EPA-600/3-77-033
March 1977
Ecological Research Series
APPLICATION OF NUMERICAL CLASSIFICATION
IN ECOLOGICAL INVESTIGATIONS OF
WATER POLLUTION
Environmental Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Corvallis, Oregon 97330
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RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development, U.S. Environmental
Protection Agency, have been grouped into five series. These five broad
categories were established to facilitate further development and application of
environmental technology. Elimination of traditional grouping was consciously
planned to foster technology transfer and a maximum interface in related fields.
The five series are:
1. Environmental Health Effects Research
2. Environmental Protection Technology
3. Ecological Research
4. Environmental Monitoring
5. Socioeconomic Environmental Studies
This report has been assigned to the ECOLOGICAL RESEARCH series. This series
describes research on the effects of pollution on humans, plant and animal
species, and materials. Problems are assessed for their long- and short-term
influences. Investigations include formation, transport, and pathway studies to
determine the fate of pollutants and their effects. This work provides the technical
basis for setting standards to minimize undesirable changes in living organisms
in the aquatic, terrestrial, and atmospheric environments.
This document is available to the public through the National Technical Informa-
tion Service, Springfield, Virginia 22161.
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EPA-600/3-77-033
March 1977
APPLICATION OF NUMERICAL CLASSIFICATION IN ECOLOGICAL
INVESTIGATIONS OF WATER POLLUTION1
by
Donald F. Boesch
Virginia Institute of Marine Science
Gloucester Point, Virginia 23062
Grant No. R803599-01-1
ROAP/TASK NO. 21 BEI
Project Officer
Richard C. Swartz
Marine and Freshwater Ecology Branch
Newport Field Station
Corvallis Environmental Research Laboratory
Newport, Oregon 97365
CORVALLIS ENVIRONMENTAL RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
CORVALLIS, OREGON 97330
1 Special Scientific Report No. 77, Virginia Institute of
Marine Science.
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DISCLAIMER
This report has been reviewed by the Corvallis Environmental
Research Laboratory, U.S. Environmental Protection Agency, and
approved for publication. Approval does not signify that the
contents necessarily reflect the views and policies of the
U.S. Environmental Protection Agency, nor does mention of trade
names or commercial products constitute endorsement or recommen-
dation for use.
11
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FOREWORD
Effective regulatory and enforcement actions by the Environmental
Protection Agency would be virtually impossible without sound
scientific data on pollutants and their impact on environmental
stability and human health. Responsibility for building this data
base has been assigned to EPA's Office of Research and Development
and its 15 major field installations, one of which is the Corvallis
Environmental Research Laboratory.
The primary mission of the Corvallis laboratory is research on the
effects of environmental pollutants on terrestrial, freshwater,
and marine ecosystems; the behavior, effects, and control of pol-
lutants in lake systems; and the development of predictive models
on the movement of pollutants in the biosphere.
This report describes classif icatory techniques for demonstrating
similarities in the distribution of species or in the composition
of biological communities. Numerical classification offers a
promising quantitative method for analyzing the impact of pollu-
tion on aquatic community structure.
r;
A. F. Bartsch, Director
Corvallis Environmental
Research Laboratory
111
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ABSTRACT
Numerical classification encompasses a variety of techniques
for the grouping of entities based on the resemblance of
their attributes according to mathematically stated cri-
teria. In ecology this usually involves classification
of collections, representing sites or sampling periods,
or classification of species. Classification can thus
simplify patterns of collection resemblance or species
distribution patterns in an instructive and efficient
manner.
Procedures of numerical classification are thoroughly re-
viewed, including data manipulations, computation of
resemblance measures and clustering methods. The importance
and effects of transformations and standardizations are
discussed. It is particularly critical to choose an
appropriate resemblance measure which best corresponds
with the investigator's concept of ecological resemblance.
Clustering methods form groups on the basis of patterns
of inter-entity similarity. Various types of clustering
methods exist but currently the most useful and best
developed are those which are exclusive, intrinsic,
hierarchical and agglomerative. Agglomerative clustering
methods which distort spatial relationships and intensely
cluster are often most useful with ecological data.
The value of post-clustering analyses in the interpretation
of the results of numerical classifications is stressed.
These include reallocation of misclassified entities,
comparison of classifications of collections with those
of species (nodal analysis), comparing alternate classi-
fications, testing differences among groups, relating
classification to extrinsic environmental factors and
interfacing classification with other multivariate
analyses.
The usefulness of numerical classification is demonstrated
for objective analysis of the data sets resulting from
field surveys and monitoring studies conducted for the
assessment of effects of pollution. However, to date few
pollution biologists have applied the more powerful classi-
ficatory techniques and post-clustering analyses.
iv
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CONTENTS
Page
Foreword iii
Abstract iv
List of Figures vii
List of Tables viii
Acknowledgments ix
Sections
I Conclusions 1
II Recommendations 2
III Introduction 3
Numerical Classification 4
Procedures of Numerical Classification 5
IV Data 10
Forms of Data 10
Data Reduction 12
Transformations 14
V Resemblance Measures 18
Normal vs. Inverse Analyses 18
General 18
Similarity Coefficients 20
Euclidean Distance 30
Correlation Coefficients 32
Information Content Measures 34
Probablistic Measures 36
VI Clustering Methods 38
General 38
v
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Sections Pagf
VI Clustering Methods (cont.)
Classification of Clustering Methods 38
Non-hierarchical Methods 42
Agglomerative Hierarchical Methods 45
Divisive Hierarchical Methods 56
VII Interpretation of Numerical Classifications 59
Stopping Rules 59
Reallocation 61
Nodal Analyses 62
Comparing Classifications 68
Testing Differences Among Groups 68
Relating Classifications to Extrinsic 70
Factors
Interfacing Classification and Ordination 71
VIII Applications of Ecological Classification 73
Designing a Classificatory Analysis 73
Application of Classification to Water 79
Pollution Problems
IX References 98
X List of Publications 114
VI
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FIGURES
No. Page
1 Sequence of Procedures in Numerical 7
Classification
2 Trellis Diagram 8
3 2x2 Contingency Table 21
4 Computation of Euclidean Distance 31
5 Classification of Clustering Methods 39
6 Recurrent Groups of Demersal Fishes 44
7 Combinatorial Computation of Inter-Group 46
Distance
8 Extensive Chaining in a Single Linkage 50
Clustering
9 Effect of Varying Cluster Intensity 54
Coefficient
10 Nodal Constancy in a Two-way Table 65
11 Nodal Fidelity in a Two-way Table 67
12 Classification of Intertidal Assemblages 84
around a Sewer Outfall
13 Classification of Intertidal Assemblages 86
Impacted by Oil Spill
14 Classification of Benthic Assemblages in 88
Hampton Roads
15 Classification of Los Angeles - Long Beach 92
Harbor Stations, November 1954
16 Nodal Constancy For Species and Site Groups, 97
Los Angeles - Long Beach Harbors
VII
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TABLES
No. Page
1 Binary Similarity Coefficients 23
2 Parameters for Combinatorial Clustering 48
Methods
3 Classification of Reish's Los Angeles - Beach 90
Harbor Sites
4 Classification of Species from Los Angeles - 93
Long Beach Harbor
vm
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ACKNOWLEDGEMENTS
Dr. Richard C. Swartz of the Environmental Protection Agency
had the vision to plan comprehensive reviews of the use of
mathematical methods of community analysis in water pollu-
tion investigations. He provided me with encouragement,
support and guidance during the preparation of this review.
My understanding and approach to numerical classification
have been strongly influenced by W. Stephenson and W. T.
Williams, two delightful Englishmen whose coincident option
for the Australian noonday sun has resulted in major impetus
to the application of numerical classification in aquatic
ecology.
Kenneth A. Dierks assisted in analysis of data and in
literature search. Finally, I thank my associates Robert
J. Diaz and David J. Hartzband for reviewing drafts of the
report and providing constructive criticism.
IX
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SECTION I
CONCLUSIONS
The wide variety of numerical classificatory techniques
available is bewildering but affords the investigator a
choice of the methods which most appropriately simulate
ecologically meaningful criteria. Guidelines for the
choice of classificatory strategies based on the efficacy
of the various techniques are given but the most appro-
priate design depends on circumstances and the questions
posed.
Currently the most useful and easiest to apply are combi-
natorial agglomerative clustering methods applied to
similarity, distance or correlation resemblance matrices.
Polythetic divisive methods are theoretically attractive
but are at this point poorly developed and are not widely
available.
Data manipulations, including reduction, transformations,
and standardizations can profoundly affect the results of
a numerical classification. Their use should be justified
and only manipulations appropriate to the ecological ques-
tions posed should be applied.
Analyses performed on the results of the numerical class-
ification greatly enhance interpretation of the classifi-
cation and thus ecological insight. In particular,
relating normal classifications (of collections) to
inverse classifications (of species) in two-way tables,
referred to here as nodal analysis, is simple and
effective.
Although numerical classification has been used effec-
tively in water pollution investigations, its use is not
widespread and most studies have not employed particularly
effective techniques. Appropriate classificatory tech-
niques applied with properly designed sampling approaches
should prove very useful in future impact assessments.
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SECTION II
RECOMMENDATIONS
Ecologists employing numerical classification should
become familiar with the wide range of methods available,
should understand the strengths and weaknesses of these
methods in the analysis of ecological data and should
design appropriate sampling and analytical approaches to
assess the environmental problem at hand. This task should
be made easier by the recent appearance of several texts
on the subject of numerical classification as well as by
this report.
Computer programs for a wide variety of numerical classi-
fication methods should be more available to and useable
by the practicing ecologist.
Methodological advances are needed in several areas,
notably in polythetic divisive clustering methods, in
objective procedures for reallocation of entities after
initial classification, and for statistically testing
differences among classificatory groups.
The use of numerical classification in water pollution
investigations should be encouraged, in particularly where
biotic assemblages are diverse and patterns of occurrence
complex. However, choice of techniques depends on the
investigator's ecological criteria and the circumstances
of the study. The approach should be rational rather
than routine.
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SECTION III
INTRODUCTION
The use of multivariate analytical techniques in community
ecology has expanded tremendously in recent years. These
techniques have the appeal of objective analysis and simpli-
fication of the complex arrays of data generated in field
studies. These data arrays typically take the form of
measures of abundance of the various species represented
in a series of collections. Mental capacity to perceive
patterns in such data arrays quickly diminishes with the
size of the array, i.e. with the number of collections and
the number of species. Thus, except in very limited studies
or in cases of extremely low species richness, reproducible
procedures for the detection and description of patterns
are indeed desirable.
The wide availability of computers have spurred a surge in
development of multivariate techniques for the analysis of
complex data sets and their wide application in ecology,
taxonomy, other biological sciences and in such disparate
fields as medicine, criminology, anthropology, geology,
remote sensing, engineering and the humanities (Anderberg
1973, Sneath and Sokal 1973, Sokal 1974). As a consequence
of the broad-based and rapid development of multivariate
analyses, the relevant literature on techniques and appli-
cations is diffuse and often obscure. A review of the
applications of multivariate analyses in aquatic ecology
shows that most practitioners were unaware of or lacked
facility with the broad range of techniques now in exist-
ence, but instead have been restricted to familiar or
readily available techniques. Also, because of the
matnematical nature of the techniques and the excessive
amount of unstandardized jargon common in the discipline,
application of multivariate techniques is often more
obfuscating than illuminating to the non-specialist.
The use of multivariate analyses in field ecological
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research on man's impacts and in environmental baseline
studies is rapidly increasing. Need for a compilation,
review and evaluation of the various techniques available
was recognized in the development of the Environmental_
Protection Agency's research program on Biological Indices
for Marine Ecosystems. Thus, this critical review was
commissioned to assist the Agency in evaluating and con-
ducting research on environmental impacts and to serve
as a reference for practicing aquatic ecologists. This
report constitutes a general review of numerical classi-
fication or cluster analysis. Subsequent reports resulting
from research on a new grant (R 804127-01-0) will focus on
new developments in numerical classification, on ordination
and related techniques, and on applications of techniques.
Several important texts and reviews treating numerical
classification have recently been published, some while
this report was in preparation. The reader is especially
directed to books by Clifford and Stephenson (1975) and
Orloci (1975) on ecological applications of multivariate
analyses, Sneath and Sokal (1973) on biological (chiefly
taxonomic) applications of numerical classification, and
Anderberg (1973) and Hartigan (1975) on general applications
of cluster analysis. Because of these existing references,
no attempt is made here to describe many of the techniques
in detail nor to be exhaustive in coverage. Rather an
overview summarization of techniques will be provided to-
gether with an evaluation of their application in aquatic
ecology. Since bothersome terminological differences exist
in the diverse literature, frequent cross-referencing of
terms will be made.
NUMERICAL CLASSIFICATION
In simplest terms, classification is the ordering of enti-
ties into groups or sets on the basis of the relationships
of their attributes. Classification is an important bio-
logical process which must predate man, but the science of
classification has had a fairly recent and parallel devel-
opment in several disciplines (Sokal 1974).
In ecology the entities most often classified are biolog-
ical collections or observations. The classification of
collections or observations, either conscious or subcon-
scious, is central to the ecologist's conception of commu-
nities. Ecologists also classify species on the basis of
their ecological attributes. Thus, we think of tropical
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intertidal or demersal species and carnivores or deposit-
feeders on the basis of where they occur or what they do.
Numerical classification or cluster analysis encompasses a
wide variety of techniques for ordering entities into groups
on the basis of certain formal pre-established criteria
rather than on subjective and undefined conceptions.
Numerical classifications have certain advantages over
subjective classifications, notably: (1) they can be
based on a much larger number of attributes than is allowed
by human mental capacity; and (2) once the classificatory
criteria are set, their results are repeatable by any
investigator studying the same data set.
It is important to distinguish classification from several
other processes and analyses. First, the process of
"identification", involving the allocation of additional
unidentified entities to the most appropriate class, once
such classes have been established (Dagnelie 1971, Sneath
and Sokal 1973, Sokal 1974), is here excluded from classi-
fication. The use of techniques of numerical identification
(e.g. discriminant analysis) both in reallocating members of
classes to improve classifications and in assigning new
members to classes will be considered in a future report.
On the other hand, "dissection", or the optimal splitting
of a continuous into a discontinuous series (Clifford
and Stephenson 1975), is here considered a case of
classification.
Secondly, various multivariate analyses other than numerical
classification may be applied to ecological data. These
include, in addition to various regression and correlation
approaches, a broad group of techniques referred to by
biologists as ordination. In ordination the relationships
among entities are expressed in a simplified spatial model
of a few dimensions, with no attempt to group or draw
boundaries between classes (Pielou 1969, Whittaker 1967,
Whittaker and Gauch 1973, Sneath and Sokal 1973, Orloci
1975) . Ordination includes such techniques as principal
components analysis, factor analysis, principal coordinates
analysis, correspondence analysis, and multidimensional
scaling.
PROCEDURES OF NUMERICAL CLASSIFICATION
To orient the reader to the following sections, a brief
description of the chain of procedures in numerical
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classifications is in order. Numerical classifications
are generally directed by a set of algebraically expressed
criteria (an algorithm). This chain of operations begins
with the original data, in one or more forms which may be
further transformed to conform to certain preconditions.
In ecological applications the original data are generally
in the form of a matrix of some measure of abundance of
each species in a series of collections (Pig. 1). Section
IV considers the different forms data may take and reduc-
tions or transformations which may be performed before
proceeding with a clustering algorithm.
From the original or transformed data matrix most numerical
classifications then require the computation of a resem-
blance measure between all pairs of entities being class-
ified. This is a numerical expression of the degree of
similarity, or, conversely, dissimilarity, between the
entities on the basis of their attributes. In ecology,
the entities being classified may be collections (repre-
senting sites, stations, or temporal intervals) with
species content as the attributes. This may be referred
to as a normal class if ication as opposed to an inverse
classification of species as entities with their presence
or abundance in the collections as attributes (Williams
and Lambert 1961a). "Normal" and "inverse" are synonymous
with the widely used terms "Q analysis" and "R analysis,"
respectively. However the Q/R distinction has been confused
in the past (Ivimey-Cook, Proctor and Wigston 1969) and the
normal/inverse terminology is fast becoming standard in
ecology. The wide variety of resemblance measures used or
proposed are reviewed in Section V.
Matrices of inter-entity resemblance measures are usually
required to perform normal or inverse analyses (Fig. 1).
These matrices are symmetric in that one corner is the
mirror image of the other across the "self-match" diagonal
and thus it is necessary to display only half the matrix,
as in "Fig. 1, .as the excluded portion is repetitious. A
familiar type of resemblance half-matrix is an inter-city
distance finder commonly found on road maps. Resemblance
matrices are often presented, sometimes as familiar shade-
coded "trellis diagrams" (Fig. 2), in the ecological liter-
ature (Macfadyen 1963). From the resemblance matrix one
can go further and seek to group entities into groups on
the basis of their patterns of resemblance (Fig. 1). This
is the essence of clustering. The great variety of
clustering methods available are summarized in Section
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Fig. 1. Illustration of the sequence of procedures in
numerical classification.
7
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IRB I5R I4R 9RB7RB 4RA8RA
0<30%
30< 50% 50 < 70%
HOMOGENEITY
70.-100%
Fig. 2. Example of a "trellis diagram" or a rearranged
resemblance matrix with degree of resemblance
shade coded (from Sanders, I960),
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All too frequently, the results of numerical classification
are presented with painfully little interpretation. Rec-
ognizing that classificatory techniques attempt only to
simplify complex data sets and not to provide ecological
interpretations, post-clustering analyses and interpretive
techniques are emphasized in Section VII.
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SECTION IV
DATA
Despite the recent proliferation of texts on mathematical
ecology (Pielou 1969, 1974, 1975, Sokal and Rohlf 1969,
Poole 1974) there is a paucity of comprehensive treatments
of the problems of ecological data and data manipulation
appropriate to applications of numerical classification.
Clifford and Stephenson (1975, Chapters 5 and 7) present
a thorough discussion of the types of data and data manip-
ulations ecologists are likely to use with numerical class-
ification. The following discussion of data problems is
intended to be complementary to their treatment.
FORMS OF DATA
The usual form of ecological data to which numerical class-
ificatory techniques are applied is the presence or some
quantitative measure of importance (numerical abundance,
biomass, productivity, rank, etc.) of taxa in collections.
However, entities may be classified on the basis of other
ecological attributes, for example classifying sites on the
basis of abiotic environmental variables.
