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

Be careful when choosing the allowable range for each factor
 

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

Consider adding some center points to your two-level design
 
 
 

Matrix notation for describing an experiment
 
 
 
 
 
 
 
 
 

Scaling or coding
 

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.

The most popular experimental designs are called 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. 

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

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 where 4 trials (or runs) were conducted with each factor set to high or low during a run according to whether the matrix had +1 or -1 set for that 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". 

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

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 get what is known as the model or analysis matrix for this simple experiment, which is shown below. The analysis matrix for a three factor experiment is shown later in this section.
 

The model for this experiment is 

    Y = b₀ + b₁X1 + b₂X2 + b₁₂X1*X2 + experimental error

and the "I" column of the design matrix has all 1's because the b₀ term appears with coefficient "1" in each trial. The X1*X2 column is formed by multiplying the the "X1" and "X2" columns together, row element by row element. This column gives the multiplier of the b₁₂ interaction term for each trial. 

In matrix notation, we can summarize this experiment by

Y = Xb + experimental error

Scaling is sometime called "orthogonal scaling" since all the columns of a scaled 2-factor design matrix (except the "I" column) are typically orthogonal.