8.4.3.1 The Podani Family of Indices
The indices of replacement and richness (or abundance) difference described by the
Podani group of authors are presented in Table 8.4 with numerical examples based
on Tables 8.2 and 8.3. The indices are built with the numerators described above,
scaled by denominators that depend on the group of indices.
Observe that in all cases the replacement and richness (or abundance) difference
indices add up to the corresponding dissimilarity index.
Podani and Schmera (2011) also proposed an index of nestedness defined as
N ¼ a + |b – c| if a > 0 and N ¼ 0 if a ¼ 0. N can be relativized (scaled between 0 and
1) by dividing it by (a + b + c). The authors’ argument for this index is that richness
Table 8.4 Formulae of the Podani group of indices for presence-absence and quantitative data,
with examples based on the fictitious values of Tables 8.2 and 8.3
Presence-absence
Abundance
Numerator
Replacement
Examples
2 Â min(b, c)
2 Â 3 ¼ 6
2 Â min(B, C)
2 Â 37 ¼ 74
Richn. or abund. Diff.
Examples
|b – c|
|5–3| ¼ 2
|B – C|
|37–71| ¼ 34
Dissimilarity
Examples
(b + c)
5 + 3 ¼ 8
(B + C)
37 + 71 ¼ 108
Jaccard – Ružička group
Denominator
Examples
(a + b + c)
4 + 5 + 3 ¼ 12
(A + B + C)
41 + 37 + 71 ¼ 149
Dissimilarity
Examples
D J ¼ (b + c)/(a + b + c)
D J ¼ 8 / 12 ¼ 0.667
D Ru ¼ (B + C)/(A + B + C)
D Ru ¼ 108 / 149 ¼ 0.725
Replacement
Examples
Repl J ¼ 2 Â min(b,c)/(a + b + c)
Repl J ¼ 6 / 12 ¼ 0.500
Repl Ru ¼ 2 Â min(B,C)/(A + B + C)
Repl Ru ¼ 74 / 149 ¼ 0.497
Richn. or abund. Diff.
Examples
RichDiff J ¼ |b – c|/(a + b + c)
RichDiff J ¼ 2 / 12 ¼ 0.167
RichDiff Ru ¼ |B – C|/(A + B + C)
RichDiff Ru ¼ 34 / 149 ¼ 0.228
Sørensen – Percentage difference group
Denominator
Examples
(2a + b + c)
2 Â 4 + 5 + 3 ¼ 16
(2A + B + C)
2 Â 41 + 37 + 71 ¼ 190
Dissimilarity
Examples
D S ¼ (b + c)/(2a + b + c)
D S ¼ 8 / 16 ¼ 0.500
D 14 ¼ (B + C)/(2A + B + C)
D 14 ¼ 108 / 190 ¼ 0.568
Replacement
Examples
Repl S ¼
2 Â min(b, c)/(2a + b + c)
Repl S ¼ 6 / 16 ¼ 0.375
Repl D14 ¼
2 Â min(B,C)/(2A + B + C)
Repl D14 ¼ 74 / 190 ¼ 0.389
Richn. or abund. Diff.
Examples
RichDiff S ¼ |b–c|/(2a + b + c)
RichDiff S ¼ 2 / 16 ¼ 0.125
RichDiff D14 ¼ |B–C|/(2A + B + C)
RichDiff D14 ¼ 34 / 190 ¼ 0.179
D J ¼ 1 – S 7 ; D S ¼ 1 – S 8
8.4 Beta Diversity
391
The indices of replacement and richness (or abundance) difference described by the
Podani group of authors are presented in Table 8.4 with numerical examples based
on Tables 8.2 and 8.3. The indices are built with the numerators described above,
scaled by denominators that depend on the group of indices.
Observe that in all cases the replacement and richness (or abundance) difference
indices add up to the corresponding dissimilarity index.
Podani and Schmera (2011) also proposed an index of nestedness defined as
N ¼ a + |b – c| if a > 0 and N ¼ 0 if a ¼ 0. N can be relativized (scaled between 0 and
1) by dividing it by (a + b + c). The authors’ argument for this index is that richness
Table 8.4 Formulae of the Podani group of indices for presence-absence and quantitative data,
with examples based on the fictitious values of Tables 8.2 and 8.3
Presence-absence
Abundance
Numerator
Replacement
Examples
2 Â min(b, c)
2 Â 3 ¼ 6
2 Â min(B, C)
2 Â 37 ¼ 74
Richn. or abund. Diff.
Examples
|b – c|
|5–3| ¼ 2
|B – C|
|37–71| ¼ 34
Dissimilarity
Examples
(b + c)
5 + 3 ¼ 8
(B + C)
37 + 71 ¼ 108
Jaccard – Ružička group
Denominator
Examples
(a + b + c)
4 + 5 + 3 ¼ 12
(A + B + C)
41 + 37 + 71 ¼ 149
Dissimilarity
Examples
D J ¼ (b + c)/(a + b + c)
D J ¼ 8 / 12 ¼ 0.667
D Ru ¼ (B + C)/(A + B + C)
D Ru ¼ 108 / 149 ¼ 0.725
Replacement
Examples
Repl J ¼ 2 Â min(b,c)/(a + b + c)
Repl J ¼ 6 / 12 ¼ 0.500
Repl Ru ¼ 2 Â min(B,C)/(A + B + C)
Repl Ru ¼ 74 / 149 ¼ 0.497
Richn. or abund. Diff.
Examples
RichDiff J ¼ |b – c|/(a + b + c)
RichDiff J ¼ 2 / 12 ¼ 0.167
RichDiff Ru ¼ |B – C|/(A + B + C)
RichDiff Ru ¼ 34 / 149 ¼ 0.228
Sørensen – Percentage difference group
Denominator
Examples
(2a + b + c)
2 Â 4 + 5 + 3 ¼ 16
(2A + B + C)
2 Â 41 + 37 + 71 ¼ 190
Dissimilarity
Examples
D S ¼ (b + c)/(2a + b + c)
D S ¼ 8 / 16 ¼ 0.500
D 14 ¼ (B + C)/(2A + B + C)
D 14 ¼ 108 / 190 ¼ 0.568
Replacement
Examples
Repl S ¼
2 Â min(b, c)/(2a + b + c)
Repl S ¼ 6 / 16 ¼ 0.375
Repl D14 ¼
2 Â min(B,C)/(2A + B + C)
Repl D14 ¼ 74 / 190 ¼ 0.389
Richn. or abund. Diff.
Examples
RichDiff S ¼ |b–c|/(2a + b + c)
RichDiff S ¼ 2 / 16 ¼ 0.125
RichDiff D14 ¼ |B–C|/(2A + B + C)
RichDiff D14 ¼ 34 / 190 ¼ 0.179
D J ¼ 1 – S 7 ; D S ¼ 1 – S 8
8.4 Beta Diversity
391