7.1 - Blocking in an Unreplicated Design

We begin with a very simple replicated example of blocking. Here we have \(2^2\) treatments and we have n = 3 blocks. In the graphic below the treatments are labeled using the standard Yates notation. Here the \(2^2\) treatments are the full set of treatment combinations so we can simply put each replicate within a block and assign them in this way.

We can use the Minitab software to construct this design as seen in the video below.

Now let’s consider the case when we don't have any replicates, hence when we only have one set of treatment combinations. We go back to the definition of effects that we defined before. We did this using following table, where {(1), a, b, ab} is the set of treatment combinations, and A, B, and AB are the effect contrasts:

The question is: what if we want to block this experiment? Or, more to the point, when it is necessary to use blocks, how would we block this experiment?

If our block size is less than four we are only going to consider, in this context of \(2^k\) treatments, block sizes in the same family, i.e. \(2^p\) number of blocks. So in the case of this example let's use blocks of size 2, which is \(2^1\). If we have blocks of size two then we must put two treatments in each block. One example would be twin studies where you have two sheep from each ewe. The twins would have homogeneous genetics and the block size would be two for the two animals. Another example might be two-color micro-arrays where you have only two colors in each micro-array.

So now the question: How do we assign our four treatments to our blocks of size two?

In our example each block will be composed of two treatments. The usual rule is to pick an effect you are least interested in, and this is usually the highest order interaction, as a means of specifying how to do blocking. In this case it is the AB effect that we will use to determine our blocks. As you can see in the table below we have used the high level of AB to denote Block 1, and the low-level of AB to denote Block 2. This determines our design.

Now, using this design we can assign treatments to blocks. In this case treatment (1) and treatment ab will be in the first block, and treatment a and treatment b will be in the second block.

Blocks of size 2

This design confounds blocks with the AB interaction. You can see this by these contrasts - the comparison between block 1 and Block 2 is the same comparison as the AB contrast. Note that the A effect and the B effect are orthogonal to the AB effect. This design gives you complete information on the A and the B main effects, but it totally confounds the AB interaction effect with the block effect.

Although our block size is fixed at size = 2 we still might want to replicate this experiment in addition. What we have above is two blocks which is one unit of the experiment. We could replicate this design additionally let's say r times and each replicate of the design would be 2 blocks of the design laid out in this way.

We show how to construct this with four replicates. Review the movie below to see how this occurs in Minitab.