Page 1

Appendix C. Analyses of sensitivity distributions f or estimation of acute hazard concentrations to aquatic animals

Matthew Etterson US Environmental Protection Agency
National Health and Environmental Effects Research Laboratory Mid-Continent Ecology Division Duluth, MN

20 December 2011Page 2



## Contents

1. Introduction ...................................

................................................... .................................................. 5

1.1. Data .........................................

................................................... .................................................. 5
1.2. Distribution fitting ..........................

................................................... ........................................... 6

1.2.1. Parameter estimation ........................

................................................... ................................. 6

1.2.2. Goodness-of-fit .............................

................................................... ...................................... 7

1.3. Inference from the fitted distribution .......

................................................... ................................. 8
1.3.1. Sampling variance ..........................

................................................... ..................................... 8

1.3.2. Confidence limits ..........................

................................................... ...................................... 9

1.3.3. Model selection & multidistributional inferen ce ................................................ ..................... 9
1.4. Objectives ...................................

................................................... ............................................... 9

2. Methods ........................................

................................................... ................................................. 1 0
2.1. Data & statistical comparisons .................

................................................... ................................ 10

2.1.1. Simulation data ............................

................................................... ..................................... 10
2.1.2. Empirical data .............................

................................................... ...................................... 11
2.1.3. Statistical comparisons ......................

................................................... ................................ 12

2.2. Statistical methods ...........................

................................................... ....................................... 13

2.2.1. Random variates ............................

................................................... ................................... 13

2.2.2. The HC 5 .................................................. ................................................... ........................... 13

2.2.3. Maximum likelihood estimation ...............

................................................... ......................... 13

2.2.4. Moment estimation ...........................

................................................... ............................... 14
2.2.5. Graphical estimation .......................

................................................... .................................. 14

2.2.6. Transformations .............................

................................................... ................................... 15

2.2.7. Bias .......................................

................................................... ............................................ 15
2.2.8. Bootstrap sampling ...........................

................................................... ................................ 16

2.2.9. Goodness-of-fit .............................

................................................... .................................... 16
2.2.10. Sampling variance .........................

................................................... .................................. 17

2.2.11. Confidence limits .........................

................................................... .................................... 17

2.2.12. Model selection.............................

................................................... .................................. 17
2.2.13. Model averaging ...........................

................................................... .................................. 18Page 3

2.2.14. Software & distributions....................

................................................... .............................. 18
2.2.15. Equations for Distributions .................

................................................... ............................. 20

3. Results & Discussion ..............................

................................................... ......................................... 27
3.1. How does sample size influence bias and variance of the estimated HC 5 ? ................................... 27

3.1.1. Bias .......................................

................................................... ............................................ 27

3.1.2. Variance ...................................

................................................... ......................................... 30
3.2. How do minimum data requirements influence bias and variance of the estimated HC 5 ? ............ 31

3.2.1. Bias .......................................

................................................... ............................................ 32

3.2.2. Variance ...................................

................................................... ......................................... 39

3.3. How do different estimation methods influence bi as and variance of the estimated HC 5 ? ........... 42

3.3.1. Bias .......................................

................................................... ............................................ 42

3.3.2. Variance ...................................

................................................... ......................................... 43
3.4. Are measures of fit (AIC, goodness-of-fit) reliab le indicators of performance? ...................... ...... 43

3.4.1. Akaike’s Information Criterion ..............

................................................... ............................. 43

3.4.2. Goodness-of-fit .............................

................................................... .................................... 44
3.5. What are actual coverage rates of 95% confidenc e limits for subsamples? ............................ ...... 47
3.6. How do different candidate distributions perform relative to each other? .......................... ........ 49
3.7. Does model-averaging across distributions improve e stimates of the HC 5 ? ................................. 51
4. Implications of these results for the use of SSD s in estimating the HC 5 ............................................... 53
4.1. Implications of sample size results for estimating the HC 5 .................................................. ......... 53
4.2. Implications of MDRs for estimating the HC 5 .................................................. ............................. 54
4.3. Implications of choice of estimation method for estimating the HC 5 ........................................... 54
4.4. Implications of performance of fit measures for e stimating the HC 5 ............................................ 54
4.5. Implications of the performance of 95% confiden ce limits for estimating the HC 5 ....................... 54
4.5. Implications of choice of distribution for estim ating the HC 5 .................................................. ..... 55
4.6. Implications of multidistributional inference fo r estimating the HC 5 ............................................ 55
5. What future research is suggested by these results? .

................................................... ...................... 55

5.1. Hierarchical models ..........................

................................................... ....................................... 56

5.2. Order statistics ...............................

................................................... .......................................... 57
5.3. Bayesian methods ..............................

................................................... ...................................... 58
6. Acknowledgments ................................

................................................... .......................................... 58

7. Literature cited ...............................

................................................... ................................................ 58Page 4

8. Supplementary results ...........................

................................................... ......................................... 61

8.1. HC 5 s estimated using all available GMAVs ..............

................................................... .................. 61

8.2. HC 5 s estimated using all available SMAVs...............

................................................... .................. 63
8.3. Comparison plots of analyses at n = 8 (ALC MDRs imposed versus not imposed) ......................... 65
8.4. Variance plots of analyses at n = 8 (ALC MDRs impo sed versus not imposed) .............................. 71
8.5. Comparison plots of analyses at n = 3 (random su bset versus 3 typical OPP species) ................... 82
8.6. Variance plots of analyses at n = 3 using random SMAVs ............................................. ................ 90

8.7. Goodness-of-fit results for n = 20 and n = 50. ............................................ .................................. 99

8.8. Model selection results for SSDs fit to GMAVs when N = 8 and ALC MDRs are met .................... 100

8.9. Model selection results for SSDs fit to GMAVs when N = 8 regardless of ALC MDRs................... 102

8.9. Preliminary analysis of the use of MOA-specific parameters for the logistic distribution ........... 104Page 5



## 1. Introduction
Species sensitivity distributions are a common tool u sed for setting safe limits on chemical concentrations in surface waters (Posthuma et al. 2002 , Suter 2002, Chapman et al. 2007, TenBrook et al. 2010). Although the analysis and interpretation of species sensitivity distributions varies widely, the basic methodology is quite general and can be summar ized as a three-step procedure. First, results from separate toxicity tests on a given chemical usin g various aquatic animal species are compiled. Second, a statistical distribution to which the test results are thought to conform is chosen and fit to t he data. Third, the fitted distribution is used to infer a concentration that will be protective of a desir ed proportion of species in an hypothetical aquatic comm unity. The procedure described above necessarily relies upon p olicy decisions, for example, concerning the proportion of species that should be protected and the necessary level of confidence with which the protective concentration is identified. This repor t focuses on fitting statistical distributions to the resu lts of toxicity tests. As such its content concerns the three steps laid out in the previous paragraph, each of which can be accomplished in different ways, with dif ferent results. This report does not consider the underlying policy decisions required for application of species sensitivity distributions to regulatory decision making. The following subsections give some hi storical background regarding data compilation, distribution fitting, and inference from fitted sensi tivity distributions, in particular as they relate to the development of a common risk assessment methodology f or use by USEPAs Office of Pesticide Programs (OPP)and Office of Water (OW). The focus of analyses in this report is on fitting SSDs to chemicals for which the minimum data are available to fulfill FIFRA requirements; howeve r, insufficient data are available to derive water quality criteria according to the methods described in the 1985 Guidelines. The inferential endpoint fo r all analyses is taken to be the fifth percentile of t he fitted SSD, or HC 5 . However, when data are limited,

HC 5 estimates may be subject to varying degrees of uncert ainty. Therefore, the focus of these analyses is on the bias and uncertainty of using SSDs to derive H C 5 values at varying sample sizes.



### 1.1. Data For this report, data are assumed to be the results of acute toxicity tests using aquatic animals. These are EC50s and LC50s from 96-hour continuous exp osures (48 hrs for Cladocerans and a few other species; USEPA 1985, ASTM 2011). An important consid eration in data compilation is the distribution of available data across species. Strictly speaking, an a ssumption of all the methods considered below is that the data (test results) pertain to a random sample of species from the community for which the analysis is intended to apply. Obviously this assumption is violated, in part due to constraints on our knowledge of husbandry of aquatic animals. Thus a r elatively limited subset of aquatic animals makes up the greater part of all aquatic toxicity tests. N evertheless, several important taxonomic requirements are commonly applied to the compilation and analysi s of multi-taxon toxicity test results. For this report, two data subsets are of particular interest, t he eight minimum data requirements (MDRs) of the 1985 Guidelines (USEPA 1985) and the three typical sp ecies used for OPP benchmark analysis. For freshwater tests the latter are rainbow trout ( Oncorhynchus mykiss ), bluegill sunfish ( LepomisPage 6

macrochirus ) and Daphnia magna . For saltwater tests the three typical OPP benchmar k species are sheepshead minnow ( Cyprinodon variegatus ), mysid shrimp ( Americamysis bahia ), and Eastern oyster ( Crassostrea virginica ).



### 1.2. Distribution fitting Many statistical distributions have been used for fitt ing SSDs ( e.g , log-normal, log-logistic, Burr III , etc.); however, several analyses have shown that no one distribution is preferred across datasets (Newman et al. 2000, Zajdlik & Associates 2005, Chap man et al. 2007). Deciding which statistical
distribution to fit to a set of data has been describ ed as one of the most important and difficult choices in the use of species sensitivity distributions (Chapman et al. 2007). Two important decisions must be made when fitting a distribution to empirical data. The first concerns how the distribution will be fit t o the data, which in this report is equivalent to the problem of parameter estimation. The second choice concerns how to assess the quality or accuracy of the f itted distribution as a general representation of the data, or goodness-of-fit. A related concern inv olves deciding how to choose among the fitted distributions when multiple distributions are fit to th e same data. Also, for many types of distributions ( e.g., normal, logistic, triangular) data are transformed pr ior to analysis, most frequently using the common log (log 10 ) transformation. This complicates comparisons between distributions fit to
transformed versus untransformed data. Three methods commonly used for estimating the paramet ers of SSDs are considered below. These are maximum likelihood, moment estimators, and graphical methods (these methods are described in more detail in Methods subsections 2.2.3, 2.2.4, and 2.2.5 below). It should be noted that
not all methods can be used with all distributions. I t is likely that some methods will perform better
than others when data are limited and when the chose n statistical distribution is a poor approximation of the underlying data distribution. Further, it is likely that rank of performance will change under different data conditions, perhaps in predictable way s. Newman et al. (2000) recommended a non-parametric method for fitting SSDs using empirical bootstrapping (Section 2.4.2 below). However, given our current interest in the fifth percentile of acu te values (see Section 1.3, below), bootstrap estimation does not seem feasible because it would require at least 20 data points to estimate the fifth percentile of the empirical distribution function. Bootstrap methods are used below to test goodness-of-fit and to e stimate sampling variance. Both Chapman et al. (2007) and TenBrook et al. (20 09) emphasize the importance of visual inspection of fitted distributions against the empiric al data to which they are fit. Such methods are omitted from this report because the focus of this expl oration is on general patterns of fit that emerge from fitting thousands of replicate data sets. Howe ver, visual inspection of data and fit would be an important step in any particular application of SSDs to empirical data.

1.2.1. Parameter estimation
Maximum likelihood methods for SSDs were first tested by Kooijman (1987) for the logistic distribution who reported substantial bias in estimation of the scale parameter ( β ) with sample sizes ≤ 5. Shao (2000) described maximum likelihood estimators f or the Burr III distribution and these estimators are implemented by the software BurrliOZ (Campbell et al. 2000). Several recent reports on the application and analysis of species sensitivity distri butions have employed maximum likelihood withPage 7

other distributions (Zajdlik and Associates 2005, Chapm an et al. 2007), often citing the first of the following desirable properties. First, when data fi t the assumed distribution, maximum likelihood parameter estimates are the most efficient parameter estimates possible ( i.e. , the estimates with the least sampling variance, Edwards 1992). Second, the u se of maximum likelihood allows the fit of different distributions to be compared using informat ion theoretic methods for comparing models (Burnham and Anderson 2002). Third, use of maximum l ikelihood allows model-averaging of estimated quantiles (Burnham and Anderson 2002). Fourth, maxi mum likelihood and restricted maximum likelihood (Harville 1977), allow specification of h ierarchical models that may otherwise be very diffic ult to fit. Fifth, the development of likelihood-based e stimators provides a link towards the development of Bayesian methods for SSDs. Finally, maximum likeliho od methods offer a generalized means for extending the advantages of the graphical methods (be low) to any continuous distribution. The latter three points are the subject of proposed future work ( Section 5). Moment estimators are a common method for fitting SS Ds (Kooijman 1987, Van Straalen and Deneman 1989). In practice, they work by equating the mean and variance of a sample to the parametric mean and variance of a chosen distributio n, which are functions of the parameters of that distribution. This creates two equations in two unkno wns, which can then be solved for the unknown parameters. The resulting solution is an estimate of t he parameters of the distribution expressed as functions of the sample mean and sample variance. Alt hough this procedure has been described in terms of the mean and variance (the first two moments) , it could be extended to higher moments as well if a distribution ( e.g. , Burr) has more than two parameters. Graphical methods for use in SSDs were described by E rickson and Stephan (1988). They are a subset of the general theory of order statistics (Arnold et al. 2008) and have two unique attributes that make them very attractive for use in fitting SSDs (Te nBrook et al. 2008). First, once data have been ordered and the empirical percentiles obtained, the linear estimation model can be weighted toward the lower tail of the distribution, which is generall y the portion of the distribution of interest for regulation and risk assessment. This can alleviate po tential biases resulting from right-skewed toxicity distributions. Second and related to the first, test results that are right-censored (known only to be greater than the highest tested concentration) can o ften be accommodated (Erickson and Stephan 1988). Graphical methods can be used on any distribu tion in the location/scale family. Of the seven distributions described below (Section 2.2.13), the n ormal, logistic, and triangular are location/scale distributions. Thursby et al. (Appendix F) show how si milar linearization methods can be extended to distributions outside the location/scale family. Secti on 5 (future work) contains descriptions of some potential generalizations of ordered methods that wou ld extend their applicability to any continuous
distribution.