In general terms, data may be considered to be of one of
five basic types (Clifford and Stephenson 1975):
(1) Binary - possessed of two character states, in ecology
generally species present or absent.
(2) Disordered multistage - possessing three or more con-
trasting forms each ranking equal, e.g. red, white, blue.
(3) Ordered multistate - possessing a hierarchy of con-
trasting forms, which encompasses the total variation in
the range of entities under study, e.g. abundant, common,
rare.
(4^ Ranked - graded within a collection, e.g. most abun-
dant, second most abundant, etc.
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(5) Quantitative - quantitative data may be meristic
(counts) or continuous (size).
Binary Data
In most ecological applications data will be binary or
quantitative. The use of binary data was generally the
rule in early ecological applications of multivariate
analyses, but the use of quantitative data is growing fast.
However, use of binary data is still quite common and in
certain applications, e.g. in biogeography, may be the only
practical approach. Many ecologists have generally dis-
dained the use of binary data in situations where quanti-
tative data may be collected (Grieg-Smith 1964, p. 160,
Clifford and Stephenson 1975, p. 39, Stephenson 1973).
Others have noted that, especially if there are many zeros
(species absences) in the data matrix, use of binary rather
than quantitative data involves loss of relatively little
information (Lance and Williams 1967b, Williams et al.
1973) .
Actually, the choice between the use of binary or quantita-
tive data involves a decision as to the ecological question
asked by the analyst. In a normal analysis resemblance
measures based on binary data ask "How similar are the
species lists of two collections?" In an inverse analysis
the question is "What is the degree of co-occurrence of two
species?" Collections may have identical species lists, but
vast differences in the relative abundances or dominance of
the species, and species may be continuously sympatric, but
have distinct habitat preferences.
Quantitative Data
Various types of quantitative data may be used, although the
most common are counts or densities (meristic) and biomass
(continuous). Other continuous data forms such as produc-
tivity, respiration, or cover may also be used. If many
replicate samples are taken, frequency of species occurrence
in the replicates may be used as an importance measure.
Again, the choice of data type is an ecological rather
than an analytical question and often the data form is
dictated by circumstances. The use of different types
of quantitative data, e.g. numerical density versus biomass,
may yield vastly different classifications (Clifford and
Stephenson 1975, p. 44) .
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In some ecological situations where analyses of samples is
quantitative but sampling effort is inconsistent or unquant-
ified, e.g. with dredge hauls,.resulting data may not be
quantitatively comparable between collections. In such
cases, the investigator not content with basing a classifi-
cation solely on binary data may express the data as ordered
multistate or ranked by using a numerical scoring system.
Alternately the data may be collection-standardized (see
below) by expressing species importance as percent of the
total in the collection.
The non-random form and typically great inequalities of
quantitative data bare frequent problems. Thus, one often
has to compare very large, and sometimes aberrant, quanti-
ties with small quantities to determine resemblance. Data
transformations of various types are often used to alleviate
this problem. Transformations are increasingly routine in
ecological classification, but their application is fre-
quently unthinking or arbitrary and their effects on class-
ifications poorly understood.
DATA REDUCTION
Ecological surveys often generate very large data matrices,
due in part to the great abundance of relatively rare
species in many communities. Large data matrices are
commonly reduced before performing numerical classifica-
tions. This is done by the elimination or amalgamation
of certain collections or by the elimination of certain
species.
Clifford and Stephenson (1975) list three reasons why data
reduction may be desirable: (1) to reduce the number of
computations, and therefore the resultant expense; (2) to
permit the use of certain classificatory strategies which
would not otherwise be available because of the mass of
data; and' (3) to exclude data which have little or no
biological meaning.
Most commonly, data matrices are reduced by elimination of
species. The simplest and most widely used criterion for
elimination is frequency in the collections. Thus, one may
eliminate species occurring only once, twice, etc. The
rationale is that since the probability of occurrence of
very rare species in any given collection is small, co-
occurrence relationships of these species may be due more
to chance than to similar habitat requirements. The
12
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occurrence of very rare species is often patternless, at
least within the limits of reasonable sampling effort.
However, before excluding species occurring in less than
some arbitrary frequency, the data should be studied for
rare species which seem to be habitat-restricted. These
should be retained if possible. Alternately, some investi-
gators have excluded species whose overall or maximum
abundance fell below a given level (e.g. Day, Field and
Montgomery 1971).
Other criteria for exclusion are also possible. Boesch
(1976) excluded species on the basis of habitat-constancy.
Only species which exceeded a minimum level of overall
constancy in the seasonally replicated samples at a site
were included in the analysis. Stephenson and his associ-
ates have used several more complicated techniques to
decide on the elimination of species. These have included:
(1) establishing a minimum inter-species resemblance level
(i.e. a species must at least have a certain resemblance to
another species to be included) (Stephenson, Williams and
Lance 1970); (2) sorting out of species which do not con-
tribute much to the overall "pattern" by a divisive mono-
thetic clustering method (see Section VI) (Stephenson,
Williams and Lance 1970, Stephenson, Williams and Cook
1972); (3) assessing the contribution of a species to the
variance of the data matrix (Williams and Stephenson 1973,
Stephenson, Williams and Cook 1974); and (4) testing the
conformity of species to predetermined collection groups
(Williams and Stephenson 1973; Stephenson, Williams and
Cook 1974). Each of Stephenson's techniques tend to
accentuate habitat-specificity at the expense of ubiquity.
There is a danger of excluding moderately common, ubigui-
tous species from the analysis, thus yielding an exaggerated
"sharpness" in the classification.
In the past, ecologists applying classification to collec-
tion data have often been far too cavalier and arbitrary in
the elimination of species from data sets. Exclusion cri-
teria ultimately depend on the ecological question one is
attempting to pose in the analysis. The intuitive criteria
in most cases are themselves multivariate, thus it is rea-
sonable to impose several criteria in making decisions on
exclusion. An elaborate attempt to incorporate a variety
of criteria was made by Grigal and Ohmann (1975) who ranked
species according to six different criteria, including
overall frequency, mean abundance, deviation of the standard
deviation of their abundance from that predicted from the
mean, information content of binary occurrence, contribution
13
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to inter-collection differences, and sums of loadings in an
inverse principal components analysis.
Reduction of the data matrix by elimination of collections
may be made on more straightforward bases. Collections of
doubtful quality, i.e. "bad hauls," may be eliminated on
practical grounds. Adjacent samples may be combined if
suitable homogeneity exists. Temporal samples from a
station may be combined if the primary aim of the analysis
is to elucidate spatial patterns while ignoring temporal
interactions and, conversely, contemporary samples may be
combined over a series of stations to examine overall
temporal patterns (Stephenson, Williams and Cook 1974).
TRANSFORMATIONS
Transformations of original data may be suggested because
of one or several of the following reasons: (1) ecological
collections usually produce large numbers (or biomass) of a
few species and small numbers of many; (2) the distribution
of species abundance tends to be non-normal; and (3) sam-
pling effort may be inconsistent. It is important to dis-
tinguish between two basic types of "transformations":
trans formations (sensu stricto) and standardizations.
Trans formations are alterations to the attribute scores
(species abundance) of entities without reference to the
range of scores within the population as a whole. Common
transformations are square root, logarithmic and arcsine
(Sokal and Rohlf 1969). Standardizations are alterations
which depend on some property of the array of scores under
consideration. A common standardization is the conversion
of values to percentages, e.g. percent of the total number
of individuals in a sample by each species.
Trans formations
Perhaps the most common transformation is conversion of
species scores into logarithms. Usually, because of the
presence of zero scores, the transformation takes the form
log (x+1). This transformation may be applied when the mean
population estimates are positively correlated with their
variance to normalize the distribution of sample estimates
(Sokal and Rohlf 1969). Logarithmic transformation has the
other very important effect in numerical classification of
reducing the discrepancy between large and small values in
the computation of resemblance measures. In ecological
14
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terms this reduces the relative contribution of very abun-
dant species to inter-collection resemblance and reduces the
relative contribution of high density occurrences to inter-
species resemblance. Clifford and Stephenson (1975) present
a detailed discussion of the effects of transformations on
commonly used resemblance measures.
Other types of transformations are exponential (e.g. x1/2,
(x+c)l/2, where c is a small number, xl/3, etc.) and arcsine
or angular (especially appropriate to percentages or pro-
portions) transformations. Another type of transformation
which has been used with the Canberra metric resemblance
measure (see Section V) involves the addition of a small
number to all species scores (Stephenson et al. 1972,
Boesch 1973) to decrease the relative contribution of 1/0
matches to resemblance.
Standardizations
The most common standardization is by collection total,
(x.j_/£ Xj_, where Xj_ is the importance of the i-th species
in a collection such that the original species scores become
proportions or percentages of the total. Collection total
standardization is implicit in the widely used "percentage
similarity" (also known to marine ecologists as index of
affinity or dominance affinity) as a measure of resemblance
(Sanders 1960, Goodall 1973). Collection total standardi-
zation is most appropriate when unequal sampling effort
disallows direct comparison of absolute abundance data.
Alternately, values may be standardized by species total,
i.e. species abundance values are divided by the total
abundance of the species in all collections XJ/E Xj, where
xj is the importance of the species in question in the j-th
collection. Clifford and Stephenson (1975) discuss in
detail the reasons for applying collection and species
standardization and the effects of standardizations on
resemblance measures.
Other standardizations which have also been used include
(Noy-Meir 1971, Burr 1968): centering by expression of
species scores as deviates from the mean quantity of the
species in all collections; division by species norm
(Z Xj2)l/2 or collection norm (E Xj_2)l/2; division by
collection or species maximum, range, mean or standard
deviation; and double standardization by totals or norms
of both species and collections. All this may seem less
15
-------
Confusing when one realizes that the familiar product -moment
correlation coefficient (Sokal and Rohlf 1969) is entity-
mean centered and entity-norm standardized.
Double standardizations have intriguing properties in that
they may alleviate scale problems (i.e. comparing large
numbers with small) in both normal and inverse analyses
using the same transformed data set (Boesch 1973) . Bray
and Curtis (1957) used a two-step (successive) standardi-
zation involving division of scores by the maximum for
that species followed by division of the new scores by
the total for the collection. Simultaneous double standard-
izations have been applied by Benzecri (1969) , Austin and
Noy-Meir (1971) , and Boesch (1973) . The double standardi-
zation used by Austin and Noy-Meir and Boesch produced
transformed elements,
x
(? Xi-; E Xi-
D i J
where XJLJ is the unstandardized value of the i-th species
in the j-th collection.
Classifications of the same data with different standardi-
zations can yield strikingly different results (Austin and
Grieg-Smith 1968) . Standardization involves weighting
information from different species or collections in the
overall multivariate analysis. The choice of standardi-
zation in any particular study is therefore critical and
should be based on consideration of the purposes of the
classification and the nature of the data, rather than a
"•cookbook formula" (Noy-Meir 1971) .
A case in point is the frequent use of collection total
standardizations in "percentage similarity" comparisons.
It is common in ecological data sets for abundant species
to vary widely in abundance and to be periodically collected
in unusually high numbers. The effect of such variations
is to cause artificial inter-collection differences in the
standardized values of species whose absolute abundances
are fairly evenly distributed. Thus standardization by
collection total only seems appropriate where sampling
effort is variable or unquantified (e.g. with dredge and
trawl hauls or an unmetered plankton tow) , where there are
considerable concordant differences in the abundances of
16
-------
most species within the collection set, or where monopoli'
zation of a habitat is an important ecological criterion
(e.g. space cover on rocky shores or fouling plates).
17
-------
SECTION V
RESEMBLANCE MEASURES
NORMAL VS. INVERSE ANALYSES
The ecological questions posed by normal and inverse anal-
yses are substantially different. A normal resemblance
measure expresses the degree of overall "likeness" between
assemblages of organisms, and an inverse resemblance mea-
sure reflects the similarity in the distribution patterns
(spatial or temporal) between species. However, class-
ificatory algorithms can proceed identically for both
normal and inverse analyses. Thus, while some authors
consider normal resemblance measures separate from inverse
measures (Goodall 1973, Anderberg 1973), I will describe
the two simultaneously and identically by referring to
entities and attributes, with the implicit understanding
that in normal analyses collections are the entities and
species are the attributes and in inverse analyses species
are entities and collections are attributes.
GENERAL
Large numbers of resemblance measures have been proposed in
the literature and many have been more or less restricted to
certain disciplines, e.g. numerical taxonomy, social sci-
ences, etc. It far exceeds the scope of this report to list
even all of those that have been used in ecology. Instead,
only those measures which have been used in aquatic ecolog-
ical investigations or show promise for application are
treated and reference is made to their application in the
literature. This section should serve only as a starting
point for the reader interested in application of one1 or
more resemblance measures. For more exhaustive discussions
one should consult Sneath and Sokal (1973, Chap. 4),
Anderberg (1973, Chaps. 4 and 5), Goodall (1973), Clifford
and Stephenson (1975, Chap. 6), and Orloci (1975, Chap. II).
18
-------
As orientation to the notation used in this summarization
of resemblance measures, consider the following m x n data
matrix, whose n columns represent the n entities to be
grouped on the basis of resemblances and whose m rows are
m unit attributes. Each entry x-^j in such a matrix is the
score of entity j for attribute i.
ATTRIBUTES ENTITIES
(Normal:species;
Inverse:collections) (Normal:collections; Inverse:species)
1 2 . . . n
1 xi,i xi,2 . . . xi,n
2 X2,l X2,2 . . . X2,n
m xm,1 xm,2 ... xmn
Other authors use different symbolism and terminology, thus
the expression of similarity measures contained herein may
appear different in other sources. Note that the entity-
attribute terminology is consistent with that of Clifford
and Stephenson (1975) except that they refer to inverse
classifications as clustering of attributes, whereas I
prefer to switch the entity-attribute distinction depending
on the type of analysis. "Entity" and "attribute" may be
considered equivalent to Sneath and Sokal's (1973) "OTU"
and "character," respectively, and Anderberg's (1973) "data
unit" and "variable," respectively.
Various taxonomies of resemblance measures are also used in
the texts listed above. In most, divisions among some of
the types of measures is rather arbitrary and some of the
authors apply identical terms to different types of mea-
sures. The terminology used here is modified from Clifford
and Stephenson (1975) by referring to their "coefficients
of association" as "correlation coefficients" to remove
the ambiguity with Sneath and Sokal's (1973) use of "asso-
ciation coefficients." Thus, I refer to (1) similarity
coefficients as those measures constrained between 0 and 1,
fTJcorrelation coefficients as those constrained between
-1 and 1,(3) Euclidean distance, (4) information content
measures, and (5)probabilistic~measures.
19
-------
SIMILARITY COEFFICIENTS
As used here, similarity coefficients are those resemblance
measures which are 1 (or very close to it) when entities are
identical and 0 (or very close to it) when entities have no
attributes in common. Many authors have expressed similar-
ity in percentages, in which case the value range from 100
to 0. The complement of similarity (1-S, if S is a simi-
larity coefficient) is dissimilarity (D). Some investi-
gators use the concept of dissimilarity rather than
similarity and compute dissimilarity coefficients for
the sake of operational ease. Also, dissimilarity can
be considered analagous to inter-entity distance, allowing
the use of dissimilarity measures in certain .clustering and
ordination techniques based on Euclidean distances (see
below). I use similarity rather than dissimilarity here
because it seems intuitively clearer to most ecologists,
but it is a very simple matter to convert one to the other,
e.g. if S = 0.6, D = 0.4 and vice versa.
Qualitative Similarity Coefficients^
Coefficients of comparison of entities based on binary data
(i.e., species presence or absence) can be conveniently
explained using the symbolism of a 2 x 2 contingency table
which lists the frequencies of agreement and disagreement
of their binary attributes. The general form of the 2x2
contingency and the meanings of its elements in ecological
terms in both normal or inverse analysis are given in Fig.
3. Note that the sum a-fc is the total number of positive
attributes (occurrences) for entity 2, the sum a+b is the
total number of positive attributes for entity 1, and the
sum a+b+c+d is the total number of attributes for which
entities have been compared.
Table 1 lists the commonly used similarity coefficients for
binary data and some of their properties and constraints.
The first coefficient, the simple matching coefficient,
differs from the others in the inclusion in the expression
of d, the number of joint absences or "double zero matches."
As Clifford and Stephenson (1975) point out, in many circum-
stances it would seem ridiculous to regard two entities as
similar largely on the basis of them both lacking something.
With most ecological data sets joint absences of species
has relatively little meaning, given the rarity and con-
tagious distribution of some species, and for this reason
similarity coefficients involving conjoint absences are
20
-------
2x2 CONTINGENCY TABLE
ENTITY I
ENTITY 2
GENERAL
COLLECTION B
0
COLLECTION A
1
NO. SPECIES
IN COMMON
NO. SPECIES
IN A BUT
NOT B
V
NO. SPECIES
IN B BUT
NOT A
NO. SPECIES
NOT REPRE-
SENTED IN
A OR B
NORMAL
ANALYSIS
SPECIES 2
SPECIES I
1
3
1
NO. OF CO-
OCCURRENCES
NO. OF
OCCURRENCES
OF 1 WITH -
OUT 2
V
NO OF
OCCURRENCES
OF 2 WITH-
OUT 1
NO. OF TIMES
NEITHER 1
OR 2 OC-
CURRED
INVERSE
ANALYSIS
Fig. 3. 2x2 contingency tables showing elements a, b,
and d used in computation of binary similarity
coefficients.
21
-------
generally not used in ecology (Green 1971, Field 1971).
However, when most species are common or where there is a
high degree of species fidelity to particular collection
types, the simple matching coefficient may be useful.
Only three of the coefficients iisted have been frequently
used in aquatic ecology—Jaccard, Dice and Fager coeffi-
cients. The Jaccard and Dice coefficients are simple and
similar, with the difference that the Dice measure doubly
weights shared positive attributes (joint presences), and
thus will always be greater than or equal to the Jaccard
measure. Column 4 in Table 1 suggests that in the case
of disparate number of positive attributes, the Dice co-
efficient yields more intuitively accurate values. Further-
more, Clifford and Stephenson (1975) offer that in cases
where there are relatively few conjoint presences in the
data set the Dice coefficient is more attractive, and with
relatively many conjoint presences the Jaccard coefficient
is more attractive because it will give a wider spread of
values in the upper end of the range. Goodall (1973) shows
that the sampling distribution of the Dice coefficient is
slightly more biased than that for the Jaccard coefficient.
