A nonparametric method for the measurement of size diversity with emphasis on data standardization

The most suitable method for estimation of size diversity is investigated. Size diversity is computed on the basis of the Shannon diversity expression adapted for continuous variables, such as size. It takes the form of an integral involving the probability density function (pdf) of the size of the individuals. Different approaches for the estimation of pdf are compared: parametric methods, assuming that data come from a determinate family of pdfs, and nonparametric methods, where pdf is estimated using some kind of local evaluation. Exponential, generalized Pareto, normal, and log-normal distributions have been used to generate simulated samples using estimated parameters from real samples. Nonparametric methods include discrete computation of data histograms based on size intervals and continuous kernel estimation of pdf. Kernel approach gives accurate estimation of size diversity, whilst parametric methods are only useful when the reference distribution have similar shape to the real one. Special attention is given for data standardization. The division of data by the sample geometric mean is proposed as the most suitable standardization method, which shows additional advantages: the same size diversity value is obtained when using original size or log-transformed data, and size measurements with different dimensionality (longitudes, areas, volumes or biomasses) may be immediately compared with the simple addition of ln k where k is the dimensionality (1, 2, or 3, respectively). Thus, the kernel estimation, after data standardization by division of sample geometric mean, arises as the most reliable and generalizable method of size diversity evaluation. *E-mail: [email protected]; telephone: + 34 972 418167; fax: + 34 972 418150 Acknowledgments This work was supported by the grants from the Ministerio de Ciencia y Tecnología of the Spanish Government, Programa Nacional de Biodiversidad, Ciencias de la Tierra y Cambio Global (ref. CGL200405433 / BOS) and the project MEASURE (MTM2006-03040/). Limnol. Oceanogr.: Methods 6, 2008, 75–86 © 2008, by the American Society of Limnology and Oceanography, Inc. LIMNOLOGY and OCEANOGRAPHY: METHODS Several investigators have used size diversity to describe the shape of size distributions (Parsons 1969; Piontkovski and van der Spoel 1998; Quiroga et al. 2005; Brucet et al. 2006; Ichinokawa and Takahashi 2006). A size diversity measure gives a unique value per size distribution and has the advantage of a more intuitive interpretation of its ecological meaning, since the concept of diversity is well established. However, as Ruiz (1994) pointed out, a methodological problem emerges when measuring size diversity by clustering the different body sizes into size classes, because the division of a continuum variable (size) into an arbitrarily selected number of size classes is needed. As a result, different size diversity values for the same community can be obtained depending on the number of size intervals chosen. In order to overcome these problems, Lurie and Wagensberg (1983, 1984) proposed a measure of size diversity index suitable for continuous variables, parallel to Shannon entropy used in information theory or species diversity (e.g., Good 1953). This index is based on the probability density function (pdf) of the size of individuals, and it takes an integral form, better than the discrete summation used in traditional diversity computations. However, the difficulties of describing natural size distributions by means of a simple pdf are evident. Several authors have modeled the relationship between size and abundance as a consequence of allometry in physiological processes and in competitive or predator-prey interactions (Platt and Denman 1977; Dickie et al. 1987; Thiebaux and Dickie 1993; Quiñones 1994; Han and Straskraba 1998), but discontinuities in this parametric relationship are frequent and of great ecological importance (Rodríguez 1994; Havlicek and Carpenter 2001). Vidondo et al. (1997) also proposed the use of the Pareto distribution arguing that most distributions in the nature follow this distribution, and this Pareto distribution has been used for size diversity measurements (Brucet et al. 2006). However, there is no reason to assume that the pdf of an actual size distribution always have to fit to a determinate parametric model. As a consequence of these difficulties with the parametric approach, some authors have proposed the use of a nonparametric kernel estimation of the pdf, especially when they investigate the existence of lumps or gaps in the size spectrum (Havlicek and Carpenter 2001; Ruiz et al. 2002). Nonparametric approaches, such as the kernel estimation, have the advantage that a whole functional expression of pdf is not required, thus, being applicable for most size distributions. The present goal is two fold. The first one is to propose a suitable way to estimate size diversity adapted to a broad class of pdfs. With the aim to obtain a general method for estimation of size diversity, several parametric and nonparametric approaches were tested using simulated samples. Their fit to different size distributions found in the nature was also analyzed. The second objective is to define a normalization to make the index of different samples and communities as comparable as possible. This is attained by a double standardization: a first one to make the size-data adimensional, and a second one to make size data in different dimensions (length, weight, volume, etc.) comparable. Materials and procedures Shannon size diversity index—Let X be a random variable representing the size of individuals, with pX(x) representing its pdf. A sample from X is available and denoted by x1, x2, ..., xn. The goal is to estimate the Shannon diversity index corresponding to pX(x) from the available sample. Shannon entropy is to be used as a diversity index of sizes (Good 1953; Lurie and Wagensberg 1983, 1984), and it is defined as