1.2.2. Goodness-of-fit
Goodness-of-fit is a measure of how well an assumed dist ribution fits a set of data, given the data and the values of the estimated distributional p arameters. Goodness-of-fit tests are applied after a
distribution has been fit to a set of data to evaluate the extent to which the fitted distribution is a g ood representation of the observed data. Tests for goodness -of-fit can be divided into parametric and nonparametric methods. In either case, the test begins with the definition of a test statistic that can be reliably predicted to increase in magnitude with lac k of fit (e.g., the summed discrepancy between thePage 8

empirical distribution function and the cumulative distribution function, evaluated at all data points, is used below). With parametric goodness-of-fit tests, a theoretical distribution for the test statistic can be derived, and probabilities are estimated from that theoretical distribution. In general, the derivati on of the theoretical distribution of the test statistic depe nds on asymptotic convergence and upon the hypothesized distribution for the data. Therefore pa rametric goodness-of-fit tests tend not to work well at small sample sizes and generally apply to only one distribution (often the normal distribution). In contrast, non-parametric methods often work by statisti cal resampling methods (Efron and Tibshirani 1994) and probabilities are assessed as simple ranks of o bserved statistics among a set of simulated statistics. They can be applied to any continuous distr ibution and are valid regardless of sample size. However, both parametric and non-parametric methods lack power at small sample sizes. Luttik and Aldenberg (1997), Aldenberg and Luttik ( 2002), and Newman et al. (2000) all considered parametric goodness-of-fit tests. Zajdlik & Associates (2005) recommended the AndersonDarling test for all distributions, except the norma l and log-normal distributions for which they recommended the Shapiro-Wilks test. Chapman et al. (2 007) carried out extensive simulations of the power properties of goodness-of-fit tests for the norm al distribution and concluded that power to detect non-normality (lack of fit) was extremely lo w, especially at sample sizes < 20, which correspond to both cases of particular interest in this document: n = 3 and n = 8). Newman et al. (2000) and Shao (2000) employed non-p arametric goodness-of-fit tests based on empirical bootstrap sampling (Efron and Tibshirani 19 94, Manly 1997). Chapman et al. (2007) describe a parametric bootstrap procedure (also described by Efr on and Tibshirani 1994), but do not apply it to goodness-of-fit testing for SSDs. Because of their ut ility at all sample sizes, and applicability to all continuous distributions, only bootstrap methods are con sidered below.



### 1.3. Inference from the fitted distribution All analyses of parametric species sensitivity distrib utions begin by estimating the parameters of the distribution (see above). Thus the distributional parameters are a universal inferential endpoint (excluding non-parametric SSDs, Newman et al. 2000) . Once the parameters are estimated, a given percentile (p) of the distribution is often chosen to represent the concentration at which (no more than) p% of species will be at risk of adverse effects, refe rred to herein as the HC p . Regardless of how the distribution is fit, an HC p is easily estimated using the quantile function for the fitted distribution. However, estimates of percentiles are subject to bia s (if the distribution doesn’t fit the data very well ) and uncertainty (especially when the number of test results are limited). Methods for handling these aspects of distribution fitting vary widely in the SS D literature. Erickson and Stephan (1988) also point ed out that, by Jensen’s inequality, an unbiased estimator for the HC p , might be a biased estimator of the percentile (intended to be p) of species protected a t the estimated HC p if the quantile function is nonlinear in p (as is generally the case).

1.3.1. Sampling variance When distributional parameters are estimated from em pirical data, the estimated parameters are subject to sampling variance. In other words, giv en data that conform to a specified distribution, if equal sized (but different) sets of data are drawn fr om the same distribution, the parameter estimates will differ with each set of data, resulting in a dist ribution of parameter estimates. The variances of thePage 9

distributions of parameter estimates are termed the sam pling variances of the parameters. If the estimation procedure is unbiased the average paramete r estimates will be arbitrarily close to the ‘true’ values if the procedure is repeated a sufficient numbe r of times. However, the expected variance in these parameter estimates may be quite large and is ge nerally inversely related to the size of the sample. This sampling variance is present in all three fitting techniques described above (maximum likelihood, moment estimators and graphical methods) , though it may differ among techniques. Sampling variance of parameter estimates translates d irectly (though not necessarily linearly) into sampling variance around quantiles of a distribution ( i.e. , the estimated HC 5 ). Common methods for estimating sampling variance around quantiles in an SS D include the delta method (Seber 1982, Shao 2000) and the bootstrap (Newman et al. 2000).

1.3.2. Confidence limits Confidence limits for an estimated percentile are an alternative expression of sampling variance in the model parameters. These can be one-sided exp ressions of confidence that the true HC p is greater than a specified concentration (Kooijman 1987, van S traalen and Deneman 1989, Aldenberg and Slob 1993) or two-sided limits related to the probability that the region defined by a lower and upper bound
would contain the true HC p (Shao 2000, Newman et al. 2000). Many methods for confidence interval development rely on either asymptotic properties of e stimators or standard (mean zero and variance 1) forms of distributions. Only one-sided confidence in tervals generated using extrapolation constants derived from standard forms of the logistic, normal, and triangular distributions (Aldenberg and Slob 1993, Aldenberg and Jaworska 2000, Pennington 2003) are considered below.

1.3.3. Model selection & multidistributional infere nce Many researchers have discussed the important (and diff icult) choice of which distribution to employ for an SSD (Newman et al. 2000, Zajdlik & Asso ciates 2005, Chapman et al. 2007). Assessing the goodness-of-fit of a distribution provides only limit ed information for comparing distributions because discrepancies of fit will generally decrease monotoni cally with increasing number of estimated parameters. Yet an over-parameterized model may hav e poor predictive ability. Formal model selection criteria impose a penalty for each estimate d parameter, which creates a tradeoff between parsimony and fit. The tendency for species sensitivit y distributions to have two estimated parameters (though the Burr III distribution has three) alleviates this concern somewh at. However, model selection methods are also useful for ranking the performance of alternative distributions and for formally averaging model predictions when multiple models are fit (Burnham and Anderson 2002). Elphick (2011) used AIC to compare several candidate distributions, i ncluding log-normal, log-logistic, log-gumbel and Weibull and report similar performance.



### 1.4. Objectives The primary objective of the work described in this report is to compare the performance of different distributions and estimation techniques comm only used in SSD analyses. In particular, performance is evaluated as the ability to accuratel y estimate the HC p (in particular the HC 5 ). Accuracy is assumed to incorporate both bias and sampling varianc e. While a formal definition of accuracy specifies that accuracy is the sum of sampling variance and squared bias (Williams et al. 2002), the term is used more loosely here to indicate that increases i n either bias or sampling variance of an estimate ofPage 10

the HC 5 will decrease the accuracy of that estimate. The wo rk described below can be divided into two parts. First, the performance of different estimatio n methods at different sample sizes is evaluated
when the data are derived from a known distribution using simulated data with a known HC 5 (Statistical comparison 2.3.1). Second, seven distributions are fit using three fitting methods to empirical toxicity data on acetylcholinesterase inhibitors (Statistical comparisons 2.3.2 – 2.3.6) for which the true
distribution and HC 5 are unknown. Presentation of the results of these ana lyses is organized around specific questions that will help to inform the use of species sensitivity distributions in the context of work done by the Office of Water and the Office of Pesticide Programs. These are:

1. How does sample size influence accuracy (bias and var iance) of the estimated HC 5 ?

2. How do minimum data requirements influence accuracy of the estimated HC 5 ?

3. How do different estimation methods influence accura cy of the estimated HC 5 ?

4. Are measures of fit (goodness-of-fit, Akaike’s Informa tion Criterion) reliable indicators of performance?

5. What are actual coverage rates of 95% confidence lim its for subsamples?

6. How do different candidate distributions perform rela tive to each other?

7. Does model-averaging across distributions improve estima tes of the HC 5 ?

8. What future research is suggested by these results?

Specific comparisons and analyses designed to answer th ese questions are described in detail below.



## 2. Methods

2.1. Data & statistical comparisons

2.1.1. Simulation data To evaluate the performance of distributions and fitt ing methods under known conditions, data were simulated from known parametric forms of seven distributions. Simulated sample sizes were 3, 8, 20, 50, and 100 mean acute values. Each simulated d ata set was replicated 10,000 times. Approximate parameter values for generating all simulated data a re presented in Table 1.

Table 1. Approximate parameters used to simulate acu te toxicity data

Distribution Parameters Fifth percentile
normal μ = 2.77, σ = 1.42 2.65 logistic α = 2.78, β = 0.85 1.94 triangular a =
- 0.59, b = 5.92 2.75 Gumbel μ = 2.05, β = 1.36 3.61 Weibull λ = 2940.47, k =0.34 0.43
Pareto x m = 0.70, α =0.15 0.99
Burr a = 334.71, b = 0.48, c = 1.20 2.35

The first three distributions (normal, logistic, and triangular) are probably the most common distributions used in SSDs and have a familiar sigmoid cumulative distribution function (Fig. 1). ThePage 11

Gumbel distribution is similar to the previous three, but slightly more right-skewed (Fig. 1). The Weibu ll and Pareto distributions were chosen to represent ‘out lier’ distributions that might be expected a priori to perform poorly. The Weibull is left-skewed relat ive to the other distributions, and the Pareto is shaped quite differently, with a monotonically decli ning slope to its cumulative distribution function (Fig. 1). Finally, the Burr distribution is a thre e parameter distribution with a sigmoid cumulative distribution function and is the primary distribution behind the BurrliOZ software (Campbell et al. 2000)

Figure 1. Plots of the cumulative distribution functi ons for seven distributions using the parameters from Table 1, a. in the region of the HC 5 , and b. over the full distribution. Note the log 10 scale of the
horizontal axis.

2.1.2. Empirical data Acute toxicity test results for seven organophosphate i nsecticides and three carbamates were obtained from Web-ICE (Raimondo et al. 2010). These chemicals were selected because their data sets fulfilled the data requirements defined under FIFRA and the 1985 Guidelines. Results are separated by
freshwater versus saltwater animals. Pesticide names an d sample sizes are given in Table 2. For empirical analyses, these data can be summarized at tw o levels, the species and the genus. When more than one toxicity test result is available for a given species, the datum for that species is taken to be the geometric mean of all such results (hereafter the speci es mean acute value or SMAV). When analysis is performed on genera, and more than one SMAV exists fo r a given genus, then the datum for that genus is taken to be the geometric mean of all such SMAVs (h ereafter the genus mean acute value, or GMAV).

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 0 0.02 0.04 0.06 0.08 0.1Cumulative Probability a. triangular normal logistic gumbel Pareto Weibull Burr

-5 0 5 10 15 0 0.2 0.4 0.6 0.8 1

Log 10 LC50Cumulative Probability b.Page 12

Summarizing the test results by genus has the effect of reducing the influence of closely related test species on the estimated HC 5 . These methods for summarizing taxonomic replication have a strong historical background in the development of U.S. aqu atic life water quality criteria (ALC) (USEPA 1985)

. However, they ignore some information when taxonomic replication is available. Alternative methods for handling replicate test results for a given taxon are suggested in Section 5 (Future analyses). It should be noted that in separate analyses with thes e data, Erickson (Appendix D) has found some errors and inconsistencies. Thus, these analyses sho uld be considered preliminary pending additional data quality review.

Table 2. Acute test results available for analysis

Class Chemical Water Test Results SMAVs GMAVs

Organophosphates malathion FW 174 71 57

diazinon FW 84 33 28 chlorpyrifos FW 100 34 30
fenitrothion FW 113 36 30 methyl parathion FW 71 35 29
dichlorvos FW 55 25 22 chlorpyrifos SW 108 26 21
fenthion SW 22 13 12

Carbamates carbaryl FW 290 75 55

carbaryl SW 23 11 9

propoxur FW 21 13 13 methomyl FW 56 20 16

2.1.3. Statistical comparisons The empirical and simulation data described above we re used to carry out numerous evaluations of components of SSD methodology. The sp ecific intent of each type of comparison is described below, and the primary inferential endpoin t is the estimated HC 5 . 1. Performance when the data conform to a known distrib ution. For this evaluation,

only the parametric simulation data (Table 1, Secti on 2.2.1 above) were used. All seven distributions using all available fitting method s (Table 3) were fit to every replicate sample of simulated data.

2. Baseline empirical performance for the maximal sets of SMAVs and GMAVs. For this

evaluation the empirical data (Table 2) were used. All seven distributions using all available fitting methods (Table 3) were fit to ever y available combination of pesticide and water type. The resulting HC 5 estimates are the reference HC 5 s against which the

HC 5 estimates generated under comparisons 3-6 (immediate ly below) are compared.

