Step-by-Step Calculation and Spreadsheet Tools for Predicting Stressor Levels that Extirpate Genera and Species

Abstract

In 2011, the U.S. Environmental Protection Agency (USEPA) released a field-based method for estimating the extirpation of freshwater aquatic benthic invertebrates by ionic mixtures dominated by HCO₃⁻, SO₄²⁻, and Ca²⁺ measured as specific conductivity (SC). The estimate of extirpation was SC at the 95^th centile (XC₉₅) of a weighted cumulative frequency distribution (CFD) of a genus or species over a range of SC. A CFD of XC₉₅ values was used to predict the SC at which 5% of genera were likely to be extirpated. Because there are many uses for XC₉₅ values and many data sets that could be analyzed using this method, we laid out a step-by-step method for calculating XC₉₅ values and the stressor level that predicts a 5% extirpation of genera (HC₀₅). Although the calculations can be done with a hand-held calculator, we provide access to 2 downloadable Microsoft Excel® spread sheet calculation tools that are easy to use to calculate XC₉₅ values, to plot a taxon’s XC₉₅ cumulative frequency distribution with increasing SC, and to plot probabilities of observing a taxon at a particular SC. They also plot cumulative frequency distributions of XC₉₅ values and calculate HC₀₅ values. In addition to the tools, we share an example and the output of XC₉₅ values for 176 distinct aquatic benthic invertebrates in Central Appalachia.

INTRODUCTION

Recently, eco-epidemiological methods have been developed for estimating the extirpation of freshwater aquatic benthic invertebrates by ionic mixtures measured as specific conductivity (SC) (USEPA 2011, 2016b). The effect is extirpation, the depletion of a population of a species or genus to the point that it is no longer a viable resource or is unlikely to fulfill its function in the ecosystem (USEPA, 2016a). This measure was selected because it was an unambiguously adverse effect. The threshold for extirpation was estimated as the SC at the 95^th centile extirpation concentration (XC₉₅) of a weighted cumulative frequency distribution (CFD) of a genus or species. A CFD of more than 100 different genus-level XC₉₅ values was used to predict the SC at which 5% of benthic invertebrate genera were likely to be extirpated, that is, a hazardous concentration extirpating 5% of genera (HC₀₅).

A distribution of XC₉₅ values (XCD) is a form of species sensitivity distribution (SSD) (Posthuma et al. 2002). There are several available tools for developing an SSD using laboratory toxicity test data. The USEPA SSD Generator produces SSDs by fitting the most commonly applied distribution, the log-probit, to toxicity data (Shaw-Allen 2010). An alternative is the Dutch program ETX which provides one method for estimating the affected fraction and the output is a log-probit plot (van Vlaardingen et al. 2004). The Environment Canada SSD Master allows for two methods to estimate the fraction affected using 5 potential curve fits: log-normal, log-logistic, extreme value, Weibull, and Gumbell (CCME 2013).

The biological extirpation analysis tools (BEAT) presented in this paper differ from the above tools in that they use the same equations used to develop field-based benchmarks (USEPA 2011) and criteria (USEPA 2016b). Those calculations involve several arithmetic manipulations that are tedious to perform by hand. Because large data sets and a variety of analyses are key to the development of the field-based method, the original work to develop the method used the open source statistical software R (R Core Team 2014). However, some researchers may not be familiar with R and not every application requires the flexibility of R. In fact, the calculations can be done with a hand-held calculator. However, a spreadsheet tool makes it easier.

To enable a wide array of users and applications, we developed 2 downloadable Excel® spread sheet tools, XC₉₅ BEAT and HC₀₅ BEAT. The XC₉₅ BEAT enables users to calculate the exposure that is predicted to extirpate a species or genera from a location or area. It generates an XC₉₅ value, a plot of a taxon’s proportional occurrence with respect to a stressor, and a plot of a taxon’s probability of being observed within discrete stressor ranges or bins. The HC₀₅ BEAT generates a cumulative frequency distribution of XC₉₅ values (XCD) and calculates the stressor level predicted to extirpate 5% of taxa. The HC₀₅ BEAT enables analysis of a set of species or genera along a stressor gradient. By changing the inclusion criteria for data sets, one can estimate benchmarks (USEPA 2011, 2016b), assess local causes (Coffey et al, 2014, Cormier et al. 2012) and risks, and evaluate potential confounders (Suter and Cormier, 2013, USEPA 2011, 2016b). This brief communication describes the tools and their uses. The tools and supporting information in Table 1 are available on line as a set of appendices on GitHub, accessible through the link in the Data Accessibility section and as supplemental material with this document. We also discuss limitations of this method to help avoid misinterpretations and potential inferential errors.

