The importance of grouping experimental units into blocks and allocating treatments in a randomised block design cannot be overemphasised.

Another illustration of the benefits from using blocks

The example below arises from a randomised block experiment with:

For each comparison of a treatment to the control, the data are a paired sample of the form that was analysed in the previous section. In particular, we can calculate the difference between the treatment and control within each block and find a 95% confidence interval for the difference.

If the experiment had been conducted as a completely randomised experiment, we would have simply had two independent samples to use for each comparison — the set of response measurements for the control and those for the treatment that we are comparing to it. A two-sample 95% confidence interval for the difference between the means would be appropriate.

In the example, we show that the confidence intervals are much narrower when using paired differences, supporting our claim that the randomised block design is more accurate.


Codeine and acupuncture for dental pain relief

In a randomised block experiment, 32 subjects were grouped into blocks of four according to an initial assessment of their tolerance to pain. Four pain relief treatments were randomly allocated to the four patients in each group and pain relief scores were recorded from each subject two hours after dental treatment.

There is a control treatment (the placebo with no codeine and inactive acupuncture points) so we will compare the three other treatments to this one.

Incorrect analysis based on independent samples
The diagram initially shows 95% confidence intervals for the increase in mean pain relief score based on a naive analysis that treats the data as four independent samples — i.e. it shows two-sample confidence intervals to compare each pair of treatments.

  • With this incorrect analysis, all confidence intervals are wide. There seems to be considerable uncertainty about how much the treatments improve pain relief.
Correct analysis using paired differences
The samples are not independent — the 4 subjects in a block have similar pain relief scores. Click Show blocks of subjects then select Correct analysis using paired differences from the pop-up menu.

  • There is a similar pattern within each block of subjects.
  • The 95% confidence intervals are much narrower and give strong evidence that the three treatments do improve pain relief.