IV.3 Designs to Minimize
Variability
Background
 An Example

– Design Steps
– Transformations
– The Analysis

A Case Study
Background
Accuracy/Precision

Factors Can Affect Response Variable by
Either
– Changing Its Average Value (Accuracy)
– Changing Its Variation (Precision) or
– BOTH
Background
Example 4 - Example I.2.3 Revisited

Which Factors Affect
– Accuracy?
– Precision?
1
2
Background
Analysis for Changes in Variability
For studying Variability, we can use ALL
the designs, ALL the ideas that we used
when studying changes in mean response
level.
 However,

– Smaller Variability is ALWAYS better
– We MUST work with replicated experiments
– We will need to transform the response s
Example 5
Mounting an Integrated Circuit on Substrate
Figure 5 - Factor Level
Lochner and Matar - Figure 5.11

Response: bond strength
Factor
A. Adhesive Type
B. Conductor Material
C. Cure Time (at 90C)
D. I.C. Post Coating
Low Level
D2A
Copper
90 min.
Tin
High Level
H-1-E
Nickel
120 min.
Silver
Example 5 - Design Steps
Selecting the Design
Figure 6 - The Experimental Design
Lochner and Matar - Figure 5.12

1. Select an appropriate
experimental design
Standard
Order
1
2
3
4
5
6
7
8
Adhesive
Type
D2A
D2A
D2A
D2A
H-1-E
H-1-E
H-1-E
H-1-E
Conductor
Material
copper
copper
nickel
nickel
copper
copper
nickel
nickel
Cure
Time
90
120
90
120
90
120
90
120
I.C.
Post
Coating
tin
silver
silver
tin
silver
tin
tin
silver
Example 5 - Design Steps
Replication and Randomization

2. Determine number of replicates to be used
– Consider at Least 5 (up to 10)
– In Example 5:
5 replicates, 40 trials

3. Randomize order of ALL trials
– Replicates Run Sequentially Often Have Less
Variation Than True Process Variation
– This May Be Inconvenient!
Example 5 - Design Steps
Collecting the Data
Figure 7 - The Data


Lochner and Matar - Figure 5.13
4. Perform experiment; record data
5. Group data for each factor level combination
and calculate s.
Standard
Order
1
2
3
4
5
6
7
8
y1
73.0
87.7
80.5
79.8
85.2
78.0
78.4
90.2
y2
73.2
86.4
81.4
77.8
85.0
75.5
72.8
87.4
y3
72.8
86.9
82.6
81.3
80.4
83.1
80.5
92.9
y4
72.2
87.9
81.3
79.8
85.2
81.2
78.4
90.0
y5
76.2
86.4
82.1
78.2
83.6
79.9
67.9
91.1
y
73.48
87.06
81.58
79.38
83.88
79.54
75.60
90.32
s
1.57
0.71
0.80
1.41
2.06
2.93
5.17
1.99
Log(s)
0.196
-0.149
-0.097
0.149
0.314
0.467
0.713
0.299
Example 5 - Design Steps
The Analysis
6. Calculate logarithms of standard
deviations obtained in 5. Record these.
 7. Analyze log s as the response.

Transformations
Why transform s?
If the data follow a bell-shaped curve, then so
do the cell means and the factor effects for the
means. However, the cell standard deviations
and factor effects of the standard deviations do
not follow a bell-shaped curve.
 If we plot such data on our normal plotting
paper, we would obtain a graph that indicates
important or unusual factor effects in the
absence of any real effect. The log
transformation ‘normalizes’ the data.

