Factorial design with equal replicates
As in experiments with two factors, if the effects of three factors are to be simultaneously assessed, the most efficient design uses the same number of replicates for all possible combinations of levels of the factors (treatments) and randomly allocates these treatments to the available experimental units.
There are two main advantages to factorial designs:
In a factorial experiment with r replicates in which factor B has nB levels, C has nC levels and D has nD levels, the total number of experimental units required is
r × nB × nC × nD
To keep the expense of the experiment low, most factorial experiments involving three (or more) factors therefore usually only use 2 or 3 levels for each and there may only be a single replicate.
Soft drink bottling
The fill level of bottles of soft drink produced by a filling machine varies from bottle to bottle. A soft drink bottler wants to discover how characteristics of the filling process affect the closeness of the level to a target height and conducts an experiment in which three variables are controlled.
There are 12 combinations of factor levels and two replicates, giving a total of 24 runs of the experiment, conducted in random order. The response is the Fill height deviation— positive if the bottle is overfilled and negative if it is underfilled.
Carbonation percent | ||||
---|---|---|---|---|
Pressure | Line speed | 10% | 12% | 14% |
25 psi | 200 bpm | -3, -1 | 0, 1 | 5, 4 |
250 bpm | -1, 0 | 2, 1 | 7, 6 | |
30 psi | 200 bpm | -1, 0 | 2, 3 | 7, 9 |
250 bpm | 1, 1 | 6, 5 | 10, 11 |
The data are displayed graphically in the 3-dimensional diagram below. The colour of the crosses represents the variable Line speed.
Rotate the diagram either by clicking Spin or dragging the centre of the diagram. Observe that Fill height deviation seems highest when carbonation, pressure and line speed are all high.