Reducing the number of runs

Fractional factorial designs require a number of runs that is a power of 2. This means that:

Some experiments are so expensive to conduct that it is necessary to estimate the main effects of say 8 factors in fewer than 16 runs.

Plackett-Burman design for 12 runs

A different class of resolution III designs called Plackett-Burman designs can be constructed with fewer runs. The table below describes a Plackett-Burman design that can be used for up to 11 factors in 12 runs.

    Factor  
Run    A     B     C     D     E     F     G     H     I     J     K  
1 +1 +1 -1 +1 +1 +1 -1 -1 -1 +1 -1
2 -1 +1 +1 -1 +1 +1 +1 -1 -1 -1 +1
3 +1 -1 +1 +1 -1 +1 +1 +1 -1 -1 -1
4 -1 +1 -1 +1 +1 -1 +1 +1 +1 -1 -1
5 -1 -1 +1 -1 +1 +1 -1 +1 +1 +1 -1
6 -1 -1 -1 +1 -1 +1 +1 -1 +1 +1 +1
7 +1 -1 -1 -1 +1 -1 +1 +1 -1 +1 +1
8 +1 +1 -1 -1 -1 +1 -1 +1 +1 -1 +1
9 +1 +1 +1 -1 -1 -1 +1 -1 +1 +1 -1
10 -1 +1 +1 +1 -1 -1 -1 +1 -1 +1 +1
11 +1 -1 +1 +1 +1 -1 -1 -1 +1 -1 +1
12 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

There is a pattern in the table — each row is identical to the one above, but shifted one place to the right, except for the final 12th row which is all -1 (necessary to ensure the same number of +1 and -1 for all factors).

In this design, all factor main effects are orthogonal to each other — the columns of ±1 are uncorrelated.

Fewer than 11 factors

The above Plackett-Burman design shows the factor levels for 11 factors in 12 runs such that all factors are orthogonal. Any subset of these 11 factors are also orthogonal, so the design can be modified to give a Resolution III design for fewer than 11 factors by omitting arbitrary columns of the design matrix.

This design can therefore be used to estimate the main effects of between 8 and 11 factors in 12 runs instead of the 16 runs that would be needed for a fractional factorial design.