# Documentation

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## Full Factorial Designs

### Multilevel Designs

To systematically vary experimental factors, assign each factor a discrete set of levels. Full factorial designs measure response variables using every treatment (combination of the factor levels). A full factorial design for n factors with N1, ..., Nn levels requires N1 × ... × Nn experimental runs—one for each treatment. While advantageous for separating individual effects, full factorial designs can make large demands on data collection.

As an example, suppose a machine shop has three machines and four operators. If the same operator always uses the same machine, it is impossible to determine if a machine or an operator is the cause of variation in production. By allowing every operator to use every machine, effects are separated. A full factorial list of treatments is generated by the Statistics and Machine Learning Toolbox™ function `fullfact`:

```dFF = fullfact([3,4]) dFF = 1 1 2 1 3 1 1 2 2 2 3 2 1 3 2 3 3 3 1 4 2 4 3 4```

Each of the 3×4 = 12 rows of `dFF` represent one machine/operator combination.

### Two-Level Designs

Many experiments can be conducted with two-level factors, using two-level designs. For example, suppose the machine shop in the previous example always keeps the same operator on the same machine, but wants to measure production effects that depend on the composition of the day and night shifts. The Statistics and Machine Learning Toolbox function `ff2n` generates a full factorial list of treatments:

```dFF2 = ff2n(4) dFF2 = 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1```

Each of the 24 = 16 rows of `dFF2` represent one schedule of operators for the day (`0`) and night (`1`) shifts.