The Fager coefficient has been widely used in marine ecology
primarily because its author was active in that field. The
Fager coefficient is the Ochiai coefficient (Table 1) mod-
ified by subtraction of a "correction factor" which means
that the measure is -not constrained between 0 and 1; rather
with no shared positive attributes the measure is slightly
less than 0 and with identical entities the measure is
slightly less than 1. Because of these and other undesir-
able properties Field (1971) and Clifford and Stephenson
(1975) raise objections to the use of the Fager coefficient.
The incorporation of a geometric mean term in the denomi-
nator of the Ochiai coefficient does make this "uncorrected"
form of the Fager coefficient more attractive than the
Jaccard coefficient when the entities have a disparate
number of positive attributes (Sepkoski and Rex 1974).
In summary, the most attractive similarity measures for
binary ecological data appear to be the Jaccard, Dice and
Ochiai coefficients. The selection of the most appropriate
coefficient depends on the nature of the data. If the task
is to discriminate relationships among closely similar
entities one might choose the Jaccard coefficient. If,
on the other hand, the entities vary widely in their number
of positive attributes (e.g. rich and poor collections in
a normal analysis or common and rare species in an inverse
22
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Table 1. COMMONLY USED BINARY SIMILARITY COEFFICIENTS AND THEIR
PROPERTIES. VARIABLES AS IN FIG. 3. EXPRESSIONS IN
COLUMN 1 RESULT WHEN TWO ENTITIES HAVE THE SAME NUMBER
OF POSITIVE ATTRIBUTES; THOSE IN COLUMN 2 WHEN THEY
SHARE NO POSITIVE ATTRIBUTES; THOSE IN COLUMN 3 WHEN
THEY ARE IDENTICAL; AND THOSE IN COLUMN 4 WHEN ONE
SAMPLE HAS TWICE AS MANY POSITIVE ATTRIBUTES AS THE
OTHER AND THE NUMBER OF ATTRIBUTES IN COMMON IS ONE-
HALF THE NUMBER IN THE ENTITY WITH THE FEWER POSITIVE
ATTRIBUTES. ASSUME b>_ c. MODIFIED FROM VALENTINE
(1973) AND CLIFFORD AND STEPHENSON (1975).
NJ
Coefficient
123
a+c = a+b a = 0 a = a+b+c
4
a+c
If a+b=l/2
a
and a+c=l/2
(1) Simple matching
a+d
a+b+c+d
(2) Jaccard (=Iverson)
a
a+b+c
(3) Dice (=S0rensen,
Czekanowski)
2a
2a+b+c
(4) Kulczynski first
a
b+c
a+d
a+2c+d
a+2c
a+c
_a
2c
b+c+d
a+d
5a+d [if a=d, then=l/3]
1/5
1/3
1/4
-------
Table 1 (continued) . COMMONLY USED BINARY SIMILARITY COEFFICIENTS AND
THEIR PROPERTIES.
to
*>.
(5)
(6)
(7)
(8)
(9)
Coefficient
Kulczynski second
all
2 (a+b a+c)
Simpson
a
c
Braun-Blanquet
a
b
Ochiai (=0tsuka)
a
"(a+b) (a+c)
Fager
a 1
/(a+b) Ca+c") 2 /a+b
123 4
a+c
a+c = a+b a = 0 a = a+b+c a+b and
a
a+c 0 1 3/8
a
aTc~ 0 1 1/2
a
a+c" 0 1 V4
a
a+c 0 1 I/ /a
1 1 i X
a+c 2/c- 2 /^b 2 /a+b U/ '«J
a
a+c=l/2
- (1/4 /i
-------
analysis) one should choose the Dice or Ochiai. Another
advantage of the Dice coefficient is that it is the binary
equivalent of the most commonly used quantitative simi-
larity measure, the Bray-Curtis or Czekanowski coefficient
(see below).
Quantitative Similarity Coefficients
As in the case of binary similarity measures, many quanti-
tative similarity coefficients have been proposed or
employed, although only a handful have been applied in
aquatic ecology. An important class of quantitative simi-
larity coefficients are derivatives of metric distance
functions (Minkowski metrics) whose general form can be
stated as
Djk. ~ (? lxij xi]
In particular, coefficients are derived from the Manhattan
metric in which p = 1, thus
D ., = E Ix. .-x..
jk i1 ij ik
Metrics in their basic forms are unconstrained (they range
from zero to infinity) and are distance rather than simi-
larity measures. The metric derivatives discussed here
are expressed as constrained similarity/dissimilarity
coefficients.
Bray-Curtis Coefficient -
The Bray-Curtis similarity coefficient (Clifford and
Stephenson 1975) is perhaps the most widely employed
quantitative measure in ecology. It can be expressed
as a similarity or dissimilarity measure:
2 E min (x.., x. )
Sjk = i 1] = 1 - Djk (12)
E (xij+xik)
? |X.:-X,k|
i 1J lk _ , q (13)
i-i ._ _ _ — -L ~ o .,
25
-------
This measure has been often referred to as the Czekanowski
coefficient (Field 1971, Day, Field & Montgomery 1971) as
it is a quantitative extension of the binary similarity
coefficient used by Czekanowski (1909) and referred to
above as the Dice coefficient. If the scores are stand-
ardized by entity-total (i.e. expressed as proportion or
percent) , Bray Curtis similarity becomes "percentage sim-
ilarity" widely used in American plant ecology and made
popular in marine ecology by Sanders' (1960) application
in the study of a marine benthic community. (Sanders refers
to the coefficient as "dominance affinity") . If the scores
are expressed as proportion of the total-for the entity
(pij = xij / * xij) ' then
Sjk = E min (Pij,Pik) (14)
or the sum of the minimum proportions (or percentages) of
each attribute.
The Bray-Curtis coefficient both in its unstandardized and
"percent standardized" forms has been extensively used in
marine ecology. Some examples are Barnard (1970), Bloom,
Simon and Hunter (1972), Day et al . (1971), Eagle (1973,
1975), Field (1970, 1971), Field and MacFarlane (1968),
Gage (1974) , Hartzband and Hummon (1974) , Kay and Knights
(1975), Markle and Musi ck (1974), Mauchline (1972),
Nichols (1970), Sanders (1960), Sanders and Hessler (1969),
Santos and Simon (1974) , Stephenson and Williams (1971) ,
Stephenson, Williams and Cook (1972) , Wade (1972) , Ward
(1973) , and Warwick and Gage (1975) .
Ruzicka Coefficient -
A variant of the Bray-Curtis coefficient was propBsed by
Ruzicka (1958) and is expressed as
2 min (x-M/xik) ^ min
sjk = = i
I (xij+xik} - I min (xij'xik} I max (xij'xik)
The dissimilarity measure thus becomes
26
-------
X. . -X.,
ID lk
? max (xij+xik)
-,
jk .
The difference between this and the Bray-Curtis coefficient
is that the Ruzicka measure divides the sum of the minimum
shared attributes by the sum of the maximum attribute
scores whereas in the Bray-Curtis measure the sum of the
minimums is divided by the sum of the average (between the
two entities) attribute scores. Because of this the
Ruzicka coefficient is more affected by large differences,
thus high attribute scores, which makes it less sensitive
in the middle range of resemblance than the Bray Curtis
coefficient. Despite the drawback the coefficient has
recently been used by Dutch marine phytoecologists (Colijn
and Koeman 1975, Van den Hoek, Cortel-Breeman and Wanders
1975) .
Canberra metric Coefficient -
A principal difference between the Bray-Curtis similarity
coefficient and the aforementioned binary similarity
coefficients is the effect of size of the score on the
measure. In the Bray-Curtis coefficient and many other
quantitative resemblance measures, attributes with high
scores largely determine the value of the measure whereas
attributes with low scores are relatively unimportant.
In ecological terms this means that abundant species
largely determine inter-collection (normal) resemblance
and dense occurrences largely determine inter-species
(inverse) resemblance. Indeed in many ecological circum-
stances this might be an intuitively appealing character-
istic, but in others it may be tantamount to basing
inter-collection resemblance on only one or two species.
To overcome this characteristic of quantitative metric
and correlation measures Lance and Williams (1966, 1967b)
proposed the Canberra metric coefficient which is usually
expressed in its dissimilarity form
- . .- d7)
m i (Xij+Xik)
-27-
-------
The similarity form of the coefficient is
= l _
Thus, the Canberra metric is the average of a series of
fractions representing the inter-entity agreement of each
attribute and, as such, has a built-in attribute stand-
ardization. An outstandingly large attribute score can
contribute to only one of the summed fractions and so does
not dominate the coefficient. In this regard the Canberra
metric coefficient can be considered intermediate between
other quantitative similarity, distance and correlation
measures and binary resemblance measures.
The incorporation of zero scores in the Canberra metric is
subject to certain conventions (Clifford and Stephenson
1975). Double zero matches (i.e. when attribute scores of
both entities being compared are zero) are usually ignored
for the same reasons that binary coefficients incorporating
the joint absence contingency are disfavored. Thus the
appropriate divisor is not m, the total number of attri-
butes, but m-r where r is the^' number of double zero compar-
isons. Secondly, since when one of the attribute scores
is zero the fraction contributed to the sum is one, small
numbers may be substituted for zero in the case of single
zero matches to ensure a greater contribution to dissim-
ilarity of an attribute difference of 1000 to 0 than of a
difference of 1 to 0 , for example.
If applied to binary data, with the supression of double-
zero matches, the Canberra metric reduces to S=a/a+b+c,
i.e. the Jaccard coefficient.
Use of the Canberra metric in aquatic ecology to date has
been confined to associates of the Canberra (Australia)
school of numerical classification, e.g. Boesch (1973) and
Stephenson et al . (1972) .
Morisita Coefficient -
The Morisita coefficient (Morisita 1959) is not derived
from metric distance functions, rather it is related to
both correlation and information content resemblance mea-
sures. However, since it ranges from 0 (no resemblance)
28
-------
to + 1 (identity) it is here considered a similarity coeffi-
cient.
given by
cient. The coefficient, often referred to as CT, or CK is
A 0
2 S x. .x.,
_ . 11 ik
jk ~ _ i _ (19)
( Aj+Ak) Z x±j Z xik
where
Z Xij (Xij - I)
i (20)
[? x. . ((Z x. .) - 1) ]
J. -L-J 1
and
Z x., (x-
xik « xik) - 1)]
(21)
The terms A. and Ak are the Simpson (1949) diversity mea-
sures for the attributes of entities j and k, respectively.
The basic term of this coefficient, as in other correlation
coefficients, is the product of the two attribute scores
being compared rather than the differences in the two
scores as in coefficients derived from metrics. This leads
to a heavier weighting of the importance of attributes with
high schores than, say, the Bray-Curtis coefficient. Thus,
the Morisita coefficient can be expected to reflect pri-
marily the resemblance of scores of the most abundant spe-
cies in a normal analysis and the resemblance of outstanding
abundances of species in an inverse analysis. On the other
hand, correlations, in general, are less influenced by
scale differences between entities than are the "metric"
expressions, which are based on differences in attribute
scores. In ecological terms this means that in an inverse
comparison a usually abundant species will have low resem-
blance to a species which is usually not very abundant,
29
-------
even if their abundances are correlated, when a "metric"
derived measure is used but may have high resemblance on
the basis of the Morisita coefficient.
The Morisita coefficient has been used in marine ecology by
Barnard (1970), Bloom et al. (1972), Livingston (1975) and
Ono (1961).
EUCLIDEAN DISTANCE
If an entity is construed to be represented by a point in
an m dimensional space each dimension of which corresponds
to an attribute and is orthagonal (at a right angle) to the
other dimensions (or axes), the Euclidean distance is the
linear distance between any two points (entities) in that
hyperspace. The coordinates of the m axes are the scores
of the attributes represented by the axes and the distance
between two points can be computed as the square root of
the sums of the squared differences between attribute
scores,
Djk = [£ (x.. - xik)2]l/2 (22)
You may recognize this as a Minkowski metric (Equation 10)
where p = 2. Euclidean distance may, of course, range from
0 (when entities are identical) to infinity. Either
Euclidean distance itself or its square may be used as the
distance measure.
The concept and computation of Euclidean distance may be
made clearer by consideration of Fig. 4 which depicts the
spatial relationship of three points (entities) in three
dimensions. The distance between any two points can be
computed by squaring the difference of their coordinates
on each axis, summing those squared values and taking the
square root of the sum. This can be expanded to additional
dimensions with the addition of attributes. The squared
differences between the scores of each additional attribute
can simply be added on to the squared distance.
Because differences between attribute scores are squared,
Euclidean distance heavily weights attributes with high
scores and worsens the scale problem between high scoring
and low scoring entities compared to the Bray-Curtis
30
-------
X2,a ' xs,a '
dab2
xl|b )-i- (x2i(
x3)b)
Fig. 4. Illustrations of the computation of Euclidean dis-
tance between entities defined by coordinates
representing three attributes.
31
-------
coefficient (Orloci 1967, 1973, 1975, Clifford and
Stephenson 1975). Thus, in ecological applications
Euclidean distance may place overemphasis on dominance
or outstanding abundances and may result in artificially
high resemblance between entities which do not have many
attributes in common but whose attribute scores are low.
To overcome these weaknesses transformations or standardi-
zations are usually applied to the data. Williams and
Stephenson (1973) and Stephenson, Williams and Cook (1974)
used a cube-root transformation. Standardizations by
entity-norm (Orloci 1967, Pielou 1969, Noy Mier 1971)
and by species variance (Hughes and Thomas 1971a, b) are
frequently used.
With binary data the Euclidean distance reduces to D = /b+c,
using the notation of the 2x2 contingency table (Clifford
and Stephenson 1975).
Marine ecological applications of Euclidean distance as a
resemblance measure include Holland and Dean (1976), Hughes
and Thomas (1971a, b) , Polgar (1975), Stephenson et al.
(1974) and Williams and Stephenson (1973).
CORRELATION COEFFICIENTS
Most correlation coefficients range from -1 (completely
dissimilar) to +1 (completely similar). Many, but not all,
are based on a probabilistic model, offering the potential
advantage of testing the significance of resemblance. How-
ever, it is only appropriate to apply tests of significant
correlation between species and not collections, because of
the assumptions of independence in the tests. Even then
assumptions of the parametric significance tests (normality,
randomness, etc.) are seldom met and one should sl*ow caution
in interpreting the results of tests of significance of
interspecies correlations.
Binary Correlation Coefficients
Two different binary correlation coefficients have been used
in aquatic ecological investigations. The point correlation
coefficient (also referred to as Kendall's coefficient of
association) (Looman and Campbell 1960, Goodall 1973) as
given in the standard terms of the 2x2 contingency is
32
-------
r = ad - be (23)
[(a+b) (c+d) (a+c) (b+d) ] V2
The significance of r can be tested by a Chi-square compar-
ison (Looman and Campbell 1960} but as discussed earlier
the meaning of this significance test is dubious, partic-
ularly since the test assumes that a+b, a+c and d are
known, when these variables are in practice subject to
sampling error. The point correlation coefficient was
employed by Lie and Kelley (1970) and Nichols (1970) in
studies of marine benthic communities.
The second binary correlation coefficient was proposed by
McConnaughy (1964) in an analysis of planktonic communities:
r = a2 - be
(a+b) (a+c)
(24)
Quantitative Correlation Coefficients
A commonly used resemblance measure, particularly in
inverse analyses, is the product-moment correlation coeffi-
cient (Sneath and Sokal 1973, Goodall 1973, Clifford and
Stephenson 1975)
k i T •"• 1 /
^ * (25)
[Z (x^-i - Xl)2 S (xik - xk)2]V2
where x. and x, are the mean values of all m attributes
of entilies j and k, respectively. This is the entity-mean
centered and entity-norm standardized form of the general
correlation expression and other forms of quantitative
correlation are possible (Noy-Meir 1973a).
In summary, although the product-moment correlation coeffi-
cient is useful in expressing the relationship of the shape
of species distribution patterns over a series of collec-
tions in an inverse analysis, the correlation coefficient
suffers from several undesirable characteristics (Clifford
33
-------
and Stephenson 1975, Field 1970, Orloci 1967, 1973, Sneath
and Sokal 1973). It tends to exaggerate the contribution
of outstandingly large scores to resemblance. It can sug-
gest spurious patterns of resemblance if "there are many
zero values in the data matrix (a common condition except
when only abundant or ubiquitous species are included in
the analysis). Finally, since it is a "shape" measure and
not a "size" measure, perfect correlations can occur between
nonidentical entities. Furthermore, tests of significance
of inter-entity correlation should be applied with caution.
In particular, it is inappropriate to apply probablistic
tests to inter-collection correlation because the attributes
(species) do not represent a "variable" in the statistical
sense and they may not be independent (Sneath and Sokal
1973, Clifford and Stephenson 1975). Moreover, tests of
significance of species correlation assume normal frequency
distributions and linear relationships between species
scores--conditions which often do not obtain in ecological
data.
The product-moment correlation coefficient has been used
extensively in marine ecological applications of numerical
classification, e.g. Angel and Fasham (1973), Chardy
(1970), Ebeling et al. (1970), Eisma (1966), Jones (1969),
and Mauchline (1972) and has received even wider use in
applications of principal components and factor analyses
(types of ordination).
INFORMATION CONTENT MEASURES
The term information is here used in a strictly technical
context and relates more to the degree of uncertainty or
surprise than to knowledge (Orloci 1969, 1971). The mea-
sures discussed here have the same information theoretical
basis as the familiar Shannon diversity measure. We can
express the information content of the attribute scores of
an entity as
I. = (Ex..) log (Ex..) - Z x log x (26)
J i 13 i -"-J i •LJ -"-J
using the same notation, x^•, for the elements of the data
matrix. The information content,Ik, of another entity k
can be similarly computed. The information content of the
combined pair of entities can be expressed:
34
-------
xij+xik)
log (x+x)
The increase in information from that represented by I-; and
Ik to that represented by Ij+k can be used as a distance
measure
AI
j,k = Jj+k ~ (Ij + Jk) • (28)
The common or mutual information between the two entities
can alternately be expressed:
V = Jj + Jk - Vk •
Similarily, the information content of arrays of binary
attributes may also be computed by one of several methods
(Lambert and Williams 1966, Dale and Anderson 1972,
Clifford and Stephenson 1975) and information gain or,
conversely, mutual information can be calculated.
Inter-entity matrices of information content resemblance
measures may be passed directly to clustering algorithms
which form groups on the basis of the resemblance matrix
alone (combinatorial clustering strategies). Alternately,
clustering may take place by procedures which require
recomputation, following consultation of the data matrix,
of the information gain for each clustering iteration
(non-combinatorial strategies) . These clustering strategies
are more fully discussed in Section VI.
