Randomised block experiments
In previous randomised block experiments, all treatments have had one or more replicate within each block. If there are missing treatments within some blocks, there will be no information to estimate some interaction parameters.
However interactions between blocks and the controlled factors are of less interest in many experiments and it is more reasonable to assume that these interactions do not exist. Provided there are not too many missing treatments within blocks it is often possible to estimate interactions between the factors of interest.
Inference
Since we are usually only interested in models with a block term and are rarely interested in testing it, the term for blocks is always added first to an analysis of variance table. In a simple randomised block experiment in which there is only a single factor, we would therefore only consider a single analysis of variance table with blocks added before the factor. Analysis therefore usually proceeds in exactly the same way as for balanced designs.
Confounding
The only exception arises if the blocks and treatments are confounded as shown in the example below.
Replicates | ||
---|---|---|
Block 1 | Block 2 | |
Treatment 1 | 3 | 0 |
Treatment 2 | 3 | 0 |
Treatment 3 | 0 | 3 |
Treatment 4 | 0 | 3 |
Treatment 5 | 0 | 3 |
Although we can compare treatments 1 and 2, differences between the mean response for treatments 1 and 3 could be caused either by different treatment effects or differences between the two blocks. There is no information from the experimental data that could allow us to distinguish.
Always try to design experiments to avoid confounding blocks and treatments.
Water quality
The following data were collected to monitor the effect of a development on water quality in a river passing through the development. Water samples were taken upstream of the development, at the development and downstream from it during four different storm events. However for unrelated reasons, some samples could not be collected. Total suspended solids (TSS) were measured from each sample.
Total suspended solids (TSS) | ||||
---|---|---|---|---|
Storm 1 | Storm 2 | Storm 3 | Storm 4 | |
Upstream | 25 | 20 | ||
At development | 51 | 100 | ||
Downstream | 173 | 137 | 170 | 110 |
For this example, we will treat the storms as being blocks — there is no interest in assessing differences between the storms — we know that some will result in more suspended solids than others.
The above analysis of variance table allows reordering of the two model terms (for storms and location) to be changed. Although the explained sums of squares depend on the order of adding the terms, it only makes sense to add storms first since we know that they will affect TSS and are not interested in testing it.
From the p-value associated with location, 0.0867, we would conclude that there is only mild evidence of any difference between TSS at the different locations.
Logarithms
With quantity data such as these TSS values, the effects of the factors are often proportional changes in the response instead of additive changes. (The effect of the development may be to increase TSS by 10% downstream rather than by say 20.) It therefore makes sense to analyse the logarithms of the TSS values rather than the raw measurements.
Select Raw data from the pop-up menu above the anova table. The p-value for comparing the locations is now 0.0382, so we would conclude from this analysis that there is moderately strong evidence that the locations affect TSS.
Note that the decision about whether to analyse the logarithms should be based on knowledge of the problem, not on the results of the analyses.
Warning
This is example is really observational, not an experiment — it is difficult to consider the locations as being 'applied to experimental units'. It is therefore important to consider other possible reasons for there being more suspended solids downstream of the development. During storms, all rivers may collect more suspended solids as they flow down so the increase could be natural, not the result of the development.