Separate experiments to assess the effects of two factors

Sometimes the response of interest is influenced by two or more factors, each of which can be controlled in an experiment.

The simplest way to study two factors is with two separate completely randomised experiments. In each of these experiments, one factor is kept constant (e.g. the colour of an artificial flower) and the other factor is varied (e.g. the flower's shape). However...

Assessing each factor in a separate experiment is inefficient.

Adding a second factor to a 1-factor design

Consider an experiment to compare the three levels of a single factor with 6 replicates — each factor level is applied to a randomly selected 6 of the 18 experimental units. The table below illustrates the type of data that would arise from this experiment.

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

Now consider a modification to this experiment that also varies a second factor, Y. The table below describes an experiment with 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.

It might initially seem that each factor will be estimated less accurately because the other factor is also varied, but this is not so.

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

A complete factorial experiment therefore estimates the effects of two factors as accurately as two completely randomised experiments involving double the total number of experimental units.

Blood pressure after an operation

The diagram below simulates a completely randomised experiment in which two surgical procedures (operations by keyhole surgery and a standard surgical method) are compared. Initially, all patients are given the same dose of a drug that is intended to reduce their blood pressure after the operation. The response variable is the systolic blood pressure of the patients two hours later.

Click Repeat experiment to randomise the patients, perform the surgery and record blood pressures. Click Accumulate then repeat the experiment several times to see the variability in the estimated difference between blood pressures using keyhole and standard operations.

Initially all patients receive the same drug dose, so the experiment is completely randomised with only a single factor (the type of operation). Drag the slider to vary the amount of drug, effectively turning the experiment into a factorial design with two factors — operation type and amount of drug.

Reducing the amount of drug for 3 patients getting each operation type increases their blood pressure and increasing the drug dose for 3 others decreases their blood pressure. However since this happens for the same number of patients getting each operation type,

The difference between the mean blood pressures for the two operation types remains the same.

With the Accumulate checkbox still selected, repeat the experiment several more times. (Unchecking the Animation checkbox speeds up the simulations.) Observe that the variability (accuracy) of the estimated difference between the operation types is the same as for the experiment using the same amount of drug for everyone.

The experiment can also vary the amount of drug and estimate its effect without affecting the accuracy of estimating the difference between the two operation types.

In a similar way, the effect of the drug on blood pressure would be estimated equally accurately whether or not two operation types were used.