# Factorial Experimental Design

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## What is Factorial Experimental Design?

Some experiments involve the study of the effects of multiple factors. For such studies, the factorial experimental design is very useful. A full factorial design, also known as fully crossed design, refers to an experimental design that consists of two or more factors, with each factor having multiple discrete possible values or “levels”.

Using this design, all the possible combinations of factor levels can be investigated in each replication. Although several factors can affect the variable being studied in factorial experiments, this design specifically aims to identify the main effects and the interaction effects among the different factors.

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## Factors, Main Effects, and Interactions

To understand the factorial experimental design, you must be well-acquainted with the following terms:

Factors: This is a broad term used to describe the independent variable that is manipulated in the experiment by the researcher or through selection.

Main Effects: The main effect of a factor refers to the change produced in response to a change in the level of the factor. Therefore the effect of factor A is the difference between the average response at A1 and A2.

Interaction: An Interaction between factors occurs when the difference in response between the levels of one factor is not the same at all the levels of the other factor.

There are three main types of interactions:

• Antagonistic Interaction: When the main effect is non-significant and interaction is significant. In this interaction, the two independent variables are likely to reverse the effect of each other.
• Synergistic Interaction: When the higher level of one independent variable enhances the effect of the other.
• Ceiling Effect Interaction: When the higher level of one independent variable lowers the differential effect of another variable.

When there is a large interaction, main effects have little practical meaning as a significant interaction often masks the significance of main effects.

## Types of Factorial Design

There are three main types of factorial designs, namely “Within Subject Factorial Design”, “Between Subject Factorial Design”, and “Mixed Factorial Design”.

1. Within Subject Factorial Design: In this factorial design, all of the independent variables are manipulated within subjects.
2. Between Subject Factorial Design: In the Between Subject Factorial Design, the subjects are assigned to different conditions and each subject only experiences one of the experimental conditions.
3. Mixed Factorial Design: This design is most commonly used in the study of psychology. It is named the ‘Mixed Factorial Design’ because it has at least one Within Subject variable and one Between Subject variable.

## Advantages of Factorial Experimental Design

The following are a few advantages of using the factorial experimental design:

• Efficient: When compared to one-factor-at-a-time (OFAT) experiments, factorial designs are significantly more efficient and can provide more information at a similar or lower cost. It can also help find optimal conditions quicker than OFAT experiments can.
• Low-Cost: When using the factorial design, additional factors can be examined without having to bear additional costs.
• Comprehensive Results: Researchers can employ the factorial design to calculate the effects of a factor as an estimate at several levels of other factors. This can yield conclusions that are valid over a range of different experimental conditions.

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## FAQs on Factorial Experimental Design

A factorial experimental design is an experimental design that is used to study two or more factors, each with multiple discrete possible values or “levels”.

When compared to the one-factor-at-a-time design (OFAT), factorial designs are less expensive, more efficient, and produce more comprehensive results.

The main effect of a factor can be defined as the change produced as a result of a change in the level of the factor.

When the difference in response between the levels of one factor is not the same at all the levels of the other factor, there is an interaction between the factors.

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