What are coenoclines?

A coenocline shows the distribution of each response variable along the gradient. The gradient can either be: 

In most community ecological data sets, response variables are the species. From a theoretical point of view, the behavior of a species along a gradient is approximately unimodal (Gauch 1982). A model frequently used to describe this pattern is the so-called Gaussian response model (Gauch 1982). An hypothetical example is presented in Figure 1. It shows the counted numbers of a particular species (dots) along the gradient temperature. 

Figure 1: Hypothetical example of a species response along the gradient temperature and the fitted curve of the Gaussian response model.

Mathematically, the Gaussian response model (which has nothing to do with the Gaussian distribution) is given by:

Y_i =

where 

In this hypothetical example, the species has an optimum value at u=20 degrees, and the maximum value is c=100. Ter Braak (1985) showed that under certain conditions the species loadings obtained by correspondence analysis are approximately equal to the optimum values u. Obviously, one can argue whether the Gaussian response model is a good model or not, and this is still a discussion point on various ecological mailing lists. However, various simulation studies have shown that this approximation is reasonably robust even if the response curves are not exactly unimodal. This gives correspondence analysis an ecological rationale in terms of (approximately) unimodal species behavior. On the other hand, if the gradient is short, one might see only a linear relationship between the species and the gradient. In this case, principal component analysis should be used (Jongman et al. 1995). To decide which ordination method to use, Brodgar produces coenoclines. These can be obtained by applying either PCA or CA (or preferably both) on the data and choosing "Tools" from the biplot menu. The user has the following options:

Technically, Brodgar takes the sample scores of the first biplot axis. This axis is then used as an explanatory variable and a LOESS smoothing function for each species is calculated:

Y_k = F(axis 1)

where F is the LOESS smoothing function for the kth species. This process is repeated for the second axis. The amount of smoothing (0.5, 0.3 or 0.8) refers to the percentage (of the total number of samples) in the nearest neighborhood algorithm used by LOESS. In general, 0.5 will do. Summarising, a coenocline is simply a smoothing function along the gradient and it is calculated for each species. If all the smoothing curves are approximately straight lines, PCA should be used. If most of the smoothing curves are approximately unimodal, CA is preferable. We advise to inspect coenoclines obtained by both PCA and CA. To illustrate this process, we show the CA biplot and coenoclines along the first 2 CA axes for the hunting spider data. Recall that this data set consists of abundances of 12 spiders measured at 28 samples. Figure 2 shows the correspondence analysis biplot and Figure 3 the corresponding coenoclines along the first axis. The shape of the smoothing curves indicate that most species behave unimodal along the first (and second) axis. This justifies the use of correspondence analysis instead of principal component analysis.

Figure 2: Correspondence analysis biplot for hunting spider data. 

Figure 3: Coenoclines for hunting spider data.

Note that Brodgar scales the axes of the CA biplot such that all loadings and scores are between -1 and 1. This is done by dividing all loadings along the first and second axis by one number, and the same is done for the scores. The re-scaling is not done for the coenoclines plot because the length of the axis has actually an ecological interpretation and is called beta-diversity (Gauch 1982).

The same story holds for redundancy analysis (RDA) and canonical correspondence analysis (CCA). If species behave approximately unimodal along the environmental variables, CCA should be used. On the other hand, RDA should be used for linear species-environmental relationships. If the samples are taken in such a way that the environmental gradients are relatively short, RDA is the preferred method as well. Table 1 shows which technique should be used depending on the data and responses.

Table 1. Linear versus unimodal responses.

Explanatory variables

Brodgar can also calculate coenoclines along the explanatory variables. These can be obtained via the "Plot data" option under the "Data exploration" menu. Choose either Coenoclines (s=0.5), Coenoclines (s=0.3) or Coenoclines (s=0.8) and click on the  "Go" button. Technically, each explanatory variable is taken in turn and LOESS smoothing curves are calculated for each response variable. The results can also be found in the ascii file: \Your_Project_Name\coenocaout.txt.

References

Gauch, HG. (1982). Multivariate analysis in community ecology. Cambridge university press.

Jongman, RHG. and Ter Braak, CJF. and van Tongeren, OFR. (1995). Data analysis in community and landscape ecology. Cambridge University Press, Cambridge.

Ter Braak, CJF. (1985).  Correspondence analysis of incidence and abundance data: properties
in terms of a unimodal response model. Biometrics, 41, 859-873.

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