Geology, Mathematical Vol. 14, No. 6, 1982
U se of the Bray-Curtis Similarity Measure in
Cluster Analysis of Foraminiferal Data j
Michael G. Michie 2
Transformation of data effectively limits the distortion by outlying values on the BrayCurtis similarity measure. It represents an effective method of using cluster analysis in
distinguishing biotopes of benthic foraminifera.
KEY WORDS: biotopes, Bray-Curtis, cluster analysis, foraminifera.
Several recent papers involving analysis of patterns in benthic populations, notably of Australian origin (e.g., Stephenson and Williams, 1971, Stephenson,
Raphael, and Cook, 1976, Stephenson and Campbell, 1977) have made use of
the Bray-Curtis similarity measure. This measure is a city-block metric derived
from Bray and Curtis (1957). In its currently used form it is expressed as
where d ij is the similarity between two operational taxonomic units (OTUs) i
and j, each defined by a set of n attributes x ik and x jk. I n this study the OTUs
are the sampling stations from a small estuary, and the attributes are the benthic
foraminiferal species found at those stations. Some authors (Stephenson, Rafael,
and Cook, 1976, Stephenson and Campbell, 1977) transform each datum x ik
t o loglo(Xik + 1). This, and other possible transformations, are commonly used
when the frequency range of the species encountered is excessive, since the
Bray-Curtis measure is sensitive to outlying values (Williams, 1976). Although
Bray and Curtis (1957) used standardized normal data in their original paper
the present measure should be used with nonnegative data.
In a previous paper (Michie, 1978) the author discussed the use of the
lManuscript received 21 September 1981; revised 15 March 1982.
~Division of Natural Sciences, Northern Territory Museum of Arts and Sciences, PO Box
4646, Darwin NT 5794, Australia.
661
0 020-5958 / 82 / 1200-0661 $03.00/0 1982 Plenum Publishing
Corporation
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Bray-Curtis Similarity with Foraminiferal Data
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E uclidean distance similarity measure with foraminiferal data. Three methods
w ere examined, with preference given to the method where the data were cent ered and normalized by site (using the standard deviation). The results obtained
b y this method are shown in Fig. 1. In this paper the Bray-Curtis measure is
b eing used with the same data as in Michie (1978). Two techniques are used,
t he first with untransformed data and the second with data which had been
t ransformed using the transformation l oglo(xik + 1). The data had been prev iously reduced to eliminate species rare either in abundance or areally. The
w eighted-pair group method (WPGMA) strategy was used in bothpapers.
T he dendrogram produced using the Bray-Curtis measure on untransformed
d ata (Fig. 2) has a cophenetic correlation coefficient, r c s , o f 0.49. The clusters
p roduced from the analysis show the same biotope groups as are shown in Fig. 1,
u sing Euclidean distance. The order of the high-level clusters has changed: the
t idal flat group (Biotope B) and the saline river channel group (Biotope C) are
g rouped together, which was not the case in Fig. 1. Internal composition of two
o f the clusters has changed, notably in Biotope A (upstream hyposaline), where
t hree subdivisions can be made of the biotope.
T he dendrogram produced with the transformed data (Fig. 3) has a cop henetic correlation coefficient of 0.71. The same three biotopes can be dis-
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Fig. 2. Dendrogram produced using the Bray-Curtis similarity measure on untransformed data ( rcs = 0.49). Numbers to the left of the dendrogram are sampiing stations, and the biotopes refer to the biotopes shown in Fig. 1.
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F ig. 3. Dendrogram produced using the Bray-Curtis similarity measure on transformed data, l oglo(Xik + 1), ( rcs = 0 .71). Biotopes refer to the biotopes shown in
Fig. 4.
t inguished, as shown in Fig. 4. Two important differences from Fig. 2 are shown:
Station 21 is transferred to Biotope B and Biotope C, and Station 14 is associated with both of the downstream biotopes, whereas it was a high level station
in Biotope A (upstream hyposaline) in Fig. 2. Station 21 is situated at the main
bend in the river opposite a small creek. This station has a high medium sand
content (20 = 91.4% by weight) and this is reflected in the presence of a large
proportion of rotaliinids in the sample (83.2%). Station 14 has a high proportion
of textulariinids (27.7%), with about 16% of the population composed of Milliamrnina fusca ( Brady).
