In a full factorial experiment, at least one trial is included for all possible combinations of factors and levels. This exhaustive approach makes it impossible for any interactions to be missed as all factor interactions are accounted for. The thoroughness of this approach, however, makes it quite expensive and time-consuming for experiments with multiple factors – and this increases exponentially with the number of factors and levels.

Factor 1 | Factor 2 | Factor 3 | ||||
---|---|---|---|---|---|---|

Trials | L_{1} |
L_{2} |
L_{1} |
L_{2} |
L_{1} |
L_{2} |

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 = F_{1} level count x F_{2} level count x … x F_{n} level count

### 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

## 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.