METHODS AND EXAMPLES

Loading the XC₉₅ BEAT and HC₀₅ BEAT

There are two separate workbooks: one for calculation of an XC₉₅ (XC₉₅ BEAT, Appendix S2) and one for calculation of an HC₀₅ (HC₀₅ BEAT, Appendix S3). The XC₉₅ and HC₀₅ BEAT are Microsoft Excel® files (*.xlsm), designed to work in 2007 or later versions, that depend on macros for operation, so you must select “enable macros” when you open the file. Directions to enable macros are included on the first worksheet (“Notes”) of the workbook. Instructions for using the workbooks are also included on the “Notes” worksheet of each workbook but are summarized below.

Entering new data into the BEAT

To use a new data set with the XC₉₅ BEAT, prepare a sample x taxon matrix work sheet using any non-zero number to indicate presence (e.g., 1) and zero (0) to indicate absence; the corresponding stressor value (e.g., “Conductivity”) also needs to be included. To use a new data set to calculate an HC₀₅, prepare a data set of taxa names and XC₉₅ values and upload to the HC₀₅ BEAT.

Running the BEAT

To run the XC₉₅ BEAT, select the taxon of interest from the pulldown list, then click on the “Calculate XC₉₅” button. The results will appear in the “Graphs” tab. The XC₉₅ values can be copied and saved to another sheet. After results are viewed, click on the “Clear Results” button to return to the Input Sheet.

To run the HC₀₅ BEAT with the preloaded XC₉₅ values in the worksheet, select a set of taxa from the list by making “Present?” equal to “TRUE” in the column titled “XC₉₅_Select.” Or, if you add your own XC₉₅ values and taxa to the same list, these additional taxa will be automatically included in the calculation if “Present?” is equal to “TRUE”. After clicking the “Calculate XCD” button the results will appear in the “XCD_Chart” worksheet.

Operations performed by the XC₉₅ BEAT

Calculation of the probability plot (Figure 1), the empirical weighted CFD (Figure 2), and the resulting XC₉₅ involves several steps, all of which are automated in the XC₉₅ BEAT. For example, for each taxon (genus or species) meeting your data-selection conditions, an empirical CFD is constructed that is weighted to correct for any potential bias from the unequal distribution of sampling of sites across the range of stressor exposures such as SC (Figure 3). Corrections for potential confounding require independent analyses and are not performed during this process (Suter and Cormier, 2013, USEPA 2011, 2016b). This weighted CFD represents the proportion of observations of a taxon that occurs below given exposure levels. The extirpation effect threshold for a taxon is defined as the 95^th centile point on the weighted CFD of the total observations of the taxon. Above this threshold, occurrences of the genus are spurious or so low that the presence of the taxon is unlikely. The two Log 10 exposure levels bracketing the 95^th centile are linearly interpolated to give an XC₉₅ for a taxon. Users wishing to display underlying worksheets may do so by selecting the following Excel ® sequence: Home>Format>Visibility>Hide & Unhide>Unhide Sheet.

Figure 1.

Probability plots of Acentrella (a) and Gammarus (b) relative to specific conductivity (SC). Probability of observing a taxon is the percentage of sites in a discrete range of SC (bin) where the individuals of a taxon were observed. For Acentrella (a), the rare occurrence of 100% probability represents 7 sites sampled within a bin from 26.8 to 29.9 μS/cm. The extirpation concentration (XC₉₅) shown by the vertical dashed line is confident because the plot clearly declines to zero probability. The XC₉₅ for Gammarus (b) is not defined because the probability increases within the measured range of SC in this data set. Gammarus’ XC₉₅ is therefore undefined and should be characterized as greater than the listed XC₉₅ value.

Figure 2.

Weighted cumulative frequency distribution (CFD) for empirical observations of Acentrella versus specific conductivity (SC). Vertical dashed blue line marks 95% of observations of Acentrella and the SC below which 95% of the genus occurred in the data set. In this example, extirpation (defined as an XC₉₅) is estimated at 947 μS/cm.

