4.2 Higher-order factorial designs

Higher-order factorial designs expand on two-factor experiments, allowing researchers to study complex relationships between multiple variables. These designs, like three-factor and four-factor setups, provide insights into main effects and interactions, offering a more comprehensive understanding of the system being studied.

As designs become more complex, issues like confounding and aliasing can arise. These challenges occur when effects of different factors or interactions become mixed, making it harder to separate their individual impacts. Researchers must carefully consider design resolution and blocking to manage these issues effectively.

Multifactor Designs

Three-factor and Four-factor Factorial Designs

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• Three-factor factorial designs involve three independent variables or factors
  • Each factor has two or more levels
  • Allows for the investigation of main effects and interactions between the three factors
  • Example: A study on the effects of temperature (low, high), pressure (low, high), and catalyst type (A, B) on yield in a chemical process
• Four-factor factorial designs include four independent variables or factors
  • Each factor has two or more levels
  • Enables the examination of main effects and interactions among the four factors
  • Example: An experiment on the impact of fertilizer type (organic, inorganic), soil pH (acidic, neutral, alkaline), watering frequency (daily, every other day), and sunlight exposure (full sun, partial shade) on plant growth

Multifactor Designs and Design Resolution

• Multifactor designs involve more than two factors
  • Allow for the study of main effects and interactions among multiple factors simultaneously
  • Provide a comprehensive understanding of the system under investigation
  • Require careful planning and consideration of the number of runs needed
• Design resolution is a measure of the degree to which main effects and interactions are confounded with each other
  • Higher resolution designs (e.g., Resolution V) allow for the estimation of main effects and two-factor interactions without confounding
  • Lower resolution designs (e.g., Resolution III) may confound main effects with two-factor interactions, making interpretation more challenging
  • The choice of design resolution depends on the objectives of the study and the resources available

Confounding and Aliasing

Confounding and Aliasing in Factorial Designs

• Confounding occurs when the effects of two or more factors or interactions are combined and cannot be estimated separately
  • Happens when the number of runs is insufficient to estimate all effects independently
  • Can be intentional (to reduce the number of runs) or unintentional (due to design limitations)
  • Example: In a 2^4 factorial design with only 8 runs, some two-factor interactions may be confounded with other two-factor interactions
• Aliasing is a consequence of confounding, where two or more effects are estimated by the same linear combination of the response values
  • Aliased effects cannot be distinguished from each other
  • The alias structure of a design depends on the design resolution and the defining relation
  • Example: In a Resolution III design, main effects may be aliased with two-factor interactions

Blocking in Factorial Designs

• Blocking is a technique used to reduce the impact of nuisance factors on the experimental results
  • Nuisance factors are sources of variability that are not of primary interest but may affect the response
  • Blocks are groups of experimental units that are expected to be more homogeneous than units across blocks
  • Example: In an agricultural experiment, blocks may represent different fields or locations
• Blocking can be used in factorial designs to improve precision and reduce confounding
  • Blocks are typically confounded with one or more high-order interactions
  • The choice of effects to be confounded with blocks depends on the objectives of the study and the expected magnitude of the effects
  • Example: In a 2^3 factorial design with two blocks, the three-factor interaction (ABC) may be confounded with blocks to allow for the estimation of main effects and two-factor interactions