Fractional Factorial Design

Introduction: Fractional Factorial Design

Fractional factorial design is a Design of Experiments method that tests a carefully chosen fraction of all possible factor combinations. It finds the main effects and the most important interactions with fewer trials. This saves time and resources while still giving sound insight.

Background

Full factorials grow very fast as you add factors. A two level design with k factors needs \( 2^{k} \) runs. A fractional design uses only a fraction \( \tfrac{1}{2}p \) of these runs, written as \( 2^{\,k-p} \). You choose generators to define which interactions are aliased with others. The set of aliases forms the defining relation and gives the resolution of the design.

Key Elements or Features

Notation. Two level fractional designs are written as \( 2^{k-p} \). For example \( 2^{5-2} = 8 \) runs

• Resolution. Resolution tells you which effects are clear of aliasing.
  Resolution III. Main effects may be aliased with two factor interactions.
  Resolution IV. Main effects are clear of two factor interactions. Two factor interactions may be aliased with each other.
  Resolution V. Main effects and two factor interactions are clear of each other.
• Aliasing. Effects that share the same alias chain cannot be separated. Choose the design so that likely small effects are aliased together.
• Generators and defining relation. Generators create the fraction. The defining relation lists all words that equal I. Its shortest word length sets the resolution.
• Centre points. Add centre points to check curvature.
• Blocking. You can block to remove known variation from nuisance sources.
• Fold over. Add a complementary fraction to break key aliases when needed.
• Randomisation and replication. Randomise run order and replicate to estimate pure error.

Applications or Examples

• Screening many process inputs in manufacturing or service settings.
• Early stage product development where budget is tight.
• Software or website experiments that test many toggles.
• Bioprocess steps such as temperature, pH, and feed rate.

Relevance or Impact

Fractional factorials give fast learning with fewer runs. Teams identify the vital few factors, confirm direction, and move to optimisation. The method supports evidence based decisions and shortens cycle time.

See also

• Design of Experiments (DOE)
• Full Factorial Design
• Analysis of Variance (ANOVA)
• Response Surface Methodology (RSM)