The Abundance-Preference Method for Assemblage Analysis: A Normalized Diagram with Algorithm Implementing the Method

Abstract

This research concerns a graphical method for analyzing species assemblages sampled in an ensemble of different habitats or sites. This approach provides a preference classification of species of an assemblage, combining two parameters in the same graphic, namely the abundance of species and their preferences for habitats, and the preferences are measured using an index adapted from the Shannon formula. In its initial version, this method provided pertinent analyses of assemblages, but it also showed some weaknesses. One weakness concerns taxonomic groups, where the available sampling methods provide only low abundances, generally converted into abundance notes or indices with low values. Therefore, a comparison between assemblages with different demographic strategies is inadequate. This research aims primarily to normalize this graphic method in the hope of minimizing the effect of demographic strategies on measuring the preference. Normalization mainly concerns the format of abundances, for which we propose alternative values that do not affect the other graphic components.

1. Introduction

Understanding the composition, organization, and functioning of species communities involves a large number of biological and ecological disciplines that use different mathematical tools [1,2,3,4,5,6,7,8,9]. Another major area of interest in ecology is the spatial distribution of living species, which is determined for each species by the availability of habitats (in terms of homogeneous landscapes or space units) in which it can fulfill its environmental requirements.

In addition to ecological factors, this distribution is also defined by biogeographical and human factors [10,11,12,13,14,15,16,17,18,19].

In practice, the most commonly used quantitative parameter in these studies is species abundance [7,18,20], which is obtained using various quantitative or semi-quantitative sampling methods [21]. For a given species, this parameter varies with habitats, depending on the factors they offer to promote facilitate the species life cycle and, more generally, its ecological niche.

Significant efforts have been made to understand the mechanisms that maintain living communities in their habitats, although these mechanisms have not been fully revealed. This effort has led to the development of diverse mathematical tools, almost all based on species abundance/dominance; most of which them aim to quantify, model, and analyze community composition, structure, or diversity [1,22,23]. They are all partially satisfactory in terms of the relevance of their results, both in the field of fundamental ecology (habitat biotypology, assessment of ecological dysfunctions, and identification of indicator species or conservation targets) and in livestock and crop development.

The method subject of this research consists of an ‘abundance preference diagram’ (APD) that offers a graphical method of characterizing species assemblages by combining their ‘degree of preference’ (DP) and their abundance. The basics of this method were established and discussed four decades ago [14], and further research tested it using real field datasets and confirmed its relevance but also some weaknesses [24]. This research addresses the major weakness of this method, which lies in the need to normalize the diagram format to make it significant for assemblages of any demographic strategy, including when species have low abundances. An algorithm is also proposed to facilitate the implementation of this method.

On the other hand, as discussed later [24], we admit that the APD method can be complemented by other existing multivariate approaches, such as OMI [25,26], but this research does not aim to compare these methods.

2. The Abundance Preference Diagram (APD) Method

The APD is a graphic tool that analyzes species assemblages by classifying species in a plan using their degree of preference (DP) as the abscissa and their abundance as the ordinates; this plan is subdivided using half-hyperbolas, which allow for the definition of preference classes in which the assemblage species are hierarchically structured [14,16].

For a species, its DP in a set of habitats is measured using Shannon’s formula [27], which was adopted in [1] to measure the abundance dispersion of species in the same assemblage (or habitat), while the DP measures this equitability for the abundance dispersion of the same species in a sample of habitats. The DP is usually calculated for several species using a matrix of ‘I species × J habitats’.

$$D_{iJ}=1-{\displaystyle \frac{S_i}{{log}_2\left(J\right)}}$$

(1)

$$S_i=-\sum {\displaystyle \frac{n_{ij}+1}{\sum (n_{ij}+1)}}{log}_2{\displaystyle \frac{n_{ij}+1}{\sum (n_{ij}+1)}}$$

(2)

n_ij is the abundance of species i in habitat j, obtained using a sampling method that makes it statistically comparable from one sample to another.

Log₂(J) is the maximum value of S_i, which is reached when all abundances (n_ij) are identical.

