# 3.7 PCO versus PCA (Clyde environmental data)

Principal components analysis (PCA) is described in detail in chapter 4 of Clarke & Warwick (2001) . As stated earlier, PCO produces an equivalent ordination to a PCA when a Euclidean distance matrix is used as the basis of the analysis. Consider the environmental data from the Firth of Clyde ( Pearson & Blackstock (1984) ), as analysed using the PCA routine in PRIMER in chapter 10 of Clarke & Gorley (2006) . These data consist of 11 environmental variables, including: depth, the concentrations of several heavy metals, percent carbon and percent nitrogen found in soft sediments from benthic grabs at each of 12 sites along an east-west transect that spans the Garroch Head sewage-sludge dumpground, with site 6 (in the middle) being closest to the actual dump site. The data are located in the file clev.pri in the ‘Clydemac’ folder of the ‘Examples v6’ directory.

**Fig. 3.9.** Ordination of 11 environmental variables from the Firth of Clyde using (a) PCA and (b) PCO on the basis of a Euclidean distance matrix. Vectors in (a) are eigenvectors from the PCA, while vectors in (b) are the raw Pearson correlations of variables with the PCO axes.

As indicated by Clarke & Gorley (2006), it is appropriate to first check the distributions of variables (e.g., for skewness and outliers) before proceeding with a PCA. This can be done, for example, by choosing **Analyse>Draftsman Plot** in PRIMER. As recommended for these data by Clarke & Gorley (2006), log-transform all of the variables except depth. This is done by highlighting all of the variables except depth and choosing **Tools > Transform(individual)** > Expression: log(0.1+V) > OK. Rename the transformed variables, as appropriate (e.g. Cu can be renamed ln Cu, and so on), and then rename the data sheet of transformed data clevt. Clearly, these variables are on quite different measurement scales and units, so a PCA on the correlation matrix is appropriate here. Normalise the transformed data by choosing **Analyse > Pre-treatment > Normalise variables**. Rename the data sheet of normalised variables clevtn. Finally, do a PCA of the normalised data by choosing **Analyse > PCA**.

Recall that PCA is simply a centred rotation of the original axes and that the resulting PC axes are therefore linear combinations of the original (in this case, transformed and normalised) variables. The ‘Eigenvectors’ in the output file from a PCA in PRIMER provide explicitly the coefficients associated with each variable in the linear combination that gives rise to each PC axis. Importantly, the vectors shown on the PCA plot are these eigenvectors, giving specific information about these linear combinations. This means, necessarily, that the lengths and positions of the vectors depend on the other variables included in the analysis^{70}.