5.
Process Improvement
5.3. Choosing an experimental design 5.3.3. How do you select an experimental design? 5.3.3.4. Fractional factorial designs


Generating relation and diagram for the 2^{83} fractional factorial design 
We considered the 2^{31} design in the previous section and
saw that its generator written in
"I = ... " form is {I = +123}. Next we look at a oneeighth fraction
of a 2^{8} design, namely the 2^{83} fractional
factorial design. Using a diagram similar to
Figure 3.5, we have the following:


2^{83} design has 32 runs  Figure 3.6 tells us that a 2^{83 }design has 32 runs, not including centerpoint runs, and eight factors. There are three generators since this is a 1/8 = 2^{3} fraction (in general, a 2^{kp} fractional factorial needs p generators which define the settings for p additional factor columns to be added to the 2^{kp} full factorial design columns  see the following detailed description for the 2^{83} design).  
How to Construct a Fractional Factorial Design From the Specification  
Rule for constructing a fractional factorial design 
In order to construct the design, we do the following:


Design generators 
We note further that the design generators, written in `I = ...' form,
for the principal 2^{83} fractional factorial design are:
These design generators result from multiplying the "6 = 345" generator by "6" to obtain "I = 3456" and so on for the other two generators. 

"Defining relation" for a fractional factorial design  The total collection of design generators for a factorial design, including all new generators that can be formed as products of these generators, is called a defining relation. There are seven "words", or strings of numbers, in the defining relation for the 2^{83} design, starting with the original three generators and adding all the new "words" that can be formed by multiplying together any two or three of these original three words. These seven turn out to be I = 3456 = 12457 = 12358 = 12367 = 12468 = 3478 = 5678. In general, there will be (2^{p} 1) words in the defining relation for a 2^{kp} fractional factorial.  
Definition of "Resolution"  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.  
Notation for resolution (Roman numerals)  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 only one word, "I = 123" (or "I = 123"), in its defining relation since there is only one design generator, and so this fractional factorial design has resolution three; that is, we may write \( 2_{III}^{31} \).  
Diagram for a 2^{83} design showing resolution 
Now Figure 3.6 may be completed by writing it as:


Resolution and confounding 
The design resolution tells us how badly the design is confounded.
Previously, in the 2^{31} design, we saw that the main effects
were confounded with twofactor interactions. However, main effects
were not confounded with other main effects. So, at worst, we have
3=12, or 2=13, etc., but we do not have 1=2, etc. In fact, a
resolution II design would be pretty useless for any purpose whatsoever!
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., some twofactor interactions are confounded with certain other twofactor interactions) etc.; but we never see anything like 2=13, or 5=34, (i.e., main effects confounded with twofactor interactions). 

The complete firstorder interaction confounding for the given 2^{83} design 
The complete confounding pattern, for confounding of up to twofactor
interactions, arising from the design given in Figure 3.7 is
35 = 46 36 = 45 37 = 48 38 = 47 57 = 68 58 = 67 

All of these relations can be easily verified by multiplying the indicated twofactor interactions by the generators. For example, to verify that 38= 47, multiply both sides of 8=1235 by 3 to get 38=125. Then, multiply 7=1245 by 4 to get 47=125. From that it follows that 38=47.  
One or two factors suspected of possibly having significant firstorder interactions can be assigned in such a way as to avoid having them aliased 
For this
\( 2_{IV}^{83} \) fractional factorial design, 15 twofactor interactions
are aliased (confounded) in pairs or in a group of three. The remaining
28  15 = 13 twofactor interactions are only aliased with higherorder
interactions (which are generally assumed to be negligible).
This is verified by noting that factors "1" and "2" never appear in
a length4 word in the defining relation. So, all 13 interactions
involving "1" and "2" are clear of aliasing with any other two factor
interaction.
If one or two factors are suspected of possibly having significant firstorder interactions, they can be assigned in such a way as to avoid having them aliased. 

Higher resoulution designs have less severe confounding, but require more runs 
A resolution IV design is "better" than a resolution III design because
we have lesssevere confounding pattern in the 'IV' than in the
'III' situation; higherorder interactions are less likely to be
significant than loworder interactions.
A higherresolution design for the same number of factors will, however, require more runs and so it is 'worse' than a lower order design in that sense. 

Resolution V designs for 8 factors  Similarly, with a resolution V design, main effects would be confounded with fourfactor (and possibly higherorder) interactions, and twofactor interactions would be confounded with certain threefactor interactions. To obtain a resolution V design for 8 factors requires more runs than the 2^{83} design. One option, if estimating all main effects and twofactor interactions is a requirement, is a \( 2_{V}^{83} \) design. However, a 48run alternative (John's 3/4 fractional factorial) is also available.  
There are many choices of fractional factorial designs  some may have the same number of runs and resolution, but different aliasing patterns.  Note: There are other \( 2_{V}^{83} \) fractional designs that can be derived starting with different choices of design generators for the "6", "7" and "8" factor columns. However, they are either equivalent (in terms of the number of words of length of length of four) to the fraction with generators 6 = 345, 7 = 1245, 8 = 1235 (obtained by relabeling the factors), or they are inferior to the fraction given because their defining relation contains more words of length four (and therefore more confounded twofactor interactions). For example, the \( 2_{V}^{83} \) design with generators 6 = 12345, 7 = 135, and 8 = 245 has five lengthfour words in the defining relation (the defining relation is I = 123456 = 1357 = 2458 = 2467 = 1368 = 123478 = 5678). As a result, this design would confound more two factorinteractions (23 out of 28 possible twofactor interactions are confounded, leaving only "12", "14", "23", "27" and "34" as estimable twofactor interactions).  
Diagram of an alternative way for generating the 2^{83} design 
As an example of an equivalent "best"
\( 2_{V}^{83} \) fractional factorial design, obtained by "relabeling",
consider the design specified in Figure 3.8.
This design is equivalent to the design specified in Figure 3.7 after relabeling the factors as follows: 1 becomes 5, 2 becomes 8, 3 becomes 1, 4 becomes 2, 5 becomes 3, 6 remains 6, 7 becomes 4 and 8 becomes 7. 

Minimum aberration  A table given later in this chapter gives a collection of useful fractional factorial designs that, for a given k and p, maximize the possible resolution and minimize the number of short words in the defining relation (which minimizes twofactor aliasing). The term for this is "minimum aberration".  
Design Resolution Summary  
Commonly used design Resolutions 
The meaning of the most prevalent resolution levels is as follows:
Main effects are confounded (aliased) with twofactor interactions.
No main effects are aliased with twofactor interactions, but twofactor interactions are aliased with each other.
No main effect or twofactor interaction is aliased with any other main effect or twofactor interaction, but twofactor interactions are aliased with threefactor interactions. 