3. Comparative performance at n = 8 without regard to ALC MDRs. For this evaluation

1,000 subsamples of 8 GMAVs from each empirical data set (Table 2) were used.Page 13

4. Comparative performance at n = 8 when ALC MDRs are met. This evaluation was

similar to (3) except that the 1,000 GMAVs were chose n so that the eight minimum data requirements (MDRs) of the 1985 guidelines (USE PA 1985) were always satisfied.

5. Comparative performance at n = 3 SMAVs without regard to OPP benchmark species.

This evaluation was similar to (3) except that the 1 ,000 subsamples were of size n = 3.

6. Comparative performance at n = 3 for the OPP benchmark species. This evaluation

was similar to (5) except that there is only one subset of data conprising the three
benchmark SMAVs.

Table 3. Distributions and fitting methods used for th e analyses below.

Distribution Maximum

Likelihood Moment

Estimators Graphical

Estimators
normal yes yes yes logistic yes yes yes triangular yes yes yes Gumbel yes yes no Weibull yes no no Pareto yes yes no Burr III yes no no



### 2.2. Statistical methods This section is provided for completeness and to all ow interested readers to evaluate the statistical methods. However, the results section has be en written to be read and understood without relying on this material.

2.2.1. Random variates Random variates were generated from a specific distr ibution (with parameters given in Table 1) by first drawing a vector of uniform random variates of the desired sample size. The uniform random variates were then used as input to the quantile func tion for the distribution together with the specifie d parameter values. The resulting output is a vector of random variates from the specified distribution. Quantile functions for all distributions used are presen ted in Section 2.2.15, below.

2.2.2. The HC 5 For the purposes of this report, the inferential endp oint for all analyses is the HC 5 . Regardless of sample size and method of estimation, the HC 5 is estimated as the output of the quantile function for the specified distribution and estimated parameters ev aluated at 0.05. Quantile functions for all distributions used are presented in Section 2.2.15, be low.

2.2.3. Maximum likelihood estimation Log-likelihood equations were developed for each of the seven distributions as the natural logarithm of the probability density function ( f ) for the distribution. The resulting log-likelihood was summed over all data points:Page 14



## ( ) ( ) 

# ( )

1 

### | ln |

n

i i 

### L f x

= 

### ∝ 

# ∑ θ X θ

Maximum likelihood estimates (MLEs) of parameters were obtained by numerically searching for the parameter values that maximized the log-likelihood o ver all data points (equivalently minimized the negative log-likelihood). MLEs were always estimated b y numerical search, even when closed-form solutions to the MLEs exist. This was done to subject M LEs for all distributions to the same (albeit minor) uncertainty associated with numerical estimati on. Likelihood expressions are presented for each
distribution in Section 2.2.15.

2.2.4. Moment estimation Moment estimators were derived, wherever possible, b y setting the expected distributional mean and variance equal to the sample mean and varia nce, and solving for the distributional parameters. For example, let 

### x and s 2 represent the sample mean and variance, respectively

(regardless of distribution). The mean and variance of a logistic distribution are α and 2 2 

### 3 π β . Setting



### x 

### α = and 2

2 2 

### 3 s π β 

### = and solving for α and β results in the two moment estimators 

### ˆ x α 

### = and



### ˆ 3 s 

### β
π = , where the circumflex over the parameter symbols i ndicates that they are estimated

quantities. For the Weibull distribution, moment est imators could not be derived because the equations for the mean and variance contain gamma fu nctions of the Weibull parameters. For the Burr distribution, moment estimators were not derived becau se the Burr distribution has three parameters, requiring three equations, but an equation for the t hird moment of the Burr distribution was not immediately available. Moment estimators for the re maining five distributions (normal, logistic, triangular, Gumbel, and Pareto) are presented in Sec tion 2.2.15, below.

2.2.5. Graphical estimation Graphical estimation works by exploiting a function al relationship between the parametric quantiles of a standard form of a distribution (para meters chosen so that the distribution has mean 0 and variance 1) to the quantiles of the empirical di stribution function. This allows the mean and variance of the sample to be estimated, from which moment esti mators may be used, in turn, to estimate the distributional parameters. This procedure is described in detail by Erickson and Stephan (1988). This procedure is only applied to distributions in the loca tion-scale family, which includes the normal, logistic, and triangular distributions among the candi date distributions considered in this report. Note that, Thursby et al. (Appendix F) apply related line arization methods to some distributions outside the location/scale family. Parameters for standard form s of normal, logistic, and triangular distributions a re presented in Section 2.2.15, below. For this report, all data points were weighted equally to minimize the number of possible explanations for differences in performance between graphical estimates and the other two methods (maximum likelihood and moment estimators).Page 15

2.2.6. Transformations When the normal, logistic, triangular, or Gumbel dist ribution was used, the data were first common-log transformed (log 10 ). When the Weibull, Pareto, or Burr distribution w as used, the data were untransformed. This complicates comparisons among distributions, especially using maximum likelihood and AIC. To solve this problem, the likel ihoods for the normal, logistic, triangular, and Gum bel distributions were reformulated as follows. First, let :



## ( ) 10 

### log y x =

. Therefore the cumulative distribution functions for t he four distributions using log 10 -transformed data

are of the form: 

## ( )

| F y θ . Thus, the probability density functions for the un transformed data ( x ) can be
calculated using the product rule. 

## ( ) ( ) ( ) ( ) ( ) 1 | | | | ln 10 d d dy f x F y F y f y

dx dy dx x = = = θ θ θ θ

In the above equation, the expressions 

## ( ) | f y θ are the probability densities for the log 10 -transformed
data for the respective distributions. When the likel ihood is maximized over the transformed data the

transformation factor 

## ( )

1
ln 10 x can be ignored for the purposes of obtaining the MLE s because the

transformation factor does not contain the parameter s of interest. However, to compare distributions using AIC the factor must be included so that the AIC statistics are on the same scale. Thus, for distributions on log 10 -transformed data:


## ( ) ( ) ( ) ( ) ( ) ( ) 

# ( )

1 1 

### 1 | ln | | ln 10 ln ln 10

ln 10 n n

i i i

i 

### L f y L y n

x

= = 

###  

∝ = − −     

# ∑ ∑ θ X θ θ Y

2.2.7. Bias Bias was estimated as the expected value of the dif ference between the estimated HC 5 and the known value. Specifically, bias was estimated as:



###  

# ( )  

# ( )

5 5 5 

### bias HC E HC HC

= −

In the above equation, the notation 

### 

5 

### HC and 5 

### HC denote the estimated and known values of the HC 5 , respectively. These were calculated differently depe nding on whether the distributions were applied to transformed versus untransformed data, respectively. Wh en a distribution was fit to untransformed data (Weibull, Pareto, Burr), the known HC 5 is:



## ( ) 1 5 

### 0.05 | HC F − 

### = θ

Above, F

-1 is the quantile function for the distribution and 

### θ is a vector of known parameters. When a distribution is fit to transformed data (normal, logis tic, triangular, Gumbel), the known HC 5 is:
( ) 1 0.05|
5 

### 10 F 

### HC −


### = θ

Similarly, when a distribution was fit to untransforme d data the expected value of the HC 5 was calculated as the simple average of the fifth perce ntiles across all iterations of the simulation:Page 16 

###  

# ( ) ( ) ( )



# ( ) 10,000 1 1 5 1 

### 1
ˆ ˆ

0.05 | 0.05 | 10,000 r r

r 

### E HC F E F − −

= 

### = = 

# ∑ θ θ

However, when a distribution on transformed data was f it, the resulting set of estimated values of the

HC 5 is non-normally distributed and the arithmetic mea n of the estimated HC 5 s is a biased estimator of the expected HC 5 . In this case the expected HC 5 was estimated as the geometric mean of HC 5 estimates,

or:



###  

# ( ) ( ) ( ) ( ) 

## ( ) 10,000 1 1 1
1 1 10,000

ˆ 10,000 0.05| ˆ

ˆ 0.05| 0.05| 10,000 5 1 

### 10 10 10 r r r r F E F F

r 

### E HC −

−

−

=

= 

###  

∑  

= = =

    

# ∏ θ θ

θ

When discussing bias in the results the term bias is use d in two ways, referring both to bias and its absolute magnitude (i.e., |bias|), where the meaning is clear from the context. Bias cannot be assessed for analyses of empirical data because the true HC 5 s are not known. Yet it is still of interest to compare the mean performan ce of the estimators when data requirements are imposed and when they are not. To do so, an HC 5 was estimated for each chemical using ‘all available
data’. To examine the effect of the ALC MDRs ‘all a vailable data’ were taken to be the set of all GMAVs for a given chemical. To examine the effect of the typical three OPP species ‘all available data’ were taken to be the set of all SMAVs for a given chemica l. Thus all calculations proceeded as described
above, but with the HC 5 calculated using all available data substituted for the known 5 

### HC

. For easier interpretation, bias is not presented as de fined above. Rather it is presented as the

ratio of the estimated HC 5 to the known HC 5 : 

### 



# ( )

5 5 

### / E HC HC . In figures, this is further presented on

the log 10 scale, in which case the y-axis tick marks represent over- or under-estimation by each successive order of magnitude. In tables, the untran sformed ratio is given.

2.2.8. Bootstrap sampling Bootstrap resampling was used to assess goodness-of-fit a nd sampling variance (next two sections). With both empirical and parametric bootstr ap sampling the methodology depends on generating replicate sets of data based on the referen ce data being analyzed (below, for demonstration purposes, 100 replicates were used). The reference da ta is the actual data set being analyzed and in this report it may be the simulation data (Table 1, Secti on 2.1.1 above) or it may be a particular set of empirical data (Table 2, Section 2.1.2). The proce ss begins after a particular distribution is fitted to t he
reference data. The new sets of data are the same si ze as the original data and are generated by drawing (with replacement) new samples from the origi nal data (empirical bootstrap) or by drawing from the fitted distribution (parametric bootstrap). To these new data sets, the same distribution is fit, generating a distribution of fits under the hypothesi s that the data fit the specified distribution with th e estimated parameter values for the original data.

2.2.9. Goodness-of-fit Goodness-of-fit was assessed using both empirical and pa rametric bootstrap resampling (Section 2.2.8. above). The summed (over all data p oints) squared distance between the empiricalPage 17

distribution function (EDF) and the cumulative distri bution function (CDF) for the fitted distribution wa s used as a summary measure of fit. Large values of this statistic indicate poorer fit, whereas small values indicate better fit. Using the bootstrapping proced ures 100 values of the fit statistic were generated t o which the observed fit statistic from the reference d ata was compared. The proportion of these bootstrapped fit statistics that were greater than or equal to the observed fit statistic for the reference data was interpreted as the P-value for lack of fit (small P-values indicate significant lack of fit). T his level of replication (100 iterations) will give poor resol ution on the P-value for any single data set, but wi ll still allow precise estimation of average behavior of the goodness-of-fit algorithm over 10,000 or 1,000 samples of simulation or empirical data, respectively

.
2.2.10. Sampling variance Sampling variance for distributional parameters and H C 5 estimates was estimated using both empirical and parametric bootstrap methods to genera te 100 values of each statistic. Sampling variance for a given estimated value was estimated as the varian ce among the 100 bootstrap samples for the

same statistic.

2.2.11. Confidence limits
One-sided 95% confidence limits for the HC 5 for the logistic, normal, and triangular distributi ons were generated using extrapolation constants provided by Aldenberg and Slob (1993), Aldenberg and Jaworska (2000) and Pennington (2003). These are in tended to be expressions of 95% confidence that
the true HC 5 is greater than the calculated confidence limit. The formula for generating the confidence
limit is: 

### L x ks = − In the above equation, L

is the lower 95% confidence limit, 

### x is the sample mean, s is the sample standard deviation, and k is an extrapolation constant that is specific to both the distribution fitted (logistic, normal, or triangular) and sample-size. T hese confidence limits were generated only for n = 3
and n = 8.

2.2.12. Model selection Whenever multiple distributions were fit using maximu m likelihood to the same set of data (whether simulated or empirical) Akaike’s Informatio n Criterion (with correction for sample size: AIC c ) was used as a relative measure of the performance of di stributions. The formula for AIC c is:

c 

### AIC 2 2 1

n L K n K   = − +  

− −  
In the above equation, L is the maximized log-likelihood function, n is the sample size, and K is the number of parameters estimated in fitting the distribu tion. The second term on the right hand side of the above equation is a penalty term. It increases t he AIC c statistic with each additional parameter estimated. Because the quotient within the parenthe ses is zero or negative whenever n ≤ K + 1, AIC c cannot be applied to such cases. Unfortunately this in cludes all cases when n = 3, corresponding to the comparisons of analyses of random sets of 3 SMAVS to tho se of the typical OPP species.Page 18

2.2.13. Model averaging Model-averaged HC 5 values were calculated as weighted averages of the H C 5 values from each
individual distribution fit to the same data set. T he weights used were Akaike weights ( w i ), as defined by Burnham and Anderson (2002:75). The formula for Aka ike weights is:

7

1 

### 1 exp 2
1 exp 2 i

i

j j 

### w

= 

###  

− ∆    

=  

− ∆     

# ∑

In the above equation, 

## ( ) ( )

c c 

### AIC distribution i min AIC i 

### ∆ = − and the summation limit of 7 indicates
that seven distributional fits were compared. Model- averaged estimates of the HC 5 were calculated as:

7

5 5 1 

### HC HC

j j j 

### w

= 

### = 

# ∑

In the above equation, the HC 5j is the estimate of the HC 5 from the j th distribution considered.

