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.

Water uptake by toads and frogs

An experiment was conducted to help identify factors affecting water uptake of amphibia. Frogs and toads were kept in moist or dry conditions prior to the experiment. Half of the animals were injected with a mammalian water balance hormone. There were therefore three factors varied in the experiment, each with two levels.

Species
Toad or frog
Moisture prior to the experiment
Moist or dry
Hormone injected
Hormone or control.

There are 8 combinations of factor levels and each was used with two animals, giving two replicates. The 16 frogs and toads were randomly allocated levels of moisture and hormone. The response measurement was the percentage increase weight after immersion in water for two hours.

   Moisture before experiment
  Animal     Injection?   Wet Dry
Toad Control 2.31
-1.59
17.68
25.23
Hormone 28.37
14.16
28.39
27.94
Frog Control 0.85
2.90
2.47
17.72
Hormone 3.82
2.86
13.71
7.38

The data are displayed graphically in the 3-dimensional diagram below. The colour of the crosses represents the variable Injection.

Rotate the diagram either by clicking Spin or dragging the centre of the diagram. Observe that the highest water uptakes seem to be by Toads kept in dry conditions before the experiment.