The document discusses the 2k factorial design, which is a special case of the general factorial design with k factors at two levels. It provides examples of using 2k factorial designs to investigate how multiple factors affect a response. For an unreplicated 2k design with no replication, there are challenges in statistical testing due to having zero degrees of freedom for error. Various methods are discussed for analyzing the effects in an unreplicated 2k design, such as normal probability plotting, Lenth's method, and conditional inference charts. Transformation of the response may also be needed to meet assumptions of the model such as equal variance.

1. Design of Engineering Experiments
Part 5 – The 2k Factorial Design

2. • Text reference, Chapter 6
• Special case of the general factorial design; k factors, all at two levels
• The two levels are usually called low and high (they could be either quantitative or
qualitative)
• Very widely used in industrial experimentation
• Form a basic “building block” for other very useful experimental designs (DNA)
• Special (short-cut) methods for analysis
• We will make use of Design-Expert

5. Chemical Process Example
A = reactant concentration, B = catalyst amount,
y = recovery

6. Analysis Procedure for a Factorial Design
Estimate factor effects
•With replication, use full model
•With an unreplicated design, use normal probability plots
Formulate model
Statistical testing (ANOVA)
Refine the model
Analyze residuals (graphical)
Interpret results

7. Estimation of Factor Effects
1
2
1
2
1
2
(1)
2 2
[ (1)]
(1)
2 2
[ (1)]
(1)
2 2
[ (1) ]
A A
n
B B
n
n
A y y
ab a b
n n
ab a b
B y y
ab b a
n n
ab b a
ab a b
AB
n n
ab a b
 
 
 
 
 
   
 
 
 
   
 
 
   
See textbook, pg. 209-210 For
manual calculations
The effect estimates are: A
= 8.33, B = -5.00, AB = 1.67
Practical interpretation?
Design-Expert analysis

8. Estimation of Factor Effects Form Tentative Model
Term Effect SumSqr % Contribution
Model Intercept
Model A 8.33333 208.333 64.4995
Model B -5 75 23.2198
Model AB 1.66667 8.33333 2.57998
Error Lack Of Fit 0 0
Error P Error 31.3333 9.70072
Lenth's ME 6.15809
Lenth's SME 7.95671

9. Statistical Testing - ANOVA
The F-test for the “model” source is testing the significance of the
overall model; that is, is either A, B, or AB or some combination of
these effects important?

10. Design-Expert output, full
model

11. Design-Expert output,
edited or reduced model

12. 12
Residuals and Diagnostic Checking

13. 13
The Response Surface

14. 14
The 23 Factorial Design

15. etc, etc, ...
A A
B B
C C
A y y
B y y
C y y
 
 
 
 
 
 
Effects in The 23 Factorial Design
Analysis
done via
computer

16. 16
An Example of a 23 Factorial Design
A = gap, B = Flow, C = Power, y = Etch Rate

17. Table of – and + Signs for the 23 Factorial Design (pg. 218)

18. • Except for column I, every column has an equal number of + and –
signs
• The sum of the product of signs in any two columns is zero
• Multiplying any column by I leaves that column unchanged (identity
element)
• The product of any two columns yields a column in the table:
• Orthogonal design
• Orthogonality is an important property shared by all factorial designs
18
Properties of the Table
2
A B AB
AB BC AB C AC
 
  

19. 19
Estimation of Factor Effects

20. ANOVA Summary – Full Model

21. Model Coefficients – Full Model

22. Refine Model – Remove Nonsignificant Factors

23. Model Coefficients – Reduced Model

24. • R2 and adjusted R2
• R2 for prediction (based on PRESS)
24
Model Summary Statistics for Reduced Model
5
2
5
2
5
5.106 10
0.9608
5.314 10
/ 20857.75/12
1 1 0.9509
/ 5.314 10 /15
Model
T
E E
Adj
T T
SS
R
SS
SS df
R
SS df

  

    

2
Pred 5
37080.44
1 1 0.9302
5.314 10T
PRESS
R
SS
    


25. Model Summary Statistics
• Standard error of model coefficients (full
model)
• Confidence interval on model coefficients
25
2
2252.56ˆ ˆ( ) ( ) 11.87
2 2 2(8)
E
k k
MS
se V
n n

     
/2, /2,
ˆ ˆ ˆ ˆ( ) ( )E Edf dft se t se        

26. The Regression Model

27. Model Interpretation
Cube plots are
often useful visual
displays of
experimental
results

28. Cube Plot of Ranges
28
What do the
large ranges
when gap and
power are at the
high level tell
you?

29. 29

30. • Section 6-4, pg. 227, Table 6-9, pg. 228
• There will be k main effects, and
The General 2k Factorial Design
two-factor interactions
2
three-factor interactions
3
1 factor interaction
k
k
k
 
 
 
 
 
 


31. • These are 2k factorial designs with one
observation at each corner of the “cube”
• An unreplicated 2k factorial design is also
sometimes called a “single replicate” of the 2k
• These designs are very widely used
• Risks…if there is only one observation at each
corner, is there a chance of unusual response
observations spoiling the results?
• Modeling “noise”?
Unreplicated 2k Factorial Designs

