Confounding different interactions in different pairs of blocks

In the design described on the previous page, the 3-factor interaction can no longer be estimated since it is confounded with the main effect of the blocks. This can be avoided by repeating the experiment in a second pair of blocks, this time confounding a different interaction with the blocks.

In such a design, all data can contribute to estimating the main effects and some interactions but the two interactions that are confounded with the pairs of blocks can only be estimated using the data from half of the experimental units. They are said to be partially confounded with the blocks.

Factorial design for 3 factors in 4 blocks of size 4

The diagram below shows rows for two repeats of the eight treatments — the first repeat is used in blocks 1 and 2 and the second repeat in blocks 3 and 4.

As in the previous page, a factor D is used to split each group of 8 treatments into two blocks. Click to alias D with the ABC interaction in the first pair of blocks, then similarly click to alias D with the BC interaction in the second pair of blocks.

With this design, each main effect can be estimated with the same accuracy as would have been obtained in a complete factorial experiment without blocking. For example, the effect of A is the difference between the mean responses of the 8 response measurements at the high level of A and those at the low level of A.

However the BC and ABC interactions can only be estimated using half of the observations. For example, the BC interaction can only be estimated from the 8 response values in blocks 3 and 4 — it is the difference between the response means of the four values with B and C at the same level and that of the four values with B and C at different levels.