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It has previously been recognised (Lande 1996; Olszewski 2004) that there is a relationship between individuals-based rarefaction curves and measures of evenness. Specifically, the initial slope of the individuals-based curve for species richness is equal to the PIE (Probability of Interspecific Encounter) index of Hurlbert (1971). The initial slope of the rarefaction curve is the difference between the expected species richness for two individuals (m = 2) and the expected species richness for one individual (m = 1), and is the probability that the second individual will be a different species from the first (Olszewski 2004). The PIE index is directly related to the Gini-Simpson index – the probability that two individuals selected at random will be different species. The difference between these two indices is in the form of random sampling – Gini-Simpson samples with replacement (thus assuming infinite population size) while PIE, just like rarefaction, samples without replacement. Following Olszewski (2004), PIE can be expressed as the following (Eq. 8) where E[S1] and E[S2] refer to the expected species richness of one and two randomly drawn individuals respectively. Note that E[S1] always equals one in this case.

When considering a sample-based curve, it is clear that the initial slope is related to the beta-diversity of the set of samples from which the curve is calculated. In this case, the difference between E[S1] and E[S2] is the expected number of species in the second sample that are not found in the first. Thus, the PIE index can be used to measure beta-diversity if applied to sample-based rarefaction. This interpretation is directly related to the additive partitioning of species diversity into alpha and beta components where alpha-diversity is the mean (expected) richness of a single sample and beta-diversity is the gain in species richness from a single sample to a larger set of samples and can be read directly from a rarefaction curve (Crist and Veech 2006).

It follows that we can also define measures of phylogenetic evenness and phylogenetic beta-diversity using the initial slope of the PD rarefaction curve, where the units of accumulation are either individuals or samples respectively (Fig. 1). In either case, the initial slope is the expected gain in PD (∆PD) when adding a second accumulation unit to the first. Further, because PD rarefaction curves can also meaningfully use species as accumulation units, we can extend this idea to include a measure of phylogenetic dispersion where the gain in PD is the expected branch length in the lineage (path from tip to root) of a second randomly selected species that is not shared with the first. Thus, we can define a general measure (∆PD) for phylogenetic evenness, phylogenetic beta-diversity or phylogenetic dispersion, depending on the accumulation units chosen (Eq. 9, see also Fig. 1). ∆PD is very similar to the ∆PDq measure of Faith (2013) although in that case, probabilities are not derived from the hypergeometric distribution. Further, ∆PDq is specifically applied to the problem of estimating loss of PD from extinction – a problem that is mathematically similar to rarefaction.

If branch lengths are measured as millions of years between branching events, then

∆PD is measured in units that make intuitive sense and allows for direct comparison across trees and systems. Alternatively, one could standardise the measure by dividing by its theoretical maximum. ∆PD will be maximum when all individuals, species or samples represent wholly distinct lineages with no shared branch lengths. For an ultrametric tree, the lineage length (path from tip to root) is invariant across species and is equal to the depth of the tree. When rarefaction is by units of individuals or species, E[PD1] is the lineage length. When rarefaction is by units of samples, E[PD1] will equal the average PD of a sample and will be equal to ∆PD in the extreme case where each sample shares no branch length with any other sample. Thus, whether referring to units of individuals, species or samples, E[PD1] represents the theoretical maximum of ∆PD and can be used to standardise the measure as follows.

 
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