32. Spacing of Factor Levels in the Unreplicated 2k Factorial Designs
If the factors are spaced too closely, it increases the chances
that the noise will overwhelm the signal in the data
More aggressive spacing is usually best

33. • Lack of replication causes potential problems in
statistical testing
– Replication admits an estimate of “pure error” (a
better phrase is an internal estimate of error)
– With no replication, fitting the full model results in zero
degrees of freedom for error
• Potential solutions to this problem
– Pooling high-order interactions to estimate error
– Normal probability plotting of effects (Daniels, 1959)
– Other methods…see text
Unreplicated 2k Factorial Designs

34. • A 24 factorial was used to investigate the
effects of four factors on the filtration rate of
a resin
• The factors are A = temperature, B = pressure,
C = mole ratio, D= stirring rate
• Experiment was performed in a pilot plant
Example of an Unreplicated 2k Design

35. The Resin Plant Experiment

36. 36
The Resin Plant Experiment

38. Estimates of the Effects

39. The Half-Normal Probability Plot of Effects

40. Design Projection: ANOVA Summary for the Model as a 23 in Factors A, C, and D

41. The Regression Model

42. Model Residuals are Satisfactory

43. Model Interpretation – Main Effects and Interactions

44. Model Interpretation – Response Surface Plots
With concentration at either the low or high level, high temperature and high
stirring rate results in high filtration rates

45. 45
Outliers: suppose that cd = 375 (instead of 75)

46. Dealing with Outliers
• Replace with an estimate
• Make the highest-order interaction zero
• In this case, estimate cd such that ABCD = 0
• Analyze only the data you have
• Now the design isn’t orthogonal
• Consequences?

48. 48
The Drilling Experiment Example 6.3
A = drill load, B = flow, C = speed, D = type of mud,
y = advance rate of the drill

49. 49
Normal Probability Plot of Effects – The Drilling Experiment

50. 50
Residual Plots
DESIGN-EXPERT Plot
adv._rate
Predicted
Residuals
Residuals vs. Predicted
-1.96375
-0.82625
0.31125
1.44875
2.58625
1.69 4.70 7.70 10.71 13.71

51. • The residual plots indicate that there are problems with the
equality of variance assumption
• The usual approach to this problem is to employ a transformation
on the response
• Power family transformations are widely used
• Transformations are typically performed to
– Stabilize variance
– Induce at least approximate normality
– Simplify the model
Residual Plots
*
y y


52. • Empirical selection of lambda
• Prior (theoretical) knowledge or experience can often suggest the
form of a transformation
• Analytical selection of lambda…the Box-Cox (1964) method
(simultaneously estimates the model parameters and the
transformation parameter lambda)
• Box-Cox method implemented in Design-Expert
52
Selecting a Transformation

54. 54
The Box-Cox Method
A log transformation is
recommended
The procedure provides a
confidence interval on
the transformation
parameter lambda
If unity is included in the
confidence interval, no
transformation would be
needed

55. Effect Estimates Following the Log Transformation
Three main effects are
large
No indication of large
interaction effects
What happened to the
interactions?

56. 56
ANOVA Following the Log Transformation

57. 57
Following the Log Transformation

58. The Log Advance Rate Model
• Is the log model “better”?
• We would generally prefer a simpler model in a transformed
scale to a more complicated model in the original metric
• What happened to the interactions?
• Sometimes transformations provide insight into the underlying
mechanism

59. Other Examples of Unreplicated 2k Designs
• The sidewall panel experiment (Example 6.4, pg. 245)
– Two factors affect the mean number of defects
– A third factor affects variability
– Residual plots were useful in identifying the dispersion effect
• The oxidation furnace experiment (Example 6.5, pg. 245)
– Replicates versus repeat (or duplicate) observations?
– Modeling within-run variability

60. Other Analysis Methods for Unreplicated 2k Designs
• Lenth’s method (see text, pg. 235)
– Analytical method for testing effects, uses an estimate of error formed by
pooling small contrasts
– Some adjustment to the critical values in the original method can be helpful
– Probably most useful as a supplement to the normal probability plot
• Conditional inference charts (pg. 236)

61. Overview of Lenth’s method
For an individual contrast, compare to the margin of error

63. Adjusted multipliers for Lenth’s method
Suggested because the original method makes too many type I errors, especially for small
designs (few contrasts)
Simulation was used to find these adjusted multipliers
Lenth’s method is a nice supplement to the normal probability plot of effects
JMP has an excellent implementation of Lenth’s method in the screening platform

65. The 2k design and design optimality
The model parameter estimates in a 2k design (and the effect estimates)
are least squares estimates. For example, for a 22 design the model is
0 1 1 2 2 12 1 2
0 1 2 12 1
0 1 2 12 2
0 1 2 12 3
0 1 2 12 4
(1) ( 1) ( 1) ( 1)( 1)
(1) ( 1) (1)( 1)
( 1) (1) ( 1)(1)
(1) (1) (1)(1)
(1) 1 1 1 1
1 1 1 1
, ,
1 1 1
y x x x x
a
b
ab
a
b
ab
    