Information content resemblance measures have been little
used in marine ecology, although they have been widely
applied in plant ecology and taxonomy (Sneath and Sokal
1973, Clifford and Stephenson 1975) . Stephenson and
Williams (1971) in a study of marine benthos attempted
the use of agglomerative information-gain clustering using
both quantitative and binary data but were dissatisfied
with their results. Stephenson et al. (1971, 1972) and
Moore (1973) used the divisive information analysis DIVINF
(Section VI) in studies of marine benthos.
35
-------
Several resemblance measures which do not express infor-
mation content per se but which incorporate information
terms have also been used in ecology. Horn (1966) proposed
a measure of inter-entity "overlap" which is an information
analog of the Morisita coefficient
Sjk=
(zx. .+Zx..)log(Zx. .+Zx.k)-x(zx. .)log(Zx. .
i -"-J i 1K- i -1 i i J i J
-(?xik)log(?xik) ] (30)
where PIJ= x^/Z x^ and P±k = x±k/Z xi]c.
The measure is constrained between 0 and 1 and is appropri-
ately classed as a similarity coefficient. Horn's overlap
coefficient was used by Kohn (1968) to study ecological
relationships among marine snails of the genus Conus and
Bloom et al. (1972) in a study of intertidal benthos.
Other information measures have been similarly used to
express "niche overlap" or "habitat overlap" (Colwell and
Futuyma 1971, Pielou 1975) between pairs of species.
Finally, Hummon (1974) formulated a complex similarity
coefficient which is a mixture of components of percentage
similarity (Bray-Curtis coefficient) , mutual information
and the Fager similarity coefficient, and applied it in a
study of marine gastrotrich taxocenes .
PROBABILISTIC MEASURES
In addition to the correlation coefficients discussed above,
several other measures may be employed to test differences
between pairs of entities. As with the correlation coeffi-
cients, however, their use as a probabilistic test of
significant differences between pairs of collections is
questionable and they are most often applied in the inverse
case of testing the significance of associations between
pairs of species.
Since, except for the correlation coefficient, no other
probabilistic measures have been used much in numerical
36
-------
classification, I will not elaborate on the methods.
Several appropriate techniques are thoroughly reviewed by
Pielou (1969, 1974). Chi-square tests of binary occurrence
data bsed on the 2x2 contingency table are the most
commonly used methods (Pielou 1969, 1974).
37
-------
SECTION VI
CLUSTERING METHODS
GENERAL
It was common in earlier years and it remains an occasional
practice in ecology to simply present resemblance matrices
or trellis diagrams as the end point of a multivariate
analysis. Frequently, the elements of the resemblance
matrix are rearranged so that the highest resemblance
scores are closest to the diagonal of the half-matrix,
i.e. to rearrange the order of entities so that they are
close to those entities they most resemble. Usually this
is done by eye, although Lie and Kelley (1970) presented
a procedure for the rearrangement of the resemblance matrix
by objective criteria. Some investigators have attempted
to draw, more or less by eye, a simple spatial model or
"plexus" of the patterns of inter-entity relationships
based on the resemblance matrix. Such matrix and plexus
techniques (Mclntosh 1973) are more appropriately con-
sidered forms of ordination rather than classification.
Rather, this section treats numerical procedures by which
entities can be objectively grouped based on their
resemblances.
CLASSIFICATION OF CLUSTERING METHODS
Various classifications of clustering methods have been
proposed (Pielou 1969, Williams 1971, Sneath and Sokal
1973) and the dichotomized scheme presented in Fig. 5
encompasses most of their salient features.
Exclusive versus Non-Exclusive
An exclusive classification is one in which an entity
may occur in only one group while in non-exclusive
38
-------
CLASSIFICATION OF CLUSTERING METHODS
I I
NONEXCLUSIVE EXCLUSIVE
I I
EXTRINSIC INTRINSIC
I 1
LO HIERARCHICAL NONHIER ARCHICAL
I
DIVISIVE AGGLOMERATIVE SERIAL SIMULTANEOUS
OPTIMIZATION OPTIMIZATION
I I
MONOTHETIC POLYTHETIC
I I
COMBINATORIAL NON-COMBINATORIAL
Fig. 5. A classification of clustering methods as discussed
in text (from Williams 1971).
-------
classifications entities may be members of more than one
group. Sneath and Sokal (1973) use the terms non-over-
lapping and overlapping as synonymous with exclusive and
non-exclusive"!Although in certain cases the use of non-
exclusive classifications in ecology may make some sense,
they have not been used except in a very few cases (e.g.
Yarranton et al. 1972) and will not be discussed further.
Extrinsic versus Intrinsic
Intrinsic classifications form groups based solely on their
attributes whereas in extrinsic classifications the re-
sulting groups, although based on internal attributes, are
required to reflect predetermined external attributes as
much as possible. In ecology only intrinsic classifi-
cations have been used but the resulting intrinsic groups
are often related to extrinsic attributes (e.g. abiotic
environmental parameters).
Hierarchical versus Non-Hierarchical
Hierarchical clustering methods optimize a route between
the individual entities to the entire set of entities by
progressive fusions or fissions. The results of hier-
archical classifications are usually expressed as a
dendrogram (Fig. 1) or tree-diagram, which depicts the
optimal route from the whole to the individual entities.
Non-hierarchical clustering methods, on the other hand,
optimize the homogeneity of the groups formed, without
defining a route between groups and their constituent
entities or between groups (Williams 1971). Hierarchical
clustering methods are better developed, more versatile
and better understood than are non-hierarchical methods.
Although most of the ensuing discussion will concern
hierarchical methods, non-hierarchical methods will also
be briefly discussed because one non-hierarchical tech-
nique^ Pager's (1957, Fager and McGowan 1963) recurrent
group analysis, has been extensively used in aquatic
ecology.
Serial versus Simultaneous Optimization
All hierarchical clustering methods are serially optimized,
but non-hierarchical methods may be serially or simulta-
neously optimized. In serially optimized non-hierarchical
40
-------
clustering, once a group is formed it is removed from the
population of entities, a second group is formed from the
remainder, and so on until all the entities are accounted
for.^ In simultaneously optimized clustering, the groups are
obtained simultaneously, usually by iterative optimization
of partitions of the population of entities.
Agglomerative versus Divisive
Agglomerative "hierarchical clustering proceeds by pro-
gressive fusion beginning with the entities and ending with
the complete population. Divisive hierarchical clustering
progressively splits the entire set of entities into
smaller and smaller groups. Agglomerative clustering
strategies are the most widely used in ecology. Williams
(1971) pointed out that agglomerative methods suffer from
some computational disadvantages and are inherently prone
to a small amount of misclassification, because they begin
at the inter-entity level, where the possibility of error
is greatest. On the other hand, most divisive clustering
strategies are monothetic (see below) which severely
handicaps their utility.
Monothetic versus Polythetic
Divisive clustering methods may be monothetic, in which
case fissions are based on a single attribute (i.e. in the
binary case the presence or absence of an attribute), or
polythetic, in which case the division is based on resem-
blance over all attributes. Clearly, monothetic methods,
which would, for example, split two collections on the
presence or absence of only one "indicator" species, are
of limited utility in ecology. However, polythetic divi-
sive strategies which appear to be the ideal hierarchical
clustering methods are poorly developed or impractical in
terms of computation time. Several new short-cut poly-
thetic divisive methods have recently been proposed and,
although none has yet been used in aquatic ecology, they
will be reviewed because of the promise they show.
Combinatorial versus Non-combinatorial
Agglomerative hierarchical clustering methods can have
combinatorial or non-combinatorial solutions (Lance and
41
-------
Williams 1967a, Williams 1971). With combinatorial methods
group/group and group/entity resemblance measures can be
calculated successively from the inter-entity resemblance
matrix and thus, once that matrix is computed, it is no
longer necessary to retain the original data matrix. Such
a procedure has obvious computational advantages over a
non-combinatorial method in which the original data matrix
must be retained for the calculation of measures required
during successive agglomerations.
By far the most widely used clustering methods are com-
binatorial, agglomerative, and hierarchical. However,
non-combinatorial agglomerative, monothetic divisive, and
serially optimized non-hierarchical methods have also been
used in aquatic ecology. Clustering methods falling in
these categories, plus the intuitively attractive poly-
thetic divisive category, are discussed below.
NON-HIERARCHICAL METHODS
Serially Optimized Methods
Recurrent Group Analysis -
The only non-hierarchical method receiving much use in
ecology is Pager's (1957) recurrent group method. Fager
(1957) gives detailed instructions for the formation of
clusters and I will only attempt an abbreviated restate-
ment. Starting with an inter-entity resemblance matrix,
it is first necessary to select an arbitrary level of
resemblance at which the investigator considers two
entities associated. Thus, the resemblance matrix is
converted into a matrix of binary attributes, "associated"
and "non-associated." One then determines the largest
group of associated entities which can be formed.» These
entities are termed the first group and are removed from
further consideration together with any other entities
which only have associations with members of the first
group. This procedure is repeated with the remaining
unclassified entities again and again until all entities
with positive associations are placed in a group. The
relationships among the groups may be indicated by the
proportion of the number of inter-entity associations
which are positive (e.g. if there were 3 positive asso-
ciations between entities in 2 groups, one with 3 members
and the other with 5, the connectivity would be 3/(3 x 5) :
0.20} .
42
-------
Recurrent group analysis has usually been employed in
inverse classifications based on the Fager binary simi-
larity coefficient (Table 1). An example of a recurrent
group classification is given in Fig. 6 and reflects the
patterns of co-occurrence of demersal fishes off Southern
California. The technique has similarly been used widely
in marine ecology in the study of plankton (Fager and
McGowan 1963, Sheard 1965, Stone 1969, Tash and Armitage
1967, Venrick 1971), benthos (Jones 1969, Lie and Kelley
1970), and fishes (Fager and Longhurst, 1968, Mearns,
1974) .
Recurrent group non-hierarchical clustering has some
serious disadvantages. The minimum resemblance required
for grouping entities must be stated a priori and is
constant for all entities and the tecEnique does not
recognizes degrees of association. Changing of the
arbitrary level of resemblance necessary for association
can produce very different results. An entity may be
"captured" by a large group early in the clustering and
may appear unassociated with entities which have high
resemblance to it but not to all other members of its
group. The analysis typically produces a few large groups
and many small remnant groups, whose entities, together
with those entities attached to, but not members of groups,
are not informatively classified. With these criticisms
in mind and with the present wide availability of superior
hierarchical clustering programs, there remains little
value in the continued use of the recurrent group analysis
and it is best considered obsolete.
Other Methods -
Various other serially optimized non-hierarchical methods
have been proposed, some of which operate on a resemblance
matrix and some of which do not involve the computation of
the entire matrix. Some methods are reviewed by Lance and
Williams (1967c) and Anderberg (1973).
Simultaneously Optimized Methods
These are of basically two types: those which operate on
an inter-entity resemblance matrix and those which operate
on subsets of entities and involve prior declaration of
the number of groups sought (Lance and Williams 1967c).
Simultaneously optimized methods generally proceed by first
partitioning the entities in some way and optimizing groups
by an iterative process of reallocation. Methods are
43
-------
PYGMY POACHER
CURLFINSOLE
NORTHERN ANCHOVY
0.50
SHALLOW
WATER
0.25
0.33
GROUP 5 (59 SAMPLES)
YELLOWCHINSCULPIN
LONGSPINE COMBFISH
o
in
ci
I^^H
GROUP 2 (49 SAMPLES)
SPECKLED SANDDAB
CALIF. TONGUEFISH
HORNYHEADTURBOT
ENGLISH SOLE
0.50
MID-DEPTH
DEEP
WATER
GROUP 1 (40 SAMPLES)
PACIFIC SANDDAB
DOVER SOLE
PLAINFIN MIDSHIPMAN
PINKSEAPERCH
SHORTSPINE COMBFISH
0.60
0.40
GROUP 4 (64 SAMPLES)
SLENDER SOLE
REX SOLE
0.50
0.50
GROUPS (19 SAMPLES)
WHITE CROAKER
QUEENFISH
WHITESEAPERCH
NOTE
GROUPS IDENTIFIED BY RECURRENT
GROUP ANALYSIS (AFFINITY INDEX
= 0.50). OTHER ASSOCIATIONS (INDI-
CATED BY CONNECTING LINES)
DETERMINED BY CONNEX ANALYSIS
STRIPETAIL ROCKFISH
BLACKBELLY EELPOUT
BLACKTIP POACHER
Fig. 6. Species groups of demersal fishes on the Southern
California shelf as defined by recurrent group
analysis using the Fager binary similarity
coefficient (from Mearns 1974).
-------
reviewed by Lance and Williams (1967c) and Anderberg
(1973).
Although non-hierarchical clustering methods offer the
attractive promise of optimization of within-group homo-
geneity, in practice the available techniques either have
serious drawbacks in performance, are limited in the types
of data or resemblance measures with which they can be
used, or are computationally difficult. Consequently, it
is recommended that the practicing ecologist use hier^-
archical methods and avoid, at least for the time being,
non-hierarchical clustering.
AGGLOMERATIVE HIERARCHICAL METHODS
Agglomerative hierarchical clustering strategies all
operate by an iterative process of fusing pairs of entities,
then pairs of groups of entities until the total population
is fused. During each fusion cycle that pair of entities
or groups most similar (or least dissimilar or distant) are
joined and new resemblances determined between the new group
and the all remaining entities and groups. Combinatorial
strategies allow the new resemblances to be computed from
the preceding resemblance matrix, while with non-combina-
torial methods, the original data matrix must be used in
the computation of the new resemblances.
Combinatorial Methods
Lance and Williams (1966, 1967a) showed that for a variety
of combinatorial strategies group/group or group/individual
resemblances can be computed by variants of a single linear
equation. The problem of defining these new resemblances
when entities or groups are fused is geometrically illu-
strated in Fig. 7. Two groups i and j are fused to form
group k, what then is the resemblance of group k to another
group, group h? Given the resemblances (expressed as dis-
similarity or distance) Dhi, Dj. and D^j , what is Dhk? The
Lance-Williams combinatorial solution is
Dhk = ai Dhi + aj Dhj + 0 DIJ + Y lDhi-Dhjl' (31)
where the parameters a-, a., 3 and y determine the nature
of the strategy. 1 3
45
-------
GROUP /
Enlilies)
GROUP h
(N Entities
A
GROUP k
( Ni +N: Entities )
GROUP j
(N Entities)
s . — C*'
hk i
ij
V hj
Fig. 7. Lance - Williams combinatorial computation of
distance between Group h and a new group, Group k,
formed by the fusion of Groups i and ^,
-------
Values of these parameters for the common combinatorial
strategies are listed in Table 2 and the strategies are
further discussed below.
Single Linkage -
In this clustering method, also referred to as "nearest
neighbor" clustering, the resemblance between two groups
is defined as the resemblance of their most similar enti-
ties, one in each group. If the resemblances are repre-
sented by distances as in Fig. 7 it can be seen that as a
group grows it must appear to move closer to some groups
or entities, but further from none, thus it is a space
contracting strategy. As a result single linkage
clustering produces excessive chaining in the hierarchical
clustering route, in which entities are fused to a few
nuclear groups one at a time rather than forming new
groups. This results in classifications in which many
entities are not effectively clustered but must be con-
sidered as individuals. A good example of an extensively
chained, single-linkage agglomeration is given in Fig. 8,
which shows a classification of marine phytoplankton
collections from the Indian Ocean. Note that the class-
ification of large numbers of collections is indeterminate
because of excessive chaining.
Jardine and Sibson (1968, Sibson 1971) defined a set of
theoretical conditions which should be met by a hier-
archical clustering method which would virtually confine
one to single-linkage clustering. However, many authors
(Williams et al. 1971, Pritchard and Anderson 1971,
Cunningham and Ogilvie 1972) have pointed out the severe
shortcomings of single-linkage clustering.
In aquatic ecology single-linkage clustering has been used
principally by British marine biologists (Field and
MacFarlane 1968, Thorrington-Smith 1971, Angel and Fasham
1973). However, because of restrictions to its utility,
the use of single-linkage clustering is not recommended.
Complete Linkage -
This method, also called furthest-neighbor clustering, is
the exact opposite of single linkage clustering in that the
resemblance between two groups is defined as the resem-
blance of their least similar entities, one in each group.
As a group grows it will recede from some groups or enti-
ties but become nearer to none, thus it is a space dilating
strategy. Whereas single-linkage agglomeration results in
47
-------
*>.
CO
Table 2. VALUES OF PARAMETERS OF THE LANCE AND WILLIAMS (19 67 a)
LINEAR SOLUTION (EQUATION 31) FOR COMPUTATION OF INTER-
GROUP RESEMBLANCE FOR EIGHT COMBINATORIAL CLUSTERING
AND n. ARE THE NUMBERS OF ENTITIES
NUMBER
OF ENTITIES IN GROUP k RESULTING FROM THE FUSION OF i
METHODS, WHERE nh, r^ AND n. ARE THE NUMBERS OF E
IN GROUPS h, i AND j, RESPECTIVELY, AND nk IS THE
AND j
Method
Single linkage
(Nearest neighbor)
Complete linkage
(Furthest neighbor)
Group average
(UPGMA)
Simple average
(WPGMA)
Centroid*
(Unweighted
centroid)
Median
(Weighted centroid)
Flexible
(i.e. nk
ai
1/2
1/2
n±/nk
1/2
ni/nk
1/2
(l-3)/2
= n± + n-j) .
oj 3
1/2 0
1/2 0
nj/nk °
1/2 0
nj/nk -aiaj
1/2 -1/4
(l-3)/2 1
Y
-1/2
1/2
0
0
0
0
0
Space distortion
contracting
dilating
conserving
conserving
conserving
conserving
B=0 , conserving
B>0, contracting
B<0, dilating
-------
Table 2.(continued). VALUES OF PARAMETERS OF THE LANCE AND WILLIAMS
(1967a) LINEAR SOLUTION (EQUATION 31).
Space
Method a^ a. g y Distortion
Incremental sums
of squares ^nh+ni^ / ^nh+nk^ ^nh+nj ^ / ^nh+nk^ ~nh^nh+n^ ° dilating
* Centroid method combinatorial only for squared Euclidean distance.
I-D
-------
50°E
60°E
70°E
-§ 0.50
o>
o
o
LO
£
Q.
