Applying an incomplete design

There are two steps to the application of an incomplete design in an actual experiment.

Treatments to use
The first decision relates to the choice of treatments to use (and not use), either overall in a completely randomised experiment or within each block of a randomised block experiment.
Randomisation
After the treatments have been picked, they must be randomly allocated to physical experimental units, either overall or within each block.

Picking treatments

In a Latin square design, there are many different ways to write letters in a square grid so that each appears in exactly once in every row and column. A good way to select one such design is to start with a square with diagonal bands for the treatments. For example, in a 6x6 Latin square, this would be:

Then:

Randomisation

Random allocation of treatments to physical experimental units depends on whether there are blocks in the experiment.

Rows and columns are both blocking variables
No further randomisation is needed — each cell of the table is effectively a separate block so it only has a single treatment.
Rows are blocks
The treatments (column-letter combinations) should be randomly allocated within each block.
No blocks
In a completely randomised experiment, all treatments (row-column-letter combinations) should be randomly allocated to the experimental units.

Winter feeding and milk production

The diagram below shows how treatments would be allocated to experimental units for the example on the previous page.

Click the buttons Permute rows, Permute cols and Permute letters to randomly permute these three variables. Since there are two blocking variables, each individual cell in the table is effectively a block so no further randomisation is needed.

Randomisation with 2 or 3 controlled factors

The diagram below shows the further randomisation that is needed if two or three of the variables are controlled factors.

In each example, click Permute rows, Permute cols and Permute letters to randomly pick the treatments that will be used in each block (or overall).

Now click the checkbox Allocate to units then click the button Randomise to randomly allocate the chosen treatments to experimental units.