Randomisation for balanced incomplete block designs

There are several stages to the choice and randomisation of a design.

Basic design
The basic design must be selected first. This will often be generated on a computer or found from tabulated designs.
Randomise treatments
The letters in the basic design template should be randomly allocated to the actual treatments.
Allocation to physical blocks
If all physical blocks are effectively equivalent, as in the pig diet example, it is sufficient to simply randomly allocate rows from the design to the blocks.
Randomisation within blocks
The final stage of the randomisation is to randomly allocate the treatments in each block to its experimental units.

Simple lattice designs

The first step above should be modified to generate a simple (unbalanced) lattice design.

Basic design
A full balanced lattice design consists of (r + 1) groups of r blocks, each containing all of the r2 treatments. For a simple lattice design with fewer blocks, a subset of these groups of r blocks should be chosen at random to form the basic design.

Groups of blocks

In some designs, the blocks can be grouped into two or more sets of blocks, each of which contains equal numbers of all treatments. In particular, lattice designs separate into groups of r blocks. If the physical blocks can be grouped into 'super-blocks' in this way, the allocation of rows of the design to the physical blocks should be modified:

Allocation to physical blocks
Randomly assign each set of rows from the design to the super-blocks, then randomly assign rows from the design within each super-block.

Illustration of randomisation

The diagram below shows the last three steps for randomisation of a balanced incomplete block design for 7 treatments in 7 blocks of size 4. The basic design that is shown below was obtained from tables.

Click Permute treatments to randomise the treatments and Permute blocks to randomly allocate rows from the basic design to the physical blocks.

Now click the checkbox Allocate to units then the button Randomise to randomly allocate the treatments within each block to its experimental units.

Further analysis

The analysis described in this section does not use all of the information available in the data about differences between the treatments.

In the next chapter, we will show that independent information about the differences between the treatments can be obtained from the block totals.

This analysis is however more complex.