An experimental unit is an item (or physical entity) that receives the treatment. Identifying the experimental unit can be a trivial task in most experiments, but there can be exceptions.

Going back to our definition, the experimental unit is the entity that receives the treatment. In this case, we have applied a water treatment to each aquarium. The fish are not the experimental units. In order for individual fish to be experimental units, somehow the investigators would have to take one fish at a time and apply the treatment independently to each fish. This would be impractical from a logistics standpoint and was not done. Instead, the water treatment levels were applied to the entire aquarium, and so the experimental unit is an aquarium with 50 fish.

Now we can determine what constitutes a replication of the experiment. Each time the full set of treatment levels (2 levels in our example) is applied, we have a complete replication. Therefore in the experiment described there is only one replication; a situation often described as an un-replicated study.

The individual fish that were caught and counted for lesions are sampling units. Sampling units are the entities from which the observations are recorded. Traditionally, to obtain a correct ANOVA, mean values of the sampling units have to be computed for each experimental unit before the calculation of the treatment SS. Failure to recognize sampling units can result in a serious problem: pseudo-replication. Pseudo-replication results from treating each sampling unit as if it were an experimental unit and inflating the error degrees of freedom. By artificially increasing the error df, we reduce the MSE and produce a larger (incorrect) F-statistic.

After identifying the experimental unit and the number of replications that will be used, the next step is to assign the treatments (i.e. factor levels or factor level combinations) to the experimental units.

In a completely randomized design (CRD), treatments are assigned to experimental units at random. This is typically done by listing the treatments and assigning a random number to each.

In the greenhouse experiment discussed in lesson 1, there was a single factor (fertilizer) with 4 levels (i.e. 4 treatments), six replications, and a total of 24 experimental units (potted plants). Suppose the image below is the greenhouse bench (viewed from above) that was used for the experiment.

We need to be able to randomly assign each of the treatment levels to 6 potted plants. To do this, we first assign numbers to the physical position of the pots on the bench.

Next, we randomly assign the 24 total treatments (4 treatment levels, replicated 6 times) to the potted plants. Examples of how this can be done using statistical software are found below.  

A CRD for the greenhouse experiment is reasonable, provided the positions on the bench are equivalent. In reality, this is rarely the case. For example, some micro-environmental variation can be expected due to the glass wall on one end, and the open walkway at the other end of the bench.

A powerful alternative to the CRD is to restrict the randomization process to form blocks. In a block design, general blocks are formed such that the experimental units are expected to be homogeneous within a block and heterogeneous between blocks. The number of experimental units within a block is called its block size.

In a randomized complete block design (RCBD), each block size is the same and is equal to the number of treatments (i.e. factor levels or factor level combinations). Furthermore, each treatment will be randomly assigned to exactly one experimental unit within every block. So if we think of the greenhouse example, in a RCBD we will have 6 blocks, each with block size of 4 (the number of fertilizer levels).

To establish an RCBD for this data, the assignments of fertilizer levels to the experimental units (the potted plants) have to be done within each block separately. Examples of how to do this using statistical software will be detailed in the section below. 

It is important to mention that blocks are usually (but not always) treated as random effects as they typically represent the population of all possible blocks. In other words, the mean comparison among specific blocks is not of interest. However, the variation between blocks must be incorporated into the model. In a RCBD, the variation between blocks is partitioned out of the MSE of the CRD, resulting in a smaller MSE for testing hypotheses about the treatments. 

The statistical model corresponding to the RCBD is similar to the two-factor studies with one observation per cell (i.e. we assume the two factors do not interact). 

Here is Dr. Shumway stepping through this experimental design in the greenhouse.

We will consider the greenhouse experiment with one factor of interest (Fertilizer). We also have the identifications for the blocks. In this example, we consider Fertilizer as a fixed effect (as we are only interested in comparing the 4 fertilizers we chose for the study) and Block as a random effect.

Therefore the statistical model would be

\(Y_{ij} = \mu + \rho_i + \tau_j + \epsilon_{ij}\)

with \(i=1,2,\dots,6\) and \(j=1,2,3,4\). \(\rho_i\)s and \(\epsilon_{ij}\) are independent random variables such that \(\rho_i \sim \mathcal{N}\left(0, \sigma^2_{\rho}\right)\) and  \(\epsilon_{ij}\sim \mathcal{N}\left(0, \sigma^2\right)\).