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

## Two-Way Nested Value-Splitting Example

Example: Operator is nested within machine. The data table below contains data collected from five different lathes, each run by two different operators. Note we are concerned here with the effect of operators, so the layout is nested. If we were concerned with shift instead of operator, the layout would be crossed. The measurement is the diameter of a turned pin.

 Machine Operator Sample 1 2 3 4 5 1 Day .125 .127 .125 .126 .128 Night .124 .128 .127 .126 .129 2 Day .118 .122 .120 .124 .119 Night .116 .125 .119 .125 .120 3 Day .123 .125 .125 .124 .126 Night .122 .121 .124 .126 .125 4 Day .126 .128 .126 .127 .129 Night .126 .129 .125 .130 .124 5 Day .118 .129 .127 .120 .121 Night .125 .123 .114 .124 .117
For the nested two-way case, just as in the crossed case, the first thing we need to do is to sweep the cell means from the data table to obtain the residual values. We then sweep the nested factor (Operator) and the top level factor (Machine) to obtain the table below.
Machine Operator Common Machine Operator
Sample

1
2
3
4
5
1 Day
.12404
.00246
-.0003
-.0012
.0008
-.0012 -.0002
.0018
Night
.0003
-.0028
.0012
.002
-.0008
.0022
2
Day
-.00324
-.0002
-.0026
.0014
-.0006
.0034
-.0016
Night
.0002
-.005
.004
-.002
.004
-.001
3 Day
.00006
.0005
-.0016
.0004
.0004
-.0006
.0014
Night
-.0005
-.0016 -.0026
.0004
.0024
.0014
4
Day
.00296
.0002
-.0012
.0008
-.0012
-.002
.0018
Night
-.0002
-.0008
.0022
-.0018
.0032
-.0028
5 Day
-.00224
.0012
-.005
.006
.004
-.003
-.002
Night
-.0012
.0044
.0024
-.0066
.0034
-.0036
What does this table tell us? By looking at the residuals we see that machines 2 and 5 have the greatest variability. There does not appear to be much of an operator effect but there is clearly a strong machine effect.
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 nested two-way page. This produces the table below. From the F-values we see that the machine effect is significant but the operator effect is not. (Here it is assumed that both factors are fixed).
 Source Sums of Squares Degrees of Freedom Mean Square F-value Machine .000303 4 .0000758 8.77 > 2.61 Operator(Machine) .0000186 5 .00000372 .428 < 2.45 Residual .000346 40 .0000087 Corrected Total .000668 49