Sampling variance of the 5



### HC was estimated using equation 4.9 of Burnham and Ande rson (2002:162): 

# ( ) ( ) 

# ( ) 7 2

5 5 5 5 1 

### var HC var HC HC HC i j j j 

### w

= 

### = + − 

# ∑

2.2.14. Software & distributions All analyses described below were accomplished using su broutines written in Matlab (hereafter, called “the SSD toolbox”). Use of the SSD toolbox requires basic Matlab software and the Matlab Optimization toolbox (Mathworks 2010). The SSD tool box contains functions for fitting seven distributions (normal, logistic, triangular, Gumbel, Weibull, Pareto, and Burr III ) using three methods (maximum likelihood, moment estimation, and graphica l estimation), though not all methods can be used with each distribution (Table 3). The SSD toolb ox also contains subroutines for performing empirical and parametric bootstrap sampling for testin g goodness-of-fit, estimating sampling variance, and performing information-theoretic model selectio n and multi-model inference. Table 4 gives some standard statistical notation used in describing the di stributions. Table 5 gives a list of distribution
functions available in the SSD toolbox.Page 19

Table 4. Statistical notation for the description of distributions tested as candidates for use in estimating
the HC p symbol description

n sample size

x sample mean:

1 

### 1 n

i i 

### x n

= 

# ∑

s sample standard deviation: 

## ( ) 2

1 

### 1
1 N

i i 

### x x n

= 

### −

− 

# ∑

exp( x ) exponential function ( e x ) X column

- vector of untransformed data (mean acute values) Y column

- vector of log 10 - transformed data (mean acute values) θ column

- vector of parameters for any given distribution f ( x| θ ) probability density at x (or y ) conditional on θ F ( x| θ ) cumulative distribution function at x (or y ) conditional on θ F - 1 ( x| θ ) quantile function at x (or y ) conditional on θ L ( θ | X ) or L log

- likelihood for θ conditional on X (or Y )

Table 5. Names of functions in the SSD toolbox

distribution pdf cdf quantile likelihood moments random variates
normal 1 normpdf 1 normcdf 1 norminv normlik normmom normrnd logistic logipdf logicdf logiinv logilik logimom logirnd

2 triangular triapdf triacdf triainv trialik trimom triarnd Gumbel gumpdf gumcdf guminv gumlik gummom gumrnd
P areto parpdf parcdf parinv parlik parmom parrnd

3 W eibull wblpdf wblcdf wblinv wbllik n/a wblrnd
B urr burpdf burcdf burinv burlik n/a burrnd

1 functions included in standard Matlab (pdf = probabil ity density function, cdf = cumulative distribution function).

2 Triangular functions written for the symmetric trian gular distribution only

3 Weibull functions are also available in Matlab’s statist ics toolbox n/a = not applicablePage 20

2.2.15. Equations for Distributions

Normal distribution

Parameters: 

### [ ; ]

μ σ 

### = θ

μ (location)

σ (scale)

Transformation: 

## ( ) 10 

### log i i 

### y x =


## ( ) ( ) 2

2 2 

### 1 | , exp 2

2 y f y μ μ σ σ

πσ 

###  

−

= −

   


## ( ) ( ) | , y 

### F y f z dz μ σ

−∞ 

### = 

# ∫

( ) ( ) 1 

### F p p − 

### = Φ


## ( ) ( ) ( ) ( ) 2

2 1 

### 1 | ln 2 ln 2 2 n

i i 

### n L n y

π σ μ

σ

= 

### = − − + −



# ∑ θ Y



## ( )

2 1 

### 1 n

i i 

### dL y d μ μ σ

= 

### = − − 

# ∑

( ) 2

2 1 

### 1 1 n

i

i 

### dL n y d 

### μ σ σ σ

= 

###  
= − + −     

# ∑

Neither the cdf ( F ) nor quantile function ( F -1 ) have explicit forms. However, both can be readil y approximated to arbitrary precision in most mathemat ical software.

Mean: μ

Variance: 2 σ

The standard normal distribution has mean 0 and unit v ariance when:



### 0

1 μ

σ 

### =

=

Maximum Likelihood Estimators:

1 

### 1

ˆ n

i

i 

### y y n μ

= 

### = = 

# ∑

1

ˆ n s n 

### σ −
=

Moment Estimators:



### ˆ y μ 

### =

ˆ s 

### σ =Page 21

Logistic distribution

Parameters: 

### [ ; ]



### α β = θ

α (location) β (scale)

Transformation: 

## ( ) 10 

### log i i 

### y x =


## ( ) ( )



# ( ) ( ) 

# ( ) 

# ( ) 2 

### exp / : , 1 exp /

y f y
y 

### α β α β β α β

− −

= + − −


## ( ) ( )



# ( ) 1 : , 1 exp / F y y 

### α β α β = + − −



## ( ) 1 

### ln 1 p F p p 

### α β

− 

###  
= +  

−  

Let:



### =

i i 

### r y 

### α

−

exp i i 

### r m 

### β  

= −    


## ( ) ( ) ( ) ( )

1 1 

### 1 | ln 2 ln 1

n n

i i i i 

### L r n m

β β

= = 

### = − − + 

# ∑ ∑ θ Y

1 

### 2
1 n i

i i 

### m dL n
d m 

### α β β

= 

###  

= −  

+   

# ∑

2 2 1 1 

### 1 2
1 n n i i i i i i



### rm dL n

r d m 

### β β β β

= = 

### = − − + 

# ∑ ∑

Mean: 

### α

Variance: 2 2 

### 3 π β

The standard logistic distribution has mean 0 and varia nce 1 when:



### 0 

### α = , 

### 3 β
π 

### =

Maximum Likelihood Estimators: Closed form estimators do not exist, but MLEs are easi ly obtained numerically.

Moment Estimators:Page 22



### ˆ y 

### α = and 

### 3

ˆ s 

### β
π =

Triangular distribution (symmetric)

Parameters: 

### [ ; ]

a b = θ

, a (minimum), b (maximum)

Transformation: 

## ( ) 10 

### log i i 

### y x =

If: 

### 2 a b a y + ≤ ≤ : If: 

### 2 a b y b + < ≤ :



## ( ) ( ) ( ) 2 

### 4 | , y a f y a b b a

−

=

− 

## ( ) ( ) ( ) 2 

### 4 | , b y f y a b b a

−

=

−


## ( ) ( ) ( ) 2

2 

### 2 | , y a F y a b b a

−

=

− 

## ( ) ( ) ( ) 2

2 

### 2 | , 1 y b F y a b b a

−

= −

− 

## ( ) ( ) ( ) ( )

1 

### | 2ln ln 4 n

i i 

### L b a y a

= 

### = − − + + −



# ∑ θ Y 

## ( ) ( ) ( ) (

1 

### | 2ln ln 4 n

i 

### L b a b y

= 

### = − − + + −



# ∑ θ Y

If: p ≤ 0.5 If: p > 0.5


## ( ) ( ) 2

1 

### 2

p b a F p a − 

### − = + 

## ( ) ( )( ) 2

1 

### 1

2

p b a F p b

− 

### − − = +

Mean:



### 2 a b +

Variance 

## ( ) 2



### 24 b a

−

The standard symmetric triangular distribution has mea n zero and variance 1 when:



### 6 a = − , and 

### 6

b =

The following moment estimators can also be used:



### ˆ ˆ 2 a y b = − and 

### ˆ 6

b y s = +Page 23

Gumbel (Gompertz) distribution

Parameters: 

## [ ]

, μ β 

### = θ

μ (location) β (scale)

Transformation: 

## ( ) 10 

### log i i 

### y x =


## ( ) ( ) 

### exp exp y 

### F y μ



### β − 

###    

= −        



## ( ) ( ) ( ) 1 exp exp

y y f y 

### μ μ β β β  

− −

 

= −  

      

## ( ) ( ) 

# ( ) 1 

### ln ln 

### F p p μ β

− 

### = − −

Let:

i

i 

### y z 

### μ β −

=

Then: 

## ( ) ( ) ( )

1 1 

### | ln exp n n

i i i i 

### L n z z

β

= = 

### = − + − 

# ∑ ∑ θ Y

Note: 

## ( )

1 exp i
i 

### dL z dz = − +

, 

### 1 i 

### dz
d 

### μ β = and 2 i i 

### dz y
d 

### μ β β −

= −



## ( ) 

# ( )

1 

### 1 1 exp n

i i 

### dL z d μ β

= 

### = − 

# ∑ and 

## ( )



# ( )

1 

### 1 1 exp n

i i i 

### dL n z z d β β

= 

### −
= − − 

# ∑

Mean: μ βγ

+ , where γ = Euler-Mascheroni constant.

Variance

2 2



### 6 

### β π

The gompertz distribution has mean 0 and variance 1 w hen :



### 6 μ γβ

β
π 

### = −

=

Moment Estimators:



### ˆ 6 s 

### β
π = and 

### ˆ

ˆ y 

### μ βγ = −Page 24

Weibull distribution

Parameters: 

## [ ]

; k λ 

### = θ

λ (scale)
k (shape)
Transformation: none


## ( ) 1 exp k



### x F x λ



###    

= − −          



## ( ) 1 

### exp

k k 

### k x x f x λ λ λ − 

###      

= −

     

     

  

## ( ) ( ) 

# ( ) 1/ 1 

### ln 1 k 

### F p p λ

− 

### = − −


## ( ) ( ) ( ) ( ) ( )

1 1 

### | ln ln 1 ln k n n i

i i i 

### x L n k kn k x

λ λ

= = 

###   = − + − −     

# ∑ ∑ θ X

1 k

n i

i 

### x dL k
n d λ λ λ

= 

###    

= − −         

# ∑

( ) ( )

1 1 

### ln ln ln k n n i i

i i i 

### x x dL n
n x dk k λ λ λ

= = 

###  

    = − + −             

# ∑ ∑

The mean and variance of the Weibull distribution ar e:



### 1 1 k λ 

###   Γ +    

2

2 

### 2 1 1 1 k k λ λ



###       Γ + − Γ +            
The gamma functions in the above two equations preven t closed-form moment estimators from being
derived.Page 25

Pareto distribution

Parameters: 

## [ ]

; m 

### x α 

### = θ

x m (scale)

α (shape)

Transformation: none


## ( ) 1 m



### x F x

x α



###  

= −    


## ( )
1

m 

### x f x

x α

α 

### α

+ 

### =


## ( ) ( ) ( ) 

# ( ) ( ) ( )

1 

### | ln ln 1 ln n

m i

i 

### L n x x

α α α

= 

### = + − + 

# ∑ θ X

m m 

### dL n
dx x α



### =



## ( ) ( )

1 

### 1 ln ln

n

m i

i 

### dL
n x x d α α

= 

###   = + −     

# ∑

Note that the derivative 

### /

m



### dL dx is necessarily negative because x m , α > 0 . Therefore the negative loglikelihood continues to get smaller with larger value s of x m and so standard numerical optimization of
the negative log-likelihood will result in 

### ˆ

m 

### x → ∞ . The solution to this is to assume that x m is known and
equal to the minimum observed value. In this case, th ere is also a simple MLE for α , which can be found by setting 

### / 0

dL d 

### α = . Thus the MLEs are: 

## ( ) ˆ min m i 

### x x =

1 

### ˆ

ln

ˆ n i

i m 

### n

x

x 

### α

= 

### =       

# ∑

The following moment estimators can also be used:



## ( ) 1

m 

### x x 

### α

α −

=

2

2 

### 1 1 x

s 

### α = + +Page 26

Burr III distribution (Note: this is the Burr III distribution from Shao 2000, not the Burr XII distribution from Shao 2003)

Parameters: 

## [ ]

; ; b c k = θ


## ( ) 1

1 k

c 

### F x b

x =     +          



## ( ) 1

1 

### 1 c

k

c 

### b
kc x f x b b

x +

+ 

###      

=     +           .

Note that Shao (2000, Eq. 8) incorrectly gives the p df as: 

## ( ) 1

1 1 

### 1 c

k

c 

### b
kc x f x b b

x +

+ + 

###      

=     +          



## ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 1 

### | ln ln ln 1 ln 1 ln 1 c n n

i i i i 

### b L n c n k cn b c x k

x

= = 

###       = + + − + − + +         

# ∑ ∑ θ X

Let:



### 1 c

i
i 

### b z x  
= +    

Then:

c

i

i 

### dz c b
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### 1 1 n i

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### dz dL cn k db b z db

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## 3. Results & Discussion
Results below are organized around seven of the eig ht questions outlined in Objectives (Section 1.4). The eighth question, concerning future analys es, is described in Section 5. For brevity only a subse t of results that are illustrative of general patterns are presented. Full results are available in section 8, Supplemental Results. As described above, increases in either bias or sampling variance reduce the accuracy of an estimate of the HC 5 . Thus patterns of both bias and sampling variance in relation to each question are presented. Bias is depicted as ratio of the estimated HC 5 to the known HC 5 in two ways. In figures the ratio is plotted with the vertical axis on the log 10 scale. In tables, the log 10 value of the ratio is given. Thus, in figures, the y-value gives the average relative over- or under-estimate at any given sample size, method and distribution. In tables positive values of the rat io depict a tendency to overestimate the HC 5 (positive bias) and negative values of the ratio dep ict a tendency to underestimate the HC 5 (negative bias). Calculating bias necessarily implies that the tr ue value of the HC 5 is known. However, this is only the case for the simulated data. For empirical ana lyses, the HC 5 calculated using all available data is
substituted for the known HC 5 . 