DISCUSSION
The Bray-Curtis similarity measure is used without data standardization in
the sense used by Michie (1978) and can be compared with the results of the
standardization methods used therein. In both papers, raw where untransformed
data yield low cophenetic correlation coefficients, this indicates that at least in
a mathematical sense data manipulations increase the level of correlation between raw data and the final pattern. This also shows that similarity measures
Bray-Curtis Similarity with Foraminiferal Data
665
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Fig, 4. Biotopes of benthic foraminifera in the Minnamurra River, using the resuits shown in Fig. 3 (transformed data). Station 14 is an anomolous station,
clustering with the downstream biotopes (Biotopes B and C).
are most efficient when the numbers being treated are small: this is especially
significant with the Bray-Curtis measure which is sensitive to outlying values.
As pointed out previously in Michie (1978) the various techniques of data
manipulation reflect a philosophy of use and are therefore user-biased. Transformation is strictly a numerical device to decrease the possibility, especially with
the Bray-Curtis measure, of distortion by outlying values. A more severe transformation results in closer convergence to zero, and discrete values lose their
significance. This is especially true where the logarithmic transformation is
concerned; this is demonstrated in Table 1. The effect of this transformation
is to produce a reverse distortion of high outlying values so that the relative
666
Michie
T able 1. A Comparison of the Effects of Data Transformation on Values
of X ik
R ange of transformed values
Range of values o f xik
(xik) 112
(xitc) 113
0-9
9-99
99-199
99-999
0-3
3-9.95
9.95-14.11
9.95-31.61
0-2.08
2.08-4.63
4.63-5.83
4.63-10
log:o(Xik + 1)
0-1
1-2
2-2.30
2-3
d ifferences in value are eroded for high values. The analyst is adopting a philosophy in which he considers that zeros and low values have the most significance.
This is taken to the extreme case where absence/presence data are concerned.
Transformation diminishes the effect of dominance in pattern analysis. Dominance by one species or by a group of species is discussed for standardized data
by Noy-Meir, Walker, and Williams (1975). Clustering is influenced by the dominant species, but the effect of the dominant species is greater when large variation between species numbers exists. Standardization appears to be a more useful
method of effecting data manipulation while retaining relative value, although
some distortion occurs with impoverished or enriched sites. It is up to the analyst to determine whether he wishes to maintain dominance in his data set, and
to what extent.
CONCLUSION
The use of cluster analysis in environmental situations cannot be regarded
as the solution to the problems confronting the ecologist. Deeper consideration
of the philosophical implications of community structure is necessary, as the
analyst can rationalize his mathematical approach to particular community
structures. The implications of this are that the methods must reflect some preconceived idea of the analyst, preferably derived through his research. This can
be achieved by
1. data manipulation to deform the structure of the original data to reflect a
particular philosophy of community structure
2. selection of a similarity measure which results in a predetermined outcome,
or
3. some combination of these to arrive at a desirable result.
The cophenetic correlation coefficient should not influence solely the decision of the analyst as to the validity of his results. It is related not only to the
similarity matrix but also to the methods employed to manipulate the data.
Bray-Curtis Similarity with Foraminiferal Data
667
Validity is a matter of interpretation of the results by the analyst rather than the
dependence on a mathematically derived coefficient. Ordination on the other
hand presents the outcome of the analysis in a more useful and realistic fashion
and this has become the preferred method of examination of the analytical
results.
REFERENCES
Bray, J. R. and Curtis, J. T., 1957, An ordination of the upland forest communities of
Southern Wisconsin: Ecol. Monogr., v. 27, p. 325-349.
Michie, M. G., 1978, Data manipulation in cluster analysis and the concept of zero presence:
Jour. Math. Geol., v. 10, no. 4, p. 335-345.
Noy-Meir, I., Walker, D., and Williams, W. T., 1975, Data transformations in ecological
ordination. II. On the meaning of data standardization: Jour. ecol., vol. 63, no. 3, p.
779-800.
Stephenson, W. and Campbell, B. M., 1977, The macrobenthos of Serpentine Creek: Mem,
Qd. Mus., v. 18, no. 1, p. 75-93.
Stephenson, W., Raphael, Y. I., and Cook, S. D., 1976, The macrobenthos of Bramble Bay,
Moreton Bay, Queensland: Mere. Qd. Mus., v. 17, no. 3, p. 425-447.
Stephenson, W. and Williams, W. T., 1971, A study of the benthos of soft bottoms, Sek
Harbour, New Guinea, using numerical analysis: Aust. Jour. Mar. Freshwater Res., v.
22, no. 1, p. 11-34.
Williams, W. T., 1976, Pattern analysis in agricultural science: CSIRO, Melbourne, and
Elsevier, Amsterdam, 331 p.