3.
Production
Process Characterization
3.2.
Assumptions / Prerequisites
3.2.3.
Analysis of Variance Models (ANOVA)
3.2.3.2.
TwoWay Crossed ANOVA
3.2.3.2.1.

Twoway Crossed ValueSplitting Example


Example: Coolant is completely crossed with machine

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 twoway 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


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.

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
twoway page. This gives the table below. From the Fvalues 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

Fvalue

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





