# 1.30 Designs that lack replication (Plankton net study)

From a practical perspective, for PERMANOVA to proceed with the analysis (regardless of which of the above two perspectives one chooses to take), the highest-order interaction term needs to be excluded from the analysis (see the section Pooling or excluded terms, above). This can either be done manually, or if the PERMANOVA routine detects that there is no within-cell replication, then it will issue a warning. If you choose to proceed by clicking ‘OK’, it will automatically exclude the highest-order interaction term from the model. If you receive this warning and you know that you do have within-cell replication, then there is a very good chance that you have mis-labeled your factor levels somehow^{35}.

An example of a two-way crossed design without replication is provided in a study by Winsor & Clarke (1940) Winsor & Clarke (1940) to investigate the catch of various groups of plankton by two nets hauled horizontally, with one net being 2 metres below the other. Ten hauls were made with the pair of nets at depths of 29 and 31 meters, respectively. The experimental design is:

```
Factor A: Position (fixed with a = 2 levels, either upper (U) or lower (L) depths).
Factor B: Haul (random with b = 10 levels, labeled simply 1-10).
```

There is only 1 value per combination of treatments, with no replication, so *N = a × b = 20*. This is effectively a *randomised block* design, where the hauls are “blocks”. The variables recorded correspond to five different groups of plankton. Standard deviations in the various groups were roughly proportional to the means, so data were transformed and are provided as logarithms of the catch numbers for each of the plankton groups. These data are located in the plank.pri file in the ‘Plankton’ folder of the ‘Examples add-on’ directory, and were provided by
Snedecor (1946)
.

**Fig. 1.33.** PCA of the study of plankton from ten hauls (numbered) at either 29 m depth (upper) or 31 m depth (lower).

Examination of the data (already log-transformed) reveals no zeros and that their distributions are fairly even, with no extreme values or outliers^{36}. The variables are also on similar scales and are measured in the same units; therefore, an analysis based directly on Euclidean distances would be reasonable here – prior normalisation is not necessary. For data like these, an appropriate ordination method is principal components analysis (PCA)^{37}. The first two principal components explained 83.6% of the total variance in the five variables (Fig. 1.33). Variability among the hauls is apparent in the diagram, but a clear difference in the plankton numbers due to the position of the nets (upper versus lower), if any, is not obvious.

The PERMANOVA analysis of these data on the basis of a Euclidean distance matrix has detected significant variability among the hauls, but also has detected a significant effect of the position of the net (Fig. 1.34). Notice that the output has identified the excluded term: ‘PositionxHaul’, as an important reminder that the analysis without replication is not without an additional necessary assumption in this regard.

It might seem surprising that the analysis has detected any effect of ‘Position’ at all, given the pattern seen in the PCA (Fig. 1.33). Looks can be deceiving, however. Close inspection of the plot reveals that, within almost all of the individual hauls (i.e., 1, 2, 3, 5, 7, 9, 10), the symbol for the ‘upper’ group lies to the right of the symbol for the ‘lower’ group. Only hauls 4, 6 and 8 do not conform to this pattern. We can perhaps understand the nature of this overall effect of Position by examining the averages for the ‘upper’ and ‘lower’ nets for each of the variables across all of the hauls. With the Plankton worksheet highlighted, select **Tools > Average** > (Samples •Averages for factor: Position) & (Variables •No averaging). The resulting worksheet shows that the average log(abundance) for all five of the plankton variables was larger for the nets towed at the shallower depth (the ‘upper’ nets) (Fig. 1.34).