Note also that Podani’s index’s fixed upper bound (1) is easily attained, while
Baselga’s upper bound D J ¼ (1 – S 7 ) can never reach 1. Indeed, a Jaccard dissimilarity of 1 corresponds to the case where no common species are found, and this, by
definition, leads to a Baselga nestedness Nes BJ ¼ 0.
Nestedness ¼ Jaccard dissimilarity – As shown above, Baselga’s nestedness is
equal to the Jaccard dissimilarity D J when a > 0 and either b or c ¼ 0 (case 8).
Podani’s index is equal to the Jaccard dissimilarity when a ¼ b + c – |b – c| (cases
4, 6 and 7), which means that the larger the difference between b and c is, the smaller
a has to be for the nestedness to be equal to D J ¼ (1 – S 7 ) (case 7). In other words,
Podani’s nestedness can be very high between two very different sites, not only
when a is large.
The following features emerge from these comparisons: (1) Podani and Baselga
agree on the fact that nestedness is possible only when there are species in common
(a > 0). (2) In both indices the maximum value is attained when either b or c is zero.
(3) In both indices nestedness increases when |b – c| increases. (4) Podani’s index
considers the common species a as direct contributors to nestedness, contrary to
Baselga’s Nes BJ . Consequently, when b ¼ c ¼ 0, Podani’s index reaches the
maximum value of 1, and Baselga’s Nes BJ equals zero. (5) The contribution of
a to nestedness is clear in Podani’s index, but intricate and non monotonic in
Baselga’s Nes BJ . (6) Podani’s nestedness has a fixed and attainable upper bound of
1; the maximum value of Baselga’s Nes BJ changes for every pair of sites, being equal
to the corresponding D J ¼ (1 – S 7 ) dissimilarity except for its maximum value 1.
8.4.3.4 Computing Replacement, Richness Difference and Nestedness
Using beta.div.comp() of package adespatial
Our practical application is based on the Doubs fish river data, like the one presented
in Legendre (2014). The first step is to compute the dissimilarity, replacement
(or turnover) and richness or abundance difference (or nestedness) matrices. We
will do this for Podani’s Jaccard-based indices, and then we will extract and plot
some results for further examination.
# Jaccard-based Podani indices (presence-absence data)
fish.pod.j <- beta.div.comp(spe, coef = "J", quant = FALSE)
# What is in the output object?
summary(fish.pod.j)
# Display summary statistics:
fish.pod.j$part
The output object produced by beta.div.comp() is a list containing three
matrices of class “dist”: the replacement ($repl), the richness/abundance difference (or nestedness in the case of Baselga indices) ($rich) and the chosen
dissimilarity matrix (Jaccard in the example above) ($D). Furthermore, the output
object contains a vector ($part) with the following global results: (1) BD Total ,
8.4 Beta Diversity
395
Baselga’s upper bound D J ¼ (1 – S 7 ) can never reach 1. Indeed, a Jaccard dissimilarity of 1 corresponds to the case where no common species are found, and this, by
definition, leads to a Baselga nestedness Nes BJ ¼ 0.
Nestedness ¼ Jaccard dissimilarity – As shown above, Baselga’s nestedness is
equal to the Jaccard dissimilarity D J when a > 0 and either b or c ¼ 0 (case 8).
Podani’s index is equal to the Jaccard dissimilarity when a ¼ b + c – |b – c| (cases
4, 6 and 7), which means that the larger the difference between b and c is, the smaller
a has to be for the nestedness to be equal to D J ¼ (1 – S 7 ) (case 7). In other words,
Podani’s nestedness can be very high between two very different sites, not only
when a is large.
The following features emerge from these comparisons: (1) Podani and Baselga
agree on the fact that nestedness is possible only when there are species in common
(a > 0). (2) In both indices the maximum value is attained when either b or c is zero.
(3) In both indices nestedness increases when |b – c| increases. (4) Podani’s index
considers the common species a as direct contributors to nestedness, contrary to
Baselga’s Nes BJ . Consequently, when b ¼ c ¼ 0, Podani’s index reaches the
maximum value of 1, and Baselga’s Nes BJ equals zero. (5) The contribution of
a to nestedness is clear in Podani’s index, but intricate and non monotonic in
Baselga’s Nes BJ . (6) Podani’s nestedness has a fixed and attainable upper bound of
1; the maximum value of Baselga’s Nes BJ changes for every pair of sites, being equal
to the corresponding D J ¼ (1 – S 7 ) dissimilarity except for its maximum value 1.
8.4.3.4 Computing Replacement, Richness Difference and Nestedness
Using beta.div.comp() of package adespatial
Our practical application is based on the Doubs fish river data, like the one presented
in Legendre (2014). The first step is to compute the dissimilarity, replacement
(or turnover) and richness or abundance difference (or nestedness) matrices. We
will do this for Podani’s Jaccard-based indices, and then we will extract and plot
some results for further examination.
# Jaccard-based Podani indices (presence-absence data)
fish.pod.j <- beta.div.comp(spe, coef = "J", quant = FALSE)
# What is in the output object?
summary(fish.pod.j)
# Display summary statistics:
fish.pod.j$part
The output object produced by beta.div.comp() is a list containing three
matrices of class “dist”: the replacement ($repl), the richness/abundance difference (or nestedness in the case of Baselga indices) ($rich) and the chosen
dissimilarity matrix (Jaccard in the example above) ($D). Furthermore, the output
object contains a vector ($part) with the following global results: (1) BD Total ,
8.4 Beta Diversity
395