Figure 3.

Histogram of the frequencies of observed specific conductivity (SC) values in samples from Ecoregions 69 and 70 in West Virginia, USA, sampled between 1997 and 2010. Bins are each 0.017 (1/60) of the range of log10 SC units wide.

An example probability plot (Figure 1a) generated by the XC₉₅ BEAT shows the distribution of the genus Acentrella (presence at N = 1,291), a mayfly nymph, from 3,734 sampled sites collected along a SC gradient. The y-axis shows the probability of observing a taxon: the fraction of sampling sites within a given bin of SC values in which this genus was observed. This probability plot is used to assess whether an XC₉₅ is defined or greater than the XC₉₅ calculated by the CFD. The decreasing scatter plot for Acentrella with a clear y-intercept of zero indicates that the XC₉₅ is well defined. In contrast, Figure 1b shows an example of a scatter plot for Gammarus, amphipods (presence at N = 317). The trend of the scatter plot is increasing and therefore the probability of observing the genus is not unequivocally near zero; therefore, the XC₉₅ for Gammarus is not defined within the sampling range of SC in the data set. This is likely to occur because different species within the genus Gammarus have widely different optima ranging from obligate freshwater species to marine species (Beadle and Cragg 1940); therefore, genus level identification may not be informative for identifying a relevant SC tolerance range for Gammarus and the XC₉₅ is greater than the estimated value, that is, it remains undefined. For a quantitative method, statistical software such as R can be used to fit a model to the scatter plot with defined confidence bounds (USEPA 2011, 2016b). The XC₉₅ BEAT does not perform this analysis but example generalized additive models of the scatter plots are provided in Appendix S7 and the confidence in the XC₉₅ values were defined using those plots and the methods described in USEPA (2011, 2016b) and provided in Appendix S1. The alternative is to assess the shape of the scatter by inspection and determine the occurrences of a genus at SC values greater than the calculated XC₉₅. There is greater confidence in XC₉₅ values followed by more bins with zero occurrences.

Each point in Figure 2 represents the cumulative fraction of observations of Acentrella relative to the SC gradient. It depicts the same data set of the weighted observations of Acentrella in Figure 1a, but without calculating probabilities. The SC value for the interpolated 95^th centile point (947 μS/cm) is the estimated XC₉₅ and is shown at the 95^th centile of the CFD as a vertical dashed line (Figure 2). For comparison, the XC₉₅ is also shown on the scatter plot but it is not calculated from that plot (Figure 1a, b).

The XC₉₅ BEAT automatically calculates the probability plot, the empirical weighted CFD, and the resulting XC_95, which if done manually, involves several steps. First, equally-sized bins are defined to compute weights for each sample. The selection of bin size depends on the size of the data set and requires balancing the requirements of sufficient observations in a bin to define the proportion and sufficient bins to define the form of the response. The default number of bins in the XC₉₅ BEAT is 60. For the example, each of the 60 bins is assigned a width equal to 0.017 (1/60) multiplied by the range of the log10 transformed SC values within the data set (Figure 3). The tool provides the user with an option for changing the number of bins on the data input worksheet.

Next, the bins are weighted to correct for any potential bias from unequal distribution of sampling of sites across the range of exposure concentrations. The assigned weight for each sample within a given bin is w_i = 1/n_i, where n_i is the number of samples in the i^th bin. The weighting ensures that sites in bins with many observations are not overly influential. The value of the weighted cumulative frequency distribution, F(x), is computed using the following equation for each unique observed value (x_ij) of the stressor associated with observations of a particular taxon:

$$F(x)=\frac{\sum _{i=1}^{N_b}{w}_i\sum _{j=1}^{M_i}I(x_{ij}<xand{G}_{ij})}{\sum _{i=1}^{N_b}{w}_i\sum _{j=1}^{M_i}I(G_{ij})}$$ Eq. 1

where

x_ij is the stressor value in the j^th sample of bin i,

N_b is the total number of bins,

M_i is the number of samples in the i^th bin,

G_ij is true if the taxon of interest is observed in j^th sample of bin i,

I is an indicator function that equals 1 if the indicated conditions are true, and 0 otherwise.