0.40
0.30
Stations (54
-];|:^1
'',;>:,-"r-'-.V-T
? v- "?jr''&:'-a
y^"t>j; •*'*>•:
~ \ i* .1^ .•'".. - »•
*' i->
.?:>V
:-l;-,!;
e;.-,!..--l,-^
) 90 62 87 21 67 23 50 80 73 31 52
93 25 99 79 59 33 54 82 04 06
Fig. 8. Dendrogram from a single-linkage clustering showing
excessive chaining. Example is from Thorrington-
Smith's (1971) study of phytoplankton assemblages
in the Indian Ocean off Madagascar.
-50-
-------
chaining, complete-linkage agglomeration typically results
in intense clustering by forming discrete groups.
Although intense clustering is often a desirable property,
one often desires a cluster intensity intermediate between
that of single- and complete linkage. Furthermore, it is
desirable to base inter-group resemblance on more infor-
mation than just maximum or minimum resemblance between
entities in the two groups. For these reasons, the com-
binatorial strategies yet to be discussed are generally
preferred.
Group Average -
In this method inter-group resemblance is defined as the
mean of all resemblances between members of one group to
members of another. This solution is widely referred to
by numerical taxonomists and American biologists as the
"unweighted pair-group method using arithmetic averages"
or UPGMA (Sneath and Sokal 1973). Group average clustering
has no marked tendencies to space contraction or dilation,
and thus may be regarded as space conserving. Hence, it
produces only moderately sharp clustering but introduces
relatively little distortion to the relationships orig-
inally expressed in the inter-entity resemblance matrix
(Cunningham and Ogilvie 1972).
Group average agglomeration is now the most widely used
clustering method in ecology and it has been extensively
employed in aquatic ecology. A non-exhaustive list of
applications include: Boesch (1973), Bowman (1971),
Cairns and Kaesler (1969, 1971), Cairns, Kaesler and
Patrick (1970) , Grossman, Kaesler and Cairns (1974) , Day
et al. (1971), Eagle (1973, 1975), Ebeling et al. (1970),
Field (1970, 1971), Gage (1974), Kaesler and Cairns (1972),
Kaesler, Cairns and Bates (1971) , Kay and Knights (1975) ,
Loya (1972) , Jones (1969) , Roback, Cairns and Kaesler
(1969), Santos and Simon (1974), Stephenson et al. (1972),
Ward (1973), and Warwick and Gage (1975).
Simple Average -
This method is equivalent to Sneath and Sokal's (1973)
"weighted pair-group method using arithmetic averages"
or WPGMA and it differs from group average clustering by
weighting the entities most recently admitted to a group
equal with all previous members. In practice the results
of simple average agglomeration are quite similar to those
produced by group average clustering. The method is space-
conserving and introduces slightly more distortion to the
51
-------
actual resemblances than does the group average method.
Centroid -
In this strategy entities are considered defined as points
in Euclidean space and when grouped defined by the co-
ordinates of the centroid, or geometric center of the
points in the group. Centroid clustering is combinatorial
only for squared Euclidean distance and variance-covariance
used as resemblance measures. For correlation coefficients
and similarity measures the original data must be retained
for computation of centroids.
Centroid clustering is space conserving but is somewhat
prone to distortion (Cunningham and Ogilvie 1972). It
suffers from a particular problem in that reversals in the
agglomeration may be produced. That is, two groups may
fuse at a given level of resemblance and may be subse-
quently fused with a third group at a higher level of
resemblance than the first fusion (for example see Fig.
5-9 in Sneath and Sokal (1973) and Fig. 8.4 in Clifford
and Stephenson (1975). Largely for this reason centroid
clustering has been recently disfavored. It has been
employed in marine ecology by Popham and Ellis (1971) ,
Colijn and Koeman (1975) and Van den Hoek et al. (1975) .
Median -
This method, referred to by Sneath and Sokal (1973) as
"weighted centroid," weights fused groups as co-equal
despite differences in sizes of the groups in a similar
fashion as the simple average method. Thus the centroid
of the fused group is the centroid of the centroids of the
precursor groups. ' Its properties are more or less similar
to that of the centroid method, including the lack of
monotonicity of the sequence of fusion levels which
results in reversals.
Flexible -
The development of a linear equation for inter-group dis-
tance in combinatorial clustering strategies allows the
use of continuously variables coefficients in the equation,
effectively creating an infinite number of clustering
strategies. Lance and Williams (1967a) proposed a flexible
strategy based on Equation (31) with the following con-
straints (&j_ + ctj + 3 = 1; ot-j_ = a4; 3<1; Y = 0) . By
varying 3 (the cluster intensity coefficient) one can
purposefully cause space distortion, as 3 increases from
0 the strategy is space-contracting and as 3 decreases from
52
-------
0 it is space dilating. Fig. 9 shows the effect of varying
on the clustering of entities defined by the same resem-
blance matrix.
Flexible sorting with 3 = -0.25 has produced satisfactory
results with a wide range of data sets and this value has
become more or less conventional (Williams 1971, Clifford
and Stephenson 1975). At this level of 3, flexible
clustering is an intensely clustering, moderately space-
dilating strategy. In practical terms, this means that
as agglomerations are made, there is a bias against an
entity or group joining an already large group and a bias
in favor of entities or small groups joining to form
separate branches of the hierarchy, i.e. it is group-size
dependent. It is important to keep in mind, however, that
3 can be varied to simulate any level of cluster intensity,
although there is little point in using 8>0.
Flexible sorting has been criticized on the grounds that
objectivity is lost if that cluster intensity is chosen
which most closely fits preconceptions about the data
(Sneath and Sokal 1973). However, the use of a variably
space-dilating strategy seems sensible in some ecological
contexts. For example, a common feature of many^ecological
data sets is high resemblance among the common or abundant
species and much lower resemblance among the rarer species.
It seems reasonable to accept a significantly lower resem-
blance between rare species than between common ones, and
in practice intense flexible sorting often compensates for
this discrepancy by forming groups of rare species which
would be chained on to larger nuclear groups in space-
conserving clustering. Intense clustering strategies are
often prone to misclassifications and one often has to
choose between non-classifications due to weakly clustering
strategies or misclassifications due to intensely clustering
strategies. The best approach depends on the data set, but
with large data sets, especially in inverse analyses, it is
often better to use an intensely clustering strategy fol-
lowed by reallocation of misclassified entities.
Marine ecological applications of flexible clustering
include Stephenson et al. (1970, 1972, 1974), Stephenson
and Williams (1971), Williams and Stephenson (1973) and
Boesch (1973). Another enlightening application of
flexible clustering was by Williams et al. (1973) in a
study of pattern in rain-forests.
53
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(3 = +0.
0=+0.50
= -0.25
P=-0.50
n
ml
Fig. 9- Effect on agglomerative hierarchy of varying the
cluster intensity coefficient in flexible clustering,
(after Lance and Williams 1967a) .
54
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Incremental Sums of Squares -
It can be seen in Fig. 4 that squared Euclidean distance-
CD2) is an additive function, as is variance and total
information content. Several authors (Ward 1963, Orloci
1967, Burr 1970) have proposed clustering methods in which
entities or groups are successively agglomerated so that
fusion will cause the smallest possible increment in the
sums of squares of Euclidean distances. Burr (1970) showed
that this strategy is combinatorial using the constants as
listed in Table 2. The strategy can also be applied to
other distance measures, but in practice is usually used
with Euclidean distance or standardized Euclidean distance
measures.
The incremental sums of squares strategy is an intensely
clustering, group-size dependent method. Thus the tech-
nique is powerful in imposing structure in relatively
patternless data, but, like other space-dilating strategies,
is prone to misclassification and may produce clusters of
entities which have relatively little in common except their
paucity of attributes.
This strategy has been applied by Hughes and Thomas (1971a,
b), Hughes et al. (1972), Polgar (1975) and Holland and
Dean (1976) in studies of marine and estuarine benthic
communities.
Non-combinatorial Methods
In general, non-combinatorial clustering methods have a
serious drawback in efficiency because the original data
must be retained for computation of resemblance matrices
after each fusion cycle. Thus they are likely to be
impractical for large data sets.
Centroid -
As noted earlier, centroid clustering is combinatorial only
for squared Euclidean distance. For other distance mea-
sures, distances between the centroid of a newly formed
group and the centroid of another group must be recal-
culated based on the average scores of all attributes for
the two groups. Since centroid distances are mainly useful
only for Euclidean metrics (for which a combinatorial
solution is available) and because of the drawbacks of
centroid clustering mentioned above, centroid methods are
seldom used for non-Euclidean distances.
55
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Information Gain -
As mentioned in the discussion of information content
measures in Section V, agglomerative clustering can be
based on the minimum information gain in fusion of entities
or groups. Methods exist for clustering based on binary
or multistate data (Williams, Lambert and Lance 1966) and
quantitative data (Lance and Williams 1967b, Orloci 1969,
1971, 1975, Dale, Lance and Albrecht 1971). Information
gain agglomeration is an intensely clustering strategy and
thus suffers attendant disadvantages.
The application of agglomerative information-gain clustering
to plant ecological problems has been thoroughly reviewed
by Dale (1971), Dale and Anderson (1972) ' and Williams et
al. (1973). Stephenson and Williams (1971) attempted
information-gain agglomeration in a study of marine benthos
using both quantitative and binary data but were dissat-
isfied with their results. Similar techniques were applied
by Jeffrey and Carpenter (1974) on ranked abundance data
in a study of seasonal succession of coastal phytoplankton.
DIVISIVE HIERARCHICAL METHODS
Divisive clustering methods offer the obvious advantage of
starting with the whole, when total information available
is maximum, and then subdividing along natural breaks in
the whole data set. In practice, though, the most devel-
oped and practical divisive methods are monothetic, i.e.
divisions are based on the presence or absence of one
attribute, and are thus of limited usefulness with most
ecological data sets.
Monothetic Methods
Association Analysis -
With this method the divisions are made on the basis of
chi2 values summed over the attributes, such that the
attribute with the greatest contribution to chi2 is used
as a basis of division of each successive set into two
groups, one of the entities possessing the attribute and
the other lacking it (Williams and Lambert 1959, 1961a,
Lance and Williams 1968).
Moore (1973) used both normal and inverse association
analysis in a study of communities associated with kelp
56
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holdfasts and the effects of turbidity and pollution
thereon. Although extensively used by plant ecologists
through the early 1960's, association analysis has been
more or less displaced by monothetic divisive methods
based on information content.
Information Content -
Entities are successively divided on the basis of presence
or absence of attributes when such divisions result in the
maximal reduction in information (Lambert and Williams
1966, Lance and Williams 1968). That is, if Ic is the
information content of the total population to be divided,
and if Ia and 1^ are two monothetically determined subsets,
then the value Ic - (Ia + I]-,) is maximized during each
fusion cycle. A similar divisive method capable of using
multistate and continuous data in addition to binary data
has been developed by Lance and Williams (1971).
Monothetic divisive information clustering using the
Australian CSIRO program DIVINF has been applied to marine
ecological problems by Stephenson et al. (1971, 1972),
Moore (1973) and Jeffrey and Carpenter (1974).
Polythetic Methods
Polythetic divisive clustering methods are theoretically
the optimal hierarchical strategies (Williams 1971).
Unfortunately, their development has lagged due to the
computational difficulties arising from the very large
(2n-l -1) number of dichotomous splits for each subdi-
vision. For example, Edwards and Cavalli-Sforza (1965)
proposed a method by which divisions are made so that the
sum of squares of Euclidean distances between subdivisions
is maximum. But for hierarchical division of more than 16
entities the computation time is indeed enormous (Gower
1967). Recently, several new polythetic divisive methods
have been proposed which are based on short-cut solutions
to the problems of examining all possible subdivisions in
fission sequence. These are of two main types: those
which base subdivisions on an ordination model, and those
involving some form of directed search to limit the number
of splits which must be examined.
57
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Methods Based on Ordination -
Lambert (1971) and Noy-Meir (1973b) developed methods of
optimized division based on principal components scores
followed by various iterations to further optimize the
divisive structure of the classifications. With the use
of principal components analysis the inter-entity resem-
blances are Euclidean distances. However, the use of
other ordination methods allows resemblances to be expressed
by other distance measures (e.g. similarity coefficients).
Another divisive classificatory method based on an ordi-
nation model has been proposed by Hill, Bunce and Shaw
(1975) under the name "indicator species analysis." The
method proceeds by ordinating entities using correspondence
(reciprocal averaging) ordination, and then successively
divides the population of entities based on scores of a
few indicator attributes which are most responsible for
the ordination structure.
Polythetic divisive classifications based on ordination
models show a great deal of promise, but it is too early
to say which of the proposed methods is best. However, all
such methods may be subject to some of the disadvantages of
ordination approaches. Furthermore, all of the extant
methods seem to have a bias toward forming subdivisions
of approximately equivalent size during each division, when
arrays of ecological entities are often not so symmetrical.
Methods Based on Directed Search -
Lambert et al. (1973, Smartt, Meacock and Lambert 1974)
developed two methods for polythetic divisive classifi-
cation which seek to form a preliminary split in the
population of entities and then examine the robustness
of that split by iterative examinations. In AXOR,»the
initial strategy is to extract principal component or
principal coordinate axes and then investigate all n-1
"ordered" splits on the axis to find the best division.
Improvements in the split are then made by relocation of
entities one at a time in the second and subsequent axes
of the ordination until the consideration of a new axis
gives no further improvement. In MONIT, the population
is first split monothetically and improvements in the
split are made by relocation of individuals one at a time
until further iterations produce no improvement. Lambert
et al. (1973) and Smartt et al. (1974) report consistently
better results with these strategies than with various mono-
thetic divisive and polythetic agglomerative strategies.
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SECTION VII
INTERPRETATION OF NUMERICAL CLASSIFICATIONS
Frequently ecological applications of numerical classi-
fication end with the definition of clusters or classi-
ficatory hierarchies or, at best, with a brief description
of the relationship of the classification to spatial or
temporal distributions of collections or species. However,
numerical classification is best viewed as a method for
simplifying complex data sets, allowing ecological analysis
to proceed more efficiently, rather than as an end in
itself. Furthermore, numerical classifications should
be viewed critically. The various clustering algorithms
discussed in the preceding sections are only, algebraic
approximations of ecological criteria for classification
and the great variety of methods available produce variable
results. Thus, further refinements of the objectively
produced classifications are frequently needed.
STOPPING RULES
A common problem in the interpretation of hierarchical
classifications is the determination of operational groups
within the hierarchy. If we consider the results of a
hierarchical classification as a dendrogram, the question
is which branches are considered groups with reasonable
internal resemblance. Frequently, investigators have
drawn a line across the dendrogram at some given level
of resemblance and stipulated that each branch crossing
that line represents a group. Thus, the "stopping rule"
is fixed. The fixed resemblance level may be independently
determined based on some assumed level of "significance"
or it may be based on the resemblance level at which a
given number of branches exist in the dendrogram.
Alternately, other investigators have used a variable
59
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stopping rule for definition of operational groups.
Usually this involves studying the dendrogram, often
in consultation with the original data matrix or two-way
tables, to determine "reasonable" groups. Thus, one may
specify two groups which cluster together at a higher
level of resemblance than that found within a third group.
Although application of a fixed stopping rule obviously
lessens subjectivity in interpretation of the classifi-
cation, there are two characteristics which suggest
variable stopping rules are often more appropriate. The
first concerns the group size dependence and space distor-
tion properties of some hierarchical clustering methods.
There is little justification for a stopping rule of fixed
resemblance when inter-group and entity-group resemblance
depends on the size of the group. Thus, there seems little
sense in applying a fixed stopping rule with classifi-
cations formed by intensely clustering methods such as
incremental sums of squares and flexible (with negative 3)
clustering. The second characteristic concerns the nature
of ecological data and.is particularly important in inverse
analyses. Most data sets include species which are more or
less ubiquitous and species which are much more rare. It
seems reasonable to require higher intra-group resemblance
in groups of ubiquitous species than in groups of rare
species for which the probability of cooccurrence is low.
A parallel problem to the definition of stopping rules in
hierarchical classification is the definition of required
intra-group homogeneity with some non-hierarchical methods.
For example, Pager's (1957) recurrent group analysis
requires the setting of a minimum resemblance level that
an entity must have with all members of a group to be
included in that group. Varying this minimum resemblance
level can severely affect the classification which is
produced (Jones 1969) . Fager and McGowan (1963) fcfund
that a minimum value of 0.5 using the Fager similarity
coefficient for binary data produced satisfactory results,
but efficacy is dependent on the resemblance measure used
and the nature of the data. Selection of a fixed level of
intra-group homogeneity is also subject to the same criti-
cism as fixed stopping-rules for hierarchical clustering,
namely that intuitive ecological criteria for grouping are
not necessarily fixed.
60
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REALLOCATION
A "misclassification" occurs when an entity is placed in
one group by a numerical classification when it would
improve within-group homogeneity if it were placed in
another. With many non-hierarchical clustering methods
misclassification problems are minimal because homogeneity
is implicitly optimized. Rather, the difficulties with
non-hierarchical methods are often quite the opposite --
entities are not effectively classified rather than
misclassified.
However, misclassifications are frequent problems in hier-
archical clustering. They can occur in divisive clustering
because similar entities may be separated in early divi-
sions or in agglomerative clustering where an entity may
resemble only one member of a group in which it is included
because of an early fusion. As discussed earlier, misclass-
ifications are most frequent in space-dilating hierarchical
methods, which tend otherwise to be particularly useful
with complex ecological data.
In the case of misclassifications, reallocation of entities
from one group to another is appropriate. Although many
ecologists using numerical classification have noted obvious
misclassifications, relatively few have attempted reallo-
cation of entities. Subjective reallocation nullifying the
objectivity of the analysis has apparently troubled many
users. On the other hand, rather casual reallocation of
entities to conform to preconceptions extrinsic environ-
mental characteristics, or visual inspection of the data
or resemblance matrices has sometimes been practiced
(Stephenson et al. 1972, Boesch 1973, Clifford and
Stephenson 1975).
One may be able to detect misclassifications by examination
of the inter-entity resemblance matrix to uncover the
presence of entities which have average resemblance to
another group higher than that in which it has been
classified.
Another convenient way to detect misclassifications is to
rearrange the original data matrix both by collection and
species groups as determined by normal and inverse numerical
classifications, respectively. This allows the examination
of the concentration of species occurrence or abundance
within the "cells," or coincidences of collection groups
with species groups, in this "two-way coincidence table"
61
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(Clifford and Stephenson 1975). Interpretive analyses
based on the coincidence of collection and species groups
are discussed below under the heading of "nodal analysis."
