Known differences between the experimental units

When nothing is known about the differences between the experimental units before the experiment is conducted, we can do no better than to randomise allocation of treatments to the units. Randomisation avoids systematic over- or under-estimation of treatment effects. However...

When more is known about the differences between the experimental units, we can improve on randomisation.

Randomised block design

The greater the variability between experimental units, the more the resulting variability within treatments (noise) tends to mask differences between the treatments (signal).

In practice, a researcher often has little influence on the choice of experimental units and they can very variable. More accurate estimates are obtained if they can be grouped together before the experiment is conducted in groups of similar units called blocks. A separate experiment is conducted within each block with an equal number of treatments of each type randomly allocated to the experimental units in the block. Although all data are analysed together, the lower variability of experimental units with each block means that differences between the treatments can be more accurately estimated.

Since treatments are randomly allocated within blocks, this design is called a randomised block design.

Effect of irrigation on grass growth

A researcher wants to conduct an experiment to assess how irrigation affects grass growth. Three levels of irrigation will be used (none, a little or a lot) and they can only be applied to whole fields. As a result, the experimental units are fields and the available fields close to the research station differ in soil type and fertility.

We will initially examine a completely randomised design. The 36 pictures above represent the 36 available fields. Click Randomise treatments a few times to show how the 12 fields getting each level of irrigation are randomly allocated to the 36 fields.

If the researcher has some prior knowledge of the each of these fields, they can be grouped together in blocks. Select Randomised block from the pop-up menu to group similar fields together into three groups.

Click Randomise treatments a few times to randomly allocate each of the treatments to 4 of the 12 fields within each block. Note that the randomisation is applied within each block.

Each treatment is applied the same number of times within each block, so if the experimental units within one block tend to have a higher mean response all treatments are affected equally. As a result, differences between the treatments are more accurately estimated.

Uptake of amino acids by fish

In an investigation of the effect of sodium cyanide (NaCN) on the uptake in vitro of a particular amino acid by intestinal preparations of a certain species of fish, it was found that each fish would give only about six preparations.

Since there could be sizeable differences between individual fish, the fish were treated as blocks (of size 6). The table below shows uptake of the amino acid (µmol per g dry weight in a 20 minute period) for four fish. The two treatments were randomly allocated to the six intestinal preparations from each fish in a randomised block design.

  Block  
Treatment Fish 1 Fish 2 Fish 3 Fish 4 Mean
Without NaCN
1.54
1.92
2.26
1.52
2.02
1.91
1.00
1.12
1.13
1.58
1.78
1.52
1.608
With NaCN
1.10
1.42
1.04
1.31
1.15
1.51
0.79
0.84
0.86
1.24
0.81
1.32
1.116

Fish 3 has a mean uptake that is considerably lower than the other fish. However since each treatment is used with the same number of samples from this fish, its lower uptake levels equally affect the two blue treatment means on the left of the table. The use of a fish with such a low amino acid uptake therefore does not affect the relative values of the two blue means.

Differences between the fish (blocks) do not therefore affect the accuracy of comparisons between the treatments — this gives us more confidence that the use of NaCN decreases amino acid uptake by around 0.5.

Comparison of completely randomised and randomised block designs

If we know anything about differences between the experimental units before the experiment is conducted, it is always worthwhile to group them into blocks and conduct a randomised block design — the effects of the treatments will be estimated more accurately.

The benefits are greatest when there are large differences between the mean response in different blocks — if all blocks are essentially the same, there is nothing to be gained from using a randomised block design.

Comparing tender and auction of houses

Many cities built estates of low-cost rental housing in the 1950s and 1960s and are now selling these to tennants. The diagram below simulates the sale of 24 houses of identical design. In order to assess whether offering the houses for tender (with the highest bid by a fixed date buying the house) or auction achieves the best sale price, an experiment is conducted, with 12 of the houses being sold by tender and the others by auction.

The top half of the diagram conducts a completely randomised experiment and the bottom half conducts a randomised block experiment. Both experiments use the same 24 houses and, in each case, half are sold by auction and half by tender — the ticks on houses represent those being sold by auction. Click Repeat several times for each of the experiments.

Observe that the estimated effect of auctioning (the difference between the price for auction and tender) is much more variable for the completely randomised experiment. The randomised block experiment therefore provides a more consistent (and accurate) estimate of the effect of auctioning.

The three districts (blocks) are initially very different — there is a strong block effect on the house prices due to the different locations. Select Districts are essentially the same from the pop-up menu, then repeat the experiments. Observe that there is now no advantage in using a randomised block experiment — it no longer estimates the effect of auctioning any more accurately than the completely randomised design.