Missing treatments in completely randomised experiments

Up to this point, we have mainly considered experiments in which all treatments are present, even if replicated in an unbalanced way.

If there are missing treatments, some model terms cannot be estimated. However except in extreme situations it is only high-order interactions that cannot be estimated. If these high-order interactions are assumed to be negligible, analysis can proceed.

A full discussion of the analysis of experiments with missing treatments will need to wait until the start of the next chapter.

Confounding

If there are enough missing treatments however, it may not be possible to estimate the effects of some important factors at all.

For example, if an experiment for two factors A (with 4 levels) and B (with 5 levels) results in the replicates shown in the table below, each level of factor B is only used in combination with one level of factor A. It is therefore impossible to separate out the effects of A and B and they are said to be confounded. From an experiment with this design, we can report no information about the separate effects of A and B.

  Replicates
   A level 1   A level 2   A level 3   A level 4 
B level 1   5 0 0 0
B level 2   5 0 0 0
B level 3   0 5 0 0
B level 4   0 0 5 0
B level 5   0 0 0 5

Confounding in randomised block experiments

Confounding can also occur in a randomised block experiment, as illustrated in the example below. Although we can compare treatments 1 and 2 since they are used in the same block, differences between the mean response for treatments 1 and 3 could be caused either by different treatment effects or differences between the two blocks. There is no information from the experimental data that could allow us to distinguish.

  Replicates
   Block 1   Block 2 
Treatment 1   3 0
Treatment 2   3 0
Treatment 3   0 3
Treatment 4   0 3
Treatment 5   0 3

Always design experiments to avoid confounding important effects.