One can then reallocate collections or species by appro-
priately adjusting the rows and columns of the two-way
coincidence table to sharpen the classifications by
increasing the "cell density" of scores. This is equiv-
alent to the elaborate table rearrangements used in the
Braun-Blanquet or Zurich-Montpellier approach of European
plant ecologists (Westhoff and van der Maarel 1973,
Mueller-Dombois and Ellenberg 1974, Popham and Ellis 1971).
Although reallocation based on the two-way coincidence
table has usually been done visually, objective criteria
for reallocation can also be employed. For example, Boesch
(1973) reallocated species if the average constancy of
species in a group within collection groups could be
increased in dense cells or decreased in sparse cells. He
further reallocated some species based on an interpretation
of an ordination model. Even then, some discretion is
involved. Ceska and Roemer (1971) proposed an objective
and automated technique for the rearrangement of two-way
tables which shows some promise for future application in
reallocation.
The development of better and more objective reallocation
methods is sorely needed. Lance and Williams (1967c)
attractively suggest that simultaneously-optimized non-
hierarchical clustering serve to reallocate entities in
groups based on an initial hierarchical clustering. Grigal
and Ohmann (1975) used multiple discriminant analysis
(referred to as a canonical analysis) in the reallocation
of entities into groups in order to resolve differences
among four different classifications of the data set.
However, most suggested procedures for reallocation do
not allow the use of the same resemblance function*that
was initially used to classify the entities.
NODAL ANALYSES
Most investigators who have applied numerical classifi-
cation to aquatic ecological problems have classified
sites (i.e. normal analysis) only. A few investigators,
particularly those applying "recurrent group analysis,"
have classified species (i.e. inverse analysis) only.
Relatively few have conducted both normal and inverse
62
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analysis of the same data set (e.g. Stephenson and Williams
1971, Stephenson et al. 1972, 1974, Hughes and Thomas
1971a, b, Boesch 1973, Moore 1973, Sepkoski and Rex 1974).
Relating normal and inverse classifications greatly enhances
the interpretation of the results of numerical classifi-
cation and is recommended as a routine post-clustering
analysis.
Normal-inverse coincidences can be conveniently examined in
a two-way table (Clifford and Stephenson 1975) which is
simply the original data matrix rearranged by collection
and species groups. As described above, two-way tables
are most helpful in identifying misclassifications and in
assisting in reallocation. But beyond that they are
extremely useful in assisting ecological interpretation
of the classifications. Differences among collection
groups can be conveniently described on the basis of
frequency or abundance of members of the species groups.
Conversely differences in the distribution patterns of
species groups can be elucidated by the relative frequency
or abundance of the species in the various collection
groups.
Williams and Lambert (1961b, Lambert and Williams 1962)
termed this approach "nodal analysis" since one attempts
to describe and interpret the dense cells or "nodes" of
the data matrix in which a group of species and group of
collections coincide. This concept of nodal analysis is
further expanded by Noy-Mier (1971) who developed pro-
cedures for the inter-relationship of normal and inverse
ordinations.
Further nodal analysis interpretations can be made in
expression of the degree of collection group and species
group coincidence by using the classic ecological concepts
of dominance, constancy and fidelity (Fager 1963, Westhoff
and van der Maarel 1973). Stephenson et al. (1972) and
Boesch (1973) expressed the pattern'of constancy of species
belonging to particular species groups in particular
collection groups as relative densities of cells of the
two-way table. Constancy was arbitrarily graded as high,
medium, low, etc. based on percentages or proportions of
the number of occurrences of species in the collection
group to the total possible number of such occurrences.
Algebraically this constancy index can be expressed as
63
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(32)
where a-ji is the actual number of occurrences of members of
species group i in collection group j and the n^ and nj are
the numbers of entities in the respective groups. The
index will take a value of 1 when all species occurred in
all collections in the group and 0 when none of the species
occurred in the collections.
Fig. 10 gives an example of a nodal constancy diagram which
also shows the hierarchical relationships of the collection
(site) and species groups. The underlying reasons for the
classifications and the relationships of the groups are
clearly apparent in terms of the patterns of species group
constancy. The analysis was based on a data set repre-
senting the abundances of 68 species of macrobenthic
animals collected from 47 sites on the shallow continental
shelf off Virginia (D. F. Boesch, in prep.). The site
classification strongly reflects substrate differences
among the sites with groups A, B and C consisting of muddy-
sand sites, group D consisting of hard-packed fine sand
sites and groups E, F, G and H consisting of the coarser
sand sites. The nodal constancy patterns conveniently
demonstrate the faunal differences between collection
groups. One can see, for example, that both the muddier
sites (groups A and B) as well as the sites with coarser
sediments (groups F, G and H) are characterized by species
which are constant there and not elsewhere, but that the
sites with intermediate sediment grain size are character-
ized by species (e.g., groups 5 and 6) which, while highly
constant at those sites, are widely distributed with
respect to sediment type.
Similarily, one can examine the fidelity of species groups
to collection groups in order to give an indication of
the degree to which species "select" or are limited to
collection types (habitats, seasons or whatever). A simple
index of fidelity is an expression of the constancy of
species in a collection group compared to the constancy
over all collections . Thus, the fidelity of species group
i in collection group j can be defined as
64
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cr\
01
SPECIES GROUPS
Fig. 10.
CONSTANCY
7 8 9IOIMS2
in a two-way table of species
Nodal constancy in * u£rom an analysis of
g^SS"*111*
- °'7
Very High
@ - °'5
High
WL - °'3
Moderate
> o.
Low
< 0.
Very Low
-------
using the same terms as in the constancy index. This index
is unity when the constancy of a species group in a site
group is equivalent to its overall constancy, greater than
1 when its constancy in that collection group is greater
than that overall and less than 1 when its constancy is
less than its overall constancy. Values of the index
greater than 2 suggest strong "preference" of species in
a group for a collection group, i.e. indicating that the
average frequency of occurrence of those species in those
collections is twice what it is considering all collections.
Values of the index much less than 1 suggest "avoidance" of
the spatial or temporal habitats represented by the collec-
tion group or negative fidelity.
Fig. 11 shows nodal fidelity patterns for the same two-way
table as in Fig. 10. Note that some species groups (e.g.
5 and 6), although highly constant in some collection
groups, are not very faithful in any. Also some species
groups (e.g. 3), although not highly constant in any col-
lection group, are highly faithful to some groups.
Using quantitative data one can also express the concen-
tration of abundance of species in the collection groups.
For each species the average abundance in the collection
group is divided by its average abundance overall. These
ratios can be averaged over all species in the species
group to reflect the average concentration of abundance
for the node.
Alternate approaches have been taken by Stephenson and his
associates (Williams and Stephenson 1973, Stephenson et al.
1974, Clifford and Stephenson 1975) in relating species
distribution patterns to collection groups. He has used
various tests of "conformity" of individual species to
collection groups. In this sense conformity is analagous
to fidelity or concentration of abundance, as used above.
Species conformity can be tested probabilistically using
F-tests or non-parametric tests of significance. The
contribution of a species to the collection classification
can then be described in terms of its conformity and
importance (i.e. relative abundance).
If the nodal constancy diagram is drawn with the width of
the rows and columns proportional to the number of entities
in the respective collection and species groups as in Fig.
10 the diagram is also useful in explaining gross differ-
ences in the species richness of collections. For example,
it is clear from Fig. 10 that collections in site groups F,
66
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SPECIES GROUPS
7 8 9 10 II 12
FIDELITY
-
High
Moderate
Low
a
-------
G and H generally contain fewer species than collections in
group C. Furthermore, it is possible to directly explain
these differences in terms of species composition of the
collections and patterns of species distribution. Boesch
(1973) used such an interpretation of a nodal constancy
table to explain patterns of species richness in estuarine
benthos in a polluted harbor.
COMPARING CLASSIFICATIONS
It is frequently useful to apply several clustering algo-
rithms to the same data set and compare the results of the
alternate classifications. This is helpful not only in
selecting the most appropriate classification, but in
interpretation of the nature of the patterns exhibited in
the data, e.g. qualitative versus quantitative patterns.
In addition to simple comparisons by subjective visual
examination, a variety of quantitative methods have been
proposed to measure the congruence of two or more classi-
fications. Rohlf (1974) reviewed the methods of comparing
classifications so comprehensively that further elaboration
here is not necessary. Sufffice it to say,.that for the
more common hierarchical classifications most methods
involve correlating matrices either of the original resem-
blance measures or of new resemblances based on separation
of entities in the classificatory hierarchy.
TESTING DIFFERENCES AMONG GROUPS
For certain purposes it may be desirable to test the
reality of the groupings of a classification by application
of tests of significant differences among the groups.
Statistical techniques for this purpose have not been
extensively developed but several different approaches
have been used.
As discussed above, Stephenson and his associates have
variously tested the conformity of species to normal
classifications. However, these are essentially uni-
variate tests and do not constitute tests of differences
among either species or collection groups.
Field (1969) proposed a test of differences between
clusters based on the information gained by each fusion
68
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in agglomerative clustering. A transform of the infor-
mation gain, 2 I, is tested for significance with the
degrees of freedom based on the number of attributes
possessed by the entities or groups being fused. Unfor-
tunately, as proposed, the test is limited to comparisons
of binary attributes, considerably reducing the usefulness
of the test. The information gain statistic has been used
as a test of internal homogeneity of classified groups by
Field (1971), Day et al. (1971) and Santos and Simon (1974).
In the previous section on correlation coefficients I
outlined several reasons why they were inappropriate for
expressing significant relationships among classified
groups. Nonetheless, correlation tests have been fre-
quently applied in this manner.
Mountford (1971) developed a probability model describing
the joint distribution of resemblance measures which allows
a conservative test of significance of clusters defined by
internal criteria. It appears not to have been applied
subsequently, thus it is difficult to assess its usefulness,
Mountford's model, as in the case of others, predicts that
the resemblance measure, or a transform of the measure, is
normally distributed. However, the sampling distributions
of most measures are unknown and Mountford concluded that
his test is more readily applicable to indices of simi-
larity constructed according to probability considerations.
The use of multiple discriminant analysis (Cooley and
Lohnes 1971) in the test of significance of resemblance
among groups of entities shows some promise (Goldstein
and Grigal 1972b, Grigal and Ohmann 1975, Polgar 1975).
When groups are compared by discriminant analysis, the
between-groups sums of squares are maximized with respect
to the within-groups sums of squares. The maximization
procedure extracts canonical axes onto which each entity
can be mapped as a point. The distances among entities
can be computed and tested for significance. However
applications of the tests do require certain assumptions
about the data (e.g. homogeneity of variance, independence
of variables and equality of group size) which may not be
met by the data.
From this discussion it is clear that further research is
needed on the sampling distributions of resemblance mea-
sures and on tests of significance among clusters. Numer-
ical classification methods are hypothesis generating
69
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rather than hypothesis testing techniques. They provide
hypothetical generalizations on the structure of multi-
variate data. Testing the constructed hypotheses could
be greatly assisted by the availability of non-parametric
multivariate tests of significance which impose few
assumptions about the nature of the data or the resemblance
measures.
RELATING CLASSIFICATIONS TO EXTRINSIC FACTORS
Ways to relate numerical classifications to extrinsic
factors (abiotic environmental variables, etc.) are limited
only by the imagination of the investigator. Approaches
range from plotting the distribution of site (collection)
or species groups on maps of the sampling area to statis-
tical comparison of the extrinsic factors corresponding to
the groups. With regard to statistical analyses, non-
parametric tests may be more appropriate than parametric
tests because of problems regarding homogeneity of variance
and unequal group size. Extrinsic variables are usually
individually related to the classification but multivariate
techniques of canonical correlation and multiple regression
may be useful (Dagnelie 1971).
One approach which has been only infrequently used is to
independently classify or ordinate collections based solely
on their associated abiotic factors. The abiotic factor
classification of collections can then be compared to the
biotic intrinsic classification. Smith (1973), in a study
of benthos along a transect in the vicinity of waste dis-
charges, ordinated sites on the basis of water and sediment
quality parameters and plotted the distribution of species
groups (as determined by a numerical classification) on
this ordination.
A frequent problem in the application of numerical classifi-
cations in ecology is in the analysis of collections taken
over both space and time, e.g. from a series of sites which
are sampled seasonally. Several approaches have been used
to classify such collections. Some investigators have
chosen to classify the collections from each season sepa-
rately (Field 1971), while others classified the combined
temporal collections for each site to elucidate spatial
patterns and the combined collections made during each
sampling period to elucidate temporal patterns (Jones
1973, Stephenson et al. 1974, Raphael 1974). Boesch (1973)
70
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classified the entire set of collections from sites sampled
over time and felt this approach allowed a better under-
standing of the spatial-temporal interactions which were
important in his study.
Williams and Stephenson (1973) developed a technique for
the classificatory analysis of the three dimensional data
matrix (sites x species x sampling periods). Using
Euclidean distance as a resemblance measure, one is able
to partition squared distance (as with variance in an
analysis of variance) to produce site/species classifi-
cations eliminating the effect of temporal changes, and
sampling period/species classifications eliminating the
effect of spatial differences. Similarly, this method
allows the classification of species based solely on either
their spatial patterns of occurrence or on their temporal
patterns. One is also able to judge the relative impor-
tance of spatial patterns, temporal patterns and spatial-
temporal interactions. The method is further discussed by
Stephenson et al. (1974) and Clifford and Stephenson
(1975).
INTERFACING CLASSIFICATION AND ORDINATION
There has been much debate among plant ecologists regarding
the most appropriate type of multivariate analysis of eco-
logical data — classification or ordination (Anderson
1965, Goodall 1970, Whittaker 1973). On one hand, ecol-
ogists interested in describing vegetation units or com-
munities have tended to use classification approaches.
On the other, those believing that species are distributed
more or less independently preferred ordination. Anderson
(1965) discusses the controversy and concludes that the
problem is non-existent. Numerical classification and
ordination are both useful tools although one may be more
relevant in certain circumstances than the other. Classi-
fication is more useful in simplifying large, complex data
sets. Ordination may be more useful in the analysis of
smaller, more homogeneous data sets when one is more
interested in interpreting the detailed relationships
among entities.
Moreover, ordination may be useful in the interpretation
of classification and vice versa. Classification and
ordination can be interfaced in several ways. The distri-
bution of member entities of classificatory groups in
ordination space can be plotted in order to show the
71
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integrity of the groups or, alternately, their overlap (Lie
and Kelly 1970, Hughes and Thomas 1971a, b, Hughes et al.
1972, Boesch 1973). In this fashion ordination can also be
used in the reallocation of entities to new groups to
sharpen the classification (Boesch 1973, Grigal and Ohmann
1975). Alternately, classificatory groups can be ordinated,
either as the centroids of the spatial cluster of their
constituent entities or by ordination of the groups as
defined by their aggregate attributes (Stephenson and
Williams 1971). Multidimensional ordinations of groups
may provide more accurate depictions of the inter-group
relationships than classificatory hierarchies, which are
essentially one-dimensional. As discussed above, ordi-
nation of groups of entities in multiple discriminant
space may allow tests of significance among clusters.
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SECTION VIII
APPLICATIONS OF ECOLOGICAL CLASSIFICATION
DESIGNING A CLASSIFICATORY ANALYSIS
Questions Addressed by Classification
The first step in any analysis of ecological data should be
a statement of the objectives of the analysis. One should
clearly pose the ecological question(s) of interest before
selecting an analytical approach. Numerical classification
is simply a technique for optimal grouping of entities
according to the resemblance of their attributes as ex-
pressed by given criteria. It is not a panacea for all
data analysis problems nor a procedure by which a computer
can do ecology.
Stephenson (1973) lists three reasons why an investigator
might wish to apply numerical classification: (1) to
appear "up-to-date," (2) to try out methods for application
in another context and (3) to attempt to analyze data too
complex for adequate consideration by "common sense" tech-
niques. The point of his admonition is that there is an
apparent trend to uncritically use numerical classification
and other multivariate analyses simply because they are
currently popular. For example, there may be little to
gain in the application of numerical classification in
the analysis of small data sets in which the patterns of
entity relationships are clearly apparent.
It must also be remembered that numerical classification is
most appropriately a hypothesis generating technique and,
with minor exceptions, significance tests which would allow
hypothesis testing are not inherent in classificatory tech-
niques. The classificatory algorithm in effect develops a
hypothesis, based on predetermined criteria, about the
nature of the data. One must then use other techniques to
73
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evaluate hypotheses regarding the homogeneity of the groups
formed, the differences among the groups or the relation-
ships of the groups to extrinsic factors.
Alternatives to Classification
Depending on the questions posed, other mathematical
techniques may be more appropriate in the analysis of
ecological data than classification. If one is primarily
interested in relating biotic entities (e.g. species) to
abiotic attributes, various correlation or regression
techniques, both univariate and multivariate, may be
apropos. If one is more interested in the degree of
relationships among ecological entities rather than a
simplification of a large number of entities into a smaller
number of entities (groups) which can be studied more
effectively, ordination may be a more appropriate analysis.
However, ordination methods are not without their pitfalls
and potentials for misuse. As a general rule, ordination
become less useful as the data set becomes larger and more
complex, whereas numerical classification becomes more
attractive under these circumstances. However, there is a
wide range of circumstances where both classification and
ordination approaches may be useful and often complementary.
There are other ecological problems requiring multivariate
pattern seeking for which both classification and ordi-
nation may be inappropriate. A good case in point is in
the analysis of patterns along an ecological gradient or
ecocline. Classification may be appropriately applied if
the question posed is "how is the ecocline optimally
dissected into zones," but in gradient studies the question
often of most interest is "what are the relative rates of
biotic change along the gradient?" On first consideration,
it would seem that ordination is ideally suited for
addressing the latter question, allowing the coenocline
(biotic part of the ecocline) to be expressed as a spatial
(hopefully linear) model which can be directly compared to
the extrinsic gradient vector. However, it has been shown
that most, if not all, ordination techniques produce con-
siderably distorted models of coenoclines (Austin and
Noy-Meir 1971, Whittaker and Gauch 1973). Terborgh (1971)
proposed a simple graphical approach to the analysis of
coenocline patterns in which inter-site resemblances are
plotted on an abscissa representing the extrinsic environ-
mental gradient. This technique and its modifications have
been successfully applied to study the distribution of
74
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benthos along estuarine gradients (Boesch 1976) and the
distribution of stream benthos along an elevation gradient
(Allan 1975). This technique, which I have termed
"coenocline similarity projection" (Boesch 1976), may
be useful in analyzing patterns along near source away
from source transects often sampled in pollution studies.