It should be noted that within the observed ranges of SC, the weighted CFDs of some genera demonstrate a response to SC (e.g., Acentrella) as shown by a steep slope and asymptote well below the maximum exposures (Figure 2). Conversely, genera unaffected within this SC range have a steady increase or remain flat over the entire range of measured exposure and do not reach a clear asymptote. It is somewhat easier to evaluate genera with undefined XC₉₅ values from the plots of the probability of observing a taxon. For example, an XC₉₅ is undefined in the plots of the probabilities of observing a taxon that increase (e.g., Figure 1b, Gammarus) or shows no directional response as SC increases (e.g., Nigronia, Appendix S7). In such cases, the XC₉₅ should be considered as greater than the value calculated.

In summary, the XC₉₅ value is defined as the stressor value corresponding to F(x) = 0.95. Equation 1 is an empirical weighted cumulative frequency distribution, and the output is the proportion of observations of the taxon that occur at or below a given exposure level where the individual observations are weighted to account for the uneven distribution of observations across the range of stressor values.

Operation for calculating an HC₀₅

Given a data set with various taxa and their related XC₉₅ values for a particular stressor, the HC₀₅ BEAT- calculates XCD and HC₀₅ for the stressor. The XCD is a cumulative distribution plot showing the proportion of taxa extirpated as a function of the stressor value. The HC₀₅ is the estimated value of the stressor at which 5% of species are predicted to become extirpated.

For example, the inset (Figure 4a) shows the XCD for SC (μS/cm) based upon macroinvertebrate data from Ecoregion 69 and 70 from West Virginia. This XCD was calculated and plotted automatically by HC₀₅ BEAT, given a sample data set of 176 genera and their associated XC₉₅ values for SC. For this example data set, the HC₀₅ indicates that approximately 5% of species will be extirpated at a SC of 245 μS/cm.

Figure 4.

Cumulative frequency distribution of XC₉₅ values (XCD) for specific conductivity (SC) (μS/cm). Each dot is an XC₉₅ value for 1 unique benthic invertebrate genus. The SC at the intersection of the XCD and the 5th centile (horizontal line) is the HC₀₅ (244.7 μS/cm). N = 176 XC₉₅ (a); N = 88 (b).

In data sets from freshwater systems, it is not uncommon that the tolerance range of half of the taxa will be undefined, such that the species is not extirpated at the highest measured SC. But, that does not influence the HC₀₅ because the XC₉₅ values for these taxa are not in the SC range near the 5^th centile of the distribution of XC₉₅ values. For other applications such as specific site evaluations, the list of XC₉₅ values should be consulted to determine where the plotted points represent XC₉₅ values greater than the calculated value. To avoid misinterpretation of an XCD constructed from other data sets, a reasonable heuristic is to restrict predictions of the proportion of affected genera to the lower third of the curve. For example, because the XC₉₅ values at SC greater than 1,500 μS/cm have been characterized as greater than the assigned XC₉₅ value in the example data set, the upper portion of the XCD in Figure 4a does not predict the proportion of genera likely to be extirpated at those concentrations. About half of the XC₉₅ values are undefined in Figure 4a, and therefore, we also show Figure 4b that depicts a portion of the XCD from the range of XC₉₅ values that are defined in the sampled range of the data set.

To estimate an HC₀₅ without the HC₀₅ BEAT, one simply counts the number of taxa and ranks their XC₉₅ values from least to greatest value. These can be plotted as a CFD with the XC₉₅ value on the x-axis and the “Proportion of Taxa” extirpated on the y-axis. The HC₀₅ is the SC at the 5^th centile of the distribution of XC₉₅ values. To identify the rank at the 5^th centile, one identifies the XC₉₅ bracketing the 5^th centile rank and interpolates the Log10 SC at the 5^th centile of the ranked XC₉₅ values.

The XC₉₅ values for the 176 genera are provided in Appendix S6. For genera with XC₉₅ values >1,500 μS/cm, the probability of observing those genera often increases as SC increases without an apparent optimum in the measured SC range of the data set.

