ICES Journal of Marine Science: Journal du Conseil Advance Access originally published online on July 30, 2008
ICES Journal of Marine Science: Journal du Conseil 2008 65(8):1456-1461; doi:10.1093/icesjms/fsn119
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This article appears in the following ICES Journal of Marine Science issue: Marine Environmental Indicators: Utility in Meeting Regulatory Needs [View the issue table of contents]
Use of simulated data as a tool for testing the performance of diversity indices in response to an organic enrichment event
Centre for Environment, Fisheries and Aquaculture Science, Lowestoft Laboratory, Pakefield Road, Lowestoft, Suffolk NR33 OHT, UK
Correspondence to J. Barry: tel: +44 1524 844113; fax: +44 1502 524569; e-mail: jon.barry{at}cefas.co.uk
Barry, J., and Rees, H. L. 2008. Use of simulated data as a tool for testing the performance of diversity indices in response to an organic enrichment event. – ICES Journal of Marine Science, 65: 1456–1461.We demonstrate how data on macrobenthic species numbers and abundance after an organic enrichment event can be simulated using the empirical Pearson–Rosenberg model in combination with further plausible ecological assumptions. The simulations were programmed in the statistical package R, using an ecological framework that included classification of species into opportunistic, tolerant, and sensitive types, together with probabilities for the occurrence of these types at any particular point in the event history. The simulations also included assumptions about the dominance of species types. The exercise was successful in that realistic, simulated datasets could be produced quickly and, because of the stochastic nature of parts of the simulation process, repeat simulations allowed variation of selected diversity indices calculated on the series to be assessed. The approach could provide a useful tool to evaluate both existing and new indicators.
Keywords: benthic communities, diversity indicators, organic enrichment, Pearson–Rosenberg, simulation
Received 23 November 2007; accepted 9 May 2008; advance access publication 30 July 2008.
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Introduction |
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We describe the creation of simulated datasets of benthic organisms, based on a temporal progression of the effect on the number of species and their abundance after an incident of organic enrichment. The approach could be adapted to represent spatial changes but, for ease of explanation, is limited to temporal changes. An outline is provided of how the Pearson and Rosenberg (1978) model has been adapted to fit into our simulation scenarios. In particular, the ecological parameters that need to be set are defined, and the values for the assessment of environmental impact are chosen. Next, the characteristics of the simulated data are demonstrated and, finally, an illustration is provided of how a range of diversity indices performs on these data.
There is a broad level of consensus on the criteria governing the practical utility of environmental indicators (Rees et al., 2006). Although some criteria are relatively easy to evaluate objectively against empirical outputs, others, such as the ease with which findings can be communicated, are necessarily more subjective and dependent on the target audience.
The advantages of employing a range of indicators to address different components of an ecosystem or to address changes occurring over different spatial scales are also widely recognized, especially in field assessments (Backer, 2008; Helslenfeld and Enserink, 2008; Johnson, 2008). However, selecting a subset of suitable indicators to meet management needs can be challenging, given the wide array of choices available (Rice and Rochet, 2005; Whomersley et al., 2008). For species/abundance data, formulations have traditionally summarized species-richness or dominance components of diversity (Pielou, 1975; Magurran, 2004). There is an element of overlap among the proposed indices within each component, and selection can be as much a function of convention or personal preference as of objective scientific discrimination.
The logical starting point for an initial evaluation of their suitability as indicators of anthropogenic impact is peer-reviewed literature. However, this can be a daunting task because the debate on utility can often be protracted and sometimes contentious (Gray and Mirza, 1979; Warwick, 1981; Lambshead and Platt, 1985; Raffaelli, 1987; Greenstreet and Rogers, 2006; Dauvin, 2007; Fleischer et al., 2007), especially for newly derived measures. Currently, renewed interest in indicators to underpin policy or regulatory needs is proceeding at a pace that requires dependable and cost-effective choices to be made on much shorter time-scales than hitherto (Rees et al., 2006; Borja et al., 2007).
Clearly, the most important test of an index is whether or not it works in practice in meeting the objectives of its use. We explore the role of simulated data along an organic-enrichment time gradient as a possible "standard" against which indicator utility might be benchmarked. Such a role would not be to the exclusion of testing against empirical data, but as a means to improve transparency and consistency in the selection process. The approach may therefore be viewed as a contribution to the quality assurance of indicators to meet specified purposes. In view of the range of attributes determining indicator utility, the use of simulated data for performance assessment does not eliminate the accompanying need for expert judgement. However, the inclusion of a standardized simulated approach alongside data from empirical sources has the potential to improve the selection process.