### 3.1. How does sample size influence bias and varian ce of the estimated HC 5 

### ?

3.1.1. Bias For all analyses of simulation data, bias in the estim ated HC 5 decreased with increasing sample size, regardless of the method used for fitting the di stribution (Fig. 2, Table 6). In general, the four distributions fit to log 10 -transformed data were less biased than the three untr ansformed distributions. Given the magnitude of sampling variance (see immedi ately below) it is possible that some of the observed bias may also be comprised of some residual sam pling variance. At sample sizes of n = 3 and n = 8, bias in distributions fit using maximum likelihood and moment estimators was large and always positive.Page 28

Figure 2. Ratio of the average estimated HC 5 to the true HC 5 for simulated data plotted as a function of sample size for simulated log-triangular data and si mulated Burr data.

Table 6. Log 10 ratios of the average estimated HC 5 to the true HC 5 for seven distributions fit to five sample sizes of simulated log-triangular toxicity dat a using maximum likelihood

. Sample Size
Distribution 3 8 20 50 100 triangular 0.813 0.373 0.177 0.085 0.050
normal 0.617 0.243 0.118 0.071 0.056 logistic 0.462 0.138 0.018

- 0.031

- 0.046

G umbel 0.821 0.421 0.269 0.210 0.193

Pareto 2.172 0.996 0.245

- 0.218

- 0.427
Weibull 1.785 0.431

- 0.297

- 0.635

- 0.752

Burr 1.944 0.924 0.426 0.203 0.122

3 8 20 50 100 0.01 0.1 1 10 100 Generating distribution = triangular

Estimation Method = maximum likelihood triangular normal logistic gumbel pareto weibull burr

3 8 20 50 100 0.01 0.1 1 10 100 Generating distribution = burr

Estimation Method = maximum likelihood

3 8 20 50 100 0.01 0.1 1 10 100Ratio of the average estimated HC5to the true HC5

Generating distribution = triangular

Estimation Method = moment estimators

3 8 20 50 100 0.01 0.1 1 10 100 Generating distribution = burr

Estimation Method = moment estimators

3 8 20 50 100 0.01 0.1 1 10 100

Sample Size

Generating distribution = triangular Estimation Method = graphical estimators

3 8 20 50 100 0.01 0.1 1 10 100

Sample Size

Generating distribution = burr Estimation Method = graphical estimatorsPage 29

Table 7. Log 10 ratios of the average estimated HC 5 to the true HC 5 for five distributions fit to five sample sizes of simulated log-triangular toxicity data usin g moment estimators

. Sample Size
Distribution 3 8 20 50 100 triangular 0.219 0.067 0.024 0.009 0.005
normal 0.255 0.106 0.063 0.049 0.045 logistic 0.281 0.134 0.092 0.077 0.073

G umbel 0.662 0.544 0.510 0.498 0.495

Pareto 2.160 0.952 0.167

- 0.324

- 0.546

Table 8. Log 10 ratios of the average estimated HC 5 to the true HC 5 for three distributions fit to five sample sizes of simulated log-triangular toxicity dat a using graphical methods

. Sample Size
Distribution 3 8 20 50 100 triangular

- 0.572

- 0.296

- 0.141

- 0.064

- 0.033
normal

- 0.697

- 0.402

- 0.201

- 0.085

- 0.033 logistic

- 0.987

- 0.576

- 0.298

- 0.132

- 0.055

Table 9. Log 10 ratios of the average estimated HC 5 to the true HC 5 for seven distributions fit to five sample sizes of simulated Burr toxicity data using maximum likelihood

. Sample Size
Distribution 3 8 20 50 100 triangular 0.849 0.261

- 0.099

- 0.440

- 0.718
normal 0.633 0.163 0.011

- 0.049

- 0.079 logistic 0.453 0.090

- 0.019

- 0.057

- 0.079

G umbel 0.872 0.383 0.183 0.064 0.002

Pareto 2.412 1.161 0.352

- 0.369

- 0.907
Weibull 1.947 3.102 3.397 2.793 1.877

Burr 2.116 1.044 0.534 0.255 0.131

Table 10. Log 10 ratios of the average estimated HC 5 to the true HC 5 for five distributions fit to five sample sizes of simulated Burr toxicity data using moment estimators

. Sample Size
Distribution 3 8 20 50 100 triangular 0.194

- 0.036

- 0.097

- 0.120

- 0.137
normal 0.233 0.008

- 0.052

- 0.074

- 0.091 logistic 0.262 0.039

- 0.019

- 0.042

- 0.058

G umbel 0.683 0.504 0.459 0.440 0.426

Pareto 2.394 1.114 0.273

- 0.481

- 1.046Page 30

Table 11. Log 10 ratios of the average estimated HC 5 to the true HC 5 for three distributions fit to five sample sizes of simulated Burr toxicity data using graphical estimators

. Sample Size
Distribution 3 8 20 50 100 triangular

- 0.681

- 0.448

- 0.285

- 0.203

- 0.181
normal

- 0.818

- 0.567

- 0.353

- 0.227

- 0.181 logistic

- 1.138

- 0.765

- 0.465

- 0.281

- 0.206

3.1.2. Variance
Variance in the estimated HC 5 decreased steeply with sample size up to about n = 50 in analyses of simulated data (Fig. 3, note log 10 -scale of vertical axis). As with bias, the normal, l ogistic, and triangular distributions performed very similarly ac ross sample sizes and fitting methods. At large sample sizes, distributions on untransformed data ofte n showed smaller sampling variances, especially the Pareto distribution (Fig. 3). The Weibull distri bution fit to Burr III -distributed data using maximum
likelihood showed an anomalous and enormous increase in sampling variance between n = 3 and n = 100 (not shown), suggesting instability in the numerical a lgorithms for estimating parameters of that distribution. This pattern of instability in the Weib ull was repeated across other generating distributions as well (see Section 8, Supplementary Results).Page 31

Figure 3. Width of 95% confidence interval in conce ntration units for 10,000 replicates of simulated data plotted as a function of sample size. Simulated data are log-triangular (column 1: panels a, c, and e) a nd

Burr III (column 2: panels b, d, and f). Distributions fit usi ng maximum likelihood (row 1: panels a and b), moment estimators (row 2: panels c and d) and graphica l methods (row 3: panels e and f). In summary, over all estimation methods, both bias a nd variance decreased with increasing sample size. The slope of this relationship was quite steep up to about n = 20, suggesting that each additional data point results in substantially improved performance at limited sample sizes. Maximum likelihood estimates exhibited the greatest variabil ity, moment estimates were generally the least biased, and graphical methods tended to produce the m ost conservative estimates. Even at the smallest sample sizes all methods produced estimates that were of ten, on average, within an order of magnitude
of the true HC 5 . However, this result should be interpreted with ca ution, because at limited sample sizes
the estimated HC 5 s may range over two or three orders of magnitude whe n data requirements are not imposed. The precision of estimated HC 5 s improves considerably with imposition of data requir ements (see below).



### 3.2. How do minimum data requirements influence bia s and variance of the
estimated HC 5 

### ?

Results of the simulation exercises do not inform the effects of the imposition of data requirements because the randomly chosen data points fr om theoretical distributions did not possess
taxonomic identities. This section describes results o f the imposition of data requirements to analyses

3 8 20 50 100 1 10 100 1,000 10,000

a.

triangular normal logistic gumbel pareto weibull burr

3 8 20 50 100 1 10 100 1,000 10,000 b.

3 8 20 50 100 1 10 100 1,000 10,000

c.Width of 95% CI around the HC5

3 8 20 50 100 1 10 100 1,000 10,000 d.

3 8 20 50 100 1 10 100 1,000 10,000

e.

Sample Size 3 8 20 50 100 1 10 100 1,000 10,000 f.

Sample SizePage 32

at similar sample sizes with no data requirements consi dered. Two such comparisons are of particular interest. The first is to compare results at n = 8 (GMAVs) when the ALC MDRs are met to the case i n which n = 8 without regard to the MDRs. The second compariso n is the similarly paired analyses at n = 3 for OPP’s typical test species compared to a random sam ple of 3 SMAVs. In the latter case, there is only one way to choose the minimum available SMAVs for the three typical species, so there is no replication associated with that half of the comparison (hence no estimate of variance among draws). As described in the methods section on bias (2.2.7) bias was estima ted in empirical data using reference values of the

HC 5 estimated using all available data and deviations f rom this value are referred to below as bias. This
is for ease of discussion and convenience and should not be confused with true statistical bias.

3.2.1. Bias For malathion the imposition of the eight ALC MDRs usually resulted in the average HC 5 being smaller (indicating greater toxicity) than it would be if a random set of eight GMAVs were used (Fig. 4, Table 12). This pattern held for all chemicals ( e.g., see carbaryl, Fig. 5, Table 13). In general the
estimated HC 5 s were within an order of magnitude of the reference value, though the distributions on untransformed data often showed greater differences i n average estimated HC 5 among the 1,000 iterations. The latter result is partly due to a f ew iterations producing extremely large HC 5 estimates. As in previous results, the triangular, normal, and logis tic distributions showed very similar performance to each other, with the Gumbel distribution (the other distribution on transformed data) being an outlier in this group. When the eight MDRs were imposed, the HC 5 s also tended to be smaller than the reference value (HC 5 calculated from all available GMAVs). In contrast, when eight randomly chosen GMAVs were used, the estimated HC 5 was often larger than the reference value (Figs. 4, 5). An exception to this pattern occurred with graphical methods (Fig. 3c).Page 33

Figure 4. Average ratio of estimated HC 5 s for malathion from 1,000 draws of empirical data ( n = 8 GMAVs) to the HC 5 estimated from all available GMAVs (n = 57). Panels represent a. maximum likelihood estimates, b. moment estimates, c. graphical estimates. Note log 10 scale of Y-axis. normal logistic triangular Gumbel Pareto Weibull Burr 0.0001 0.001 0.01 0.1 1 10 100 a. 8 GMAVs chosen at random 8 GMAVs meeting ALC MDRs

normal logistic triangular Gumbel Pareto 0.0001 0.001 0.01 0.1 1 10 100Ratio of HC5at n = 8 to HC5estimated using 57 GMAVs

b.

normal logistic triangular 0.0001 0.001 0.01 0.1 1 10 100

Distribution c.Page 34

Table 12. Ratio of HC 5 s for malathion for n = 8 with and without MDRs compared to a refe rence value ( i.e., HC 5 calculated from n = 57 GMAVs)

Distribution MDRs Maximum Likelihood Moment Estimators Graphical Estimators

normal no 0.246 0.054

- 0.273

yes

- 0.369

- 0.666

- 1.012

logistic no

- 0.096 0.051

- 0.340

yes

- 1.025

- 0.610

- 1.100

triangular no 0.144 0.045

- 0.488

yes

- 0.572

- 0.634

- 1.271

Gumbel no

- 1.929 0.034 n/a

yes

- 1.391

- 0.598 n/a

Pareto no 1.279 1.383 n/a

yes 0.222 0.252 n/a

Weibull no 1.237 n/a n/a

yes

- 0.159 n/a n/a

Burr no 1.083 n/a n/a

yes

- 0.070 n/a n/a

Figure 5. Average ratio of estimated HC 5 s for carbaryl from 1,000 draws of empirical data ( n = 8 GMAVs) normal logistic triangular Gumbel Pareto Weibull Burr 0.0001 0.001 0.01 0.1 1 10 100 a. 8 GMAVs chosen at random 8 GMAVs meeting ALC MDRs

normal logistic triangular Gumbel Pareto 0.0001 0.001 0.01 0.1 1 10 100Ratio of HC5at n = 8 to HC5estimated using 55 GMAVs

b.

normal logistic triangular 0.0001 0.001 0.01 0.1 1 10 100

Distribution c.Page 35

to the HC 5 estimated from all available freshwater GMAVs (n = 5 5). Panels represent a. maximum likelihood estimates, b. moment estimates, c. graphical estimates. Note log 10 scale of Y-axis.