Example calculation using the HC₀₅ BEAT

Details of the manual calculation of the HC₀₅ are shown in an example (Appendix S1). The detailed steps are performed automatically by the spreadsheet tool. A simple quantile-based method is performed. Users may load XC₉₅ values that they have calculated using the XC₉₅ BEAT. Users who choose to use the preloaded values in the XC₉₅ BEAT (Appendix S6), from Ecoregions 69 and 70 (Omernik 1987) should keep in mind that the species comprising the genera in their region may differ. As a result, regional XC₉₅ values at the genus level may be higher or lower because different species with different tolerances and different proportions of species may be represented in a genus XC₉₅ value (Griffith et al., 2018). The BEAT HC₀₅ reports the HC₀₅ and provides a plot of the cumulative proportion of genera versus XC₉₅.

DISCUSSION

The step-by-step description of the derivation of an XC₉₅ value, an XCD, and an HC₀₅ is intended to make these effect endpoints accessible to a broad variety of interested entities. The spreadsheet calculation tool will save time for water quality practitioners and researchers who would like to calculate XC₉₅ values for their own data sets and applications but prefer not to develop a calculation tool.

An example using the method was provided for a combined area in the Central Appalachians using freshwater aquatic benthic invertebrate genera (USEPA 2011, Cormier and Suter 2013, Cormier et al. 2013). However, this general method can be applied to either a species or genus level of taxonomic identification and is not limited to invertebrates. For example, the method was applied using data for freshwater fish species in the same region (USEPA 2016b). Although not tested, it may be applicable to other taxonomic or functional groups and other ecosystems including terrestrial ones.

The (USEPA 2011) field-based method can be used for other applications in addition to the development of effect levels or benchmarks. The XC₉₅ values of macroinvertebrates in streams have been shown to be useful for determining the causes of altered biological communities. For example, Cormier et al. (2012) showed that increased ionic concentration was a likely cause of altered benthic invertebrate communities in the Ten Mile Creek watershed in southwestern Pennsylvania. Conversely, Coffey et al. (2014) used XC₉₅ values to show that ionic concentration was less likely than organic enrichment as the cause of biological impairments identified by the Tennessee Department of Environmental Conservation in Pigeon Roost Creek in central Tennessee, USA.

Because the XC₉₅ represents the limit of a taxon’s tolerance and not a central tendency like a lethal concentration that kills 50% of test animals (LC₅₀), there are risks that the BEAT could be misapplied. Thus, the following considerations should be carefully evaluated when using the tools. Some recommended heuristics are listed below.

• Measurements of the agent(s) are paired in space and time with biological sampling;

• High-quality or reference sites are included in the data set;

• Background SC levels are similar throughout the region;

• Characteristics of the agent (i.e., ionic composition) are similar across the region for the paired data (i.e., other mixtures or agents may occur but they are analyzed separately);

• Some biological sampling occurs when intolerant genera are likely to be collected (e.g., March through June in Appalachia) and where they are likely to occur (e.g., leaf packs, riffles);

• The exposure gradient is broad enough to include no effects, weak effects, and strong effects;

• Sensitivity analyses are performed to evaluate effect of bin size, sample size, and other methodological parameters.

• Data are available and sufficient to evaluate potential confounding factors; and

• An independent data set or statistical models validate the results.

It is also important to keep in mind that a formula will yield a value whether the data are appropriate or not. One mistake is to assume that differences between XCD plots are due to biological or ecological causes when there is some other explanation. For example, divergences of XCD curves above the 50^th centile are often due to differences in the exposure range of particular data sets. The XC₉₅ values > the 50^th centile of the XCD are often not precisely estimated because the range of conductivity values of most data sets do not include the full range of SC tolerated by every organism. That is, the SC tolerated by a particular organism is greater than the highest measured SC in the data set. Therefore, the SC range is not wide enough to measure XC₉₅ values for some taxa, there are too few samples at the higher SC ranges, or there are other confounding factors affecting those values. For example, a comparison of XCDs with similar background SC from Hun-Tai River Basin in Northeast China and Central Appalachia areas in Kentucky and West Virginia of the United States are statistically different, however, the narrower range of SC in the Hun-Tai River Basin data set does not measure any XC₉₅ values greater than 1,160 μS/cm (Zhao et al. 2016), whereas, XC₉₅ values that can potentially be measured at the maxima SC for data sets are 2,390 μS/cm in Kentucky and 11,646 μS/cm in West Virginia (USEPA 2011). Consequently, the maximum XC₉₅ and maximum of the XCD is limited to the highest measured SC in each data set, and so XCDs diverge when the measured ranges of SC are substantially different.