$D_{iJ}$ has its minimum value (zero) when S_i = Log₂(J) for a species with a theoretical flat profile, whereas it tends towards 1 for very abundant species that are present in a single habitat (so-called exclusive species) or in very few habitats.

Each assemblage, belonging to a given habitat, can be represented in an APD (Figure 1), where each species is positioned by its DP as abscissa and its abundance as ordinate, this latter being transformed into a base-2 logarithm. In this diagram, two types of ‘standard curves’ are used to give meaning to the species positions, the exclusive species curve (ESC), as defined by Formula (3), and five half-hyperbolas whose focal axis is parallel to the diagonal of the diagram. For practical reasons, the hyperbolas are positioned on the ESC, making their vertices coincide with the chosen points of this curve.

$$S_i={\displaystyle \frac{n_{ij}+1}{n_{ij}+J}}\left({\displaystyle \frac{n_{ij}+1}{n_{ij}+J}}\right)+\left(J-1\right)\ast {\displaystyle \frac{1}{n_{ij}+J}}\left({\displaystyle \frac{1}{n_{ij}+J}}\right)$$

(3)

$${\displaystyle \frac{dDP}{d{n}_{ij}}}={\displaystyle \frac{-1}{{log}_2\left(J\right)}}\ast {\displaystyle \frac{d{S}_i}{d{n}_{ij}}}={\displaystyle \frac{J-1}{{log}_2\left(J\right){\left(n_{ij}+J\right)}^2}}lo{g}_2\left(n_{ij}+1\right)$$

The inflection point of the ESC is provided at 0 value in the second derivative of Formula (4):

$${\displaystyle \frac{d^{2}DP}{d{n}_{ij}^2}}={\displaystyle \frac{-1}{ln \left(2\right){log}_2 \left(J\right)}}[{\displaystyle \frac{2 (n_{ij}+1)+2 (J-1)}{{\left(n_{ij}+J\right)}^3}}-{\displaystyle \frac{1}{{\left(n_{ij}+J\right)}^2}}]$$

(4)

The half-hyperbolas allow for the delimitation of six preference classes, named using qualifiers commonly employed in community analyses; a diagram can be subdivided in two compartments separating species having more or less preference for the considered habitat/assemblage (characteristic, elective, and preferential) and species that statistically have no evident preference for this habitat (transgressive, foreign, and accidental).

This terminology, mainly borrowed from phytoecology, remains relatively specific to the APD method in the sense that each term qualifies a species in a habitat. However, each class name has a precise ecological significance. Indeed, a given habitat is not clearly preferred by a transgressive species, as it could be equally or more abundant in other habitats, while this non-preference is more flagrant for ‘foreign’ species. Similarly, a characteristic species can be qualified as an indicator of the considered habitat, while such a qualifier is not applicable to an elective species and even less to a preferential species, even if they have their highest abundances in this habitat.

It should be noted that when the number of sampled habitats (J) decreases, the ESC shows an atypical shape at low abundances by moving far from the y-axis [24]. As a low number of sampled habitats provides high uncertainty on the preference of a species, mainly when its abundance is low, it was recommended to base the APD method on a minimum of 10 sampled habitats [24], but it is always advisable to have a larger habitat sample size.

3. Methods

We first presented the APD method, in which normalization is addressed in three steps, covering both theoretical and practical aspects.

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To make this tool applicable to any assemblage of any systematic group, we introduced two adaptations: the first consists of reducing the scale of the ordinate axis (abundances) to the interval [0–1], identically to the abscissas axis; the second adaptation concerns the equation of the hyperbola proposed for ranking the assemblage species according to their preference, more precisely its opening, the number of hyperbolas required, and their positions in relation to the curve of exclusive species (ESC);

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Developing an algorithm for implementing the method: this tool is intended to calculate the DP and to plot the APD in both its original and normalized forms.

To illustrate the different essays of adapting the APD, we used experimental data provided by studies carried out in Morocco on two different groups of animals, namely running water insects in the Upper Sebou (Middle Atlas) and waterbird wintering in Morocco during the period of 2018–2022. These data are provided in the Supplementary Materials.

