5.3. Choosing an experimental design
|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
|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
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".
|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
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
in this section.
|Model for the experiment
The model for
this experiment is
|Model in matrix notation
In matrix notation, we can
summarize this experiment by
|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.