Task F
Work through these exercises after reading the Principal Coordinates Analysis chapter. The point of PCoA is that you choose the dissimilarity, so the task explores how much that choice matters, using the Doubs fish and environmental data. Four exercises are hands-on calculations and two are short conceptual questions. A worked answer is given under each exercise; try it yourself before opening it.
capscale(spe ~ 1, distance = "bray"). Report the proportion of variation captured by the first two axes, and check the eigenvalues for negative (“imaginary”) values.library(tidyverse)
library(vegan)
load(here::here(
"data",
"BCB743",
"NEwR-2ed_code_data",
"NEwR2-Data",
"Doubs.RData"
))
keep <- rowSums(spe) > 0
spe <- spe[keep, ]
env <- env[keep, ]
d_bray <- vegdist(spe, method = "bray")
pcoa_bray <- capscale(spe ~ 1, distance = "bray") # PCoA (capscale keeps the real axes)
var_pcoa <- round(
100 * eigenvals(pcoa_bray)[1:2] / sum(eigenvals(pcoa_bray)),
1
)
var_pcoa # % variation on MDS1, MDS2 (real axes)MDS1 MDS2
52.4 15.6 [1] 11
MDS1 captures 52.4% of the variation and MDS2 15.6% on the real axes, with MDS1 again ordering the sites along the river. capscale() returns only the real axes, but the full PCoA spectrum (from cmdscale()) contains 11 negative (“imaginary”) eigenvalues: Bray-Curtis is not a Euclidean distance, so it cannot be embedded perfectly in real coordinate space, and the leftover, non-embeddable part shows up as those negative eigenvalues. This is why the proportion of variation from a non-Euclidean PCoA has to be read with care, and why the corrections in Exercise 5 are sometimes applied.
Both ordinations recover the same dominant upstream-downstream gradient, so the broad story is robust to the choice of dissimilarity. The configurations differ in detail: the abundance-based Bray-Curtis ordination is pulled by the dominant species and spaces sites by how much their abundant taxa differ, whereas the presence-absence Jaccard ordination weights every species equally and is driven by which species are gained or lost. Sites that share their common species but differ in rare ones move between the two pictures. The choice of dissimilarity is therefore an ecological decision, not a technical default.
cluster::daisy(..., metric = "gower")) on a table that mixes variable types — for example the Doubs environmental data together with one or more categorical variables of your own making. Run a PCoA on the result and explain why PCA and CA could not have analysed this table directly.MDS1 MDS2
69.3 13.2 
The Gower coefficient combines standardised numeric variables with the categorical reach factor into a single dissimilarity, and PCoA then ordinates that matrix. PCA could not handle this table because it works on Euclidean distances of numeric variables and has no way to use a factor; CA could not either, because it expects a frequency/abundance table of non-negative counts. PCoA’s strength is exactly this flexibility: choose a dissimilarity appropriate to the data (here one that accepts mixed types), and PCoA will ordinate it.
[1] 0.07429342
The two configurations are very similar, both showing the river gradient, but they answer different optimisation problems. PCoA finds axes that reproduce the dissimilarities metrically and reports its fit through the eigenvalues (the proportion of variation on the first axes). nMDS reproduces only the rank order of the dissimilarities and reports its fit through stress (here 0.074, indicating a faithful two-dimensional representation). When a low-stress nMDS and a high-variance PCoA agree, as here, that agreement is itself reassuring: the gradient is strong enough that the method choice barely matters.
PCoA seeks a set of real Euclidean coordinates whose pairwise distances reproduce the input dissimilarities. That is only possible if the dissimilarity is itself Euclidean. Many ecologically sensible measures (Bray-Curtis, Jaccard) are not Euclidean: there is no arrangement of points in real space whose straight-line distances equal them exactly. The part that cannot be embedded shows up as negative eigenvalues, the “imaginary” axes. Two standard fixes add a constant to the dissimilarities (the Cailliez correction) or to their squares (the Lingoes correction) so that the adjusted matrix becomes Euclidean and all eigenvalues turn non-negative. The alternative is simply to report proportions against the sum of absolute eigenvalues and to keep the negative values in view, which is what capscale does by default.
Choose PCoA when the dissimilarity measure is the thing you care about and you want a metric, eigenvalue-based ordination of it: it accepts any distance, including non-Euclidean ones (Bray-Curtis, Jaccard) and Gower distances on mixed-type data, which PCA and CA cannot. Choose PCA when the data are continuous variables with roughly linear relationships and Euclidean distance is appropriate. Choose CA when the data are an abundance table with unimodal species responses along a long gradient. Choose nMDS when you care more about preserving the rank order of dissimilarities than about metric axes, and are willing to trade eigenvalue interpretability for a lower-stress, flexible two-dimensional map. PCoA is the natural choice whenever the right description of difference is a specific dissimilarity rather than raw variables.
This Task is not formally assessed. It is built around four hands-on analyses (Exercises 1–4) and two short conceptual questions (Exercises 5–6); work through all six and bring your annotated Quarto document to class for discussion.
@online{smit2026,
author = {Smit, A. J.},
title = {10: {Principal} {Coordinates} {Analysis} {(PCoA)}},
date = {2026-06-14},
url = {https://tangledbank.netlify.app/BCB743/tasks/Task_F.html},
langid = {en}
}