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.