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


Confounding means we have lost the ability to estimate some effects and/or interactions  One price we pay for using the design table column X_{1}*X_{2} to obtain column X_{3} in Table 3.14 is, clearly, our inability to obtain an estimate of the interaction effect for X_{1}*X_{2} (i.e., c_{12}) that is separate from an estimate of the main effect for X_{3}. In other words, we have confounded the main effect estimate for factor X_{3} (i.e., c_{3}) with the estimate of the interaction effect for X_{1} and X_{2} (i.e., with c_{12}). The whole issue of confounding is fundamental to the construction of fractional factorial designs, and we will spend time discussing it below.  
Sparsity of effects assumption  In using the 2^{31 }design, we also assume that c_{12} is small compared to c_{3}; this is called a 'sparsity of effects' assumption. Our computation of c_{3} is in fact a computation of c_{3} + c_{12}. If the desired effects are only confounded with nonsignificant interactions, then we are OK.  
A Notation and Method for Generating Confounding or Aliasing  
A short way of writing factor column multiplication  A short way of writing 'X_{3} = X_{1}*X_{2}' (understanding that we are talking about multiplying columns of the design table together) is: '3 = 12' (similarly 3 = 12 refers to X_{3} = X_{1}*X_{2}). Note that '12' refers to column multiplication of the kind we are using to construct the fractional design and any column multiplied by itself gives the identity column of all 1's.  
Next we multiply both sides of 3=12 by 3 and obtain 33=123, or I=123 since 33=I (or a column of all 1's). Playing around with this "algebra", we see that 2I=2123, or 2=2123, or 2=1223, or 2=13 (since 2I=2, 22=I, and 1I3=13). Similarly, 1=23.  
Definition of "design generator" or "generating relation" and "defining relation"  I=123 is called a design generator or a generating relation for this 2^{31}design (the darkshaded corners of Figure 3.4). Since there is only one design generator for this design, it is also the defining relation for the design. Equally, I=123 is the design generator (and defining relation) for the lightshaded corners of Figure 3.4. We call I=123 the defining relation for the 2^{31 }design because with it we can generate (by "multiplication") the complete confounding pattern for the design. That is, given I=123, we can generate the set of {1=23, 2=13, 3=12, I=123}, which is the complete set of aliases, as they are called, for this 2^{31} fractional factorial design. With I=123, we can easily generate all the columns of the halffraction design 2^{31}.  
Principal fraction  Note: We can replace any design generator by its negative counterpart and have an equivalent, but different fractional design. The fraction generated by positive design generators is sometimes called the principal fraction.  
All main effects of 2^{31} design confounded with twofactor interactions  The confounding pattern described by 1=23, 2=13, and 3=12 tells us that all the main effects of the 2^{31} design are confounded with twofactor interactions. That is the price we pay for using this fractional design. Other fractional designs have different confounding patterns; for example, in the typical quarterfraction of a 2^{6} design, i.e., in a 2^{62} design, main effects are confounded with threefactor interactions (e.g., 5=123) and so on. In the case of 5=123, we can also readily see that 15=23 (etc.), which alerts us to the fact that certain twofactor interactions of a 2^{62} are confounded with other twofactor interactions.  
A useful summary diagram for a fractional factorial design 
Summary: A convenient summary diagram of the discussion so far
about the 2^{31} design is as follows:
