just at one station, B the number just at the other, and C the number at both, then J = C/(A + B + C).
The proportion of stations with both species present is taken as an index of their similarity. Both
Sorensen’s and 1 − J are distances in a species space with all axes running from 0 to 1. Clearly, the
two indices weight the importance of relative abundance differently – important vs. not important.(^) The computation proceeds by a more efficient method analogous to sliding an arbitrary plane (or X-
space) through the S-space, then calculating the sum of squares of all the distances that the points
must be moved to sit on it, termed the “stress” for that plane. That sum is stored for later reference
and another plane is tried, then another and another. After some sufficiently small stress values are
found, the algorithm offers the plane with the lowest value as its result, and it plots the stations where
they project onto it (shortest distance). That is the ordination product. The station clusters may or may
not be obvious across it. Unlike PCA (and many other schemes), the first axis is arbitrary, not
necessarily the one “relieving” the most stress. Thus, if just one axis is fitted, and then independently
two, the second may “explain” more of the distance separating the station points than did the one
fitted alone. The so-called first axis is, thus, arbitrary. Three axis nMDS are possible and can be
useful if substantially more stress is “relieved”. The more-efficient methods start with an arbitrary
plane (a PCA plane might be used to start close to a useful result) and then iteratively manipulate the
points to determine better locations for the ordination plane. They run the risk that a local minimum
of stress can trap the result, requiring that more than one starting plane be tried.
(^) Methods for “testing” the reality of the cluster distinctions appearing in cluster analyses and
ordinations have been developed; most are “bootstrap” methods. They depend upon running the
clustering or ordination routine many times (say, B [for “bootstrap”] = 1000), each time randomizing
the abundance estimates for each species among the stations. A measure of the strength of clustering,
Crand, is calculated for each run to compare with that of the data, Cdata. If the fraction of measures
for which [Crand < Cdata] is smaller than a chosen probability (α), perhaps α = 0.05, the clustering is
said to be “significant”. These probabilities are indicative, but they cannot encompass the entire
universe of community-abundance estimates that could be found in the field. To fully test the
hypothesis about community structure variation embodied in an ordination or clustering, it is
necessary to repeat the entire sampling and taxonomic exercise. That is consistently impractical, and
results are judged on the subjective reasonableness of the relation of clusters to environmental
variation.
Fig. 14.7 Positions of station clusters (A to D) in a “species space” projected onto a
best-fit plane defined by canonical correlation analysis. These clusters are unusually
tight for such an analysis.
(^) (After Bilyard & Carey 1979.)
Fig. 14.8 Positions of Beaufort Sea stations identified by polychaete-based cluster
designations (A to D) on a ternary diagram of sediment composition.
(After Bilyard & Carey 1979.)