For our simulations, the non-linear empirical model of Pearson and Rosenberg (1978) is followed. This model is defined by "rules" governing successional changes in the composition of the macrobenthic community and can be visualized in terms of changes in total densities and numbers of species present. Pearson and Rosenberg's model has been widely applied and validated in many marine benthic studies and is therefore highly appropriate as a template for data simulations.
Built in to the model are three characteristics of the underlying populations: two summary indices (abundance and number of species) and the component species whose individual changes in densities and, ultimately, presence or absence, account for changes in the summary measures and hence drive the model. The simulated dataset is designed to evaluate the relative performance of other derivations (especially diversity indices) in terms of their plausibility and precision in expressing changes, predicated upon the benchmark of conformity of the dataset with the empirical enrichment model.
Software has been developed using the freely available R statistical programming package (R-Development Core Team, 2006) that allows the user to specify various ecological parameters before data are simulated. To illustrate the approach, the choice of parameters has been based on ecologically realistic options (see below).
Stochastic elements in the simulation process ensure that, even if the same parameters are used on successive runs, the output will not be the same. Conditional on the underlying framework for abundance and species numbers laid down by the Pearson–Rosenberg formulation, intervals of variation for the diversity indices can be constructed. This approach to simulating data can be useful when, for example, exploring the estimation bias in diversity indices from samples when compared with their true population values.
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Methods |
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TopIntroductionMethodsResultsDiscussionReferences |
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Adapting the Pearson–Rosenberg model
The response curves of Pearson and Rosenberg (1978) have been used as a basis for choosing the abundance and the number of taxa (species richness) at any particular time after an organic enrichment event (Figure 1). In our simulation scenario, the x-axis has been divided into 101 units between 0 and 100. From left to right, this axis can represent the time after the event, or the magnitude of the enrichment or effect present in the ecosystem. We assume that time and effect are perfectly negatively related so that time = 100–effect.
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The highest level of the effect represents the level immediately after the elimination of all taxa. The taxa and abundance curves are represented by a Gaussian distribution, with means 67 and 30, and variances 1000 and 300, respectively. To attach numbers to these curves, the abundance curve has been defined to have a maximum of 5000 and a minimum of 200, and the taxa curve to have a maximum of 40 and a minimum of 4.
The general approach used in generating our simulated datasets is to use information from the Pearson–Rosenberg formulation to divide the post-enrichment period into stages from 0 to 100. At each stage, the appropriate number of taxa is introduced such that their overall abundance agrees with the relevant point of the Pearson–Rosenberg curve. However, to be more realistic, the concept has been introduced that species are opportunistic, tolerant, or sensitive, and rules govern the probability that any of these taxon types will be present at any particular post-enrichment point. Three types of taxa are distinguished:
- Opportunistic species are typically small and appear in high abundance at high levels of organic enrichment. They are generally present throughout the period after an event, but other types predominate as enrichment is reduced.
- Tolerant species are present at much the same level throughout the period after an event, except at very high levels of enrichment.
- Sensitive species are typically larger and are present at relatively low abundances, but only when enrichment levels are low.
Pearson and Rosenberg (1978; pp. 299 and 234) describe a number of phases and transition points in benthic communities along a gradient of organic enrichment. The phases chosen here are based loosely on some of their definitions, the dividing points being:
- start of effect (highest level of enrichment);
- peak of opportunists, where the abundance curve reaches a maximum and is characterized by a few, highly abundant opportunists;
- start of transitional period, where species richness is at its highest level;
- start of normal period, where total abundance becomes relatively constant at a low level and the ecosystem moves towards natural equilibrium;
- end of effect, i.e. equilibrium (status quo) level is achieved. In principle, organic enrichment is reversible and hence, given time and assuming a purely organic discharge, the community will revert to its predischarge status.
These five phases have been used to allocate relative probabilities of being present to each of the three taxon types (Figure 2). These demonstrate the relative probability of, say, an opportunistic species being introduced into the simulated dataset when compared with a tolerant species. The way that these relative probabilities are converted into actual probabilities is demonstrated at point (iv) in the next section. When the enrichment effect is at its highest level, the tolerant and sensitive species have zero probability of being present. However, when the effect is <20, both tolerant and sensitive species are twice as likely to be present as the opportunistic species. Although the choice of these probabilities is subjective, they are founded on ecological knowledge in accordance with Pearson and Rosenberg (1978).
