A full factorial is a design in which every possible combination of the factors defined, and their respective levels, is used. This type of design is more powerful than measuring one-factor-at-a-time (OFAT), as it allows the rapid characterisation of optimal levels. These designs are also conceptually easier to understand, and any effect you may want to investigate is free from interference from any other possible effect. However, the number of experimental runs rapidly increases as additional factors are added to the design.
The number of runs required for a full factorial design can be defined by the following equation:
runs = x^(k), where x is the number of levels and k is the number of factors.
Hence, the simplest factorial experiment would consist of 2 factors, with 2 levels each. For example, if you were to study the effect of different concentrations of 2 media components as experimental factors (e.g., IPTG and glycerol), with levels of 1 and 2 mM each - the experimental runs would be as shown in Table 1.
Run number | IPTG (mM) | Glycerol (mM) |
1 | 1 | 1 |
2 | 1 | 2 |
3 | 2 | 1 |
4 | 2 | 2 |
Table 1. A 2-factor, 2-level full factorial design, using IPTG and glycerol as factors, set at 1 and 2 mM.
In reality, you may wish to examine more than 2 factors, and more than 2 levels. A full factorial design with 5 factors, with 3 levels each, results in 243 runs. If we increase the number of factors to 7, the number of runs needed is 2187. This can quickly become hard to implement, and so, if an experiment requires many factors, a D-Optimal or space-filling design might be more appropriate.
To learn about the statistics behind full factorial designs, click here.
To learn how to calculate a full factorial design, click here.
To learn about other design types, click here.