6. Process or Product Monitoring and Control
6.6. Case Studies in Process Monitoring
6.6.1. Lithography Process

6.6.1.3.

Subgroup Analysis

Control charts for subgroups

The resulting classical Shewhart control charts for each possible subgroup are shown below.

Site as subgroup

The first pair of control charts use the site as the subgroup. However, since site has a subgroup size of one we use the control charts for individual measurements. A moving average and a moving range chart are shown.

Moving average control chart

Moving range control chart

Wafer as subgroup

The next pair of control charts use the wafer as the subgroup. In this case, that results in a subgroup size of 5. A mean and a standard deviation control chart are shown.

Mean control chart

SD control chart

Note that there is no LCL here because of the small subgroup size.

Cassette as subgroup

The next pair of control charts use the cassette as the subgroup. In this case, that results in a subgroup size of 15. A mean and a standard deviation control chart are shown.

Mean control chart

SD control chart

Interpretation

Which of these subgroupings of the data is correct? As you can see, each sugrouping produces a different chart. Part of the answer lies in the manufacturing requirements for this process. Another aspect that can be statistically determined is the magnitude of each of the sources of variation. In order to understand our data structure and how much variation each of our sources contribute, we need to perform a variance component analysis. The variance component analysis for this data set is shown below.

Component of variance table

Component	Variance Component Estimate
Cassette	0.2645
Wafer	0.0500
Site	0.1755

Equating mean squares with expected values

If your software does not generate the variance components directly, they can be computed from a standard analysis of variance output by equating means squares (MSS) to expected mean squares (EMS).

JMP ANOVA output

Below we show SAS JMP 4 output for this dataset that gives the SS, MSS, and components of variance (the model entered into JMP is a nested, random factors model). The EMS table contains the coefficients needed to write the equations setting MSS values equal to their EMS's. This is further described below.

Variance Components Estimation

From the ANOVA table, labelled "Tests wrt to Random Effects" in the JMP output, we can make the following variance component calculations: 4.3932 = (3*5)*Var(cassettes) + 5*Var(wafer) + Var(site) 0.42535 = 5*Var(wafer) + Var(site) 0.1755 = Var(site) Solving these equations we obtain the variance component estimates 0.2645, 0.04997 and 0.1755 for cassettes, wafers and sites, respectively.