8.9 - Randomized Block Design: Two-way MANOVA

Within randomized block designs, we have two factors:

  1. Blocks, and
  2. Treatments

A randomized complete block design with a treatments and b blocks is constructed in two steps:

  1. The experimental units (the units to which our treatments are going to be applied) are partitioned into b blocks, each comprised of a units.
  2. Treatments are randomly assigned to the experimental units in such a way that each treatment appears once in each block.

Randomized block designs are often applied in agricultural settings. The example below will make this clearer.

In general, the blocks should be partitioned so that:

  • Units within blocks are as uniform as possible.
  • Differences between blocks are as large as possible.

These conditions will generally give you the most powerful results.

Example 8-10: Rice Data (Experimental Design) Section

Let us look at an example of such a design involving rice.

We have four different varieties of rice; varieties A, B, C, and D. And, we have five different blocks in our study. So, imagine each of these blocks as a rice field or patty on a farm somewhere. These blocks are just different patches of land, and each block is partitioned into four plots. Then we randomly assign which variety goes into which plot in each block. You will note that variety A appears once in each block, as does each of the other varieties. This is how the randomized block design experiment is set up.

Block 1
D  C 
B  A 
Block 2
A  D 
B  C 
Block 3
B  D 
C  A 
Block 4
D  B 
A  C 
Block 5
A  C 
D  B 

A randomized block design with the following layout was used to compare 4 varieties of rice in 5 blocks.

This type of experimental design is also used in medical trials where people with similar characteristics are in each block. This may be people who weigh about the same, are of the same sex, same age, or whatever factor is deemed important for that particular experiment. So generally, what you want is for people within each of the blocks to be similar to one another.

Back to the rice data... In each of the partitions within each of the five blocks, one of the four varieties of rice would be planted. In this experiment, the height of the plant and the number of tillers per plant were measured six weeks after transplanting. Both of these measurements are indicators of how vigorous the growth is. The taller the plant and the greater number of tillers, the healthier the plant is, which should lead to a higher rice yield.

In general, randomized block design data should look like this:

Table of randomized block design data
  1 2 \(\cdots\) b
Treatment 1 \(\mathbf{Y_{11}} = \begin{pmatrix} Y_{111} \\ Y_{112} \\ \vdots \\ Y_{11p} \end{pmatrix}\) \(\mathbf{Y_{12}} = \begin{pmatrix} Y_{121} \\ Y_{122} \\ \vdots \\ Y_{12p} \end{pmatrix}\) \(\cdots\) \(\mathbf{Y_{1b}} = \begin{pmatrix} Y_{1b1} \\ Y_{1b2} \\ \vdots \\ Y_{1bp} \end{pmatrix}\)
Treatment 2 \(\mathbf{Y_{21}} = \begin{pmatrix} Y_{211} \\ Y_{212} \\ \vdots \\ Y_{21p} \end{pmatrix}\) \(\mathbf{Y_{22}} = \begin{pmatrix} Y_{221} \\ Y_{222} \\ \vdots \\ Y_{22p} \end{pmatrix}\) \(\cdots\) \(\mathbf{Y_{2b}} = \begin{pmatrix} Y_{2b1} \\ Y_{2b2} \\ \vdots \\ Y_{2bp} \end{pmatrix}\)
Treatment a \(\mathbf{Y_{a1}} = \begin{pmatrix} Y_{a11} \\ Y_{a12} \\ \vdots \\ Y_{a1p} \end{pmatrix}\) \(\mathbf{Y_{a2}} = \begin{pmatrix} Y_{a21} \\ Y_{a22} \\ \vdots \\ Y_{a2p} \end{pmatrix}\) \(\cdots\) \(\mathbf{Y_{ab}} = \begin{pmatrix} Y_{ab1} \\ Y_{ab2} \\ \vdots \\ Y_{abp} \end{pmatrix}\)

We have a rows for the a treatments. In this case, we would have four rows, one for each of the four varieties of rice. We also set up b columns for b blocks. In this case, we have five columns, one for each of the five blocks. In each block, for each treatment, we are going to observe a vector of variables.

Our notation is as follows:

  • Let \(Y_{ijk}\) = observation for variable k from block j in treatment i
  • We will then collect these into a vector \(\mathbf{Y_{ij}}\) which looks like this:

    \(\mathbf{Y_{ij}} = \left(\begin{array}{c}Y_{ij1}\\Y_{ij2}\\\vdots \\ Y_{ijp}\end{array}\right)\)

  • a = Number of Treatments
  • b = Number of Blocks