# What’s Behind Aliasing in Fractional-Factorial Designs

Shari Kraber on March 8, 2022

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 2 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 2^3, 8-run factorial design. These 8 runs can be used to estimate all possible factor effects including the main effects A, B, 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.

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 2 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 2^3, 8-run factorial design. These 8 runs can be used to estimate all possible factor effects including the main effects A, B, 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). Likewise, B=AC and C=AB. Finally, I=ABC. These paired columns are said to be “aliased” with each other.

In the half-fraction, the effect of A (and likewise BC) is calculated like this:

When the effect calculations are done on the half-fraction, one mathematical calculation represents each pair of terms. They are no longer unique. Software may label the pair only by the first term name, but the effect is really all the real effects combined. The alias structure is written as:

I = ABC
[A] = A+BC
[B] = B+AC
[C] = C+AB

Looking back at the original data, the A effect was -1 and the BC effect was -21.5. When the design is cut in half and the aliasing formed, the new combined effect is:

A+BC = -1 + (-21.5) = -22.5

The aliased effect is the linear combination of the real effects in the system. Aliasing of main effects with two-factor interactions (2FI) is problematic because 2FI’s are fairly likely to be significant in today’s complex systems. If a 2FI is physically present in the system under study, it will bias the main effect calculation. Any system that involves temperature, for instance, is extremely likely to have interactions of other factors with temperature. Therefore, it would be critical to use a design table that has the main effect calculations separated (not aliased) from the 2FI calculations.

What type of fractional-factorial designs are “safe” to use? It depends on the purpose of the experiment. Screening designs are generally run to correctly identify significant main effects. In order to make sure that those main effects are correct (not biased by hidden 2FI’s), the aliasing of the main effects must be with three-factor interactions (3FI) or greater. The alias structure looks something like this (only main effect aliasing shown):

I = ABCD
[A] = A+BCD
[B] = B+ACD
[C] = C+ABD
[D] = D+ABC

If the experimental goal is characterization or optimization, then the aliasing pattern should ensure that both main effects and 2FI’s can be estimated well. These terms should not be aliased with other 2FI’s.

Within Design-Expert or Stat-Ease 360 software, color-coding on the factorial design selection screen provides a visual signal. Here is a guide to the colors, listed from most information to least information:

• White squares – full factorial designs (no aliasing)
• Green squares – good estimates of both main effects and 2FI’s
• Yellow squares – good estimates of main effects, unbiased from 2FI’s in the system
• Red squares – all main effects are biased by any existing 2FI’s (not a good design to properly identify effects, but acceptable to use for process validation where it is assumed there are no effects).

Good luck with your DOE data analysis!