The main idea... Principal coordinates analysis (PCoA; also known as metric multidimensional scaling) summarises and attempts to represent interobject (dis)similarity in a lowdimensional, Euclidean space (Figure 1; Gower, 1966). Rather than using raw data, PCoA takes a (dis)similarity matrix as input (Figure 1a). It is conceptually similar to principal components analysis (PCA) and correspondence analysis (CA) which preserve Euclidean and χ^{2} (chisquared) distances between objects, respectively; however, PCoA can preserve distances generated from any (dis)similarity measure allowing more flexible handling of complex ecological data. Additionally, (dis)similarity matrices calculated from quantitative, semiquantitative, qualitative, and mixed variables can be handled by PCoA. As always, the choice of (dis)similarity measure is critical and must be suitable to the data in question. The choice of measure will also, together with the number of input variables, determine the number of dimensions that comprise the PCoA solution. As an important caveat, be aware that PCoA can only fully represent Euclidean components of the matrix even if the matrix contains nonEuclidean distances. To arrive at a fully Euclidean solution, consider nonmetric multidimensional scaling (NMDS) or using data transformations.
As with other ordination techniques such as PCA and CA, PCoA produces a set of uncorrelated (orthogonal) axes to summarise the variability in the data set. Each axis has an eigenvalue whose magnitude indicates the amount of variation captured in that axis.The proportion of a given eigenvalue to the sum of all eigenvalues reveals the relative 'importance' of each axis. A successful PCoA will generate a few (23) axes with relatively large eigenvalues, capturing above 50% of the variation in the input data, with all other axes having small eigenvalues. Each object has a 'score' along each axis. The object scores provide the object coordinates in the ordination plot. Interpretation of a PCoA plot is straightforward: objects ordinated closer to one another are more similar than those ordinated further away. (Dis)similarity is defined by the measure used in the construction of the (dis)similarity matrix used as input. While PCoA is suited to handling a wide range of data, information concerning the original variables cannot be recovered. This is because PCoA takes a (dis)similarity matrix derived from the original data as input and not the original variables themselves. However, object scores along the PCoA axes may be correlated with object scores along each original variable's axis, assuming the these are either quantitative or dummy variables (Legendre & Legendre, 1998). This may be used as a measure of the original variables' contribution to a given PCoA axis. Warnings
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