4. Results

4.1. Towards the Normalized APD

4.1.1. Justification

A statistical method would be much more relevant if it produces comparable results in terms of values and format [28]. In this respect, the range of variation in the APD ordinates (abundances) differs greatly from one systematic group to another, depending on the demographic strategies of its species and the format of the abundances provided by the sampling methods. For instance, abundance indices (e.g., between 1 and 5), used in phytosociology and vertebrate ecology, give a low amplitude to the ordinate axis while generating low-value degrees of preference. In contrast, the counts of other groups, waterbirds and aquatic invertebrates, provide numbers (or densities) estimated in hundreds or thousands and high degrees of preference. These differences mainly affect the height of the y-axis and may then modify the shape of the ESC, which is considered a useful referential curve for interpreting the cloud of species in the diagram.

Considering these variations, the APD is intended to provide a reliable comparison between different assemblages by normalizing the Y-axis and the delimitation of preference classes [29,30,31,32,33].

4.1.2. Normalizing the Y-Axis (Abundance)

In light of the above justification, we suggest a normalized format diagram, which provides a comparable APD from one systematic group to another, regardless of the sampling methods and the abundance format of the species. This consists of normalizing the scale of the y-axis to the 0–1 interval by using a form of ‘relative abundance’ of species, where the ratio terms are transformed into base-2 logarithms, as indicated in Formula (5).

For a given assemblage j, in J studied assemblages, the new APD ordinates are defined by the following ratio:

A_ij = log₂(a_ij+1)/log₂(max(a_ij) + 1)

(5)

a_ij = abundance of species i in the assemblage j;

max (a_ij) = maximum abundance recorded in J assemblages (named in the figures as ‘abundance_max’).

From a statistical perspective, logarithmic transformation, by mitigating the skewness of species abundance and stabilizing their variance, reduces the influence of high-abundance values and slightly favors low-abundance species. The use of a base-2 logarithm avoids blurring the differences between these abundances. As illustrated in Figure 2, the addition of 1 to the raw abundances prevents undefined logarithm values, without changing of the ecological meaning of the preference index (DP).

4.1.3. Adapting the APD Zoning to the Normalized Y-Abscissa

As mentioned above, the APD zoning allows for the classification of the species of an assemblage in a grid of different preference classes using the ESC and half-hyperbolas, as shown in Figure 1. To draw the ESC, a vector of coordinates is constituted from DP (abscissa of APD), calculated using the original Formula (3), and the transformed abundance A_ij, as indicated by Formula (5). We recall that these same coordinates are used to project the species points on the APD in a way that their relative position remains as in the non-normalized APD.

It is important to note that with the transformed y-axis, the ESC remains the same as in the non-normalized APD, both in its general shape and its variability depending on the number of studied assemblages (Figure 3).

The half-hyperbolas, as determinant tools for zoning the APD in preference classes, are positioned on the ESC to discriminate the species simultaneously according to their DP and their abundances (Figure 4). To maximize this discrimination, we used equilateral half-hyperbolas [34,35] whose asymptotes are parallel to the APD axes. The equation of such curves should ensure equal lengths of the real and imaginary half-axes of the hyperbola (a = b); this means that the product of their slopes is −1.

$$-{\displaystyle \frac{b}{a}}\ast {\displaystyle \frac{b}{a}}=-1$$

This reference hyperbola has the following equation: $y=a^2/2x$.

In the original APD [14], the hyperbola vertices are positioned on the ESC at five points with the following abscissas: [0.01, 0.1, 0.324, 0.5, 0.75], determined according to an experimental approach. In the normalized APD, the abscissa 0.324 was substituted by the usual value of 0.3 to avoid unnecessary decimal precision without altering the ecological meaning of the preference classes. In the same way, the hyperbola at DP = 0.01 was removed, as it cannot be plotted when the number of sampled habitats (J) is low (Figure 3). The preference class of accidental species is then fused with that of foreign species, as very low abundances indicate a clear non-preference of the habitat concerned. The five preference classes are then named as in the original APD (Figure 1).

