5. Process Improvement
5.3. Choosing an experimental design

5.3.2.

How do you select and scale the process variables?

Guidelines to assist the engineering judgment process of selecting process variables for a DOE

Process variables include both inputs and outputs - i.e., factors and responses. The selection of these variables is best done as a team effort. The team should

• Include all important factors (based on engineering judgment).
• Be bold, but not foolish, in choosing the low and high factor levels.
• Check the factor settings for impractical or impossible combinations - i.e., very low pressure and very high gas flows.
• Include all relevant responses.
• Avoid using only responses that combine two or more measurements of the process. For example, if interested in selectivity (the ratio of two etch rates), measure both rates, not just the ratio.

Be careful when choosing the allowable range for each factor

We have to choose the range of the settings for input factors, and it is wise to give this some thought beforehand rather than just try extreme values. In some cases, extreme values will give runs that are not feasible; in other cases, extreme ranges might move one out of a smooth area of the response surface into some jagged region, or close to an asymptote.

Two-level designs have just a  "high" and a "low" setting for each factor

The most popular experimental designs are two-level designs. Why only two levels? There are a number of good reasons why two is the most common choice amongst engineers: one reason is that it is ideal for screening designs, simple and economical; it also gives most of the information required to go to a multilevel response surface experiment if one is needed.

Consider adding some center points to your two-level design

The term "two-level design" is something of a misnomer, however, as it is recommended to include some center points during the experiment (center points are located in the middle of the design `box').

Notation for 2-Level Designs

Matrix notation for describing an experiment

The standard layout for a 2-level design uses +1 and -1 notation to denote the "high level" and the "low level" respectively, for each factor. For example, the matrix below

	Factor 1 (X1)	Factor 2 (X2)
Trial 1	-1	-1
Trial 2	+1	-1
Trial 3	-1	+1
Trial 4	+1	+1

describes an experiment in which 4 trials (or runs) were conducted with each factor set to high or low during a run according to whether the matrix had a +1 or -1 set for the factor during that trial. If the experiment had more than 2 factors, there would be an additional column in the matrix for each additional factor.

Note: Some authors shorten the matrix notation for a two-level design by just recording the plus and minus signs, leaving out the "1's".

Coding the data

The use of +1 and -1 for the factor settings is called coding the data. This aids in the interpretation of the coefficients fit to any experimental model. After factor settings are coded, center points have the value "0". Coding is described in more detail in the DOE glossary.

The Model or Analysis Matrix

Design matrices

If we add an "I" column and an "X1*X2" column to the matrix of 4 trials for a two-factor experiment described earlier, we obtain what is known as the model or analysis matrix for this simple experiment, which is shown below. The model matrix for a three-factor experiment is shown later in this section.

I	X1	X2	X1*X2
+1	-1	-1	+1
+1	+1	-1	-1
+1	-1	+1	-1
+1	+1	+1	+1

Model for the experiment

The model for this experiment is

\( Y = \beta_{0} + \beta_{1}X_{1} + \beta_{2}X_{2} + \beta_{12}X_{1}X_{2} + \mbox{experimental error} \)

and the "I" column of the design matrix has all 1's to provide for the \(\beta\)₀ term. The X₁*X₂ column is formed by multiplying the "X₁" and "X₂" columns together, row element by row element. This column gives interaction term for each trial.

Model in matrix notation

In matrix notation, we can summarize this experiment by

Y = X\(\beta\) + experimental error

for which X is the 4 by 4 design matrix of 1's and -1's shown above, \(\beta\) is the vector of unknown model coefficients \( (\beta_{0},\beta_{1},\beta_{2},\beta_{12}) \) and Y is a vector consisting of the four trial response observations.

Orthogonal Property of Scaling in a 2-Factor Experiment

Coding produces orthogonal columns

Coding is sometime called "orthogonal coding" since all the columns of a coded 2-factor design matrix (except the "I" column) are typically orthogonal. That is, the dot product for any pair of columns is zero. For example, for X₁ and X₂: (-1)(-1) + (+1)(-1) + (-1)(+1) + (+1)(+1) = 0.