Table 13. Ratio of HC 5 s for carbaryl for n = 8 with and without MDRs compared to a refer ence value ( i.e.,

HC 5 calculated from n = 55 GMAVs)

Distribution MDRs Maximum Likelihood Moment Estimators Graphical Estimators

normal no 0.254 0.033

- 0.214

yes

- 0.193

- 0.537

- 0.872

logistic no

- 3.604 0.068

- 0.298

yes

- 1.370

- 0.518

- 0.935

triangular no

- 0.249 0.065

- 0.376

yes

- 0.592

- 0.491

- 1.099

Gumbel no

- 0.161 0.078 n/a

yes

- 0.702

- 0.479 n/a

Pareto no 1.396 1.570 n/a

yes 0.269 0.293 n/a

Weibull no 0.892 n/a n/a

yes

- 0.251 n/a n/a

Burr no 1.049 n/a n/a

yes

- 0.129 n/a n/a

Analysis of the three typical OPP test species also on av erage resulted in smaller HC 5 s (indicating greater toxicity) compared to the analysis of random subsets of 3 SMAVs (Figs. 6, 7 Tables 14,15). The pattern was similar to that resulting from imposition of the 8 ALC MDRs, except that bias ratios were much larger for the distributions on untransformed da ta, ranging close to two orders of magnitude. Similarity of performance among the distributions on untransformed data is again evident (Figs. 6, 7).
Values of the HC 5 s estimated using all available SMAVs are given in Sect ion 8. Similarly, figures showing comparison between random sets of size 3 and the three typical OPP species are given for all chemicals
considered in Section 8.Page 36

Figure 6. Average ratio of estimated HC 5 s for malathion from 1,000 draws of empirical data ( n = 3 SMAVs) to the HC 5 estimated from all available SMAVs (n = 71). Panel s represent a. maximum likelihood estimates, b. moment estimates, c. graphical estimates. Note log 10 scale of Y-axis. normal logistic triangular Gumbel Pareto Weibull Burr 0.0001 0.001 0.01 0.1 1 10 100 a. 3 SMAVs chosen at random Typical OPP species

normal logistic triangular Gumbel Pareto 0.0001 0.001 0.01 0.1 1 10 100Ratio of HC5at n = 3 to HC5estimated using 71 SMAVs

b.

normal logistic triangular 0.0001 0.001 0.01 0.1 1 10 100

Distribution c.Page 37

Table 14. Ratio of HC 5 s for malathion for n = 3 with and without MDRs compared to a refe rence value ( i.e., HC 5 calculated from n = 71 SMAVs).
Distribution MDRs Maximum Likelihood Moment Estimators Graphical Estimators

normal no 0.502

- 0.079

- 0.983

yes

- 0.767

- 0.910

- 1.271

logistic no

- 0.405

- 0.175

- 1.028

yes

- 0.757

- 0.933

- 1.321

triangular no

- 0.055

- 0.075

- 1.412

yes

- 0.745

- 0.949

- 1.434

Gumbel no 0.228

- 0.098 n/a

yes

- 0.763

- 1.191 n/a

Pareto no 1.701 1.744 n/a

yes

- 0.159

- 0.053 n/a

Weibull no 1.694 n/a n/a

yes

- 0.275 n/a n/a

Burr no 1.280 n/a n/a

yes

- 0.495 n/a n/aPage 38

Figure 7. Average ratio of estimated HC 5 s for propoxur from 1,000 draws of empirical data ( n = 3 SMAVs)
to the HC 5 estimated from all available SMAVs (n = 13). Panel s represent a. maximum likelihood estimates, b. moment estimates, c. graphical estimates. Note log 10 scale of Y-axis. normal logistic triangular Gumbel Pareto Weibull Burr 0.0001 0.001 0.01 0.1 1 10 100 a. 3 SMAVs chosen at random Typical OPP species

normal logistic triangular Gumbel Pareto 0.0001 0.001 0.01 0.1 1 10 100Ratio of HC5at n = 3 to HC5estimated using 13 SMAVs

b.

normal logistic triangular 0.0001 0.001 0.01 0.1 1 10 100

Distribution c.Page 39

Table 15. Ratio of HC 5 s for propoxur for n = 3 with and without MDRs compared to a refe rence value ( i.e., HC 5 calculated from n = 13 SMAVs).

Distribution MDRs Maximum Likelihood Moment Estimators Graphical Estimators

normal no 0.321 0.059

- 0.347

yes

- 0.441

- 0.953

- 1.759

logistic no

- 0.403 0.076

- 0.403

yes

- 0.550

- 0.940

- 1.850

triangular no 0.343 0.087

- 0.596

yes

- 0.459

- 0.931

- 2.110

Gumbel no 0.322 0.051 n/a

yes

- 0.675

- 0.793 n/a

Pareto no 0.849 0.905 n/a

yes

- 0.139

- 0.135 n/a

Weibull no 1.458 n/a n/a

yes 0.393 n/a n/a

Burr no 0.614 n/a n/a

yes

- 0.800 n/a n/a

3.2.2. Variance Imposition of the 8 ALC MDRs greatly reduced the var iance in 1,000 replicate draws of empirical data (Fig. 8, note log 10 scale of vertical axis). In general, 95% confidence intervals around estimated HC 5 s were at least twice as wide when the MDRs were not i mposed. Thus, while the imposition of MDRs often produced, on average, HC 5 s that differed only modestly from similar sample siz es with no data requirements, particular results from given iteration s differed substantially. Random subsets of size 3 (SMAVs) produced even larger variance estimates than random subsets of size 8 (Fig. 9, again note log 10 scale of vertical axis). This pattern holds across al l AChE inhibitors, but the variances are not always a s large as those observed for malathion ( e.g. , propoxur, Fig. 10). This result reflects two impor tant differences when data requirements are imposed. First , there are fewer combinations of test results that satisfy the requirements, and thus less variation in the resulting estimated HC 5 s. Second, and more importantly, the imposition of data requirements fo rces the inclusion of sensitive species in the analyses. Thus subsets of data including only test resu lts with high LC50s are excluded from
consideration. In summary, minimum data requirements greatly improv ed the quality of HC 5 estimates, regardless of the method used for estimation. Although MDRs often produced more biased estimates of
the HC 5 , they virtually always did so in a conservative way – by producing HC 5 estimates that were, on average, lower than the reference values (HC 5 s estimated using all available data). Data require ments at

n = 8 also greatly reduced the variation among HC 5 estimates. The latter result is due partly to the
forced inclusion of both sensitive and insensitive ta xa in every data sample and partly due to the reduced number of possible data combinations.Page 40

Figure 8. Width of 95% confidence interval for malathion HC 5 estimates from 1,000 draws of 8 randomly chosen GMAVs compared to when the 8 ALC MDRs are impo sed. Y-axis is plotted on log 10 scale. The three panels correspond to a. maximum likelihood, b. moment estimators, and c. graphical methods, respectively. normal logistic triangular Gumbel Pareto Weibull Burr

-2 0 2 4 a. 8 GMAVs chosen at random 8 GMAVs meeting ALC MDRs

normal logistic triangular Gumbel Pareto -2 0 2 4Width of 95% CI around the HC5

b.

normal logistic triangular -2 0 2 4

Distribution c.Page 41

Figure 9. Width of 95% confidence interval for malathion HC 5 estimates from 1,000 draws of 3 randomly chosen SMAVs. Y-axis is plotted on log 10 scale. The three panels correspond to a. maximum li kelihood, b. moment estimators, and c. graphical methods, respec tively. normal logistic triangular Gumbel Pareto Weibull Burr 0 1 2 3 4 5 a.

normal logistic triangular Gumbel Pareto 0 1 2 3 4 5Width of 95% CI around the HC5

b.

normal logistic triangular 0 1 2 3 4 5

Distribution c.Page 42

Figure 10. Variance in propoxur HC 5 estimates from 1,000 draws of 3 randomly chosen SMA Vs. Y-axis is plotted on log 10 scale. The three panels correspond to a. maximum li kelihood, b. moment estimators, and c. graphical methods, respectively.



### 3.3. How do different estimation methods influence bias and variance of the
estimated HC 5 

### ?

3.3.1. Bias In analyses of simulated data, the most notable effect of fitting method was that the graphical methods tended to produce the lowest (most conservative ) estimates of the HC 5 , especially in the range of sample sizes of concern for this project (3 ≤ n ≤ 8, Fig. 2). However, moment estimators generally produced the least biased (closest to true or referenc e HC 5 ) estimates in this range of sample sizes. Differences among methods diminished as sample size incr eased. With both moment estimators and graphical methods, bias was often still quite small, ev en at the smaller sample sizes. Within the range of sample sizes of interest maximum likelihood consistentl y overestimated the true HC 5 . The reason for this overestimation is partly due to the method (maxi mum likelihood) consistently underestimating the variance of the distribution being fit. When couple d with unbiased estimates of the mean, the
underestimate of the variance has the effect of drawi ng the HC 5 in towards the mean ( i.e. , introducing a positive bias in the estimated HC 5 ). Analyses of empirical data followed similar patterns as analyses of simulated data (above) with respect to estimation method. In analyses of repeated draws of empirical data for malathion at n = 8 GMAVs, moment estimators tended to produce closer esti mates of the HC 5 to the reference values (analysis of all available GMAVs) when the ALC MDRs w ere met (Figs. 6, 7). In analyses of random normal logistic triangular Gumbel Pareto Weibull Burr 0 1 2 3 4 a.

normal logistic triangular Gumbel Pareto 0 1 2 3 4Width of 95% CI around the HC5

b.

normal logistic triangular 0 1 2 3 4

Distribution c.Page 43

samples of size n = 3 SMAVs graphical methods also produced, on average , smaller (more conservative)
estimates of the HC 5 compared to reference values (Fig. 9, n = 71 SMAVs), than did moment estimators. Maximum likelihood consistently produced larger estim ates than the reference values (empirical data).

3.3.2. Variance Variance was very large, especially with limited dat a and similar among estimation methods. Variance among estimated HC 5 s tended to be slightly smaller with graphical metho ds, but was still very large when data was limited (Fig. 3). Similar patte rns were observed in analyses of empirical data.



### 3.4. Are measures of fit (AIC, goodness-of-fit) rel iable indicators of performance?

3.4.1. Akaike’s Information Criterion In analyses of AIC c weights for all distributions that were fit to the s imulation data for the normal distribution, AIC c showed little ability to discriminate among distribut ions when data was limited (Table 16 - 22). With simulated log-normal data, e ven at n = 50, AIC c failed to assign the most weight to the normal distribution, though by n = 100, the normal distribution was identified as the best distribution, albeit with only 39% of total weight o n average (Table 16). This pattern generally held across distributions (Tables 17-22), though AIC c showed better discriminatory ability with the Weibu ll and Pareto distributions (Table 20, 21). Note that at n = 3 AIC c cannot be calculated for the distributions
considered here.

Table 16. Average AIC c weight for each distribution fit to 10,000 samples o f log- normal data.

n normal logistic triangular Gumbel Weibull Pareto Burr

8 0.15 0.12 0.21 0.12 0.16 0.24 0.01

20 0.19 0.15 0.30 0.11 0.14 0.06 0.05

50 0.25 0.19 0.34 0.06 0.07 0.00 0.08

100 0.39 0.21 0.25 0.02 0.03 0.00 0.10

Table 17. Average AIC c weight for each distribution fit to 10,000 samples o f log- logistic data.

n normal logistic triangular Gumbel Weibull Pareto Burr

8 0.14 0.13 0.20 0.12 0.17 0.23 0.01

20 0.18 0.19 0.24 0.12 0.16 0.06 0.06

50 0.22 0.32 0.17 0.07 0.09 0.00 0.13

100 0.24 0.46 0.06 0.02 0.03 0.00 0.20

Table 18. Average AIC c weight for each distribution fit to 10,000 samples o f log- triangular data.

n normal logistic triangular Gumbel Weibull Pareto Burr

8 0.15 0.12 0.21 0.12 0.16 0.24 0.01

20 0.18 0.12 0.36 0.10 0.13 0.06 0.04

50 0.17 0.07 0.63 0.04 0.05 0.00 0.03

100 0.11 0.02 0.84 0.01 0.01 0.00 0.01Page 44

Table 19. Average AIC c weight for each distribution fit to 10,000 samples o f log- Gumbel data.

n normal logistic triangular Gumbel Weibull Pareto Burr

8 0.11 0.10 0.16 0.14 0.09 0.40 0.01

20 0.11 0.10 0.16 0.25 0.02 0.30 0.07

50 0.06 0.06 0.08 0.55 0.00 0.07 0.18

100 0.02 0.02 0.01 0.69 0.00 0.00 0.26

Table 20. Average AIC c weight for each distribution fit to 10,000 samples o f Weibull data.

n normal logistic triangular Gumbel Weibull Pareto Burr

8 0.15 0.14 0.21 0.08 0.28 0.13 0.01

20 0.13 0.13 0.19 0.02 0.49 0.01 0.03

50 0.07 0.07 0.08 0.00 0.76 0.00 0.02

100 0.03 0.02 0.01 0.00 0.93 0.00 0.01

Table21. Average AIC c weight for each distribution fit to 10,000 samples o f Pareto data.

n normal logistic triangular Gumbel Weibull Pareto Burr

8 0.06 0.05 0.08 0.11 0.03 0.66 0.01

20 0.01 0.01 0.02 0.06 0.00 0.89 0.01

50 0.00 0.00 0.00 0.01 0.00 0.99 0.00

100 0.00 0.00 0.00 0.00 0.00 1.00 0.00

Table 22. Average AIC c weight for each distribution fit to 10,000 samples o f Burr III data.

n normal logistic triangular Gumbel Weibull Pareto Burr

8 0.14 0.12 0.20 0.12 0.16 0.25 0.01

20 0.17 0.18 0.24 0.14 0.12 0.08 0.06

50 0.21 0.31 0.17 0.11 0.05 0.00 0.14

100 0.24 0.42 0.06 0.04 0.01 0.00 0.23

3.4.2. Goodness-of-fit Results of parametric goodness-of-fit tests are presen ted in a series of tables below. In all cases, the tables give P-values for the probability of det ecting lack-of-fit in a fitted distribution with a desired Type I error rate of α = 0.05. Columns represent attempts to fit a given di stribution to data generated under the distribution specified for each row. P-valu es represent the average frequency with which a distribution is rejected (at α = 0.05) by a bootstrap goodness-of-fit test. Thus, wh en the generating distribution (row) and fitted distribution (column) a re identical, the rejection rate should be close to t he nominal rate (0.05). When the generating and fitte d distributions are different, the P-values represent the power of the test to detect lack of fit. Ideally they should be elevated over the nominal rate. Eac h table is specific to a sample size and fitting method. Only the results of parametric bootstrap tests are presented; goodness-of-fit tests based on the empirical bootstrap never rejected alternative
distributions at rates above the nominal rate.

