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

5.3.3.4.3.

Confounding (also called aliasing)

Confounding means we have lost the ability to estimate some effects and/or interactions

One price we pay for using the design table column X₁*X₂ to obtain column X₃ in Table 3.14 is, clearly, our inability to obtain an estimate of the interaction effect for X₁*X₂ (i.e., c₁₂) that is separate from an estimate of the main effect for X₃. In other words, we have confounded the main effect estimate for factor X₃ (i.e., c₃) with the estimate of the interaction effect for X₁ and X₂ (i.e., with c₁₂). 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 2^3-1 design, we also assume that c₁₂ is small compared to c₃; this is called a 'sparsity of effects' assumption. Our computation of c₃ is in fact a computation of c₃ + c₁₂. If the desired effects are only confounded with non-significant 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 'X₃ = X₁*X₂' (understanding that we are talking about multiplying columns of the design table together) is: '3 = 12' (similarly 3 = -12 refers to X₃ = -X₁*X₂). 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 2^3-1design (the dark-shaded 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 light-shaded corners of Figure 3.4. We call I=123 the defining relation for the 2^3-1 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 2^3-1 fractional factorial design. With I=123, we can easily generate all the columns of the half-fraction design 2^3-1.

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 2^3-1 design confounded with two-factor interactions

The confounding pattern described by 1=23, 2=13, and 3=12 tells us that all the main effects of the 2^3-1 design are confounded with two-factor interactions. That is the price we pay for using this fractional design. Other fractional designs have different confounding patterns; for example, in the typical quarter-fraction of a 2⁶ design, i.e., in a 2^6-2 design, main effects are confounded with three-factor 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 two-factor interactions of a 2^6-2 are confounded with other two-factor interactions.

A useful summary diagram for a fractional factorial design

Summary: A convenient summary diagram of the discussion so far about the 2^3-1 design is as follows:

FIGURE 3.5: Essential Elements of a 2^3-1 Design

The next section will add one more item to the above box, and then we will be able to select the right two-level fractional factorial design for a wide range of experimental tasks.