• Therefore, although we will stick to the conventional names of the most classical
coefficients, as they can be found in textbooks, it will be implicit from now on
that all similarity measures have been converted to dissimilarities when computed
by R functions. For instance, the Jaccard (1901) community index is originally a
similarity, but the output of the computation of that coefficient in stats,
vegan, adespatial and ade4 is a dissimilarity matrix; it is produced as
D ¼ (1 – S) in stats and vegan and as D ¼ sqrt(1 – S) in adespatial and
ade4 .
3.3.1 Q Mode: Quantitative Species Data
Let us use the fish species dataset spe again. We will consider the data as
quantitative although, strictly speaking, the values do not represent raw fish
abundances.
Quantitative species data generally require asymmetrical dissimilarity measures.
In this category, frequently used coefficients are the percentage difference, often
(incorrectly) referred to as the Bray-Curtis dissimilarity D 14
6 (also known as the
reciprocal of the Steinhaus similarity index, S 17 ), the chord distance D 3 , the chisquare distance D 21 , and the Hellinger distance D 17 . Let us compute dissimilarity
matrices using some of these indices. In the process, we shall use the package
gclus for visualization.
• Percentage difference (aka Bray-Curtis) dissimilarity matrices can be computed directly from raw data, although true abundances are often log-transformed,
because D 14 gives the same importance to absolute differences in abundance
irrespective of the order of magnitude of the abundances. In this coefficient, a
difference of 5 individuals has the same weight when the abundances are 3 and
8 as when the abundances are 6203 and 6208.
• The chord distance is a Euclidean distance computed on site vectors normalized
to length 1; this normalization is called the chord transformation. The normalization is done by vegan’s function decostand(), argument normalize
7 .
The chord distance can also be computed in a single step by using function
dist.ldc()
8 of package adespatial with the argument chord.
6 In this book, symbols and numbering of similarity and dissimilarity measures are taken from
Legendre and Legendre (2012).
7 This way of presenting several distance measures (as pre-transformations followed by computation
of the Euclidean distance) was described by Legendre and Gallagher (2001). More about this topic
in Sect. 3.5.
8 The name dist.ldc refers to the authors of the article presenting the properties and use of
16 coefficients for the study of beta diversity: Legendre and De Cáceres (2013). We will revisit
this function and some of its indices in Chap. 8.
3.3 Q Mode: Computing Dissimilarity Matrices Among Objects
39
coefficients, as they can be found in textbooks, it will be implicit from now on
that all similarity measures have been converted to dissimilarities when computed
by R functions. For instance, the Jaccard (1901) community index is originally a
similarity, but the output of the computation of that coefficient in stats,
vegan, adespatial and ade4 is a dissimilarity matrix; it is produced as
D ¼ (1 – S) in stats and vegan and as D ¼ sqrt(1 – S) in adespatial and
ade4 .
3.3.1 Q Mode: Quantitative Species Data
Let us use the fish species dataset spe again. We will consider the data as
quantitative although, strictly speaking, the values do not represent raw fish
abundances.
Quantitative species data generally require asymmetrical dissimilarity measures.
In this category, frequently used coefficients are the percentage difference, often
(incorrectly) referred to as the Bray-Curtis dissimilarity D 14
6 (also known as the
reciprocal of the Steinhaus similarity index, S 17 ), the chord distance D 3 , the chisquare distance D 21 , and the Hellinger distance D 17 . Let us compute dissimilarity
matrices using some of these indices. In the process, we shall use the package
gclus for visualization.
• Percentage difference (aka Bray-Curtis) dissimilarity matrices can be computed directly from raw data, although true abundances are often log-transformed,
because D 14 gives the same importance to absolute differences in abundance
irrespective of the order of magnitude of the abundances. In this coefficient, a
difference of 5 individuals has the same weight when the abundances are 3 and
8 as when the abundances are 6203 and 6208.
• The chord distance is a Euclidean distance computed on site vectors normalized
to length 1; this normalization is called the chord transformation. The normalization is done by vegan’s function decostand(), argument normalize
7 .
The chord distance can also be computed in a single step by using function
dist.ldc()
8 of package adespatial with the argument chord.
6 In this book, symbols and numbering of similarity and dissimilarity measures are taken from
Legendre and Legendre (2012).
7 This way of presenting several distance measures (as pre-transformations followed by computation
of the Euclidean distance) was described by Legendre and Gallagher (2001). More about this topic
in Sect. 3.5.
8 The name dist.ldc refers to the authors of the article presenting the properties and use of
16 coefficients for the study of beta diversity: Legendre and De Cáceres (2013). We will revisit
this function and some of its indices in Chap. 8.
3.3 Q Mode: Computing Dissimilarity Matrices Among Objects
39