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Simulation software
To apply the above information on event stages and probabilities to create simulated datasets, a "lifetime approach" has been adopted. Initially, a taxa pool was set up for potential entry into the simulated community. This pool is made up of 15 opportunistic, 40 tolerant, and 60 sensitive species, but not all of these will necessarily be used in any one simulated dataset. Each species in the taxa pool is assigned a lifetime that defines how long a species is observed in the dataset over the period of the enrichment event. These lifetimes were defined to be Poisson random variables with means 5 (opportunists), 20 (tolerants), and 30 (sensitives) units of time, which are intended to convey expected contrasts in life-history traits. The actual lifetime for any particular species was taken as a realization from the appropriate Poisson distribution. The dominance of each species was shaped by a dominance factor. This factor was defined to be 20 for the tolerant species and 1 for the sensitive species, reflecting their relative abundances. For the opportunists, the dominance factor was tapered to allow increasing and decreasing of the populations, and set at 15 times the height of a Gaussian distribution with a standard deviation of one-quarter of the lifetime for the individual species (i.e. reflecting that the majority of a Gaussian random variable is encompassed by the mean ± 2 s.d.). Clearly, this could be changed in other studies to reflect the perceived magnitude of the effect being examined.
Once the taxa pool and the characteristics of species in this pool have been established, an organic enrichment event can be simulated. At each stage e in the event, the procedure is as follows:
- Define the current taxa as the se present in the dataset at event stage e.
- Check if any of the current taxa have reached the end of their lifetime. If so, exclude those from the dataset and put them back into the taxa pool. Decide on the number k of new taxa needed, which is determined by the difference between the number currently present in the dataset and the number assigned by the Pearson–Rosenberg curve.
- If k is negative, then there are apparently too many taxa present, and a corresponding number of current taxa that are nearest the end of their lifetime is removed from the community. If there is a tie, then exclusion is done at random. Excluded taxa are put back in the taxa pool from which new species may be selected and can therefore re-enter the community later.
- If k is positive, then choose the taxon type (opportunist, tolerant, or sensitive) at random for each of these k species, based on their probabilities for the event stage e. The probability for, say, opportunists is given by por = po/(po + pt + ps), where po, pt, and ps are the relative probabilities from the probability curves at event stage e for the opportunist, tolerant, and sensitive taxa, respectively (Figure 2). Note that in our formulation, for the first event, pt = ps = 0, so all initial taxa will be opportunists.
- Choose abundances for each of the se current taxa (after adding new species or deleting species present), based on multinomial sampling and on the abundance (ae) of individuals needed at event stage e and by the dominance (see above) for each taxon. Effectively, this means that each of the ae individuals is assigned to a taxon at random, based on a probability for each current taxon t, which is given by its dominance (dt) divided by the sum of the dominances for all the current taxa:
Although our method does not explicitly take into account the abundance at the previous stage, dependence on previous abundances is built in because the abundances are based on the Pearson–Rosenberg curve (i.e. the abundances at successive times are highly correlated) and, for the opportunists, because of the tapered dominance factor (see above).
Because of the stochastic nature of the simulation process (although constrained by the fixed nature of the abundance and number of taxa rules), the variability of items of interest, such as diversity indices, can be examined.
Diversity indices
Many diversity indices have been proposed, and good summaries can be found in Magurran (2004) and Clarke and Warwick (2001). In developing the various indices, two important concepts are equitability/dominance and species richness.
Equitability defines how similar are the abundances of species. Maximum evenness occurs when all species are equally abundant. The dominance of a particular species reflects how much the numbers for that species dominate compared with the abundances of all other species. We might say that dominance and evenness are opposite concepts. A sample with high dominance would have one or two species dominating in terms of abundances, whereas a sample with high evenness would have no dominant species.
Species richness reflects the number of species. The purest measure of richness is simply the number of species present. However, many richness indices (e.g. Shannon–Wiener) are some function of the number of species. The main problem with such indicators occurs when using samples to estimate the actual value of the indicator in the community, because the estimator is nearly always biased, i.e. the value calculated from the sample will not be the same as the value in the community. This is easy to see for the number of species present, because any sampling strategy will inevitably fail to find all species present in an area, so the sample estimate will underestimate the community value. In our simulations, this problem of bias is avoided by generating a community (e.g. m–2) and evaluating the diversity indices for the entire community, i.e. there is no sampling involved. The following indices were selected for evaluation, using S to denote the number of species, N for the overall abundance, and nj for the abundance of species j:
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The Shannon–Wiener index measures both evenness and richness. Its maximum value ln(S) occurs where all species are present in equal proportions and so reflects the ultimate evenness. However, this maximum is a direct function of species richness, and so the two concepts are somewhat confounded. The numerator of Margalef's index measures species richness, and the effect of the denominator is to reduce this value by the total abundance. Effectively, Margalef measures richness per unit level of abundance (on a natural log scale). Pielou's index is a function of the Shannon–Wiener index, but weighted by its maximum value. Thus, the index must lie between 0 and 1. Effectively, this weighting means that the index measures evenness rather than richness. Simpson's index is a function of species richness in that its numerator is summed over all species in the sample. However, for a given number of species, it measures dominance in that its value is low when there are one or two dominant species.
