THE USE O F STATISTICS IN PHYTOSOCIOLOQY
83
E. NODAL ANALYSIS
Whereas the normal analysis alone gives site-groups and the inverse
gives species-groups, the central information concerning specieslsite
relationships must clearly lie in a correlation between the two. I n any
set of data from a natural community, it is hardly to be expected that
species/species and site/site relationships will correlate perfectly : this
situation would only occur if all the sites except identicals were completely discrete from one another with no overlap of species. Instead,
it is common experience that most sites contain “accidental” species
whose main phytosociological relationships lie elsewhere in the population; conversely, species frequently occur aberrantly in sites outside
their normal ecological range. On the other hand, there may well be
groups of ecologically similar sites in which one or other of their more
typical species happens to be absent from certain of the sites ; conversely,
a mqmber of a group of phytosociologically related species may occasionally be missing from among its usual companions. These chance presences and absences in fact represent peripheral information largely
irrelevant to the central situation: a method is therefore needed to
reject these individual variations and to synthesize the central
information.
Since both sets of results derive from the same set of primary data,
some coincidence between the two is only to be expected. Theoretically,
it would appear more satisfactory to extract this central information
by means of a single analysis rather than by attempting to correlate
two independent sets of results. However, as Williams and Dale (1964)
point out, there is a fundamental dificulty in such an approach: since
the original data-matrix is not symmetrical, simultaneous manipulation
of the sites and species is impossible with any statistical method
involving the geometrical concept of alternative site- and species-axes
in Euclidean space. Methods of extracting the central information by
other statistical means are under consideration, but the difficulties
appear formidable. At present, the only statistical method which exists
for approximating to this central information is one in which coincidences are sought by dividing the sites and species with reference to
each axis in turn at each successive subdivision in “association-analysis”
(Williams and Lambert, 1961b; Lambert and Williams, 1962).
The concentrates of information eventually obtained by such a
method may be called “noda”, each of which is now defined by two sets
of parameters, one relating to the site- and the other to the speciessubdivisions. Dense, highly ordered concentrations are clearly of greater
importance than more diffuse aggregations, but the level of concentration at which aggregates are to be rejected or retained has still to be
83
E. NODAL ANALYSIS
Whereas the normal analysis alone gives site-groups and the inverse
gives species-groups, the central information concerning specieslsite
relationships must clearly lie in a correlation between the two. I n any
set of data from a natural community, it is hardly to be expected that
species/species and site/site relationships will correlate perfectly : this
situation would only occur if all the sites except identicals were completely discrete from one another with no overlap of species. Instead,
it is common experience that most sites contain “accidental” species
whose main phytosociological relationships lie elsewhere in the population; conversely, species frequently occur aberrantly in sites outside
their normal ecological range. On the other hand, there may well be
groups of ecologically similar sites in which one or other of their more
typical species happens to be absent from certain of the sites ; conversely,
a mqmber of a group of phytosociologically related species may occasionally be missing from among its usual companions. These chance presences and absences in fact represent peripheral information largely
irrelevant to the central situation: a method is therefore needed to
reject these individual variations and to synthesize the central
information.
Since both sets of results derive from the same set of primary data,
some coincidence between the two is only to be expected. Theoretically,
it would appear more satisfactory to extract this central information
by means of a single analysis rather than by attempting to correlate
two independent sets of results. However, as Williams and Dale (1964)
point out, there is a fundamental dificulty in such an approach: since
the original data-matrix is not symmetrical, simultaneous manipulation
of the sites and species is impossible with any statistical method
involving the geometrical concept of alternative site- and species-axes
in Euclidean space. Methods of extracting the central information by
other statistical means are under consideration, but the difficulties
appear formidable. At present, the only statistical method which exists
for approximating to this central information is one in which coincidences are sought by dividing the sites and species with reference to
each axis in turn at each successive subdivision in “association-analysis”
(Williams and Lambert, 1961b; Lambert and Williams, 1962).
The concentrates of information eventually obtained by such a
method may be called “noda”, each of which is now defined by two sets
of parameters, one relating to the site- and the other to the speciessubdivisions. Dense, highly ordered concentrations are clearly of greater
importance than more diffuse aggregations, but the level of concentration at which aggregates are to be rejected or retained has still to be