Factorial Designs, Study notes of Design

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Factorial Designs

Definitions and Principles

• Many experiments involve the study of the effects of two or more factors. Factorial designs are most efficient for this type of experiment.
• In a factorial design, all possible combinations of the levels of the factors are investigated in each replication.
• If there are a levels of factor A, and b levels of factor B, then each replicate contains all ab treatment combinations.

Interaction

• In some experiments we may find that the difference in response between the levels of one factor is not the same at all levels of the other factor. When this occurs, there is an interaction between the factors.
• At B 1 , the A effect is:
• At B 2 , the A effect is:

FACTOR B B 1 B (^2) FACTOR A A 1 20 40 A 2 50 12

Interaction graphs

Advantages of Factorials

• They are more efficient than one-factor-at-a-time experiments.
• A factorial design is necessary when interactions may be present to avoid misleading conclusions.
• Factorial designs allow the effects of a factor to be estimated at several levels of the other factors, yielding conclusions that are valid over a range of experimental conditions.

The Two-Factor Factorial Design

• The simplest type of factorial designs involve only two factors or sets of treatments.
• There are a levels of factor A, and b levels of factor B, and each replicate contains all ab treatment combinations.
• In general, there are n replicates.

Data

ANOVA

No Interaction analysis

Temperature: p-value= 1.24 e-

Material types: p-value= 0.

Thoughts on

Factorial vs One-Factor Designs

Interaction

• Main effect A:
  • over the 2 levels of B the A effect is different:
    • 30 vs -
    • It depends on the level chosen for the other factor, B

Thoughts

• By use of the factorial design, the interaction

can be estimated, as the AB treatment

combination

• In the 1-factor design, can only estimate main

effects A and B

• The same 4 observations can be used in the

factorial design, as in the 1-factor design, but

gain more information (e.g. on the interaction)

Additional Concepts in Factorial

Designs

Random Effects and

Degrees of Freedom

Fixed and Random Effects

Fixed Effect:

• the levels of a factor are pre-determined
• the inference will be made only on the levels

used in the experiment