Resemblance coefficients and the horseshoe effect in principal coordinates analysis
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Podani J, Miklos I
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ECOLOGY 83 (12): 3331-3343 DEC 2002   
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<b> Abstract: </b>
Although principal coordinates analysis is one of the most widely used ordination 
methods in ecology, no study had been undertaken as yet on the combined effect of 
gradient type and resemblance coefficient on the results. We examine the performance 
of principal coordinates analysis with different choices of the resemblance function 
and different types of a single, underlying gradient. Whereas unimodal species 
response to long gradients always leads to horseshoe (or arch)-shaped configurations 
in the first two dimensions, the converse is not true; curvilinear arrangements cannot 
generally be explained by the Gaussian model. Several resemblance coefficients widely 
used in ecology produce paradoxical arches from perfectly linear data. Species richness 
changes alone, may also lead to a horseshoe for even more distance functions, with the 
noted exception of Manhattan metric. The appearance of arches is a mathematical 
necessity in these cases; true artifacts are introduced only if distances are treated 
inappropriately before eigenanalysis. Examples illustrate that similar configurations 
(curves and even circles) may arise from very different data structures; therefore 
the shape of the point scatter is insufficient by itself to identify background 
ecological phenomena. The horseshoe effect may be diminished and eigenvalue extraction 
may be made more efficient if input measures are raised to high powers; but this 
operation is recommended only in combination with standard analyses, as part of a 
comparative approach. We derive a new distance function, implying standardization 
by species totals, from the chi-square distance. We found that this function improves 
gradient recovery when there is unimodal species response and. some species have 
their optima outside the range of study.
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<b>Author Keywords:</b>
arch effect, Bray-Curtis coefficient, chi-square distance, correspondence analysis, detrending, dissimilarity, gradients, linear scaling, Manhattan metric, ordination, principal coordinates analysis, resemblance coefficients and the horseshoe effect
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KeyWords Plus:</B>
DIRECT GRADIENT ANALYSIS, ECOLOGICAL DISTANCES, ORDINATION, DISSIMILARITY