Diversity and Evenness: A Unifying Notation and Its Consequences

By M. Hill. 1973

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Measuring Data (2022, background): “Diversity is also referred to as heterogeneity in many sciences (Rényi et al., 1961; Nunes et al., 2020) which have their own approaches to measure it, Hill numbers in ecology (Hill, 1973), Hannah–Kay indices in economics (Hannah and Kay, 1977).”
Diversity in Faces (2019, background): “This type of analysis comes up in multiple disciplines, including bio-diversity [76, 77], where an important objective is to quantify species diversity of ecological communities.”
Unifying Species Diversity, Phylogenetic Diversity, Functional Diversity, and Related Similarity and Differentiation Measures Through Hill Numbers (2014, background): “The parametric family that unifies various effective numbers was found by Hill (1973) and was recently reintroduced to ecologists by Jost (2006, 2007).”
Rarefaction and extrapolation with Hill numbers: a framework for sampling and estimation in species diversity studies (2014, unknown): “They were ﬁrst used in ecology by MacArthur (1965), developed by Hill (1973), and recently reintroduced to ecologists by Jost (2006, 2007).”
Dos and don'ts when inferring assembly rules from diversity patterns (2020, background): “Indices can differ in whether they account for: (a) species abundances or not (Chao, Chiu, & Jost, 2010; Hill, 1973; Leinster & Cobbold, 2012); (b) intraspecific variability or not (Pavoine & Izsak, 2014; Violle et al., 2012); (c) different phylogenetic (e.g., species, family versus order level) or…”
Insights into global diatom distribution and diversity in the world’s ocean (2016, methodology): “We next examined intragenus diversity (expressed as exponentiated Shannon Diversity Index) (42) and distribution in different oceanic contexts for the 20 most abundant genera.”
Spatiotemporal changes in microphytobenthos assemblages in a macrotidal flat (Bourgneuf Bay,France) 1 (2007, methodology): “…was also studied through univariate diversity measurement performed for each sample using Hill’s index, N1, derived from the Shannon–Wiener index, H¢ (Hill 1973): N1 ¼ eH 0 ; with H 0 ¼ Xs i¼1 ðpiÞ logeðpiÞ ð3Þ with pi being the proportion of the ith species in the sample, and s, the species…”
Declining diversity and abundance of High Arctic fly assemblages over two decades of rapid climate warming (2018, background): “Each Hill number qD calculates diversity in units of effective number of species and differ from others only by an exponent q, which controls for the sensitivity of the index to species relative abundance (Hill 1973).”
Visualizing the structure and bridges of the intellectual property management and strategy literature: a document co-citation analysis (2014, unknown): “Nevertheless, Mowery et al. (2001) argue that the Bayh–Dole Act was only one of several important factors behind the rise of university patenting and licensing activity. The act appears to have had little effect on the content of academic research, even though it hastened the entry into patenting an”
A unified index to measure ecological diversity and species rarity (2008, background): “This is not a surprise because a significant positive relationship between S and H (an index whose value is reached when q01) has been recorded (Hill 1973).”
The importance of fine-scale, vertical profiles in characterising nematode community structure (2003, methodology): “Nematode diversity was expressed as Hill indices N0 and N1 (Hill, 1973).”
Estimating diversity and entropy profiles via discovery rates of new species (2015, background): “For this reason, Hill (1973), T othm er esz (1995), Gotelli & Chao (2013), Chao et al. (2014) and others suggested that biologists should instead use all the information contained in their data, by plotting the complexity measure as a continuous function of q.”
Linking trait variation to the environment: critical issues with community‐weighted mean correlation resolved by the fourth‐corner approach (2017, methodology): “Hill (1973) used these equations in his reciprocal averaging method, which was a rediscovery of correspondence analysis.”
Harmonization is more important than experience—results of the first Nordic–Baltic diatom intercalibration exercise 2007 (stream monitoring) (2009, background): “The error did not decrease linearly with increasing similarity; even with Bray–Curtis similarities >60%, the threshold recommended by Kelly (2001) following Engelberg (1987), some IPS results deviated more than ±0.”
Zooplankton community structure and environmental conditions in a set of interconnected ponds (2004, methodology): “…variables: the exponent of the Shannon– Wiener diversity index (giving the number of taxa, but taking into account the rareness of some taxa; Hill, 1973) and the total number of individuals in each sample (adding the average number of individuals for each scoring class over all taxa for…”
Negative per capita effects of purple loosestrife and reed canary grass on plant diversity of wetland communities (2006, methodology): “The indices S, H ′ and D were transformed to Hill’s Series (N a ), where N 0 = S, N 1 = exp(H ′ ) and N 2 = (1/D) (Hill, 1973).”