At n = 3 parametric goodness-of-fit tests showed approxim ately nominal rejection rates at α = 0.05 (diagonal entries Tables 23 - 25). However, th ey also showed little power to reject distributions from which the data were not generated (off-diagona l P-values in Tables 23 - 25 rarely larger thanPage 45

nominal rejection rate: 0.05). At slightly larger sa mples ( n = 8), off-diagonal rejection rates were elevated over the nominal rate, but still reflected a low probability of rejecting alternative distribut ions as candidates (Tables 26 - 28). At large sample siz es ( n = 100), power was high (Tables 29 - 31) and alternative distributions would be rejected more ofte n than not. Power tables for intermediate sample sizes ( n = 20 and n = 50) are presented in the supplementary results (Se ction 8). Power did not depend upon the method used to fit the distribution. Howev er, rejection rates for moment estimators for the Pareto distribution suggest that they rarely produce reasonable parameter estimates.

Table 23. Power of parametric bootstrap goodness-of-f it tests for maximum likelihood at n = 3.

Fitted Distribution
normal logistic triangular Gumbel Pareto Weibull Burr

Generating
Distribution normal 0.047 0.047 0.053 0.040 0.027 0.040 0.040

logistic 0.053 0.040 0.027 0.027 0.000 0.053 0.027 triangular 0.073 0.073 0.053 0.073 0.047 0.040 0.060
Gumbel 0.087 0.100 0.073 0.073 0.053 0.067 0.040

Pareto 0.040 0.067 0.060 0.033 0.053 0.067 0.013
Weibull 0.040 0.047 0.073 0.027 0.020 0.047 0.040

Burr 0.053 0.060 0.060 0.020 0.020 0.053 0.033

Table 24. Power of parametric bootstrap goodness-of-f it tests for moment estimators at n = 3.

Fitted Distribution
normal logistic triangular Gumbel Pareto

Generating
Distribution normal 0.047 0.053 0.060 0.053 0.513

logistic 0.040 0.027 0.040 0.127 0.520 triangular 0.053 0.073 0.067 0.060 0.567
Gumbel 0.087 0.073 0.093 0.073 0.460

Pareto 0.067 0.067 0.047 0.027 0.567
Weibull 0.047 0.060 0.060 0.073 0.573

Burr 0.060 0.073 0.053 0.067 0.540

Table 25. Power of parametric bootstrap goodness-of-f it tests for graphical estimates at n = 3.

Fitted Distribution
normal logistic triangular

Generating
Distribution normal 0.053 0.040 0.060

logistic 0.033 0.040 0.047 triangular 0.060 0.080 0.060
Gumbel 0.093 0.093 0.100

Pareto 0.073 0.047 0.067
Weibull 0.047 0.053 0.060

Burr 0.053 0.053 0.067Page 46

Table 26. Power of parametric bootstrap goodness-of-f it tests for maximum likelihood at n = 8.

Fitted Distribution
normal logistic triangular gumbel Pareto Weibull Burr

Generating
Distribution normal 0.107 0.107 0.080 0.073 0.033 0.067 0.087

logistic 0.033 0.040 0.027 0.060 0.067 0.040 0.033 triangular 0.047 0.093 0.067 0.073 0.047 0.093 0.047
Gumbel 0.067 0.060 0.093 0.020 0.033 0.087 0.047

Pareto 0.300 0.187 0.313 0.020 0.013 0.120 0.147
Weibull 0.127 0.080 0.133 0.087 0.200 0.047 0.127

Burr 0.067 0.107 0.120 0.047 0.060 0.093 0.060

Table 27. Power of parametric bootstrap goodness-of-f it tests for moment estimators at n = 8.

Fitted Distribution
normal logistic triangular Gumbel Pareto

Generating
Distribution normal 0.093 0.093 0.073 0.100 0.980

logistic 0.040 0.027 0.047 0.140 0.980 triangular 0.060 0.047 0.067 0.133 0.960
Gumbel 0.087 0.073 0.113 0.033 0.973

Pareto 0.367 0.273 0.353 0.047 0.980
Weibull 0.113 0.107 0.133 0.253 0.980

Burr 0.113 0.093 0.107 0.140 0.960

Table 28. Power of parametric bootstrap goodness-of-f it tests for graphical estimates at n = 8.

Fitted Distribution
normal logistic triangular

Generating
Distribution normal 0.040 0.040 0.040

logistic 0.053 0.027 0.073 triangular 0.067 0.040 0.073
Gumbel 0.107 0.073 0.147

Pareto 0.300 0.240 0.313
Weibull 0.127 0.100 0.127

Burr 0.100 0.073 0.093

Table 29. Power of parametric bootstrap goodness-of-f it tests for maximum likelihood at n = 100.

Fitted Distribution
normal logistic triangular gumbel Pareto Weibull Burr

Generating
Distribution normal 0.067 0.147 0.453 0.567 1.000 0.753 0.127

logistic 0.167 0.073 0.860 0.547 1.000 0.820 0.120 triangular 0.093 0.227 0.093 0.627 1.000 0.733 0.173
Gumbel 0.807 0.627 1.000 0.033 1.000 0.613 0.067

Pareto 1.000 1.000 1.000 0.107 0.053 0.260 0.993
Weibull 0.773 0.600 1.000 0.220 1.000 0.040 0.853

Burr 0.200 0.047 0.880 0.487 1.000 0.867 0.053Page 47

Table 30. Power of parametric bootstrap goodness-of-f it tests for moment estimates at n = 100.
Fitted Distribution
normal logistic triangular Gumbel Pareto

Generating
Distribution normal 0.073 0.133 0.147 0.793 1.000

logistic 0.167 0.053 0.440 0.853 1.000 triangular 0.087 0.260 0.073 0.893 1.000
Gumbel 0.827 0.800 0.887 0.053 1.000

Pareto 1.000 1.000 1.000 0.887 1.000
Weibull 0.807 0.760 0.880 1.000 1.000

Burr 0.207 0.080 0.427 0.660 1.000

Table 31. Power of parametric bootstrap goodness-of-f it tests for graphical estimates at n = 100.

Fitted Distribution
normal logistic triangular

Generating
Distribution normal 0.080 0.047 0.180

logistic 0.293 0.047 0.520 triangular 0.060 0.060 0.080
Gumbel 0.853 0.760 0.913

Pareto 1.000 1.000 1.000
Weibull 0.807 0.727 0.920

Burr 0.273 0.113 0.487



### 3.5. What are actual coverage rates of 95% confiden ce limits for subsamples? With simulated data, the actual coverage rates of 9 5% lower confidence limits on the HC 5 were very close to nominal coverage rates at n = 3 and n = 8 (Tables 32 and 33). Actual coverage rates were slightly closer to nominal rates when the data were fi t to the same distribution under which it was generated, but the effect of switching distributions w as quite minor (Tables 32 and 33).

Table 32. Frequency with which the true HC 5 is greater than lower 95% confidence limit based on published extrapolation constants at N = 3.

Fitted Distribution
normal logistic triangular

Generating Distribution normal 0.948 0.953 0.948 logistic 0.943 0.951 0.944 triangular 0.951 0.956 0.952Page 48

Table 33. Frequency with which the true HC 5 is greater than lower 95% confidence limit based on published extrapolation constants at N = 8.

Fitted Distribution
normal logistic triangular

Generating Distribution normal 0.948 0.960 0.942 logistic 0.937 0.952 0.929 triangular 0.952 0.966 0.945

In contrast to simulated data, actual coverage rate s generally differed from nominal rates for empirical data, when the true HC 5 was taken to be the value estimated using all avail able data. When minimum data requirements were imposed, actual coverage rates always exceeded nominal rates (in fact actual coverage rates were often 100%). When minimum data requirements were not imposed coverage rates were often much lower than nominal rates (Tables 34, 37, 39).

Table 34. Frequency with which the reference HC 5 is greater than lower 95% confidence limit based on published extrapolation constants at N = 8, but not ne cessarily considering ALC MDRs for malathion

. Fitted Distribution
normal logistic triangular

Generating Distribution normal 0.696 0.727 0.657 logistic 0.823 0.849 0.778 triangular 0.704 0.651 0.640

Table 35. Frequency with which the reference HC 5 is greater than lower 95% confidence limit based on published extrapolation constants at N = 8, and ALC MD Rs are met for malathion

. Fitted Distribution
normal logistic triangular

Generating Distribution normal 1.000 0.999 0.998 logistic 1.000 1.000 1.000 triangular 0.999 0.997 0.989

Table 36. Frequency with which the reference HC 5 is greater than lower 95% confidence limit based on published extrapolation constants at N = 3, but not ne cessarily typical OPP species for malathion

. Fitted Distribution
normal logistic triangular

Generating Distribution normal 0.978 0.958 0.960 logistic 0.969 0.980 0.967 triangular 0.971 0.960 0.967Page 49

Table 37. Frequency with which the reference HC 5 is greater than lower 95% confidence limit based on published extrapolation constants at N = 3, but not ne cessarily considering ALC MDRs for carbaryl

. Fitted Distribution
normal logistic triangular

Generating Distribution normal 0.712 0.674 0.635 logistic 0.848 0.811 0.738 triangular 0.619 0.637 0.620

Table 38. Frequency with which the reference HC 5 is greater than lower 95% confidence limit based on published extrapolation constants at N = 8, and ALC MD Rs are met for carbaryl

. Fitted Distribution
normal logistic triangular

Generating Distribution normal 0.971 0.969 0.935 logistic 1.000 0.999 0.992 triangular 0.921 0.932 0.926

Table 39. Frequency with which the reference HC 5 is greater than lower 95% confidence limit based on published extrapolation constants at N = 3, but not ne cessarily typical OPP species for carbaryl

. Fitted Distribution
normal logistic triangular

Generating Distribution normal 0.783 0.787 0.757 logistic 0.863 0.846 0.837 triangular 0.545 0.775 0.733



### 3.6. How do different candidate distributions perfo rm relative to each other? Several patterns of variation among distributions emer ged in the previous sections, which are restated here. First, the normal, logistic, and trian gular distributions tended to perform very similarly to each other over the long run (10,000 simulated data sets, or 1,000 random draws of empirical data). This pattern generally reflected a qualitative diff erence between distributions fit to log 10 -transformed
data and those fit to untransformed data. The Gumbel distribution (log 10 -transformed data) was more similar to the other distributions on transformed data, but was an outlier in that group. Distributions on
untransformed data tended to be much more biased than distributions on transformed data especially at smaller sample sizes. At all sample sizes conside red, it was often the case that the least biased distribution was not the same distribution as the gener ating distribution. The most conspicuous effect of distribution concerned the tendency for very large positive
biases in the HC 5 at limited sample sizes in the simulation data when distributions were fit to untransformed data, especially when maximum likelihoo d was used for parameter estimation. Thus, at with limited data, fitting distributions to untransfor med data tends to result in over-estimates of the

HC 5 . A similar pattern was observed relative to the ref erence values when random subsets of data ofPage 50

size n = 8 (GMAVs) or n = 3 (SMAVs) were analyzed. Part of this result is lik ely due to uncertainty associated with the numerical fitting algorithms appli ed to the untransformed data (with very large variance). The ambiguous results observed above concerning distri butional performance in analyses of the simulation data were also evident in analyses of the e mpirical data. Regardless of whether the ALC MDRs were imposed, AIC c weights across distributions were remarkably evenly dist ributed for analyses of 12 AChE inhibitors (Tables 40, 41). The Burr di stribution tended to get slightly less weight than the other six distributions probably due to the fact that it requires three estimated parameters rather than two. The evenness of distributional performance did not depend on whether or not the ALC MDRs were imposed (Tables 40, 41).

Table 40. Average AIC c weights for GMAVs when n = 8 and ALC MDRs are met

water normal logistic triangular Gumbel Pareto Weibull Burr

malathion FW 0.15 0.16 0.16 0.14 0.17 0.13 0.08
diazinon FW 0.15 0.13 0.17 0.14 0.21 0.16 0.03 chlorpyrifos FW 0.16 0.13 0.16 0.16 0.21 0.13 0.07
fenitrothion FW 0.15 0.15 0.19 0.14 0.15 0.17 0.05 methyl parathion FW 0.16 0.13 0.16 0.13 0.21 0.15 0.06
dichlorvos FW 0.14 0.15 0.19 0.12 0.13 0.18 0.09 chlorpyrifos SW 0.15 0.14 0.13 0.16 0.22 0.11 0.10
fenthion SW 0.16 0.12 0.21 0.09 0.12 0.23 0.06 carbaryl FW 0.15 0.13 0.17 0.14 0.19 0.18 0.04 carbaryl SW 0.15 0.11 0.19 0.11 0.23 0.21 0.01 methomyl FW 0.15 0.12 0.21 0.12 0.25 0.15 0.01

propoxur FW 0.16 0.13 0.20 0.19 0.23 0.07 0.01

Table 41. Average AIC c weights for GMAVs when n = 8 without regard to ALC MDRs

water normal logistic triangular Gumbel Pareto Weibull Burr

malathion FW 0.15 0.16 0.17 0.13 0.17 0.14 0.09
diazinon FW 0.14 0.14 0.16 0.13 0.16 0.16 0.10 chlorpyrifos FW 0.16 0.14 0.15 0.13 0.19 0.14 0.09
fenitrothion FW 0.14 0.13 0.18 0.13 0.15 0.16 0.11 methyl parathion FW 0.13 0.15 0.16 0.14 0.17 0.15 0.10
dichlorvos FW 0.16 0.15 0.16 0.13 0.16 0.16 0.09 chlorpyrifos SW 0.14 0.12 0.15 0.15 0.19 0.12 0.12
fenthion SW 0.16 0.12 0.17 0.15 0.14 0.17 0.08 carbaryl FW 0.16 0.15 0.14 0.16 0.15 0.17 0.06 carbaryl SW 0.16 0.11 0.20 0.10 0.22 0.20 0.01 methomyl FW 0.16 0.12 0.20 0.13 0.17 0.19 0.03

propoxur FW 0.16 0.15 0.16 0.14 0.14 0.16 0.07Page 51

In contrast to the ambiguous results above AIC c did show good discriminatory power when applied to maximal sets of GMAVs (Table 42). At thes e sample sizes (ranging from 12 to 57, Table 2) AIC c placed the greatest weight (on average) on the trian gular and Weibull distributions, though there was also considerable variation among chemicals as to whic h distribution performed best (Table 42). Note however, that this result is due largely to the lack o f replication inherent in this comparison, whereas previous results represented average AIC weights over 1, 000 subsamples.

