Generation
In practice, experimenters typically rely on statistical reference books to supply the "standard" fractional factorial designs, consisting of the principal fraction. The principal fraction is the set of treatment combinations for which the generators evaluate to + under the treatment combination algebra. However, in some situations, experimenters may take it upon themselves to generate their own fractional design.
A fractional factorial experiment is generated from a full factorial experiment by choosing an alias structure. The alias structure determines which effects are confounded with each other. For example, the five factor 25 − 2 can be generated by using a full three factor factorial experiment involving three factors (say A, B, and C) and then choosing to confound the two remaining factors D and E with interactions generated by D = A*B and E = A*C. These two expressions are called the generators of the design. So for example, when the experiment is run and the experimenter estimates the effects for factor D, what is really being estimated is a combination of the main effect of D and the two-factor interaction involving A and B.
An important characteristic of a fractional design is the defining relation, which gives the set of interaction columns equal in the design matrix to a column of plus signs, denoted by I. For the above example, since D = AB and E = AC, then ABD and ACE are both columns of plus signs, and consequently so is BDCE. In this case the defining relation of the fractional design is I = ABD = ACE = BCDE. The defining relation allows the alias pattern of the design to be determined.
| Treatment combination | I | A | B | C | D = AB | E = AC |
|---|---|---|---|---|---|---|
| de | + | − | − | − | + | + |
| a | + | + | − | − | − | − |
| be | + | − | + | − | − | + |
| abd | + | + | + | − | + | − |
| cd | + | − | − | + | + | − |
| ace | + | + | − | + | − | + |
| bc | + | − | + | + | − | − |
| abcde | + | + | + | + | + | + |
Read more about this topic: Fractional Factorial Design
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