Adding a second factor to a 1-factor design

Consider an experiment to compare the 2 levels of a single factor with 9 replicates — each factor level is applied to a randomly selected 9 of the 18 experimental units. The table below illustrates the type of data that would arise from this experiment.

Factor X  
X = A  
xA1   xA2   xA3   xA4   xA5   xA6   xA7   xA8   xA9
X = B  
xB1   xB2   xB3   xB4   xB5   xB6   xB7   xB8   xB9

Now consider a modification to this experiment that also varies a second factor, Y. The table below describes a factorial experiment with 3 replicates for each combination of the levels of factors X and Y. This experiment uses the same number of experimental units.

  Factor Y
Factor X Y = S Y = T Y = U
X = A  
xAS1   xAS2   xAS3
xAT1   xAT2   xAT3
xAU1   xAU2   xAU3
X = B  
xBS1   xBS2   xBS3
xBT1   xBT2   xBT3
xBU1   xBU2   xBU3

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.

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.