Designs with orthogonal factor and blocks

In a randomised block design, researchers usually try to use the treatments in the same proportion in every block. In most cases, the same number of replicates are used for all treatments, but the blocks and factor are still orthogonal when a control treatment is replicated more times than the other treatments.

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

The design even remains balanced if some blocks have more experimental units than others provided the treatments are in the same proportion within each block, such as the following:

  Replicates
   Block 1   Block 2   Block 3   Block 4 
Treatment 1   10 5 5 5
Treatment 2   4 2 2 2
Treatment 3   4 2 2 2
Treatment 4   4 2 2 2
Treatment 5   4 2 2 2

Non-orthogonal designs

Orthogonality of blocks and treatments is much less important in a block design with a single factor than in completely randomised designs in which two factors are varied. Non-orthogonal designs generally mean that the sums of squares in an analysis of variance table are different for different orders of adding the model terms. However in block designs, the block effect is included in all models of interest, so it is always the first term added in an analysis of variance table.

As a result, there is a unique analysis of variance table for a randomised block design involving a single factor whatever the mix of factor levels in the different blocks.


The situation becomes more complicated for randomised block designs involving two or more factors. If the two factors are not orthogonal within any of the blocks, the sums of squares for the factors again depend on the order of adding them to the model. If possible, the factors should therefore be orthogonal within blocks.