In this page, we discuss the reasons for using randomised blocks when designing an experiment to compare several treatments.

Goal of reducing unexplained variability

When an experiment is conducted to compare several treatments, the greater the unexplained variation in the response measurement, the less accurately we can compare the treatments. In practice, there are two main ways to reduce unexplained variation:

Use experimental units that are as similar as possible.
If all experimental units were identical and the response measurement was completely determined by the treatment applied, we would only need a single repetition of the experiment for each treatment to compare the treatments. In practice, this ideal is impossible but similar experimental units give best accuracy.
Group the experimental units into blocks so that those in each block are similar.
When it is impossible to avoid variability in the experimental units, it may be possible to group them into blocks of similar units. These blocks 'explain' some of the variability, so the unexplained variability is reduced.

Randomised block experiments

The simplest way to use blocks in an experiment is with a randomised block design. In this, the block size is a multiple of the number of treatments. Each treatment is used for the same number of experimental units within each block, with the treatments randomly allocated to units within the blocks.

The examples in earlier pages of this section were randomised block experiments in which each treatment was used once within each block. Randomised block experiments generalise this to allow treatments to be used two or more times within each block.

Amino acid uptake by fish

In an investigation into the effect of NaCN (sodium cyanide) on the uptake in vitro of a particular amino acid by intestinal preparations from a certain species of fish, it was found that each fish would only give about six preparations. The measurements of amino acid uptake (expressed as µmol g-1 dry weight per 20 min period) were found to vary considerably between fish.

The experimental units are the individual 'preparations' and they arise in blocks of six. Presumably the six preparations from a single fish will be similar but there may well be differences between different fish.

A randomised block experiment was therefore conducted in which six preparations (the experimental units) were obtained from each fish (the blocks). Three of the experimental units within each block were randomly chosen to get NaCN and the other three were controls that did not get NaCN.

As in the example on the previous page, an incorrect analysis that ignores the blocks is first shown. Click on any cross to see the six measurements from that block.

Select Correct analysis taking account of blocks to see the p-value and conclusion when blocking is taken into account. Observe that the evidence for a difference between the treatments becomes much stronger when blocks are taken into account — the p-value becomes much closer to zero.

Intentional variability in the experimental units

Even for experiments in which it is possible to use very similar experimental units, it is not always desireable to do so. If all experimental units are identical, it is difficult to generalise the results to other kinds of experimental units.

Using a variety of different experimental units (grouped into blocks) allows results to be generalised more easily — we can be more confident that the results will hold for a variety of different types of experimental unit.

Penicillin production

In the example below, a process for manufacture of penicillin was being investigated. Four variants of the basic process (A, B, C and D) were being studied and the response measurement was the yield of penicillin.

An important raw material for the process was corn steep liquor, and this was quite variable. Instead of conducting the experiment with only one blend of corn steep liquor, it was therefore decided to use five different blends (blocks) in order to ensure that the results were not specific to one particular blend. Four different runs of the experiment were used with each blend, with the four process variants randomly allocated to the runs.

Again an incorrect analysis ignoring the blocks is first shown. Select Correct analysis taking account of blocks to see the p-value and conclusion when blocking is taken into account. In this example, we would still conclude that there is no evidence of any difference between the processes.