3. Production Process Characterization
3.2. Assumptions / Prerequisites
3.2.3. Analysis of Variance Models (ANOVA)
3.2.3.2. Two-Way Crossed ANOVA

## Two-way Crossed Value-Splitting 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 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
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 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