Information about treatment effects

In a randomised block experiment, there are two potential sources of information about differences between the treatments.

Differences within blocks
If two units within the same block have different treatments, the difference between their response measurements is an estimate of the difference between their treatment effects.
Differences between block means
The response averages in the different blocks may also provide information about the differences between the treatments.

These two sources of information are independent. The relative amounts of information of the two types depend on the design of the experiment.

Orthogonal randomised blocks

In Chapter 4, we described randomised block designs in which the treatments were used in the same proportions within each block. In these designs, the treatments were orthogonal to the blocks.

Differences within blocks
The standard estimate of the difference between any two treatment means can be written as an average of the within-block differences between the treatments.
Differences between block means
Since all blocks contain the same mix of treatments, the block means contain no information about differences between the treatments.

Nested experiments

The first section of this chapter described experiments in which the treatments are varied at block level. In such experiment,

Differences within blocks
Since all units within a block get the same treatment, differences between units contain no information about treatment differences.
Differences between block means
The standard estimate of the difference between any two treatment means depends on the block means of the blocks getting these treatments.

Other designs

In many experiments, the treatments are neither orthogonal to the blocks nor varied at block level. This may occur by design (e.g. lattice designs) or accident (e.g. missing values).

In these designs, some information about the treatments is available both from differences within blocks and from the block means.

A continuum of designs

The diagram below shows possible allocation of 5 treatments (A, B, C, D and E) to 50 experimental units in 10 blocks.

The diagram initially shows treatments allocated in a randomised block experiment with all treatments used once per block. The average responses in the 10 blocks are all averaged over all 5 treatments, so the block averages can hold no information about treatment differences.

Drag the slider to the right to change the design to a nested design in which the treatments are allocated at block level. Since all units in a block get the same treatment, differences between units in a block hold no information about treatment differences.

At intermediate slider positions,

  • the design is not orthogonal but there is still some information about treatment differences from differences between units within blocks.
  • some blocks get more of some treatments than other blocks, so the block means hold some information about treatment differences.