3.5.1.4. Analysis of Variance

3. Production Process Characterization
3.5. Case Studies
3.5.1. Furnace Case Study

3.5.1.4. Analysis of Variance

The next step is to confirm our interpretation of the plots in the previous section by running an analysis of variance.

In this case, we want to run a nested analysis of variance. Although Dataplot does not perform a nested analysis of variance directly, in this case we can use the Dataplot ANOVA command with some additional computations to generate the needed analysis.

The basic steps are to use a one-way ANOA to compute the appropriate values for the run variable. We then run a one-way ANOVA with all the combinations of run and furnace location to compute the "within" values. The values for furnace location nested within run are then computed as the difference between the previous two ANOVA runs.

The Dataplot macro provides the details of this computation. This computation can be summarized in the following table.

Analysis of Variance
Source Degrees of Freedom Sum of Squares Mean Square Error F Ratio Prob > F

Run 20 61,442.29 3,072.11 5.37404 0.0000001

Furnace Location [Run] 63 36,014.5 571.659 4.72864 3.85e-11

Within 84 10,155 120.893

Total 167 107,611.8 644.382

Components of Variance

From the above analysis of variance table, we can compute the components of variance. Recall that for this data set we have 2 wafers measured at 4 furnace locations for 21 runs. This leads to the following set of equations.

Solving these equations yields the following components of variance table.

Components of Variance
Component Variance Component Percent of Total Sqrt(Variance Component)

Run 312.55694 47.44 17.679

Furnace Location[Run] 225.38294 34.21 15.013

Within 120.89286 18.35 10.995