The hyperbolas are plotted using a vector made of the real coordinates in the new APD, both varying in the interval [0, 1], independently of that used for drawing the ESC and for projecting the species. As in the original APD, the four hyperbolas are drawn by translating the reference hyperbola to their respective positions, defined by the DP values [0.1, 0.3, 0.5, 0.75]. For each curve, this process starts with searching the value of a² at its intersection with the ESC. This intersection defines the coordinates (x₀, y₀), where S(x₀, y₀) corresponds to the vertex of the hyperbola, positioned as close as possible to the origin. For a given DP value, the coordinates are determined by solving the optimization problem:

$$d\left(O,S\right)= (\sqrt{\left({x_0}^2+{y_0}^2\right)} = (\sqrt{{x_0}^2+{ ({\displaystyle \frac{a^2}{2x_0}})}^2})$$

Once vertex S($x_0$,$y_0$) is determined, four hyperbolas are generated by translating the reference hyperbola to their accurate positions in the predefined locations. Translation is then achieved using a vector defined as

$$\overrightarrow{u_i}={x^{\prime }}_i\overrightarrow{i}+y_i^{\prime }\overrightarrow{j},\hspace{1em}i\in \left\{1,2,3,4\right\},$$

such that

$$u_1\left(0.1,y_1-y_0\right);u_2\left(0.3,y_2-y_0\right);u_3\left(0.5,y_3-y_0\right),u_4(0.75,y_4-y_0)$$

4.2. Application of the Normalized APD Across a Case Study

This application consists of comparing normalized and original APDs for the same dataset. Two examples are extracted from the wintering counts of waterbirds in Morocco during the five-year period of 2028–2022 (Supplementary Materials Table S1) and a bionomic study of sandy beach invertebrates from the Moroccan Atlantic coast [36]. In the first example, the number of monitored habitats/sites (J) is 171, while the second study covers only 15 habitats (Supplementary Materials Table S2). For each dataset, the comparisons are performed for one assemblage (Figure 5). For the waterbird counts, it corresponds to Wad Dr’a Mouth (habitat 5160 in the Supplementary Materials), which has a medium-rich bird population, while all beach invertebrate assemblages have low richness and population count.

In the two types of assemblages, normalization favors species with low abundance, as some of them move from one preference class to the upper one. For instance, in Dr’a Mouth, this is verified for Calidris alba (22), Charadrius hiaticula (31), Marmaronetta angustirostris (66), Pluvialis squatarola (83), etc., while in the sandy beach AZ, at least five species are favored, Donax venustus (7), Eurydice pulchra (12), Gastrosaccus sanctus (14), Lumbrineis impatiens (21), Parachiridotea panousei (25), etc. However, Donax trunculus (6), one of the most abundant species, has moved from the elective to preferential class.

In addition, the normalized APD has the same format for the two types of faunas, making it possible to compare between them if necessary.

4.3. Algorithm for Implementing the APD Method

4.3.1. Introduction

To automatically implement the APD method, a web application was made available to users (https://apd.biodiversity.ma), as indicated in [24]. This application provides two main outputs, the DP of species and the APD of assemblages, and requests data in a standard format, as Excel or CSV files, that the user is called to upload.

The algorithm described here is performed to help users develop a personal application for implementing the APD method in its two forms (non-normalized and normalized) for the same data matrix submitted by a user.

The main outputs from this algorithm are the DP of species, provided in an additional column of the data matrix, and their position in the two APD forms, as defined above (Figure 1 and Figure 4). The two APDs are provided for each habitat as images that can be downloaded.

4.3.2. Inputs

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I: Total number of species;

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J: Total number of habitats;

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Abundance: Matrix[I][J].

4.3.3. Outputs: APDs

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DP: Vector of preference indices of the species;

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APD axes;

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X-axis (abscissa): preference index (DP), ranging from 0 to 1;

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Y1-axis (ordinate):$\left(Abundance\right)$, ranging from 0 to $({Abundance}_{max}$ + 1), in non-normalized APD and Y2-axis:$(Abundance+1)$/$({Abundance}_{max}$ + 1), ranging from 0 to 1 in normalized APDs;

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Reference curves: exclusive species curve (ESC) and hyperbolas;

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Species projection for each assemblage.

