Alias Structure

The alias structure of a DOE design specifies exactly which effects are confounded (aliased) with each other — meaning they cannot be independently estimated from the experimental data. It is the complete map of what information is lost (or mixed) when using a fractional factorial design.

Why Alias Structure Matters

• In a fractional factorial, you save runs by confounding certain effects. The alias structure tells you exactly which effects are mixed together.
• Before running the experiment, you must examine the alias structure to ensure that effects you care about are not aliased with other important effects.
• If two important effects are aliased, the design is inadequate — choose a higher-resolution design or add runs.

How to Read an Alias Structure

For a $2^{4-1}$ design with generator $D = ABC$:

• $I = ABCD$ (defining relation)
• $A = BCD$
• $B = ACD$
• $C = ABD$
• $D = ABC$
• $AB = CD$
• $AC = BD$
• $AD = BC$

This means:
- Main effect A is aliased with the 3-factor interaction BCD. Since BCD is likely negligible, the A estimate is reliable.
- But 2-factor interaction AB is aliased with 2-factor interaction CD — if both could be important, this is a problem.

Alias Structure and Resolution

• Resolution III ($2^{k-p}_{III}$): Main effects aliased with 2-factor interactions. Alias structure shows pairs like $A = BC$. Risky for detailed process understanding.
• Resolution IV ($2^{k-p}_{IV}$): Main effects aliased with 3+ factor interactions (clear). But 2-factor interactions aliased with each other: $AB = CD$.
• Resolution V ($2^{k-p}_{V}$): Main effects and 2-factor interactions are clear. 2-factor interactions aliased with 3-factor interactions (usually negligible).

Using Alias Structure for Design Selection

• Step 1: List the effects you expect to be important (main effects + suspected 2-factor interactions).
• Step 2: Check the alias structure of the candidate design.
• Step 3: Verify that none of your important effects are aliased with each other.
• Step 4: If important effects are aliased, either use more runs (higher resolution) or use a different fraction.

De-Aliasing (Fold-Over)

• If the experiment reveals a significant aliased pair (e.g., $AB + CD$) and you need to separate them, a fold-over design adds runs that reverse the aliasing, independently estimating each effect.

The alias structure is the blueprint of information loss in fractional factorial designs — understanding it before running the experiment prevents the frustration of discovering ambiguous results afterward.