Saturated Designs

Very efficient analysis of a large set of factors.

Readings 8.6, 8.7, Rmarkdown

1. There are designs that let you study \(K\) factors using only \(N + 1\) samples; such designs are called saturated. This is an extremely efficient use of samples.

2. The reason we don’t use saturated designs all the time are that (1) they are not available for all choices of \(K\) and (2) the resulting aliasing structure can make definitive inferences difficult. That said, it’s worth being familiar with a few saturated designs, since it can result in dramatically reduced sampling effort in some special cases.

Resolution III Designs

3. When \(K + 1\) is a power of 2, then there are resolution III designs that use only \(N = K + 1\) samples. Here are some examples,

• \(2^{3 - 1}_{III}\): 3 factors in 4 samples
  • Generators: \(C = AB\)

A	B	C
-	-	+
-	+	-
+	-	-
+	+	+

• \(2^{7 - 4}_{III}\): 7 factors in 8 samples

  • Generators: \(D = AB, E = AC, F = BC, G = ABC\)
  • See Table VIII(h) for alias groups.

• \(2^{15 - 11}_{III}\): 15 factors in 16 samples

  • See Table VII(h) for generators and alias groups (there are many)

• Exercise: Give another example (e.g., for 31 factors?)

Plackett-Burman Designs

4. Plackett-Burman designs are a collection of design options working outside the usual fractional factorial paradigm — they don’t rely on the ideas of generators or defining relations that we’ve been using so far to subset full factorial designs into less-costly experiments.

5. We won’t describe their construction, which relies on techniques from abstract algebra. The properties that are most important to know are

• A saturated design can be found whenever \(K + 1\) is a multiple of 4, which is much more often than in the resolution III situation above
• Their aliasing structure is typically more complicated. For example, when \(K = 11\), then every main effect is aliased with every pairwise interaction not including that main effect
• Their projectivity properties are typically good.
  • E.g., resolution III Plackett-Burman has projectivity 3, while any \(2^{k - p}_{III}\) fractional factorial has only projectivity 2
  • So, we can collapse to any subset of 3 factors and get a full factorial

6. To use these designs, you have to look up the tables of +/-’s, saying which factors to keep on or off for each run. See e.g., Table 8.23/8.24. Alternatively, use the FrF2 R package.