    
    
    
    
    
        
      
      
    
  
     
   
 
 
y = Xβ + ε y X
0 1
1 2
2 3
12 4
, ,
1
1 1 1 1
 
 
 
 
    
    
      
    
    
     
β ε
The four
observations from
a 22 design

66. The least squares estimate of β is
1
0
1
4
2
12
ˆ
4 0 0 0 (1)
0 4 0 0 (1)
0 0 4 0 (1)
0 0 0 4 (1)
(1)
4ˆ
(1) (
ˆ (1)1
ˆ (1)4
(1)ˆ
a b ab
a ab b
b ab a
a b ab
a b ab
a b ab a ab b
a ab b
b ab a
a b ab





 
     
        
     
   
     
  
                         
      
-1
β = (X X) X y
I
1)
4
(1)
4
(1)
4
b ab a
a b ab
 
 
 
 
 
    
 
   
 
 
The matrix is
diagonal –
consequences of an
orthogonal design
XX
The regression
coefficient estimates
are exactly half of the
‘usual” effect estimates
The “usual” contrasts

67. The matrix has interesting and useful properties:XX
2 1
2
ˆ( ) (diagonal element of ( ) )
4
V  




X X
Minimum possible value for a four-run
design
|( ) | 256 X X
Maximum possible value for a four-run
design
Notice that these results depend on both the design that you have chosen and the model
What about predicting the response?

68. 2
1 2
1 2 1 2
2
2 2 2 2
1 2 1 2 1 2
1 2
2
1 2
1 2
2
1 2
ˆ[ ( , )]
[1, , , ]
ˆ[ ( , )] (1 )
4
The maximum prediction variance occurs when 1, 1
ˆ[ ( , )]
The prediction variance when 0 is
ˆ[ ( , )]
V y x x
x x x x
V y x x x x x x
x x
V y x x
x x
V y x x




 
 
   
   

 

-1
x (X X) x
x
4
What about prediction variance over the design space?average

69. Averageprediction variance
1 1
2
1 2 1 2
1 1
1 1
2 2 2 2 2
1 2 1 2 1 2
1 1
2
1
ˆ[ ( , ) = area of design space = 2 4
1 1
(1 )
4 4
4
9
I V y x x dx dx A
A
x x x x dx dx

 
 
 
   

 
 

70. Design-Expert® Software
Min StdErr Mean: 0.500
Max StdErr Mean: 1.000
Cuboidal
radius = 1
Points = 10000
FDS Graph
Fraction of Design Space
StdErrMean
0.00 0.25 0.50 0.75 1.00
0.000
0.250
0.500
0.750
1.000

71. 71
For the 22 and in general the 2k
• The design produces regression model coefficients that have
the smallest variances (D-optimal design)
• The design results in minimizing the maximum variance of the
predicted response over the design space (G-optimal design)
• The design results in minimizing the average variance of the
predicted response over the design space (I-optimal design)

72. 72
Optimal Designs
• These results give us some assurance that these designs are
“good” designs in some general ways
• Factorial designs typically share some (most) of these properties
• There are excellent computer routines for finding optimal
designs (JMP is outstanding)

73. • Based on the idea of replicating some of the runs in a factorial design
• Runs at the center provide an estimate of error and allow the
experimenter to distinguish between two possible models:
Addition of Center Points to a 2k Designs
0
1 1
2
0
1 1 1
First-order model (interaction)
Second-order model
k k k
i i ij i j
i i j i
k k k k
i i ij i j ii i
i i j i i
y x x x
y x x x x
   
    
  
   
   
    
 
  

75. 75
no "curvature"F Cy y 
The hypotheses are:
0
1
1
1
: 0
: 0
k
ii
i
k
ii
i
H
H








2
Pure Quad
( )F C F C
F C
n n y y
SS
n n



This sum of squares has a
single degree of freedom

76. 76
Example 6.6, Pg. 248
4Cn 
Usually between 3 and 6
center points will work
well
Design-Expert provides
the analysis, including the
F-test for pure quadratic
curvature
Refer to the original experiment shown in Table 6.10.
Suppose that four center points are added to this
experiment, and at the points x1=x2 =x3=x4=0 the
four observed filtration rates were 73, 75, 66, and 69.
The average of these four center points is 70.75, and
the average of the 16 factorial runs is 70.06. Since
are very similar, we suspect that there is no strong
curvature present.

78. 78
ANOVA for Example 6.6 (A Portion of Table 6.22)

79. If curvature is significant, augment the design with axial runs to create a
central composite design. The CCD is a very effective design for fitting a
second-order response surface model

80. Practical Use of Center Points (pg. 260)
• Use current operating conditions as the center point
• Check for “abnormal” conditions during the time the
experiment was conducted
• Check for time trends
• Use center points as the first few runs when there is little or no
information available about the magnitude of error
• Center points and qualitative factors?

81. Center Points and Qualitative Factors

82. 82
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