Distance-based
approaches rely on a square, symmetric distance matrix or similarity matrix.
See Similarity, Distance and Difference. For polar
ordination, it is necessary for data to obey the triangle inequality (i.e. the
distance between A and B plus the distance between B and C cannot exceed the
distance between A and C). Unlike methods derived from eigenanalysis, distance-based methods do not provide
species and sample scores simultaneously.

- Also known as Bray-Curtis Ordination or Wisconsin
Ordination.
- PO is perhaps the easiest ordination technique to
visualize, and it is possible to perform without a computer.
- It is possible for axes to be correlated.
- The user must pre-specify samples as end points
(alternatively, there are automatic ways endpoints can be specified).
These endpoints have a privileged position; all other samples are defined
relative to them.
- PO is the only technique where one can control the
direction of gradients (i.e. left vs. right or up vs. down).
- New samples can be added without affecting the
ordination.

- This technique has been labelled
"Multidimensional Scaling", although this term might be better
avoided, since it is now becoming synonymous with "Nonmetric Multidimensional Scaling". PCoA has also been termed "Metric
Multidimensional Scaling".
- PCoA maximizes the linear
correlation between distance measures and distance in the ordination.
- One cannot easily put new points in a PCoA.
- PCoA (and NMDS) can be
useful if one has only a distance (or similarity) matrix (e.g. for DNA
hybridization, or for plotless sampling)
- The underlying model for PCoA
(and NMDS) is that there a fixed number of gradients. In contrast, PCA,
RA, and DCA assume that there are potentially many gradients, but of
declining importance.
- Special case: for Euclidean distance, PCoA = PCA.
- PCoA can be expressed as an eigenanalysis.

- This is sometimes labeled "Multidimensional
Scaling (MDS)", although this term has been used for PCoA.
- NMDS is very computer intensive; it has only recently
become feasible for large data sets on the microcomputer.
- NMDS Maximizes rank-order correlation between distance
measures and distance in ordination space. Points are moved to minimize
"stress". Stress is a measure of the mismatch between the two
kinds of distance.
- NOTE: Gauch (1982) is WRONG
is stating that NMDS assumes species have monotonic relationships to
gradients. What it assumes is that dissimilarity is monotonically related
to ecological distance.
- The user must pre-specify number of dimensions, or
examine a plot of "stress" as a function of number of axes, and
select the number of dimensions
*a posteriori*. - The configuration will change as the number of axes
change.
- There is no guarantee that the correct (lowest stress)
solution will be found, though it is widely assumed that this is not a big
problem.
- To increase the likelihood of finding the correct
solution, a DCA is often performed first.
- A recent variant of NMDS is Local NMDS, which only
minimizes stress locally.

- It can be shown that some methods described in eigenanalysis-based ordination
methods are also special cases of distance-based methods, where the
distance is based on the chi-squared distribution. While this is true, it
is worth keeping in mind that the underlying philosophy of the eigenanalysis-based methods is fundamentally
different: these methods attempt to faithfully place species along
gradients (either inferred or directly related to measured variables), and
not to faithfully relate difference to distance.

(See also selected
references for self-education)

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