Transformations
Distributions and Normal Probability Plots of s2 and
Log(s2)
0.20
0.60
Sam pling from Norm al
Sam pling from Norm al
(n =5 si gma=1 )
(n =5 si gma=1 )
0.10
0.30
0.00
0.00
0.0
3.0
6.0
9.0
12.0
-1.0
0.0
1.0
2.0
3.0
2.5
2.5
40 obser vatio ns of s
0.0
x
x
x
x
x
x
xxx
xx
xx
xx
x
xxx
xx
x xx
x
x
xxx
x
x
x
x
2
x
xx
xx
x
x
x
2
40 obser vatio ns of Log s
0.0
x
x
x
x
x
x
xx
xx
xx
xx
xx
xx
x
x
xx
x x
x
x
x
x
x
xx
xxx
x
x
x
x
-2.5
-2.5
0.0
2.0
4.0
6.0
8.0
-0.80
0.00
0.80
1.60
2.40
Example 5 - Analysis
Figure 8 - Response Table for Mean
Lochner and Matar - Figure 5.14
Standard
Order
Sum
Divisor
Effect
y
A
B
C
AB
Bond
Strength
Adhesive
Type
Conductor
Material
Cure
Time
AC
BC
D
IC Post
Coating
1
73.48
-1
-1
-1
1
1
1
-1
2
83.88
1
-1
-1
-1
-1
1
1
3
81.58
-1
1
-1
-1
1
-1
1
4
75.6
1
1
-1
1
-1
-1
-1
5
87.06
-1
-1
1
1
-1
-1
1
6
79.54
1
-1
1
-1
1
-1
-1
7
79.38
-1
1
1
-1
-1
1
-1
8
90.32
1
1
1
1
1
1
1
650.84
7.84
2.92
21.76
2.08
-1
3.28
34.84
8
4
4
4
4
4
4
4
81.355
1.96
0.73
5.44
0.52
-0.25
0.82
8.71
Example 5 - Analysis
Figure 9 - Response Table for Log(s)
Lochner and Matar - Figure 5.15
Standard
Order
Sum
Divisor
Effect
y
A
B
C
AB
Log(s)
Adhesive
Type
Conductor
Material
Cure
Time
AC
BC
D
IC Post
Coating
1
0.196
-1
-1
-1
1
1
1
-1
2
0.314
1
-1
-1
-1
-1
1
1
3
-0.097
-1
1
-1
-1
1
-1
1
4
0.713
1
1
-1
1
-1
-1
-1
5
-0.149
-1
-1
1
1
-1
-1
1
6
0.467
1
-1
1
-1
1
-1
-1
7
0.149
-1
1
1
-1
-1
1
-1
8
0.299
1
1
1
1
1
1
1
1.892
1.694
0.236
-0.36
0.226
-0.162
0.024
-1.158
8
4
4
4
4
4
4
4
0.2365
0.4235
0.059
-0.09
0.0565
-0.041
0.006
-0.2895
Example 5 - Analysis
Figure 10 - Effects Normal Probability Plot for Mean

What Factor
Settings Favorably
Affect the Mean?
Example 5 - Analysis
Figure 11 - Effects Normal Probability Plot for
Log(s)
Lochner and Matar - Figure 5.16

What Factor
Settings Favorably
Affect Variability?
Example 5 - Interpretation
Silver IC post coating increases bond
strength and decreases variation in bond
strength.
 Adhesive D2A decreases variation in bond
strength.
 120-minute cure time increases bond
strength.

Case Study 1
Filling Weight of Dry Soup Mix - Factors and Response
Factor
Low
A: Number of Ports
B: Cooling M ethod
C: M ixing Time (se cs)
D: Batch we ight (lbs)
E: Delay (days)
High
1
Water coole d
60
1500
1
3
Air cooled
80
2000
7
Re sponse : Weight of packe t conte nts (oz)
A
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
B
1
1
1
1
2
2
2
2
1
1
1
1
2
2
2
2
C
1
1
2
2
1
1
2
2
1
1
2
2
1
1
2
2
D
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
E
2
1
1
2
1
2
2
1
1
2
2
1
2
1
1
2
Log(s)
-.0132
.2672
.0719
-.1192
.0531
-.1079
.1673
.0374
.2304
-.2076
-.0088
.3222
.0969
.1335
.1072
.0414
Case Study 1
Filling Weight of Dry Soup Mix - Effects Table

Interpret This Data
– Determine the Important
Effects
– Do the Interaction
Tables and Plots for
Significant Interactions
Effect
A
B
C
D
E
AB
AC
AD
AE
BC
BD
BE
CD
CE
DE
Value
0.04482
-0.00175
0.02088
-0.04222
-0.17175
0.01245
0.03132
0.00818
-0.04610
0.02355
-0.03780
0.13837
0.02828
0.05725
-0.11665