Separate experiments to assess the effects of two factors

The simplest way to study the effects of two factors on a response is with two separate completely randomised experiments. In each of these experiments, one factor is kept constant and the other factor is varied. However...

Assessing each factor in a separate experiment is inefficient.

Adding a second factor to a 1-factor design

The table below shows data that may have arisen from a completely randomised experiment with one 3-level factor.

Factor X
  X = A      X = B      X = C   
xA1
xA2
xA3
xA4
xA5
xA6
xB1
xB2
xB3
xB4
xB5
xB6
xC1
xC2
xC3
xC4
xC5
xC6

The table below describes results from an experiment that also varies a second factor, Y. In it, there are 3 replicates for each combination of the levels of factors X and Y. This experiment uses the same number of experimental units as the earlier experiment.

  Factor X
Factor Y   X = A      X = B      X = C   
Y = S xAS1
xAS2
xAS3
xBS1
xBS2
xBS3
xCS1
xCS2
xCS3
Y = T xAT1
xAT2
xAT3
xBT1
xBT2
xBT3
xCT1
xCT2
xCT3

Although it is not intuitively obvious, the effect of changing the levels of factor X is estimated equally accurately in both experiments.

A second factor, Y, can be added by using a factorial design without reducing the accuracy of estimating the effect of X.

In the factorial experiment however, we can also estimate the effect of changing factor Y, so the factorial design provides a 'free' estimate of the effect of Y.

In a complete factorial experiment, the effect of each factor can be estimated as accurately as in a completely randomised experiment with the same number of experimental units.