This chapter discusses factorial designs that are commonly used in designed
experiments, but are not necessarily limited to two level factors. These
designs are the Plackett-Burman designs
and Taguchi's orthogonal arrays.
It was mentioned in Chapter 7
that resolution III designs can be used as highly fractional designs to
investigate main effects using runs (provided that three factor and
higher order interaction effects are not important to the experimenter).
A limitation with these designs is that all runs in these designs have
to be a power of 2. The valid runs for these designs are 4, 8, 16, 32,
etc. Therefore, the next design after the 2 design with 4 runs is the 2design with 8 runs, and the design
after this is the 2 design with 32 runs and so on (see
Table 8.1). Plackett-Burman designs solve this problem. These designs
were proposed by R. L. Plackett and J.P. Burman (1946). These designs
also allow the estimation of main effects using runs. In these designs, runs are a
multiple of 4. The valid runs for Plackett-Burman designs are 4, 8, 12,
16, 20 and so on. When the runs are a power of 2, these designs correspond
to the resolution III two factor fractional factorial designs. Although
Plackett-Burman designs are all two level orthogonal designs, the alias
structure for these designs is complicated when runs are not a power of
2.
Table 8.1: Highly fractional designs to investigate main
effects.
As an example, consider the 12-run Plackett-Burman design shown in Figure
8.1. If 11 main effects are to be estimated using
this design, then each of these main effects is partially aliased with
all other two factor interactions not containing that main effect. For
example, the main effect is partially aliased with all two
factor interactions except , , , , , , , , and . There are 45 such two factor interactions
that are aliased with .
Due to the complex aliasing, Plackett-Burman designs involving a large
number of factors should be used with care. Some of the Plackett-Burman
designs available in DOE++
are included in Appendix B. The 12-run Plackett-Burman design shown in
Figure 8.1 can be set up using the properties displayed
in Figure 8.2.
Figure: 8.1: 12-run Plackett-Burman design.
Figure 8.2: Design properties for the 12-run Plackett-Burman
design.
main effects using 
runs (provided that three factor and
higher order interaction effects are not important to the experimenter).
A limitation with these designs is that all runs in these designs have
to be a power of 2. The valid runs for these designs are 4, 8, 16, 32,
etc. Therefore, the next design after the 2
design with 4 runs is the 2
design with 8 runs, and the design
after this is the 2
design with 32 runs and so on (see
Table 8.1). Plackett-Burman designs solve this problem. These designs
were proposed by R. L. Plackett and J.P. Burman (1946). These designs
also allow the estimation of 
main effects using 
runs. In these designs, runs are a
multiple of 4. The valid runs for Plackett-Burman designs are 4, 8, 12,
16, 20 and so on. When the runs are a power of 2, these designs correspond
to the resolution III two factor fractional factorial designs. Although
Plackett-Burman designs are all two level orthogonal designs, the alias
structure for these designs is complicated when runs are not a power of
2.
