Blocks of similar experimental units

Experiments with matched pairs or matched groups are special kinds of randomised block experiments. In many of these experiments, the grouping of experimental units is fairly subjective, perhaps based on similar weights, ages, etc.

In other situations, the experimental units naturally separate into groups. For example,

If the number of experimental units in each block is a multiple of the number of factor levels in the experiment, it is possible to randomly allocate each factor level the same number of times within each block. The resulting experiment is called a randomised block design.

Acupuncture and Codeine for dental pain relief

An anaesthetist conducted an experiment to assess the effects of codeine and acupuncture for relieving dental pain. The experiment used 32 subjects who were grouped into blocks of 4 according to an initial assessment of their tolerance to pain.

The four treatment combinations of (codeine or a sugar capsule) and (active or inactive acupuncture points) were randomly given to the four subjects in each block. Pain relief scores were recorded from each subject two hours after dental treatment. The experiment was double blind since neither the subjects nor the person assessing pain relief knew which treatment had been administered.

  Pain relief score
Tolerance
group
  Control   Codeine
only
Acupuncture
only
Codeine +
Acupuncture
1
2
3
4
5
6
7
8
0.0
0.3
0.4
0.4
0.6
0.9
1.0
1.2
0.6
0.7
0.8
0.9
1.5
1.6
1.7
1.6
0.5
0.6
0.8
0.7
1.0
1.4
1.8
1.7
1.2
1.3
1.6
1.5
1.9
2.3
2.1
2.4

In the above experiment the tolerance groups are matched blocks of subjects. In the following examples, the blocks occur naturally.

Measuring sulphur in soils

Measurements were made on the amount of sulphur (in parts per million) in soil samples using four different solvents. The soil samples were collected from five different geographical locations in Florida, USA, and represented different soil types. The soil samples are the blocks in this experiment:

The solvents used to analyse the each of the five soils were:

Each soil sample was split into four and the solvents were randomly allocated to them.

  Soil sample
       Solvent        Troup   Lakeland  Leon Chipley Norfolk
CaCl2
NH4OAc
Ca(H2PO4)3
H2O
5.07
4.43
7.09
4.48
3.31
2.74
2.32
2.35
2.54
2.09
1.09
2.70
2.34
2.07
4.38
3.85
4.71
5.29
5.70
4.98

Effect of cultivar on plant height

An experiment was conducted to compare the differences in growth among four different cultivars of a house plant. The greenhouse had three benches in different locations which form natural blocks. Two pots of each cultivar were randomly assigned to each bench for a total of six pots per cultivar and eight pots per bench.

In this experiment, the benches are blocks and there are two replicates — each treatment (cultivar) was used twice on each bench.

  Cultivar
  Bench       A       B       C       D   
1
19.3
17.2
20.1
19.4
17.4
16.6
16.6
15.7
2
16.7
15.5
21.2
20.8
14.4
13.6
13.5
12.9
3
17.7
19.8
21.0
21.9
15.8
17.4
12.8
14.7