Statistician’s corner what’s behind aliasing in fractional-factorial designs

Abstract

Aliasing in a fractional-factorial design means that it is not possible to estimate all effects because the experimental matrix has fewer unique combinations than a full-factorial design. The alias structure defines how effects are combined. When the researcher understands the basics of aliasing, they can better select a design that meets their experimental objectives. Starting with a layman’s definition of an alias, it is two or more names for one thing. Referring to a person, it could be “Fred, also known as (aliased) George.” There is only one person, but they go by two names. As will be shown shortly, in a fractional-factorial design, there will be one calculated effect estimate that is assigned multiple names (aliases). This example (Figure 1) is a 2̂ 3, 8-run factorial design. These eight runs can be used to estimate all possible factor effects including the main effects A, B, and C, followed by the interaction effects AB, AB, BC and ABC. An additional column “I” is the Identity column, representing the intercept for the polynomial. Each column in the full-factorial design is a unique set of pluses and minuses, resulting in independent estimates of the factor effects. An effect is calculated by averaging the response values where the factor is set high (+) and subtracting the average response from the rows where the term is set low ( ). Mathematically, this is written as follows: In this example, the A effect is calculated like this: The last row in Figure 1 shows the calculation result for the other main effects, 2-factor and 3-factor interactions and the Identity column. In a half-fraction design (Figure 2), only half of the runs are completed. According to standard practice, we eliminate all the runs where the ABC column has a negative sign. Now the columns are not unique—pairs of columns have the identical pattern of pluses and minuses. The effect estimates are confounded (aliased) because they are changing in exactly the same pattern. The A column is the same pattern as the BC column (A = BC).