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I've used the Shannon Wiener Diversity Index for a single ecosystem (species as categorical variables). Do you know any alternative indices, especially ones that compare diversity between different samples/ecosystems ?

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Have a look at the Wikipedia page on Diversity Index

Here are the indices that the wikipedia page describes

  • "True Diversity" Index
  • Species Richness
  • Shannon Index
  • Rényi entropy
  • Simpson index
  • Inverse Simpson index
  • Gini–Simpson index
  • Berger–Parker index

The article also links to related concepts such as

  • $\alpha$ Diversity
  • $\beta$ Diversity
  • $\gamma$ Diversity
  • Isolation Diversity
  • Relative Species Abundance

The literature is important on the question of measuring species diversity, ecosystemic function diversity and genetic diversity. What you want to use very much depends on your interest. Just to cite one other example: In the conservation literature, total sum of branch length over a phylogeny is sometimes used as measure of diversity and is sometimes used for prioritize populations (see for example Curnick et al. 2015). This method implicitely assume that more distantely related branches have different ecosystemic functions and keeping many of these functions is good for conservation purpose.

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