To illustrate the effect of a limited stressor range on the XCD, we prepared 2 data sets from the example data set. One set is the full exposure range between 15.4−11,646 μS/cm and the other is the same data set restricted to samples <1,000 μS/cm. The HC₀₅ values remain fairly similar; however, the XCDs diverge at higher proportions of affected genera (Figure 5). This statistical divergence is a data set limitation and not a measure of ecological similarity or tolerance of the biological assemblage. However, for the taxa near the 5^th centile, the range is sufficient to estimate XC₉₅ values and thus estimate the HC₀₅. It is necessary to examine the scatter plots of the probability of occurrence of taxa to ascertain if the estimated XC₉₅ values of the lower part of the XCD have been measured or if they are greater than the measured range of the SC in the data set.

Figure 5.

Cumulative frequency distribution of XC₉₅ values (XCD) for specific conductivity (SC) (μS/cm). Permutation of the same data set and same taxa occurrence within the full SC range of specific conductivity from 15 to 11,646 μS/cm (a) and 15 to 1,000 μS/cm (b). Divergence of the XCD is due to SC range, not different biological responses. In plot b, the maximum XC₉₅ is restricted to 1,000 μS/cm, causing the entire plot to shift left because higher exposures are not possible.

The original method (USEPA 2011, Cormier and Suter 2013) used confidence bounds of a generalized additive model fitted to the probability histogram and was calculated using R statistical software (Leppo 2018). This capability is not provided with the XC₉₅ BEAT due to the complexity of the coding. However, confidence in an XC₉₅ value may be assessed by examining the shape and asymptote of the plots of the probability of observing a taxon e.g., Figures 1a–b. In lieu of a more sophisticated analysis, users should assume that most XC₉₅ values in the upper portion of the XCD are greater than the calculated value.

Potential confounding can be another concern. Because the method was developed to generate an HC₀₅, methods for assessing the potential confounding of the XCD model were developed and performed (Suter and Cormier, 2013; EPA 2011, 2016b). In this data set, the HC₀₅ is not appreciably confounded by other variables (EPA 2016b). Similar analytical approaches might be applied to individual genera or species, but have not been explored by us.

SUMMARY

In summary, the extirpation of an entire genus or species is a serious consequence of exposure to the measured stressor and an extirpation of genera or species over a wide area may lead to extinction. The tools described here need to be used with care and the outputs interpreted conservatively with consideration of the full suite of scientific issues more fully explained in (USEPA 2016b). We hope that scientists and practitioners will use these calculation tools to identify areas where organisms are at a high risk of extirpation, as well as help to set protective and restoration goals for stressors.

Table 1.

Tools, Step-by-Step Calculation, and Example Outputs

Appendix	Contents	Description
S1	Step-by-Step	Describes the underlying calculation of the XC95 and HC05 values with a detailed example using specific conductivity as the stressor and benthic invertebrates as the affected taxa
S2	XC95 BEAT	Calculates exposure that is predicted to extirpate a species or genus from a location or area (XC95 value) Plots a taxon’s cumulative proportional occurrence with respect to a stressor Plots a taxon’s probability of being observed within discrete stressor ranges or bin.
S3	HC05 BEAT	Generates a cumulative frequency distribution of XC95 values (XCD) Calculates the stressor level predicted to extirpate 5% of taxa
S4	Metadata	Description of data set from Ecoregion 69 and 70
S5	Data set	Example data set from Ecoregion 69 and 70
S6	XC95 examples	Example XC95 values for 176 distinct aquatic benthic invertebrate genera
S7	Scatter plots	Example plots of probability of observing for each of 176 genera
S8	CFD plots	Example plots of cumulative proportion of occurrence of each of 176 genera
S9	Taxa maps	Geographic distributions of 462 genera from an example data set

Key points.

1. Two downloadable Excel® Biological Extirpation Analysis Tools are described.

2. They estimate extirpation concentrations (XC₉₅) and 5% extirpation of assemblage (HC₀₅).

3. We provide access to example calculations, output XC₉₅ values, biogeographical maps, and data set.

4. We discuss some limitations to help avoid misinterpretations and inferential errors.