4.3.4. Algorithm Steps

The computation begins with an initialization phase in which vectors are created to store the preference indices (DP) and sums of adjusted abundances. Reference values for the hyperbolas are then defined for both normalized and non-normalized DP forms.

The program then uses Formula (1) to calculate the degree of preference for each species.

Once the DP values have been calculated, the exclusive species curve (ESC) is drawn and reference hyperbolas representing various preference levels are created. To project species onto the DP diagram, either in its normalized or non-normalized version, the abundances are finally converted into coordinates.

The detailed pseudocode is provided below to enable other researchers to implement it independently and reproduce the results.

Initialization

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Create a vector DP[I] to store preference indices.

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Create a vector D_sum[I] to store the sum of adjusted abundances (increased by 1).

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Define Dk_n as a list of reference values [0.1, 0.3, 0.5, 0.75] for normalized APD.

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Define Dk_nn as a list of reference values [0.01, 0.1, 0.3, 0.5, 0.75] for non-normalized APD.

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Initialize matrix A[I][J] as species abundances for projecting each habitat/assemblage (in non-normalized APD).

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Initialize matrix B[I][J] as species abundances for projecting each habitat/assemblage (In normalized APD).

Calculating Preference Indices (DP)

For each species I from 1 to I:

Compute the sum of abundances adjusted by adding 1:

$$D_{sum\left[i\right]}\leftarrow \sum _{j=1}^J\left(Abundances\left[i\right]\left[j\right]+1\right)$$

Calculate entropy $S_i$ for species i:

$$S_i\leftarrow \sum _{j=1}^J({\displaystyle \frac{Abundances\left[i\right]\left[j\right]+1}{D_{sum\left[i\right]}}})\times {log}_2({\displaystyle \frac{Abundances\left[i\right]\left[j\right]+1}{D_{sum\left[i\right]}}})$$

Compute the preference index:

$$DP[i]\leftarrow 1-{\displaystyle \frac{S_i}{{log}_2\left(J\right)}}$$

Drawing the Exclusive Species Curve (ESC)

For each abundance a from 0 to the maximum value of the data with a step size of 0.01,

Calculate the exclusive species entropy $S_i$ (exclusive):

$$\begin{gathered} S_i\left(exclusive\right)\leftarrow -{\displaystyle \frac{a+1}{a+J}}\times {log}_2\left({\displaystyle \frac{a+1}{a+J}}\right)+\left(J-1\right)\times {\displaystyle \frac{1}{a+J}}\times {log}_2\left({\displaystyle \frac{1}{a+J}}\right) \\ DP\left[i\right]\leftarrow 1-{\displaystyle \frac{Si\left(exclusive\right)}{{log}_2\left(J\right)}} \end{gathered}$$

Plot ESC, with $\left(Abundance\right)$ or $(Abundance+1)$/$({Abundance}_{max}$ + 1) on the y-axis and DP on the X-axis.

Calculate and Plot Hyperbolas

Non-normalized APD

For each reference value in Dk_n:

Compute the translation_vector to translate hyperbolas

For values of x in the range [0, 1]:

Plot the four hyperbolas by translating (using translation_vector) the reference hyperbola, defined by the equation: Y ←$a^2/2x$

Normalized APD

For each reference value in Dk_nn:

Compute the translation_vector to translate hyperbolas.

For values of x in the range [0, 1] with a step size of 0.01:

Plot the four hyperbolas by translating (using translation_vector) the reference hyperbola, defined by the equation: $y= (ax+b)/ (cx+d)$.

