Update: A non‐parametric method for the measurement of size diversity, with emphasis on data standardization. The measurement of the size evenness

Abstract

A method for the measurement of the size diversity based on the classical Shannon-Wiener expression was proposed as a proxy of the shape of the size distribution. The summatory of probabilities of a discrete variable (such as species relative abundances) in the original Shannon-Wiener expression was substituted by an integral of the probability density function of a continuous variable (such as body size). Here, we propose an update of this method by including the measurement of the size e-evenness, just dividing the exponential of the size diversity by its possible maximum for a given size range. Assuming a domain of the size range of (0,1), for a given logarithmic mean (m ln ) and a logarithmic standard deviation r ln ð Þ, the distribution with the highest diversity is the Log-Normal. The size e-evenness ranges between 0 and 1 because of the division by the maximum exponential diversity. Size e-evenness is useful to discriminate whether variations in size diversity are due to changes in the shape of the size distribution or caused by differences in size dispersion. Quintana et al. (2008) proposed a nonparametric method for the measurement of the size diversity as a proxy of the shape of the size distribution. Size diversity is computed on the basis of the Shannon-Wiener diversity expression (Pielou 1969) and adapted for continuous variables, such as size. The use of the size diversity for the analysis of the shape of a size distribution has several advantages (Quintana et al. 2008): (1) it integrates sizes of individuals and a size-density in the same way that Shannon-Wiener species diversity integrates the number of species and their relative abundance; (2) after data standardization, samples measured with different units, such as length, weight or volume, are comparable; and (3) its meaning is easy to interpret, since the concept of diversity is well established (Hurlbert 1971): a high size diversity means a wide size range and/or similar proportions of the different sizes along the size distribution. For the size diversity measurement the summatory of probabilities of discrete variables of the classical Shannon-Wiener expression (such as species relative abundances), is substituted by an integral involving the probability density function of the size X ð Þ of the individuals (p X x ð Þ) and takes the form

Size‐based interactions across trophic levels in food webs of shallow Mediterranean lakes

“…We used two non‐taxonomic size metrics: body size diversity and body size evenness. We calculated size diversity and size evenness (Brucet et al., ; Quintana et al., , ) for each fish, zooplankton and phytoplankton sample using individual size measurements as proposed by Quintana et al. ().…”

“…We used two non-taxonomic size metrics: body size diversity and body size evenness. We calculated size diversity and size evenness (Brucet et al, 2006;Quintana et al, 2008Quintana et al, , 2016 for each fish, zooplankton and phytoplankton sample using individual size measurements as proposed by Quintana et al (2008). For zooplankton and phytoplankton, we obtained the actual distribution of individual body sizes in each lake by multiplying the proportion of individuals of a given body size for a given species (or life stage in the case of copepods) by the total number of this species (or life stage in the case of copepods).…”

Size‐based interactions and trophic transfer efficiency are modified by fish predation and cyanobacteria blooms in Lake Mývatn, Iceland

“…Size diversity is very useful and easy to interpret as it defines a single value that is comparable across studies and represents the size range and evenness of a size distribution. High size diversity indicates a broad size range with equal distribution of the different sizes within a size spectrum, whereas low size diversity specifies a narrow size range with high dominance of certain sizes (Emmrich, Brucet, Ritterbusch, & Mehner, 2011;Hurlbert, 1971;Quintana et al, 2016). Since size diversity is calculated as the relative contribution of different sizes along the size distribution, the relationship between size diversity and total biomass is not a result of spurious correlation but of the ecosystem processes (Garc ıa-Comas et al, 2016; Ye et al, 2013).…”

Bayes Hilbert Spaces

“…However, there are cases in which centring can be usefully applied to probability densities with respect to the Lebesgue measure. For instance, in Quintana et al (2008) centring was proposed for entropy or diversity computation.…”