Alpha diversity

Alpha diversity

Measures diversity within a single sample

Captures two key aspects:

• Richness: how many unique species are present
  
• Evenness: how evenly distributed they are

Counting richness and evenness

Finite sampling

github.com/mblstamps/stamps2019

Applications

Alpha diversity, ageing and body-mass index across the life span.

Data: HITChip Atlas, Lahti et al. Nat Comm 2014

Applications

Richness

\[ R = \text{Number of unique species} \]

Observed richness

• observed richness: counting the number of unique species

True richness to be estimated based on a finite sample (Sanders, 1968).

• Chao1: is more sensitive to sampling effort

• ACE: more stable, emphasizes information from rare taxa

Shannon diversity

\[ H' = - \sum_{i=1}^{R} p_i \log(p_i) \]

where:

• \(p_i\) = relative abundance of species \(i\)
  

• \(R\) = total number of species

Combines richness and evenness in one metric.

Pielou’s evenness

\[ E = \frac{Shannon\ diversity}{ln(Obs.\ Richness)} = \frac{H}{ln(S)}\]

Berger-Parker dominance

Maximum relative abundance in the sample; or relative abundance \(p_i\) of the most abundant type.

\[ max_i (p_i) \]

Simpson’s dominance index

Simpson index \(\lambda\) measures how likely two randomly picked individuals represent the same species.

\[\lambda = \sum_{i=1}^{R} p_i^2\]

where:

• \(p_i\) = relative abundance of species \(i\)
  

• \(R\) = richness, or the number of unique species

Simpson’s diversity?

However, the original Simpson index \(\lambda\) decreases with diversity. Therefore we typically use:

• Inverse Simpson: \(1 / \lambda\) (the most common case of Simpson’s diversity)

• Reciprocal Simpson (or Gini-Simpson): \(1 - \lambda\)

Hill’s diversity: a unifying concept

True diversity, the effective number of types, refers to the number of equally abundant types needed for the average proportional abundance of the types to equal what is observed in the dataset of interest.

\[ ^qD = \frac{1}{M_{q-1}} = (\sum_{i=1}^R p_i^q)^\frac{1}{1-q}, \]

where (again)

• \(p_i\) is relative abundance of species \(i\),
• \(R\) is the richness (number of species)
• \(q^D\) is the Hill number of order \(q\), the effective number of species
• Denominator \(M_{q-1}\): average relative abundance, with weighted generalized mean with exponent \(q − 1\).

Hill’s diversity: a unifying concept

Common diversity measures are special cases:

• \(q = 0\): Observed Richness
• \(q = 1\): Shannon diversity
• \(q = 2\): (Inverse) Simpson diversity
• \(q ≠ 1\): Renyi entropy
• \(q = \infty\): \(\frac{1}{{^{\infty}}D}\): Berger-Parker index

Faith phylogenetic diversity

Faith index incorporates phylogenetic differences (tree) to diversity: sum of branch lengths in a phylogenetic tree that connects all species present in a sample.

Faith phylogenetic diversity

Phylogenetic diversity provides increased statistical power to differentiate age groups in shotgun metagenomics but not in 16S rRNA sequencing.

Armstrong et al. (2021), Genome Research.