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Results |
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TopIntroductionMethodsResultsDiscussionReferences |
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As an example of the output from the simulation model, Figure 3 shows the abundances of six species (two of each type) from a single simulation run. As designed in the simulations, the opportunistic species occur in high abundances towards the beginning of the effect (high enrichment) for short periods, and the sensitive species enter at lower levels of the effect.
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Figure 4 shows the diversity indices calculated from a single realization together with the cumulative abundance and richness indices for each of the three taxon types. Figure 5 shows the envelopes within which 95% of the values for the four diversity indices lie, based on 100 repeat realizations of the simulated dataset (the envelope representing the 3rd and 98th highest values of the index at each stage of the event).
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The indices generally perform as expected. The Shannon–Wiener and Margalef indices roughly follow the shape of the taxa curve, which indicates that they largely measure richness. Because Margalef is a purer measure of richness, its curve follows the taxa curve the most closely. Initially, at high levels of organic enrichment, the Simpson index broadly follows the change in species richness. However, the Simpson index does not fall away to reflect the reduced number of taxa at lower levels of enrichment. This is because, at these lower levels, the community is characterized by a higher evenness, reflecting the decline in opportunistic species and the increase in the less-abundant tolerant and sensitive species. The results for the Pielou index seem the most interesting. The higher variability at elevated levels of enrichment may be attributable to the reduced numbers of species present at these levels. Overall, the index is reflecting, as expected, that evenness is higher at lower levels of enrichment. Figure 5 shows that the variability of the Pielou index is even more marked over repeat simulations. Note, however, that the overall variation of the Pielou index is over a relatively limited range.
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Discussion |
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TopIntroductionMethodsResultsDiscussionReferences |
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The aim of the study was to develop a tool for the evaluation of indicator performance for a specified impact type, through the generation of simulated data. In this initial exercise, the model of Pearson and Rosenberg (1978) for successional changes in macro-infauna along a temporal gradient of organic enrichment was employed. The model has the advantage of being empirically derived and widely validated, typically for inputs to relatively quiescent environments supporting stable soft-sediment assemblages.
Further refinements to the simulation of the successional model are necessary. In particular, more data are needed on realistic lifespans for the three taxa types. For example, a distribution other than the Poisson may be more appropriate for specifying the lifetimes for species of a particular taxon type. Also, more empirical evidence is needed for choosing the means for the taxa types. Similarly, their relative dominances and, hence, abundances could be improved upon, especially if simulations were required for a particular situation. A type of Markov structure might be considered where the abundance at a particular point is more explicitly dependent on the abundance at the previous point. Our model generates implicit dependence because abundances depend on the other taxa currently present, and these would not change much between successive event points and, for the opportunist species, because dominance is tapered according to a Gaussian distribution.
There may be merit to identifying one or more "real" datasets as standards against which indicators may be assessed. However, this will require a process of consultation to promote a wide consensus. Such standards might be especially appropriate for impact scenarios where there is currently no universally accepted response model.
What would be the benefits of such a standard testing framework? An indicator that reflects changes well should provide a good incentive for its future use in circumstances where the impact represented by the simulation might be expected in the field (e.g. for a study of sewage discharges). However, it does not follow from such a selection that the preferred indicator(s) will continue to perform effectively in response to changing human or natural influences at a location, or that new formulations might not supplant established ones. Also, among the wide variety of available indicators for testing, the properties measured are variable, and the outcome might typically be the identification of a subset of indicators that represent the most plausible expressions of those of special interest. In our use of simulated data, we clearly do not eliminate subjectivity (i.e. the requirement for expert judgement) in the interpretation of performance. However, the approach does enhance transparency through comparative testing against a singular and established model of population responses to an enrichment event, and this was a key objective of the work. Therefore, the application of a standard tool in evaluations of the efficacy of indicators to meet particular needs might reduce the risk for the investigator being influenced by what might (delicately) be termed fashion or advocacy, while still not binding the investigator to any preordained outcome.
The Pearson–Rosenberg simulation scenario does not allow the evaluation of indices such as taxonomic distinctness (Warwick and Clarke, 1995). However, such indices might be included with more sophisticated simulation techniques.
We conclude by asserting that the adopted approach represents the start of a process of quality assurance of environmental indicator applications, which we consider to be especially important in view of the increasing interest in their use as explicit enforcement tools in regulatory processes. Further refinements are necessary, along with a widening of the scope of the exercise to increase the array of indicator classes and impact scenarios that can be tested by the model. This may be better done against real rather than simulated data, depending on the nature of an indicator or the level of confidence in a dose/response model. However, the development of a consensus view on a standard testing framework will depend on contributions from the wider scientific community.
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