Design of Engineering Experiments
Part 5 – The 2k Factorial Design
• 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
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The Simplest Case: The 22
“-” and “+” denote the low and
high levels of a factor,
respectively
• Low and high are arbitrary
terms
• Geometrically, the four runs
form the corners of a square
• Factors can be quantitative or
qualitative, although their
treatment in the final model
will be different
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Chemical Process Example
A = reactant concentration, B = catalyst amount,
y = recovery
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Analysis Procedure for a
Factorial Design
• Estimate factor effects
• Formulate model
– With replication, use full model
– With an unreplicated design, use normal probability
plots
•
•
•
•
Statistical testing (ANOVA)
Refine the model
Analyze residuals (graphical)
Interpret results
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Estimation of Factor Effects
=
A y A+ − y A −
ab + a b + (1)
=
−
2n
2n
= 21n [ab + a − b − (1)]
=
B yB + − yB −
ab + b a + (1)
=
−
2n
2n
= 21n [ab + b − a − (1)]
See textbook, pg. 235-236 for
manual calculations
The effect estimates are:
A
= 8.33, B = -5.00, AB = 1.67
Practical interpretation?
Design-Expert analysis
ab + (1) a + b
=
−
AB
2n
2n
= 21n [ab + (1) − a − b]
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Estimation of Factor Effects
Form Tentative Model
Model
Model
Model
Model
Error
Error
Term
Effect
SumSqr
% Contribution
Intercept
A
8.33333
208.333
64.4995
B
-5
75
23.2198
AB
1.66667
8.33333
2.57998
Lack Of Fit 0
0
P Error
31.3333
9.70072
Lenth's ME
Lenth's SME
Chapter 6
6.15809
7.95671
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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?
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Design-Expert output, full model
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Design-Expert output, edited
or reduced model
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Residuals and Diagnostic Checking
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The Response Surface
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The 23 Factorial Design
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Effects in The 23 Factorial Design
=
A y A+ − y A −
=
B yB + − yB −
=
C yC + − yC −
etc, etc, ...
Analysis
done via
computer
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An Example of a 23 Factorial Design
A = gap, B = Flow, C = Power, y = Etch Rate
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Table of – and + Signs for the 23 Factorial Design (pg. 218)
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Properties of the Table
• 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:
A× B =
AB
AB × BC= AB 2C= AC
• Orthogonal design
• Orthogonality is an important property shared by all factorial designs
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Estimation of Factor Effects
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ANOVA Summary – Full Model
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Model Coefficients – Full Model
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Refine Model – Remove Nonsignificant Factors
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Model Coefficients – Reduced Model
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Model Summary Statistics for Reduced Model
• R2 and adjusted R2
SS Model 5.106 ×10
=
=
=
0.9608
R
5
5.314 ×10
SST
5
2
R
2
Adj
SS E / df E
20857.75 /12
=
=
=
1−
1−
0.9509
5
5.314 ×10 /15
SST / dfT
• R2 for prediction (based on PRESS)
PRESS
37080.44
2
RPred =
1−
=
1−
=
0.9302
5
SST
5.314 × 10
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Model Summary Statistics
• Standard error of model coefficients (full
model)
se
( βˆ )
=
V=
( βˆ )
σ2
=
n2k
MS E
=
n2k
2252.56
= 11.87
2(8)
• Confidence interval on model coefficients
βˆ − tα / 2,df se( βˆ ) ≤ β ≤ βˆ + tα / 2,df se( βˆ )
E
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The Regression Model
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Model Interpretation
Cube plots are
often useful visual
displays of
experimental
results
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Cube Plot of Ranges
What do the
large ranges
when gap and
power are at the
high level tell
you?
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The General 2k Factorial Design
• Section 6-4, pg. 253, Table 6-9, pg. 25
• There will be k main effects, and
k 
  two-factor interactions
 2
k 
  three-factor interactions
 3

1 k − factor interaction
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6.5 Unreplicated 2k Factorial Designs
• 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”?
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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
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Unreplicated 2k Factorial Designs
• 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
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Example of an Unreplicated 2k Design
• 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
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The Resin Plant Experiment
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The Resin Plant Experiment
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Estimates of the Effects
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The Half-Normal Probability Plot of Effects
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Design Projection: ANOVA Summary for
the Model as a 23 in Factors A, C, and D
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The Regression Model
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Model Residuals are Satisfactory
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Model Interpretation – Main Effects
and Interactions
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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
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Outliers: suppose that cd = 375 (instead of 75)
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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?
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The Drilling Experiment
Example 6.3
A = drill load, B = flow, C = speed, D = type of mud,
y = advance rate of the drill
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Normal Probability Plot of Effects –
The Drilling Experiment
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Residual Plots
DESIGN-EXPERT Plot
adv._rate
R e s id ua ls vs . P re d ic te d
2.58625
R e s id u a ls
1.44875
0.31125
- 0.82625
- 1.96375
1.69
4.70
7.70
10.71
13.71
P re d ic te d
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Residual Plots
• 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
y =y
*
λ
• Transformations are typically performed to
– Stabilize variance
– Induce at least approximate normality
– Simplify the model
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Selecting a Transformation
• 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
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(15.1)
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The Box-Cox Method
DESIGN-EXPERT Plot
adv._rate
A log transformation is
recommended
6.85
L n (R e s id u a lS S )
Lambda
Current = 1
Best = -0.23
Low C.I. = -0.79
High C.I. = 0.32
B o x-C o x P lo t fo r P o w e r T ra ns fo rm
The procedure provides a
confidence interval on
the transformation
parameter lambda
5.40
Recommend transform:
Log
(Lambda = 0)
3.95
If unity is included in the
confidence interval, no
transformation would be
needed
2.50
1.05
-3
-2
-1
0
1
2
3
Lam bda
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Effect Estimates Following the
Log Transformation
Three main effects are
large
No indication of large
interaction effects
What happened to the
interactions?
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ANOVA Following the Log Transformation
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Following the Log Transformation
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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
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Other Examples of
Unreplicated 2k Designs
• The sidewall panel experiment (Example 6.4, pg. 274)
– 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
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• Example 6.6, Credit Card Marketing, page
278
– Using DOX in marketing and marketing
research, a growing application
– Analysis is with the JMP screening platform
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Other Analysis Methods for
Unreplicated 2k Designs
• Lenth’s method (see text, pg. 262)
– 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. 264)
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Overview of Lenth’s method
For an individual contrast, compare to the margin of error
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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
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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
y =β 0 + β1 x1 + β 2 x2 + β12 x1 x2 + ε
(1) = β 0 + β1 (−1) + β 2 (−1) + β12 (−1)(−1) + ε1
a = β 0 + β1 (1) + β 2 (−1) + β12 (1)(−1) + ε 2
b = β 0 + β1 (−1) + β 2 (1) + β12 (−1)(1) + ε 3
ab =
β 0 + β1 (1) + β 2 (1) + β12 (1)(1) + ε 4
 (1) 
a
,X
y = Xβ=
+ ε, y =
b
 
