Lecture 37: Three Factors - RCBD
Treatment Combinations
As the number of factors
increase so do the number of
treatment combinations.
With a large number of
treatment combinations it
becomes more difficult to have
adequate replication.
Example
Cutting Speed: 4 levels
Tool Geometry: 3 levels
Cutting Angle: 4 levels
Treatment combinations: 48
Minimum of 96 experimental
units.
1
Factors at 2 levels
2
Tool Wear Experiment
Factor A: Cutting Speed
50 rpm (–1) and 100 rpm (+1)
One way to reduce the
number of treatment
combinations is to have
factors at only 2 levels.
Factor B: Tool Geometry
Type 1 (–1) and Type 2 (+1)
Factor C: Cutting Angle
5 degrees (–1) and 10 degrees (+1)
3
Block Design
4
Informal Analysis
The experimental material
consists of metal bars.
Metal bars have different
properties depending on the
heat that produced them.
Form block by sorting on heat.
A: Cutting Speed
 50 rpm (–1): Mean = 59.25
100 rpm (+1): Mean = 58.75
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1
Lecture 37: Three Factors - RCBD
Informal Analysis
B: Tool Geometry
 Type 1 (–1): Mean = 64.75
 Type 2 (+1): Mean = 53.25
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8
Informal Analysis
C: Cutting Angle
 5 degrees (–1): Mean =
59.25
10 degrees (+1): Mean =
58.75
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10
Summary
There is very little effect due to
cutting speed.
Type 2 tool geometry has much
lower average wear.
Increasing the cutting angle
tends to decrease the average
wear.
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12
2
Lecture 37: Three Factors - RCBD
Speed by Angle
When speed is 50 rpm,
increasing the angle tends to
decrease the wear a lot.
When speed is 100 rpm,
increasing the angle tends to
increase the wear a little.
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Other Interactions
JMP Data Table
The other interaction plots
show indication of some
interaction but not as much
as between Speed and
Angle.
A: Speed
50
100
50
100
50
B: Tool
1
1
2
2
1
C: Angle
5
5
5
5
10
Heat
I
I
I
I
I
100
2
10
III
Wear
78
68
65
55
57
61
15
Fit Model
16
Analysis of Variance
Y: Wear
Highlight all three factors
under Select Columns.
Macros – Full Factorial.
Add Heat
Source
Treatment
Heat
Error
C. Total
17
df
SS
7 1608.0
2 271.0
14 203.0
23 2082.0
MS
229.7
135.5
14.5
F
15.84
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3
Lecture 37: Three Factors - RCBD
Effect Tests
Source
A: Speed
B: Tool
AB
C: Angle
AC
BC
ABC
df
1
1
1
1
1
1
1
SS
1.5
793.5
24.0
294.0
433.5
37.5
24.0
Bonferroni Correction
MS
F Prob>F
1.5 0.1034 0.7525
793.5 54.7241 <.0001
24.0 1.6552 0.2191
294.0 20.2759 0.0005
433.5 29.8966 <.0001
37.5 2.5862 0.1301
24.0 1.6552 0.2191
Because we are doing
multiple F tests, we should
choose a lower cut off for
when a P-value indicates
statistical significance.
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20
Bonferroni Correction
Significant Effects
A P-value must be less than
0.05/(# tests) before we will
declare statistical significance.
0.05/7 = 0.007
B: Tool – F = 54.7241,
P-value < 0.0001
C: Angle – F = 20.2759,
P-value = 0.0005
AC – F = 29.8966,
P-value < 0.0001
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Alternative Analysis
JMP Data Table
Use coded levels of the
factors and multiple
regression.
Low level: –1
High level: +1
23
XA
–1
+1
–1
+1
–1
XB
–1
–1
+1
+1
–1
XC
–1
–1
–1
–1
+1
Heat
I
I
I
I
I
+1
+1
+1
III
Wear
78
68
65
55
57
61
24
4
Lecture 37: Three Factors - RCBD
Fit Model
Parameter Estimates
Source
XA
XB
XA*XB
XC
XA*XC
XB*XC
XA*XB*XC
Y: Wear
Highlight all three coded
factors under Select Columns.
Macros – Full Factorial.
Add Heat
Estimate Std Error T Ratio
–0.25
1.5
–0.32
–5.75
793.5
–7.40
1.00
24.0
1.29
–3.50
294.0
–4.50
4.25
433.5
5.47
1.25
37.5
1.61
1.00
24.0
1.29
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Comment
Prob>F
0.7525
<.0001
0.2191
0.0005
<.0001
0.1301
0.2191
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Significant Effects
XB: t = –7.40, P-value < 0.0001
XC: t = –4.50, P-value = 0.0005
XA*XC: t = 5.47, P-value <
0.0001
The P-values associated
with each of the coded
factors and interactions are
the same as with the
previous analysis.
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Prediction Equation
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Minimize Wear
XB = +1, Type 2 Tool
XC = +1, 10 degree Angle
XA = –1, 50 rpm Speed
Predicted Wear = 59.0 –
5.75*XB – 3.50*XC +
4.25*XA*XC
XA*XC = (+1)*(–1) = –1
Predicted Wear = 45.5
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