Calculate A and B

For each habitat j from 1 to J:

For each species I from 1 to I:

Calculate the ordinates of species projection:

$$\begin{gathered} A\left[i\right]\left[j\right]\leftarrow {log}_2\left(Abundances\left[i\right]\left[j\right]+1\right) \\ B[i][j]\leftarrow {\displaystyle \frac{{log}_2(Abundances\left[i\right]\left[j\right]+1)}{{log}_2\left({Abundance}_{max}\right[i\left]\right[j]+1)}} \end{gathered}$$

Project the points with the given coordinates (DP[i], $A\left[i\right]\left[j\right])$ for non-normalized APD, and (DP[i], $B\left[i\right]\left[j\right])$ for normalized APD.

5. Discussion

Ecological preferences are a major area of investigation in both fundamental and applied fields. Numerous indices have been developed to quantify them, but the great majority of them have aimed to investigate the factors that control the development of exploitable species, more specifically, the study of trophic parameters. The selected studies provided in this article give a summarized overview of their diversity [37,38,39,40,41,42,43].

The DP is another index that measures the degree of habitat preference of a species, based on the dispersion (entropy) of their abundances in a sample of habitats [14].

The organization of assemblages within ecosystems is another priority area of research that is constantly improving, thanks to an ever-expanding arsenal of methods that have led to significant advances in our ecological knowledge. Among these, the DP introduces a relevant approach to studying the assemblage organization, mainly through its use of the APD method [14]. Almost all these methods are based on species abundance; however, for characterizing assemblages, especially in selecting its characteristic species, this parameter appears misleading. The APD makes it possible to overcome this weakness by associating the criteria of habitat preference of species with their abundance and by its zoning in preference classes, providing more strength to this diagram in describing and analyzing assemblages.

The normalization of the APD, as the objective of this research, has made it possible to extend its applicability to various types of natural assemblages, regardless of their demographic strategies. In addition, the square shape of this diagram, obtained by using relative abundances varying between 0 and 1, and its invariable zoning scheme can lead to fruitful comparisons between assemblages with different demographic strategies.

In addition, the APD implementation is based on a data matrix similar to that used for different multivariate analyses and automatic classifications, and its methods remain complementary to the APD. For instance, factorial analysis (FCA), shown in Supplementary Materials Table S2, provides similarities between habitats/assemblages (Figure 6). In this concrete case, assemblages KH and TF are very close on the F2xF3 plan. However, using the APD method (Figure 7), these two assemblages, supposed to be very similar, have different compositions, indicating the usefulness of combining these two methods. Unlike PCA or FCA, the APD is not a dimensionality reduction method; rather, it is a complementary analytical tool. While PCA and FCA aim to reveal hidden gradients or aggregations in the data, based only on abundances, the APD combines the species preference with their abundance in splitting an assemblage in preference classes.

The algorithm developed as part of this work and presented in a web application (https://apd.biodiversity.ma) has made it much easier to apply the APD method. It performs all the calculations based on a simple raw data matrix in a commonly used format. Our aim is to remove all technical obstacles to the implementation of this method, mainly by ecologists unfamiliar with statistics.

6. Conclusions

The integration of habitat preferences in analyzing species assemblages constitutes a substantial advancement in the understanding of ecosystems. Indeed, the DP is an efficient concept that leads, through the APD method, to a better understanding of the complexity of ecological communities. However, in its original format, this tool showed some limitations, such as its especially regarding its capacity to compare between different assemblages. These limitations were removed by its normalization, which was the main purpose of this study. Indeed, with its unified Y-axis scale and its zoning in preference classes, we have made it possible to use APD results when comparing between assemblages with different demographic strategies.

The second objective of this research was to develop an algorithm that can make it easier to implement this tool by ecologists. This algorithm, briefly presented in this article, performs (through the website mentioned above) all the necessary calculations from a raw data matrix of standard format, making this method more accessible and usable in a variety of research contexts.

To achieve an objective evaluation of the APD pertinence, we currently implement it with diverse real data matrices, and the results confirm that it brings significant insights into the understanding of the organization of assemblages and useful information on the health of habitats. In addition, and in waiting for user feedback, we are improving the website interface by adding some pertinent functions, expanding the range of data types acceptable for treatment, and creating a practical guide to facilitate its use and help for interpreting its results.