Criteria for Selecting Algorithms
Data Manipulations -
There has been a tendency among those using numerical
classification not to give proper consideration to the
nature of the data and transformations of the data before
proceeding with a classificatory analysis. Data reduction,
transformation and standardization can profoundly affect
the results of the classification. Of course, to a large
measure the nature of the data is determined by the partic-
ular study or by practical limitations. For example, in
some cases truly quantitative data cannot be collected and
classifications must be based on binary or ranked data.
Data reductions may be justified for one of three reasons:
(1) the data set is too large for computational practi-
cality, (2) aberrant collections exist due to sampling
problems and (3) for the exclusion of species which are
very rare or inconsistently identified. Justifications
are too study-specific to recommend general criteria for
data reduction, but appropriate criteria should be con-
sistently applied and clearly stated.
The justifications for data transformations were discussed
in Section IV. The application of logarithmic or exponen-
tial transformations is appropriate in many ecological
cases when large variations in abundance exist. It must
be remembered, however, that in addition to "normalizing"
species distributions, such transformations may profoundly
affect inter-entity resemblance by reducing differences in
the size of abundance estimates.
Decisions concerning the use of standardizations are
difficult because of the wide range of possible standar-
dizations and uncertainty regarding the effects of their
application. The investigator is urged to consider the
effects (as discussed on p. 16) of standardizations by
collection total (percent standardization), as is fre-
quently done in ecology, and to judge whether these effects
are indeed desirable in the case at hand. A second
75
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recommendation is that species-standardizations are often
appropriate (depending on resemblance measure used) in
inverse classifications in order to reduce the scale
problem which exists between abundant and non-abundant
species. Standardizations are particularly required with
resemblance measures which heavily weight scale differences,
e.g. Euclidean distance measures.
Numerical and Ecological Resemblance -
The ecologist interested in applying numerical classifi-
cation is often confused by the bewildering variety of
resemblance measures available. Often selections are made
from convenience (e.g. based on program availability) or
because of precedent rather than rational criteria. Selec-
tion of an appropriate resemblance measure is critical, for
it is here more than anywhere else in the clustering algo-
rithm that one attempts to express ecological ideas in
algebraic expressions. Therefore, the first step in the
selection of resemblance measures should be a verbal state-
ment, in ecological terms, of the criteria for resemblance
between entities. Is the investigator more interested in
qualitative or quantitative resemblance; how important is
dominance (in normal analyses) or outstanding abundance (in
inverse analysis) in defining ecological resemblance; and
are there underlying spatial or probabilistic conceptu-
alizations in the investigator's perception of ecological
resemblance?
Although the selection of qualitative or quantitative
resemblance often depends on whether quantitative data
are available, ecologists are often interested in patterns
of both qualitative and quantitative resemblance among
entities. The controversy over whether qualitative com-
parisons are as informative as quantitative comparisons
(Grieg-Smith 1964, Dale and Anderson 1972, Moore»1974,
Clifford and Stephenson 1975) is moot because qualitative
and quantitative patterns may indeed be quite different.
Insight into distributional patterns can often be enhanced
by comparing qualitative and quantitative resemblances
(Boesch 1976, Boesch, Diaz and Virnstein in press).
If the investigator's concept of inter-collection resem-
blance is based largely on the similarity of abundance of
dominant species a variety of quantitative resemblance
measures may be used in normal classification. The Bray-
Curtis coefficient is the preferred similarity measure for
this purpose. Euclidean distance, correlation and
76
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information content measures weight dominance even more.
However, there may be reasons for use of Euclidean distance
and information measures because of the requirements of
clustering methods or for their additive, spatial or
probabilistic properties. One must take care to appro-
priately standardize Euclidean distances (Orloci 1975) in
order to avoid nonsense results which primarily reflect
coincidental aberrations.
If, on the other hand, the investigator's concept of inter-
collection resemblance is based on more or less equal
weighting of all the species in the collection and he
wishes to account for quantitative as well as qualitative
differences between collection, he can choose to use the
Canberra Metric coefficient or one of the aforementioned
measures after application of species-standardization.
Ecological criteria for inter-species resemblance may
likely be different than those for inter-collection
resemblance. Thus, the investigator may choose, different
data standardizations and resemblance measures for inverse
analysis than used for normal analysis.
The scalar differences in species abundances pose a problem
in inverse analyses and if the algorithm is not adjusted
for them, the ultimate classification may be one which
largely separates abundant species from those which are
not — a finding hardly worth the effort. Species-stand-
ardizations may help alleviate this effect in the computa-
tion of resemblance. As with normal analyses, similarity
coefficients derived from the Manhattan metric may be more
appropriate than Euclidean metrics or correlation measures
when very large discrepancies exist in the scale of attri-
bute scores, because Manhattan metrics are based on absolute
differences rather than squared differences or products.
If comparison of shape of the distribution patterns between
species makes ecological sense (and it often does), corre-
lation coefficients may be useful when there are not large
numbers of zero-values in the data matrix.
Classification Structure -
Once a satisfactory method of reflecting ecological resem-
blance is chosen, the investigator must then choose a
strategy for optimum grouping of entities based on these
resemblances (Section VI). Given the poor state of devel-
opment of non-hierarchical clustering methods and the
theoretical advantage but impracticality of divisive
77
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hierarchical clustering methods, the clustering methods
currently most available and useful are agglomerative and
hierarchical. Among these, the computational simplicity
of combinatorial methods make these clustering methods the
most useful. Armed with three of these clustering methods-
group average, flexible and incremental sums of squares-
the investigator would have a versatile array of methods
to suit most purposes.
Group average clustering has space conserving properties
which produce clusters with little distortion of the
actual resemblance relationships. Its advantages over
centroid clustering (also space conserving) include its
combinatorial properties for all distances measures and
the fact that it is not susceptible to reversals. However,
as discussed in Section VI, group average clustering often
does not cluster ecological data intensively enough for
effective interpretation. Thus, group average clustering
is most useful when entities are relatively few, when space
conservation in the classification is required, or as a
first look at the unaltered relationships among entities
before proceeding to a more intensively clustering method.
Flexible clustering advantageously allows continuous vari-
ability of clustering intensity and is more useful than
group average clustering when many entities are being
classified and their patterns of resemblances are complex.
As discussed in (Section VI) flexible clustering is partic-
ularly helpful in inverse classifications of large numbers
of species of varying abundance. Incremental sums of
squares clustering is an intense clustering strategy which
may have certain advantages over flexible clustering when
Euclidean distance is used as a resemblance measure.
Program Availability
A very important limitation to the custom design of
classificatory algorithms has been, and continues to be,
the availability of computer programs for their execution.
Versatile program systems for numerical classification
should include options for the use of various transfor-
mations, standardizations, resemblance measures and
clustering methods. I know of no set of programs which
has facility for application of all the techniques
described in this report.
Classificatory programs require extensive computer storage
-------
and are time-consuming in operation, thus programs are
usually written to maximize efficiency of operation at a
particular computer facility. This means most programs
are highly machine dependent and considerable reprogramming
is often required in order to make them operable at another
facility. The alternative of de_ novo programming is even
more expensive in terms of development time and cost.
-Several program systems or descriptions of available
program systems have been published (Wishart 1969, Rohlf,
Kishpaugh and Kirk 1971, Goldstein and Grigal 1972a,
Anderberg 1973, Hartigan 1975) and the only alternative
for prospective users is to attempt to modify one of these
or some other extant program system to suit their needs.
A program is currently under development at this laboratory
for combinatorial, gplythetic, a.gglomerative hierarchical
clustering (COMPAH) which will Tnclude options" for most of
the data manipulations and resemblance measures described
in this report and the eight combinatorial agglomerative
strategies in Table 2. We are attempting to make the sys-
tem relatively machine independent and plan to publish a
thoroughly documented listing of the programs.
APPLICATION OF CLASSIFICATION TO WATER POLLUTION PROBLEMS
Approaches to Assessing Effects of Pollutants on Community
Structure
Field surveys have long been conducted to assess the
effects of pollutants or other anthropogenic stresses on
aquatic ecosystems. The aim of such sampling approaches
is to assess the effect of the stress on community compo-
sition and structure. If the effects are catastrophic,
there is usually no difficulty in detecting and describing
effects based on sampling before and after the stress or
surveying impacted and control areas. However, it is more
difficult to detect and quantify effects when the level of
impact is intermediate. This is because natural communi-
ties are usually composed of populations of many species
which are often highly variable. Several approaches have
been taken to simplify the problem of interpretation of
such collection data (Swartz 1972).
Indicator Species -
Perhaps the earliest approach was to concentrate attention
on certain "indicator species" which either are partic-
ularly sensitive and thus likely to be eliminated by the
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stress or, more commonly, are particularly hardy and oppor-
tunistic and thus likely to be favored by the stress. For
example, it has long been known that organic pollution in
rivers and streams may result in the elimination of many
insect, molluscan and crustacean taxa and favor the estab-
lishment of dense populations of a few species of chirinomid
insect larvae and tubificid oligochaetes. The indicator
species approach has also been applied in studies of lakes
and marine waters, but with less success than in running
freshwater habitats. In both lacustrine and marine hab-
itats, species favored by anthropogenic stress are more
closely related to the constituents of natural assemblages
than in streams. In streams the faunal replacements are
often at the order of phylum level, whereas changes may be
at subfamily levels in lacustrine, estuarine or marine
environments.
Heavy reliance on the indicator species concept has been
widely criticized on the grounds that it discards from
consideration potentially valuable information on the
distribution of the large number of species not considered
indicators a_ priori. It has also been noted that indi-
cators presumably favored by pollution are also constit-
uents of natural communities. The so-called pollution
indicators are adapted for exploitation of resources
following disturbances, or thrive under stress conditions
which reduce biotic pressures, and they can frequently be
abundant in unpolluted situations (Grassle and Grassle
1974). Nonetheless, the indicator species concept is
fundamental to the interpretation of community data col-
lected for the assessment of impacts. Any changes in the
composition or structure of communities can only be under-
stood after consideration of the habitat preferences and
life history characteristics of species whose abundances
are affected. If extensive knowledge of these character-
istics is available for a local biota, the indicator
species approach can be effectively and meaningfully
applied. For example, it has been successful in research
on the effects of pollution on benthic communities in
various parts of the Baltic Sea (Leppakoski
1975, Anger 1975). There the extensive collective
experience of Scandinavian and German investigators has
allowed the classification of large numbers of species
into grades of progressiveness, regressiveness or
indifference with respect to the response of their
populations to pollution stress.
80
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In regions where the responses of the biota are less well
known or where community patterns are more complex than
those of the Baltic benthos, exclusive use of an indicator
species approach to interpretation of data is likely to be
less reliable. In such cases numerical classification and
other multivariate analyses should prove valuable because
they allow analysis of data on all (or at least a larger
portion) of the biota. Inverse classifications can produce
objective groupings of species corresponding to the response
of their populations to pollution, thus generating hypoth-
eses about the relative effects of the pollution on com-
ponents of the biota which may be experimentally or
empirically tested.
Species Diversity -
It has long been known that pollution stress often reduces
the species diversity of communities (Jacobs 1975). Species
diversity, in the sense of species richness or the number
of species in a community, was used as a criterion for
assessing the effects of pollution by early workers on
stream pollution. The mid 1960's witnessed an explosive
increase in interest in species diversity and the quanti-
fication of diversity in ecology which has profoundly
affected aquatic pollution ecology. A paper by Wilhm
and Dorris (1968) suggested that quantitative measures
of species diversity be incorporated as water quality
criteria and introduced many pollution biologists to
information diversity measures. Now the use of diversity
indices in investigations of the effects of pollution on
aquatic community structure is virtually universal and is
often required by contractors and regulatory agencies.
A typical approach in the use of species diversity in
pollution studies is the computation of one or more indices
of diversity and the correlation (casual or statistical) of
these indices with pollution stress and other environmental
factors. Often this is only analysis of the multispecies
data resulting from the collections. In addition to some
theoretical problems concerning the diversity indices used
(Hurlburt 1971, Peet 1974), there are severe practical
limitations to the usefulness of this approach. Summarizing
community structure in one parameter, such as a diversity
index, involves a drastic reduction in the information
contained in the multivariate entity summarized, i.e. the
collection or community. Biotic assemblages with different
numbers of taxa and concentration of dominance can,
depending on the diversity index, have similar diversity.
81
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Furthermore, assemblages without any species in common can
have the same diversity. Numerous cases have been reported
in which pollution stress resulted in changes in species
populations but actually_in-ereased the species diversity.
Frequently this is a result of increased evenness of dis-
tribution of abundance among the species rather than an
increase in species.
Multivariate Analyses -
Numerical classification and other multivariate techniques
allow simplification of patterns of multispecies distri-
bution which involve far less loss of the information
originally contained in the data than do diversity indices.
Furthermore, in classification, comparisons are based on
the identity of the species in the collections and the
species are not simply treated as strictly numerical enti-
ties as in the computation of diversity indices. As in the
case of indicator species, numerical classification may be
very useful in the interpretation of species diversity
analyses of a set of collections (e.g. Boesch 1973). To
argue for the exclusive use of one of the three approaches--
indicator species, species diversity or multivariate anal-
ysis—is foolhardy, for they are in fact complementary.
Multivariate techniques serve to provide one level of
simplification of the collection data by defining optimal
structure of the inter-entity relationships. The concep-
tualized structure of the data, whether it results from
numerical classifications or the investigators subjective
appraisal, should then be interpreted in terms of what is
known about the biology of the constituent species, which
in turn provides the basis of designation of indicator
species. Species diversity indices provide quantification
of one important aspect of the ecological structure of
communities. Multivariate analyses and biological inter-
pretations allow placing species diversity in a larger
framework of the total community structure.
Previous Applications of Classification
Despite the explosive increase in the use of numerical
classification in aquatic ecology, in particular marine
benthic ecology, classificatory techniques have not been
widely applied to water pollution problems. However,
judging from inquiries and knowledge of ongoing work,
applications of multivariate analyses will soon become
82
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widespread in pollution biology. This brief review of
published applications of numerical classification is not
exhaustive but is intended to illustrate some of the ways
classification has been applied in the assessment of water
pollution problems.
Grossman et al. (1974) used normal classifications (Jaccard
similarity coefficient, group average clustering) of
sampling sites based on presence-absence of macrobenthic
species in the assessment of effects of spills of hazardous
materials in the Clinch River, Virginia. They found the
analyses useful in documenting the downstream effects of
the spills. The analyses were also found "useful in
determining the effects of type of substrate, time of
sampling, longitudinal succession, and flooding on the
composition of the macrobenthic community." By performing
cluster analyses of stations based on various taxonomic
groups considered separately, they were able to describe
different recovery rates for insects and gastropods.
Mearns (1974) used recurrent group analysis in a study of
distribution patterns of demersal fishes off Southern
California (Fig. 6). He was particularly interested in
determining the effects of pollution from ocean outfalls
on fish distribution. Species groups defined by the anal-
ysis largely reflected depth distributions, and the effects
of the outfalls were not apparent in this analysis except
for the absence of the yellowchin sculpin (Fig. 6) in one
localized region. However, the limitations of the recurrent
group analysis (i.e. binary similarity and sequentially
optimized non-hierarchical clustering) restrict the power
and reliability of the analysis.
Littler and Murray (1975) used a normal classification
(product-moment correlation coefficient, simple average
clustering) of quadrat samples of rocky intertidal
organisms to assess the effects of a small sewage discharge
in California. Quadrat groups were identified on the
basis of the species which were cover dominants (Fig. 12).
The distribution of quadrat groups showed a modification
of the normal zonation patterns around the outfall by the
replacement of a stratified, diverse algal cover by a low
turf of blue-green, green and red algae (Gelidium and Ulva)
in the mid intertidal and by calcareous tube worms
(Serpulorbis) and calcareous red algae (Corallina) in the
lower intertidal (Fig. 12).
Cimberg, Mann and Straughan (1973) applied cluster analysis
83
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-1.0
CLUSTERED GROUPS
A BLUE-GREEN ALGAE
B BLUE-GREEN • ULVA
C GELIDIUM'PUSILLUM • ULVA
0 CORALLINA • GIGARTINA CANALICULATA I
E SERPULORBIS • CORALLINA • PTEROCLADIA
F EGREGIA
G EISENIA
H PSEUDOLITHODERMA
HYDROLITHON
NON-CLUSTERED SAMPLES
J BLUE-GREEN • CHTHAMALUS
K EGREGIA • MACROCYSTIS
L PHYLLOSPAOIX • GELIDIUM ROBUSTUM
M SARGASSUM • EISENIA
N ULVA • PTEROCLADIA
OUTFALL TERMINUS
Fig. 12. Classification of quadrat collections from
transect surveys of intertidal organisms adjacent
to and removed from a small sewage outfall.
Distribution of collection groups plotted in
lower figure (from Littler and Murray 1975).
84
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(percent standardized Bray-Curtis similarity coefficient,
group average clustering) of sites, each representing line
transect surveys of rocky intertidal organisms, in a study
of the long term effects of the Santa Barbara oil spill
along the Southern California coast (Fig. 13). They inter-
preted the results to indicate that sand coverage and sub-
strate stability were the most important environmental
factors influencing species composition, whereas oil
apparently had a minor influence. This conclusion was
based on the fact that the classification did not separate
those beaches oiled from those not. Of course there is a
danger in interpreting the classification as indicating that
there were no effects (or at most very minor effects) of
oiling on intertidal organisms. The effects of seasonal
coverage of the rocky surfaces by sand are understandably
greater than any effect oil coating might have had.
Likewise, the well documented effects of the Santa Barbara
oil spill were greatest in the higher intertidal zones.
Classifications of collections representing intertidal
zones, rather than whole transects, from beaches which
were similar in terms of sand burial and substrate stabil-
ity, but differed in the degree of oiling, would certainly
have been more instructive.
Moore (1973) used monothetic divisive analyses in the
classification of collections and species from kelp
holdfasts in northeast Britain. He found association
analysis most useful for normal classification and divisive
information analyses (DIVINF) most useful for inverse
analysis. Comparisons of the classifications in a nodal
analysis showed that turbidity was a primary factor
governing the distribution of the holdfast fauna and sites
could be characterized by the presence of turbid water
species, various groups of clean water species and turbidity
indifferent species. Moore (1973, 1974) concluded that
the effects of pollution on this fauna, reported by others
as important, were not apparent but could not be ruled out
because their "definition becomes complicated by the
intervention at lower levels of heterogeneity of other
correlating factors, e.g. holdfast morphology."
