+T4 }
CA= -T1+T2-T3+T4
EB ={(T3- T1)+ (T4- T2)}/2 =(1/2){ -T1-T2+T3+T4 }
CB= -T1-T2+T3+T4
EA B={(T4- T3)- (T2- T1)}/2 =(1/2){ T1-T2-T3+T4 }
CAB= T1-T2-T3+T4

58. Orthogonal Table Linear Contrast Treatment A B AB Total (1) -1 +1 T1 a

59. Orthogonal Table (22 Factorial)

60. 23 Factorial Experiments (3 Factors, 2 Levels each)
Yijkl =  + Ai + Bj + ABij
+ Ck+ ACik+BCjk+ABCijk+l(ijk)
i=1, 2
j=1, 2
k=1, 2
l=1, 2, …, n

61. Notations for Treatment Combinations
Factor C (0)
Factor A
Factor B
Low (0)
High (1)
A0B0C01
A1B0C0a
A0B1C0b
A1B1C0ab
Factor C (1)
Factor A
Factor B
Low (0)
High (1)
A0B0C1c
A1B0C1ac
A0B1C1bc
A1B1C1abc

62. Orthogonal Table Linear Contrast Trmt A B C AB AC BC ABC Total (1) -1+1 T1 a T2 b T3 ab T4 c T5 ac T6 bc T7
abc T8

63. Orthogonal Table (23 Factorial)

64. 2f Factorial Experiments (f Factors, 2 Levels each)

65. Fraction Factorial Experiments
Study only a fraction or subset of all the possible combinations
To reduce the total number of experiments to a practical level
Different designs exist

66. Fraction Factorial Experiments Plakett-Burman Screening Design
Basic 8 run Plakett-Burman Design A B C D E F G 1 - 2 + 3 4 5 6 7 8

67. Fraction Factorial Experiments Plakett-Burman Screening Design
Basic 12 run Plakett-Burman Design A B C D E F G H I J K 1 - 2 + 3 4 5 6 7 8 9 10 11 12

68. Fraction Factorial Experiments Taguchi DesignB C D E F G 1 - 2 + 3 4 5 6 7 8