3.
Production
Process Characterization
3.2.
Assumptions / Prerequisites
3.2.3.
Analysis of Variance Models (ANOVA)
3.2.3.2.
Two-Way Crossed ANOVA
3.2.3.2.1.
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Two-way Crossed Value-Splitting Example
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|
|
Example: Coolant is completely crossed with machine
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The data table below is five samples each collected from
five different lathes each running two different types of coolant. The
measurement is the diameter of a turned pin.
|
Machine
|
Coolant
A |
1
|
2 |
3 |
4 |
5 |
| .125 |
.118 |
.123 |
.126 |
.118 |
| .127 |
.122 |
.125 |
.128 |
.129 |
| .125 |
.120 |
.125 |
.126 |
.127 |
| .126 |
.124 |
.124 |
.127 |
.120 |
| .128 |
.119 |
.126 |
.129 |
.121 |
Coolant
B
|
.124 |
.116 |
.122 |
.126 |
.125 |
| .128 |
.125 |
.121 |
.129 |
.123 |
| .127 |
.119 |
.124 |
.125 |
.114 |
| .126 |
.125 |
.126 |
.130 |
.124 |
| .129 |
.120 |
.125 |
.124 |
.117 |
|
|
|
For the crossed two-way case, the first thing we need to
do is to sweep the cell means from the data table to obtain the
residual values. This is shown in the tables below.
|
|
The first step is to sweep out the
cell means to obtain the residuals and means
|
|
Machine
|
|
1
|
2
|
3
|
4
|
5
|
|
A
|
.1262
|
.1206
|
.1246
|
.1272
|
.123
|
|
B
|
.1268
|
.121
|
.1236
|
.1268
|
.1206
|
Coolant
A |
|
-.0012
|
-.0026 |
-.0016 |
-.0012 |
-.005
|
|
.0008
|
.0014
|
.0004
|
.0008
|
.006
|
|
-.0012
|
-.0006 |
.0004
|
-.0012 |
.004
|
|
-.0002
|
.0034
|
-.0006 |
-.0002 |
-.003
|
|
.0018
|
-.0016 |
.0014
|
.0018
|
-.002
|
Coolant
B
|
-.0028 |
-.005
|
-.0016 |
-.0008 |
.0044
|
|
.0012
|
.004
|
-.0026 |
.0022
|
.0024
|
|
.0002
|
-.002
|
.0004
|
-.0018 |
-.0066 |
| -.0008 |
.004
|
.0024
|
.0032
|
.0034
|
|
.0022
|
-.001
|
.0014
|
-.0028 |
-.0036 |
|
|
Sweep the row means
|
The next step is to sweep out the row means. This gives the table below.
|
|
|
|
|
Machine |
|
|
1
|
2 |
3 |
4 |
5 |
|
A
|
.1243 |
.0019 |
-.0037 |
.0003
|
.0029 |
-.0013 |
|
B
|
.1238
|
.003
|
-.0028 |
-.0002 |
.003 |
-.0032 |
|
|
Sweep the column means
|
Finally, we sweep the column means to obtain the grand mean, row
(coolant) effects, column (machine) effects and the interaction effects.
|
|
|
|
|
Machine |
|
|
1
|
2 |
3 |
4 |
5 |
|
.1241
|
.0025
|
-.0033 |
.00005
|
.003
|
-.0023 |
|
A
|
.0003
|
-.0006 |
-.0005 |
.00025
|
.0000
|
.001
|
|
B
|
-.0003
|
.0006
|
.0005
|
-.00025 |
.0000 |
-.001
|
|
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What do these tables tell us?
|
By looking at the table of residuals, we see that the residuals for
coolant B tend to be a little higher than for coolant A. This implies
that there may be more variability in diameter when we use coolant B.
From the effects table above, we see that machines 2 and 5 produce
smaller pin diameters than the other machines. There is also a very
slight coolant effect but the machine effect is larger. Finally, there
also appears to be slight interaction effects. For instance, machines 1
and 2 had smaller diameters with coolant A but the opposite was true
for machines 3,4 and 5.
|
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Calculate sums of squares and mean squares
|
We can calculate the values for the ANOVA table according to the
formulae in the table on the crossed
two-way page. This gives the table below. From the F-values we
see that the machine effect is significant but the coolant and the
interaction are not.
|
|
|
Source
|
Sums of Squares
|
Degrees of Freedom
|
Mean Square
|
F-value
|
|
Machine
|
.000303
|
4
|
.000076
|
8.8 > 2.61
|
|
Coolant
|
.00000392
|
1
|
.00000392
|
.45 < 4.08
|
|
Interaction
|
.00001468
|
4
|
.00000367
|
.42 < 2.61
|
|
Residual
|
.000346
|
40
|
.0000087
|
|
|
|
Corrected Total
|
.000668
|
49
|
|
|
|
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|
