Full Factorial Design leads to experiments where at least one trial is included for all possible combinations of factors and levels. This exhaustive approach makes it impossible to miss any interactions, as all factor interactions are accounted for. However, the thoroughness of this approach makes it quite expensive and time-consuming for experiments with multiple factors – this increases exponentially with the number of factors and levels.

	Factor 1	Factor 2	Factor 3
Trials	L1	L2	L1	L2	L1	L2
1	✔		✔		✔	
2	✔		✔			✔
3	✔			✔	✔	
4	✔			✔		✔
5		✔	✔		✔	
6		✔	✔			✔
7		✔		✔	✔	
8		✔		✔		✔

Sample factorial design table for a three-factor experiment with two levels per factor

Calculating the Number of Trials

The number of trials required for a full factorial experimental run is the product of the levels of each factor:

No. of trials = F₁ level count x F₂ level count x … x F_n level count

How Many trials in a Full Factorial Design?

Found by taking the number of levels as the base and the number of factors as the exponent:

Ex1. a design of 4 factors with 3 levels each would be: 3 x 3 x 3 x 3 = 3^4 = 81

Ex 2. 4 factors (A = 3, B = 2, C = 5, D = 4 levels).   3 x 2 x 5 x 4 = 120 observations.

Example

Let’s look at an experiment with four factors:

• The first factor has two possible levels.
• The second factor has five possible levels.
• The third factor has three possible levels.
• The fourth factor has six possible levels.

To cover all of the potential combinations, the experiment will need:

No. of trials = 2 x 5 x 3 x 6 = 180 trials

Full Factorial Design Notes

• Full factorial designs include all possible combinations of every level of every factor.
• Full factorial designs can require a lot of trials, which can take a lot of time.
• Full factorial designs can require a lot of trials, which can cost a lot of money.
• Requires at least one observation for every combination of factors and levels.
• Allows for the measurement of all possible interactions.
• Expensive and time-consuming.

Analyzing Full Factorial Designs

Factorial ANOVA

You can use an Analysis of Variation – ANOVA to determine the results of full factorial design experiments.

Yates Analysis

Yates analysis is used in experiments with multiple factors, all having two levels. In some circumstances, the two levels can be ‘high’ and ‘low’ data points. It can be used in both full and fractional factorial design experiments. Read more about Yates analysis.

Why You Would Use Partial or Fractional Factorial Design Instead

One of the big drawbacks of fractional factorial design is the potential to miss important interactions.

Fractional factorials (like Latin and Graeco-Latin Squares) will not allow the analysis of interactions. The interactions are confounded with other effects.

Moving from Full Factorial to Partial Factorial

• There will be fewer trials
• There will be confounding
• Resolution will decrease

Design of Experiments Factorial Design Video