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.