 ab 
Chapter 6
1 −1 −1 1 
1 1 −1 −1

 ,β
=
1 −1 1 −1


1
1
1
1


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The four
observations
from a 22 design
 β0 
β 
1 
=
,ε
 β2 
 
 β12 
 ε1 
ε 
 2
ε 3 
 
ε 4 
67
The least squares estimate of β is
βˆ = (X′X)-1 X′y
−1
(1) + a + b + ab 
 a + ab − b − (1) 


b + ab − a − (1) 


a
b
ab
(1)
−
−
+


 (1) + a + b + ab 


4
(1) + a + b + ab   a + ab − b − (1) 

 a + ab − b − (1)  


1 
4

I4
=


4 b + ab − a − (1)   b + ab − a − (1) 


4

(1)
a
b
ab
−
−
+

 
 (1) − a − b + ab 




4
4
0
=
0

0
 βˆ0 
 
 βˆ1 
 ˆ 
 β2 
ˆ 
 β12 
The “usual” contrasts
Chapter 6
0 0 0
4 0 0 
0 4 0

0 0 4
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The X′X matrix is
diagonal –
consequences of an
orthogonal design
The regression
coefficient estimates
are exactly half of the
‘usual” effect estimates
68
The matrix X′X has interesting and useful properties:
V ( βˆ ) = σ 2 (diagonal element of (X′X) −1 )
=
σ2
Minimum possible
value for a four-run
design
4
Maximum possible
value for a four-run
design
|(X′X) |= 256
Notice that these results depend on both the design that you
have chosen and the model
What about predicting the response?
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V [ yˆ ( x1 , x2 )] = σ 2 x′(X′X)-1 x
x′ = [1, x1 , x2 , x1 x2 ]
V [ yˆ ( x1 , x2 )]=
σ2
(1 + x12 + x22 + x12 x22 )
4
±1, x2 =
±1
The maximum prediction variance occurs when x1 =
V [ yˆ ( x1 , x2 )] = σ 2
x=
The prediction variance when x=
0 is
1
2
V [ yˆ ( x1 , x2 )] =
σ2
4
What about average prediction variance over the design space?
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Average prediction variance
1 1
I
1
2
ˆ
=
V
[
y
(
x
,
x
)
dx
dx
A
=
area
of
design
space
=
2
4
1
2
1
2
∫
∫
A −1 −1
1 1
1
2 1
2
2
2 2
σ
(1
+
x
+
x
+
x
1
2
1 x2 ) dx1dx2
∫
∫
4 −1 −1 4
4σ 2
=
9
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Design-Expert® Software
FDS Graph
Min StdErr Mean: 0.500
Max StdErr Mean: 1.000
Cuboidal
radius = 1
Points = 10000
1.000
StdErr Mean
0.750
0.500
0.250
0.000
0.00
0.25
0.50
0.75
1.00
Fraction of Design Space
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For the
2
2 and
in general the
k
2
• 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 (Ioptimal design)
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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)
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Addition of Center Points
to a 2k Designs
• 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:
k
k
k
First-order model (interaction) y =
β 0 + ∑ β i xi + ∑∑ β ij xi x j + ε
=i 1
k
k
=i 1 j >i
k
k
β 0 + ∑ β i xi + ∑∑ β ij xi x j + ∑ β ii xi2 + ε
Second-order model y =
i ==
1
i 1 j >i
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i=
1
75
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y=
yC ⇒ no "curvature"
F
The hypotheses are:
k
H 0 : ∑ β ii = 0
i =1
k
H1 : ∑ β ii ≠ 0
i =1
SS Pure Quad
nF nC ( yF − yC ) 2
=
nF + nC
This sum of squares has a
single degree of freedom
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Example 6.7, Pg. 286
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.
Chapter 6
nC = 4
Usually between 3
and 6 center points
will work well
Design-Expert
provides the analysis,
including the F-test
for pure quadratic
curvature
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ANOVA for Example 6.7
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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
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Practical Use of Center Points (pg. 289)
• 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?
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Center Points and Qualitative Factors
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