5.
Process Improvement
5.3. Choosing and running an experimental design 5.3.3. How do you select an experimental design? 5.3.3.2. Fractional factorial designs


Generating
relation and diagram for the 2^{83 }fractional factorial design
"Defining relation "for a factorial design Definition of "Resolution"
Notation for resolution (roman numerals)
Resolution and confounding

We have considered the 2^{31} design in the previous
section, and seen that its generating relation written in '‘I =...'’form
is { I = + 123 }. Now consider the oneeighth fraction of a 2^{8 }design,
namely the 2^{83} fractional factorial design. Using a diagram
similar to Figure 3.5, we are given the
following:
FIGURE 3.6 Specifications for a 2^{83} Design
(1) Write down a full factorial for 83 = 5 factors; that is,
a 2^{5} full factorial design. Such a design has 2^{5}
= 32 rows.
We note further that the generating relations, written in ‘I = ...’ form, for the 2^{83 }is This collection of generating relations for a factorial design is called a defining relation. There are three ‘words’ in the defining relation for the 2^{83 }design. The length of the shortest word in the defining relation is called the resolution of the design. Resolution describes the degree to which estimated main effects are aliased (or confounded) with estimated 2level interactions, 3level interactions, etc The length of the shortest word in the defining relation for the 2^{83} design is four. This is written in Roman numeral script, and subscripted as 2_{IV}^{83}. Note that the 2^{31} design has one word: ‘I = ±123’ in its defining relation, and so has resolution three; that is, we may write 2_{III}^{31}. Now Figure 3.6 may be completed by writing it as: FIGURE 3.7 Specifications for a 2^{83}, Showing Resolution
‘IV’
Similarly, in a resolution IV design, main effects are confounded with at worst threefactor interactions. We can see, in Figure 3.7, that 6=345. We also see that 36=45, 34=56, etc. (i.e. twofactor interactions are confounded with certain other twofactor interactions) etc.; but we never see the likes of 2=13, or 5=34, (main effects confounded with twofactor interactions). The complete confounding pattern, for confounding of up to twofactor interactions, arising from the design given in Figure 3.7 is 34 = 56 = 78
Note that we have deliberately left out reference to threefactor interactions, and higher than threefactor interactions, because as we have said they are assumed negligible. (This may be a wrong assumption in some rare instances). If we are interested in estimating up to two of the twofactor interactions free of confounding with other twofactor interaction when running this design, we would assign those factors of interest to columns 1 or column 2. This type of consideration is often key in choosing and setting up a design. This means that a resolution IV design is ‘better’ than a resolution III design because we have a less severe confounding pattern in the ‘IV’ than in the ‘III’ situation; higher order interactions are usually assumed to be much less significant than loworder interactions. A higher resolution design for the same number of factors will, however, require more runs and so it is ‘worse’ than a lowerorder design in that sense. Similarly, a resolution V design, main effects would be confounded with at worst fourfactor interactions, and two factor interactions would be confounded with certain threefactor interactions. Example: The 2^{83} design is of resolution IV, and requires 32 runs. The 2^{82 }design, also for eight factors, has resolution V, but requires 64 runs. The identifying relation for a design is not necessarily unique. An alternative relation that will generate a 2^{83 }design is { I = ± 1236; I = ± 1247; I = ± 23458 }. Diagramatically, this is FIGURE 3.8 Another Way of Generating the 2^{83 }Design


Commonly used design Resolutions 
Design Resolution Summary
The meaning of the most prevalent resolution levels is as follows: Resolution III Designs
Resolution V Designs