The problem of interpretation of the results of Mearns
(1974), Cimberg et al. (1973) and Moore (1973) point out
the need for design, if possible, of sampling approaches
which establish suitable controls to mitigate the effect
of overwhelming natural environmental factors on the
assessemnt of effects of pollution. Classification cannot
mysteriously decipher pollution effects in complex data
85
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SIMILARITY
00
10
20 30 40 50 60 70 80 90 100
.111 i i i i
BEACH YEAR OIL
i — ARROYO SEQUIT 1969
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• MRR^RN ifl7ft •
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. .. PARPINTFRIA 147? A
1 CARPINTFRIA 14C4 •
LAnrlNltnIA 1970 •
CUAL UIL POINT 1969 •
NDANA PRINT 1477
i CORONA DEL MAR 1972
._ ,, . . 1. _ [ACT PABBIlin mill •
EMO 1 liHBniLLU 187 U •
. . CACT TADDIlin IIIY1! A
I Ail ItABnlLLD 19/Z •
CORONA DEL MAR 1969
——~~~ EACT TADDII 1 n 107h A
j — tA»l CABnlLLD lain 9
\. — . , .RAVIDTA 1B1H A
UHVIUIA I9IU •
— ~— CAUIIITA iacn ^
^ oAVIDIA 1969 •
NDAUA DHIUT iaea
d.UANA rUINI 1969
ttflBAua nci UAB 1BTII
CUnUNA OIL MAR 197D
— . . . ,11 nAiiA niniiT IBTO
'N.UANA rUINI 18/0
__ . ri PIDITIU 1B19 A
_ ELbHrimN nit V
II , fiAVIRTfi 147? A
IL CAPITAN 19/0 •
EL CAPITAN 1969 •
•POLLUTED IN 1969 BY
SANTA BARBARA OIL SPIU
Fie. 13. Classification of transect collections of rocky
intertidal organisms from beaches along the
Southern California coast (from Cimberg et al.
1973).
-------
sets which are strongly influenced by other environmental
factors, and their inability to do so should not necessarily
be construed as proving the lack of pollution effects.
Smith (1973) used an inverse classification (Bray-Curtis
similarity coefficient, group average clustering) to group
macrobenthic species taken from seven stations aligned
along a transect from pollution sources in Los Angeles
Harbor. He plotted the distribution of species groups on
an ordination based on water and sediment quality para-
meters. He was thus able to relate species distributions
to the composite "quality" of the habitat.
Boesch (1973) applied normal and inverse classifications
(simultaneous-double standardization, Canberra metric,
flexible and group average clustering) in the study of
distributional patterns of macrobenthos in a multi-use
harbor. The classifications and nodal analysis were useful
in interpreting substrate and seasonal patterns. Normal
analysis clearly separated the collections from (Elizabeth
River) the most heavily polluted part of the harbor, from
those from other muddy-sand bottoms (Fig. 14). Inverse
analysis interpreted via two-way tables indicated the
shifts in species occurrence and abundance which were
responsible for these differences.
Smith and Greene (1976) used normal and inverse classifi-
cations (cube root transformed, species mean standardized,
Bray-Curtis similarity and flexible clustering) in the
analysis of macrobenthos on the continental shelf and
slope around a Southern California sewage outfall. Sites
around the outfall were distinctly grouped and a two-way
coincidence table showed that they were characterized by
a group of ubiquitous species which were unusually abundant
at these sites. The results of the classification were
compared with those of ordinations and related to environ-
mental variables.
It is clear from this review that relatively few of the
available techniques have yet been used in water pollution
applications of numerical classification. Some of the
analyses used, e.g. monothetic divisive methods, recurrent
group analysis, and binary similarity coefficients, suffer
severe limitations. Furthermore, few investigators have
attempted both collection and species classifications or
have interpreted classifications in two-way tables. It is
also clear that some investigators have had naive expecta-
tions about what numerical classification can show and have
87
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_Elizabeth Mud Muddy-sand
River sites sites sites
February a May -
-Sand sites-
Fig. 14. Classification of assemblages of macrobenthic
animals from the Hampton Roads area, Virginia,
showing the clear separation of the heavily
polluted Elizabeth River sites from the other
muddy-sand sites. (From Diaz, et al. 1973
after Boesch 1973).
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not designed their sampling or analytical approaches to
meaningfully assess the effects of pollution.
A New Example of a Classificatory Approach
To further illustrate the potential usefulness of numerical
classification in assessment of the effects of water pol-
lution, data from Reish's (1959) classic study of effects
of pollution in Los Angeles - Long Beach Harbors were
reanalyzed by normal and inverse classification. Reish
sampled macrobenthos at a large number of sites in the
inner and outer harbors during January, June and November
1954. He subjectively classified those sites as "healthy,"
"semi-healthy I," "semi-healthy II", "polluted" or "very
polluted" (no macrofaunal life) on the basis of the compo-
sition of the collections at the sites, relying heavily on
a few indicator species of polychaetes.
Reish identified 141 species in his study. Those species
inconsistently identified or present in fewer than five
collections were excluded from the classification analysis,
leaving 78 species. The Bray-Curtis similarity coefficient
based on log-transformed data and group average clustering
were used for both normal and inverse classifications.
This clustering algorithm is perhaps the most widely used
in marine ecology. Normal cluster analysis was run on all
collections in which animals were present for each sampling
period separately. Inverse cluster analysis was performed
on the 78 species with abundance in all collections (i.e.
at each of the stations during each of the periods) as
attributes.
Site groups selected for each of the three sampling periods
are listed in Table 3 and the hierarchical classification
for November is given in Fig. 15. The agreement of the
objective numerical classification with the subjective
classification of Reish for the November collections is
remarkable. Only one of Reish's "healthy" sites is not
clustered in Site Group 1. The numerical classificatory
separation of the "semi-healthy I and II" and "polluted"
sites was not completely congruent with Reish's classifi-
cation, but the general trends are in agreement. Species
diversity indices show the expected trend of higher diver-
sity at the "healthy sites" but the ranges of diversity
among both the numerical and Reish site groups broadly
overlap. This warns against the exclusive use of summary
89
-------
Table 3. CLASSIFICATION OF COLLECTIONS OF BENTHIC
INVERTEBRATES FROM SITES IN LOS ANGELES
AND LONG BEACH HARBORS DURING THREE
SAMPLING PERIODS FROM DATA OF REISH
(1959). SITE CLASSIFICATIONS MADE
INDEPENDENTLY FOR EACH SAMPLING PERIOD.
Site Groups Stations
JANUARY 1954
0 (no animals collected) LA 32, 35, 51
LB 5, 15
1 LA 4A, 5A, 6, 7, 8, 13, 22,
26, 48A, 55
LB 1, 2, 2A, 6, 7, 12, 18,
20, 21
2 LA 11, 30C, 41
LB 10, 11, 17, 23
3 LA 30, 30A, SOB, 31, 33, 36,
38, 40, 43A, 49A, 54
4 LA 37
5 LA 29, 43
6 LA 10
7 LA 16, 20, 29A, 34, 39, 49, 50
LB 14
8 LA 45
JUNE 1954
0 (no animals collected) LA 11, 32, 33, 34, 35, 36, 39,
49, 49A, 50, 51
LB 10, 14
1 LA 4A, 5A, 6, 7, 10, 22, 26, 55
LB 1, 2, 2A, 5, 6, 7, 12, 17,
18, 20, 21, 23
90
-------
Table 3 (continued). CLASSIFICATION OF LOS ANGELES-
LONG BEACH BENTHOS
Site Groups Stations
JUNE 1954 (cont.)
2 LA 29A, 30, 30A, 30B, 30C, 31,
37, 38, 40, 41, 43, 48A
3 LA 13
4 LA 16, 29, X3A, 45
LB 15
5 LB 11
6 LA 20, 54
NOVEMBER 1954
0 (no animals collected) LA 11, 16, 20, 32, 34, 35
36, 49, 49A, 50, 51, 54
1 LA 4A, 5A, 6, 7, 8, 22, 26,
55
LB 1, 2, 2A, 5, 6, 7, 10, 12,
17, 18, 20, 21
2 LA 10, 13
LB 11
3 LA 29, 29A, 30, 30A, 30B, 31,
37, 40, 41
LB 23
4 LA 30C, 38, 43, 43A, 45, 48A
LB 14, 15
5 LA 33, 39
91
-------
«/•»
c/»
e-s
III
111 I
mil
II
OC/It/IfcOOfctCOCiOC/tt/IC/)
-a -01-o
(=(=<=
10 O 01
£2 m
*-«-*' ^ £; S
» o
c-> ui ^ cj o
1 IS> f^> KJ
^oi—jCnis»Nj—-^
IS» *"
—i.IJtnootneT>-*'
Icn
Fig. 15.
Classification of stations in Los Angeles - Long
Beach Harbors based on data of Reish (1954) for
November, 1954. Reish's designation of degree
of pollution effects indicated for each station
as is the species diversity according to Shannon's
information measure (Pielou 1975).
-------
Table 4. CLASSIFICATION OF SPECIES OF BENTHIC
INVERTEBRATES COLLECTED BY REISH
(1959) FROM LOS ANGELES AND LONG
BEACH HARBORS. DATA FROM ALL THREE
SAMPLING PERIODS WERE INCLUDED.
ENVIRONMENTAL INDICATOR SPECIES USED
BY REISH LABELLED H (HEALTHY) SHI
(SEMIHEALTHY I) SHII (SEMIHEALTHY
II) AND P (POLLUTED).
Species
Group
Indicator
Species
H
H
H
SHII
Nereis procera
Tharyx p'arvus
Cossura Candida
Nemerteans
Chaetozone corona
Lumbrineris minima
Capitata ambiseta
Marphysa sp.
Spiophanes missionensis
Tellina buttoni
Tharyx multifilis
Paraonis gracilis
Spiochaetopterus sp.
Armandia bioculata
Hypoeulalia bilineata
Petricola californiensis
Cirriformia luxuriosa
Chione undatella
Neanthes caudata
Saxidomus nutalli
Hesperone complanata
Anaitxdes williamsi
Chone minuta
Prionospio heterobranchia
Cirriformia spirabranchra
Crepidula onyx
Glycera americana
Pherusa inflata
Tagelus californianus
93
-------
Table 4 (continued). CLASSIFICATION OF SPECIES
OF BENTHIC INVERTEBRATES
Species
Group Indicator Species
B-, SHI Poly dor a paucibranchiata
SHI Dorvillea articulata
Macoma nasuta
p Capitella capitata
Podarke pugettensis
Oligochaetes
B- Diopatra sjplendissima
Lumbrineris erecta
Polydora brachycephala
Bo Corophium acherusicum
Caprellids
C Lyonsia californica
Acteon punctocoelata
Thyasira barbarensis
Lumbrineris latreilli
Prionospio cirrifera
D-J_ Me linn a cristata
Epineb~alia sp.
Platynereis bicanaliculata
Nuculana taphria
Crenella decussata
D2 Pinnixa franciscana
Callianassa californiensis
Psephidia ovalis
Polydora cirrosa
Protothaca staminea
E Amphicteis scaphobranchiata
Pectinaria californiensis
Prionospi£ pinnata
Holothurians
Stylatula elongata
Ostracods
Terebellides stroemi
Streblospio erassibranchiata
94
-------
Table 4 (continued). CLASSIFICATION OF SPECIES
OF BENTHIC INVERTEBRATES
Species
Group
Indicator
Species
Drilonereis nuda
Lumbrineris sp. (juv..)
Diopatra sp. (j uv.)
Axiothella rubrocincta
Lumbrineris 1. j aponica
Asychis disparidentata
Laonice cirrata
1
Chione fluctifragra
Haploscoloplos elongus
Chone mollis
Fusinus kobetti
Nephtys caecoides
Eteone californica
Polydora sp.'
Acteocina magdalensis
95
-------
statistics in the description of differences in community
structure.
Species groups selected are listed in Table 4. The taxa
used are those of Reish. The three polychaetes Reish
designated as indicators of "healthy" bottoms, Nereis
procera, Tharyx parvus and Cossura Candida, are clustered
in the large Species Group A and were in fact very closely
clustered in the infra-group hierarchy. The single "semi-
healthy II" indicator, Cirriformia luxuriosa, was also
included in Species Group A. This species was broadly
distributed among the sites and occurred in very large
densities at some inner harbor stations, Reish's "semi-
healthy II" sites. The two species indicative of "semi-
healthy I" bottoms, Polydora paucibranchiata and Dorvillea
articulata, and the single polychaete indicative of
polluted bottoms, Capitella capitata, were clustered in
Species Group B-j_. Members of this group are pollution
tolerant species which largely comprised the fauna of the
inner harbor. Often only Capitella was found in the most
severely affected zones of the inner harbor, which Reish
termed "polluted." However Capitella widely cooccurred
with the other members of Group B and is understandably
grouped with them.
Normal and inverse analyses are compared in nodal constancy
diagrams in Fig. 16. Note the high constancy of Species
Group B, in the "semi-healthy" sites, moderate constancy
(due to Capitella) in the polluted sites and low to very
low constancy in the healthy sites. Also observe the
moderate constancy of Species Group A at the "healthy"
sites and low constancy at "semi-healthy" and "polluted"
sites. Most of the other species groups consist of rela-
tively inconstant species, i.e. the rarer forms, some of
which demonstrate complex spatial-temporal patterns* of
constancy. Most of the species in these groups are
largely restricted to "healthy" bottoms.
Further analyses of these data provided even more insight
into the interactions of species distribution and community
structure. However, this brief description demonstrates
the utility of the approach. The basic conclusions one
would reach via the numerical classification are similar
to those of Reish, based on his extensive experience with
the fauna, and thus the technique was efficacious. Further-
more, the numerical classificatory approach allows greater
insight into patterns than empowered simply by our limited
multivariate mental processes.
96
-------
JANUARY
JUNE
NOVEMBER
CONSTANCY
• HIGH
-50%
MODERATE
.25%
LOW
.10%
DVERY LOW
a 10%
D
ABSENT
SITE
GROUPS
SPECIES
GROUPS
r-A,
B3
C
D,
D2
•E,
3 456 7
2 3 4 56
23 45
SHIl
REISH'S HEALTHY
CLASSIFICATION
HiUC. SHItll.P K-^PIUC. — -
HEALTHY
SKItll -PlUC-»
HEALTHY UC SHI ill SHIIP F
Fig. 16. Nodal constancy in two way tables (site classifi-
cations performed separately for each sampling
period) for species groups in site groups
determined for Reish s (1954) Los Angeles -
Long Beach Harbor data.
-------
SECTION IX
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113
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SECTION X
LIST OF PUBLICATIONS
The following publications have been produced partially as
a result of Grant No. R803599-01-1:
Boesch, D. F. 1977. A new look at the zonation of benthos
along the estuarine gradient. In B. C. Coull (ed.) Ecology
of marine benthos. Univ. South Carolina Press, Columbia.
Boesch, D. F., M. L. Wass and R. W. Virnstein. 1976. The
dynamics of estuarine benthic communities, p. 177-196. In
M. L. Wiley (ed. ) . Estuarine processes, Vol. I. AcademTc"
Press, New York.
Boesch, D. F. in press. Classification and ordination:
"Objective" interpretation of complex biological data? In
J. A. Sherk (ed.). Biological indicators. U.S. Fish and~~
Wildlife Service. Tech. Paper.
114
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1. REPORT NO.
EPA-600/3-77-033
2.
4lTITLE WtffKtSON OF NUMERICAL CLASSIFICATION
IN ECOLOGICAL INVESTIGATIONS OF WATER POLLUTION
7. AUTHOR(S)
Donald F. Boesch
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Virginia Institute of Marine Science
Gloucester Point, Virginia 23062
12. SPONSORING AGENCY NAME AND ADDRESS
Corvallis Environmental Research Laboratory
U.S. Environmental Protection Agency
200 SW 35th Street
Corvallis, Oregon 97330
3. RECIPIENT'S ACCESSION" NO.
5. REPORT DATE
March 1977
6. PERFORMING ORGANIZATION CODE
8. PERFORMING ORGANIZATION REPORT NO.
10. PROGRAM ELEMENT NO.
1BA608
11. CONTRACT/GRANT NO.
R803599-01-
13. TYPE OF REPORT AND PE RIOD COVERED
final
14. SPONSORING AGENCY CODE
EPA/600-02
15. SUPPLEMENTARY NOTES
J^umefTcaY classification encompasses a variety of techniques for the grouping of entiti
based on the resemblance of their attributes according to mathematically stated criteri
In ecology this usually involves classification of collections representing sites or
sampling periods, or classification of species. Classification can thus simplify patte
of collection resemblance or species distribution patterns in an instructive and effici
manner. Procedures of numerical classification are thoroughly reviewed, including data
manipulations, computation of resemblance measures and clustering methods. The importa
and effects of transformations and standardizations are discussed. It is particularly
critical to choose an appropriate resemblance measure which best corresponds with the
investigator's concept of ecological resemblance. Clustering methods form groups on th
basis of patterns of inter-entity similarity. Various types of clustering methods exist
but currently the most useful and best developed are those which are exclusive, intrins
hierarchical and agglomerative. Agglomerative clustering methods which distort spatial
relationships and intensely cluster are often most useful with ecological data. The
usefulness of numerical classification is demonstrated for objective analysis of the
data sets resultinn from field surveys and monitoring studies conducted for the assess-
ment of effects of pollution. However, to date few pollution biologists have applied
the more powerful classificatory techniques and post-clustering analyses.
17.
a. DESCRIPTORS
species classification
species distribution
indicator species
clustering methods
numerical classification
18. DISTRIBUTION STATEMENT
RELEASE TO PUBLIC
KEY WORDS AND DOCUMENT ANALYSIS
b. IDENTIFIERS/OPEN ENDED TERMS
aquatic ecology
mathematical methods of
community analysis in
water pollution
19. SECURITY CLASS (This Report)
UNCLASSIFIED
20. SECURITY CLASS (This page)
UNCLASSIFIED
c. COSATI Field/Group
06/C,F
21. NO. OF PAGES
125
22. PRICE
ns
nt
ce
EPA Form 2220-1 (9-73)
15
ir U 5. GOVERNMENT PRINTING OFFICE: 1977-797-5821101 REGION
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