Designs based on complete factorial designs
In incomplete designs, some main effects are inevitably confounded with interactions. The best way to create such a design starts from a complete design — if there are 2k experimental units, this would be a complete design for k factors.
The levels of the extra factors should be allocated based on the columns of +/-1 for interactions between the original factors.
There are often alternative possibilities here, but we would ideally prefer designs in which the main effects are only confounded 3-factor and higher-order interactions.
A design that is obtained in this way for k factors in 2k-r experimental units is called a 2k-r fractional factorial design.
Confounding
In fractional factorial experiments, many interactions are confounded, not just the interactions that define the 'extra' factors. For example, consider a 23 factorial design for factors A, B and C in which the levels of D are defined by the 3-factor interaction between the three original factors,
D = ABC
Several other effects are confounded in this design. They can be found using arithmetic on the columns of ±1, noting that squaring any column gives +1,
A2 = B2 = C2 = D2 = 1
Therefore
A = AD2 = (AD)D = (AD)(ABC) = A2BCD = BCD
and similarly,
B = ACD
C = ABD
AB = CD
AC = BD
AD = BC
ABCD = 1
24-1 fractional factorial design
The columns on the left below show the levels for a complete 23 factorial design for factors A, B and C. We now consider varying a fourth factor, D, in this experiment..
In the diagram above, it is possible to decide which of the interactions will be confounded with factor D in the design by clicking the red heading above the table. Observe that:
Since it is more reasonable to assume that there are no 3-factor interactions,
The best design uses the column of +/-1 for the ABC interaction to define the levels of factor D.
Another way to describe the treatments
The treatments that are used in the design are displayed in a different way under the table — each 'word' describes the factors that are at their 'high' level for one of the experimental units. For example:
Click on rows of the table or 'words' under it to see how the 'words' correspond to the levels of the factors.