Table 42. Summary of AIC c weights by chemical and distribution for all GMAVs.

Chemical Water normal logistic triangular Gumbel Pareto Weibull Burr

malathion FW 0.10 0.02 0.84 0.00 0.00 0.03 0.01
diazinon FW 0.07 0.02 0.16 0.00 0.00 0.74 0.01 chlorpyrifos FW 0.11 0.03 0.62 0.11 0.09 0.02 0.03
fenitrothion FW 0.01 0.01 0.00 0.00 0.00 0.97 0.01 methyl parathion FW 0.05 0.01 0.09 0.00 0.00 0.84 0.01
dichlorvos FW 0.12 0.09 0.37 0.01 0.00 0.39 0.03 chlorpyrifos SW 0.04 0.04 0.02 0.26 0.59 0.00 0.05
fenthion SW 0.15 0.09 0.18 0.02 0.01 0.48 0.07 carbaryl FW 0.00 0.00 0.00 0.00 0.00 1.00 0.00 carbaryl SW 0.15 0.10 0.22 0.06 0.12 0.33 0.01 methomyl FW 0.15 0.08 0.28 0.03 0.01 0.41 0.03

propoxur FW 0.24 0.17 0.29 0.15 0.03 0.09 0.03
Average 0.10 0.05 0.26 0.05 0.07 0.44 0.02



### 3.7. Does model-averaging across distributions impr ove estimates of the HC 5 

### ? In analyses of the parametric simulation data, model- averaging inflated bias, often substantially (Fig. 11, compare to Fig. 2). However, model-avera ging had little effect on variance in estimates of t he

HC 5 from the 10,000 replicates of the parametric simula tion data (Fig. 12). As above, both bias and variance, declined sharply with sample size.Page 52

Figure 11. Bias in 10,000 replicates of model-average d HC 5 estimates. Legend refers to generating
distribution.

Figure 12. log 10 (variance) in 10,000 replicates of model-averaged HC 5 estimates. Legend refers to generating distribution.

In contrast to the results on simulated data, AIC c model averaging results on HC 5 s estimated from empirical data were probably the most interestin g and surprising of all. The most surprising pattern was the ability of model-averaging to insulat e the estimated HC 5 from effects of poor fitting and stochastic subsampling. For example, Table 43 shows the average estimated HC 5 s under each distribution for 1,000 sets of 8 GMAVs subsampled from t he malathion data without regard to the ALC MDRs. In particular, the first row of this table giv es the estimated HC 5 s for maximum likelihood analyses and the last column gives the AIC c model-averaged value. Note that this value is much lower than the averages of all the distributions, and within an orde r of magnitude of the value estimated from the

8 20 50 100 0.1 1 10 100

Sample SizeRatio of estimated HC5to true HC5 normal logistic triangular Gumbel Pareto Weibull Burr

8 20 50 100 -2 -1 0 1 2 3 4

Sample Sizelog10(var(HC5)) triangular normal logistic gumbel pareto weibull burrPage 53

maximal set of GMAVs (Section 8). A similar effect was observed for datasets of size n = 8 with the ALC MDRS met (Table 44). Further, this pattern was repe ated across chemicals ( e.g. , fenitrothion, Table 45). This result suggests that formal model-averaging acro ss distributions can help insulate against outlier values (though data sets that contain only large value s cannot produce a low HC 5 ).

Table 43. Mean estimated malathion HC 5 s when n = 8 GMAVs regardless of MDRs Fitting Method normal logistic triangular G umbel W eibull P areto B urr average Maximum Likelihood 16.2 18.4 22.9 21.8 8.5 30.2 23.3 4.7
Moment Estimators 15.4 16.6 14.6 30.5 26.5 Graphical Methods 6.0 4.1 7.6 15.5

Table 44. Mean estimated malathion HC 5 s when n = 8 GMAVs and ALC MDRs are met Fitting Method normal logistic triangular G umbel W eibull P areto B urr average Maximum Likelihood 1.7 1.3 2.2 2.4 0.4 2.6 2.1 1.4 Moment Estimators 1.1 1.2 1.0 3.9 2.1 Graphical Methods 0.3 0.2 0.4 1.2

Table 45. Mean estimated fenitrothion HC 5 s when n = 8 GMAVs regardless of MDRs Fitting Method normal logistic triangular G umbel W eibull P areto B urr average Maximum Likelihood 21.35 34.66 24.75 22.11 14.21 25.38 19.09 0.51

Moment Estimators 29.57 16.54 23.18 20.30 17.35 Graphical Methods 16.79 15.18 11.55 11.75



## 4. Implications of these results for the use of SSD s in estimating the HC 5 For brevity, the above results have been stated with relatively little interpretation. Below the primary conclusions with respect to the estimation of HC 5 s are presented as bulleted points. As above, the sections are organized around the first seven ques tions from the Objectives section (1.4 above). The eighth question, concerning future work, is add ressed in Section 5 below.



### 4.1. Implications of sample size results for estima ting the HC 5
1. When SSDs are fit to samples within the range of con cern for this project (3 ≤ n ≤ 8), bias is likely to be large. However, when distributions are fit to l og 10 -transformed data, bias was often less than an order of magnitude. Within the range of int erest, direction of bias depended primarily on which estimation method was used (see 4.3 below).

2. When SSDs are fit to sample sizes less than 50 varianc e is likely to be very large.

3. The above two points apply to all three fitting met hods, but are most worrisome with maximum
likelihood.

4. To control large biases and variances with limited d ata, additional constraints must be imposed on the analyses ( e.g. , minimum data requirements – see next section).Page 54



### 4.2. Implications of MDRs for estimating the HC 5
1. Imposition of ALC MDRs and limitation of analysis to n = 8 often introduces a small negative bias ( i.e. , within an order of magnitude or less) compared to t he HC 5 derived from a maximal set of GMAVs for a given chemical.

2. Analysis restricted to the three typical OPP species al so introduces a small negative bias compared to the HC 5 derived from a maximal set of GMAVs for a given che mical.

3. Imposition of ALC MDRs greatly reduces the variance i n the HC 5 estimated from random subsets of data. This occurs for two reasons. First, data samp les are forced to include both sensitive and insensitive species. Second, there are fewer pot ential data combinations.

4. The latter result is partly due to a reduction in the number of possible data combinations.

5. Although the imposition of MDRs often produces, on ave rage, HC 5 s that differed only modestly from similar sample sizes with no data requirements, p articular results from given iterations differed substantially.



### 4.3. Implications of choice of estimation method fo r estimating the HC 5
1. Within the sample-size range of interest for this pro ject (3 ≤ n ≤ 8), maximum likelihood methods appear to be positively biased.

2. Positive bias with maximum likelihood can often be q uite large, but is minimized with
distributions on untransformed data.

3. Moment estimators were generally the least biased met hod.

4. Graphical methods, on average, produced the lowest va lues of the HC 5 . 5. Variance in the estimated HC 5 did not differ much between methods.



### 4.4. Implications of performance of fit measures fo r estimating the HC 5
1. AIC c is not likely to be useful for discriminating among c ompeting distributions at the sample sizes of interest for this project (3 ≤ n ≤ 8).

2. Goodness-of-fit tests are not likely to be useful for rejecting alternative distributions the sample sizes of interest for this project.

3. Both AIC c and parametric bootstrap goodness-of-fit tests would be quite useful at larger sample

sizes.

4. Empirical bootstrap sampling showed little utility at any sample size.

5. Goodness-of-fit tests applied to the Pareto distributio n fit using moment estimators suggest that
those estimators are not accurate.



### 4.5. Implications of the performance of 95% confide nce limits for estimating
the HC 5
1. Extrapolation constants produce confidence limits at n ominal coverage rates when data are
simulated from a known distribution.

2. In empirical analyses, the HC 5 estimated using all available data was lower than th e 95% confidence limit based on subsamples of n = 3 and n = 8 with no data requirements more often than expected. True coverage rates were between 60% and 90%. Thus, this procedure might result in a lower confidence that the number of speci es at risk was 5% or less. Conversely, itPage 55

could be argued that the reference value (the HC 5 estimated using all available data) is itself a
biased estimator of the true HC 5 if the sample of tested species is biased towards insens itive species.

3. In contrast to (2) above, the HC 5 estimated using all available data was greater than the 95% confidence limit based on subsamples of n = 3 and n = 8 when data requirements were imposed more often than expected. True coverage rates were very close to 1. Thus, this procedure might result in over-protection.



### 4.5. Implications of choice of distribution for est imating the HC 5
1. In general, distributions on transformed data performe d better than those on untransformed
data.

2. Of the distributions on transformed data, the triangu lar distribution performed best across all 9 organophosphates.

3. Of the distributions on untransformed data, the Burr III distribution seemed to perform best.

4. Regardless of points 1 and 2 above, there remains consid erable uncertainty about which distribution is ‘best’ and the results do not support th e exclusive use of any one distribution.



### 4.6. Implications of multidistributional inference for estimating the HC 5
1. Calculating the average HC 5 across distributions for simulated data did not help p roduce a better
estimate of the HC 5 . 2. In contrast to (1), calculating the AIC c model-averaged HC 5 for empirical data sets greatly
reduced bias in estimates of the HC 5 , even when, on average, every distribution was biased

. 3. The reasons for the different conclusions resulting fro m simulated versus empirical data are not
clear and should be studied further before a distribut ion-averaging approach is adopted.



## 5. What future research is suggested by these resul ts?
The results of analyses of both simulation and empiric al analyses indicate several endemic problems with fitting species sensitivity distributions. While all fitting methods tend to be unbiased, especially with large sample sizes, large variability hampers our ability to make non-trivial statements
about the location of the HC 5 with limited data. Further, no single distribution reliably performs best even for analyses of random subsets of test results on the same chemical. Another problem (not explored in the above results) is that goodness-of-fit of a full distribution may not be a reliable indica tor of goodness-of-fit in the region of the HC 5 . The latter two points seem to argue in favor of dif ferent fitting methods, with the goodness-of-fit issue suggest ing graphical methods, which can be weighted toward the region of the HC 5 , whereas the variable performance of distributions argues for the use of maximum likelihood whereby distributional performanc e can be easily compared and formal averaging techniques may be used across distributions. The follo wing section describes how various extensions of maximum likelihood can be brought to bear on all of these issues.Page 56



### 5.1. Hierarchical models A hierarchical model, in modern statistical usage, is one in which a model parameter is specified to be (in turn) the prediction of some further model . For example, de Zwart (2002) developed a hierarchical model of the logistic distribution para meter β upon chemical mode of action. He also provided a table of estimated β values for use in fitting log-logistic SSDs, if a che mical’s mode of action is known (de Zwart 2002: Table 8.4). In that case all available data for a given chemical may be used to

estimate α (the other logistic distribution parameter) and the value of β can be assumed known. Similarly, interspecies correlation extrapolation (IC E) models exploit correlations among species test results across previously tested species to predict the to xicity for untested species based on values for tested species (Raimondo et al. 2010). These predicte d values can then be used as surrogate test results in the development of SSDs (Appendix B). These meth ods are vulnerable to the criticism that they do not properly take into account the uncertainty associ ated with the lower-level model predictions. Furthermore, preliminary analyses of de Zwart’s (2002) method using different data suggest considerable disagreement among estimates of β , especially with respect to the AChEI MOA (see Sectio n 8.9 in Supplementary Results). Thus caution is warrant ed in consideration of adoption of this method. Modern hierarchical model fitting methods attempt to simultaneously fit all hierarchically linked models and to properly account for the associated model vari ance. Hierarchical models, in the context of SSDs, could be specified in many ways, which are briefly ou tlined below. The most important limitation to the use of SSDs is sam ple size, which could lead to large biases, substantial uncertainty, and uninformative measures of fit. One way to bring more data to the table would be to investigate methods for “borrowing” infor mation on toxicity from related chemicals, for example using information on mode of action (de Zwa rt 2002) or adverse outcome pathway (AOP, Ankley et al. 2010). Such a model might look very m uch like de Zwart’s (2002) regression model, but the regression would be fit simultaneously with the SSD, using numerical optimization methods and maximum likelihood. The resulting model might be ab le to exploit some of the many benefits of maximum likelihood estimation (Section 1.2.1), inclu ding the ability to use formal model selection methods (Section 1.3.3), without suffering the large biases and variance associated with limited data, if the pool of available data for a given mode of acti on is large. Furthermore, the resulting model could be updated as new test results become available.
Another common feature in modern hierarchical model s is the specification of random effects ( i.e. , a model parameter is itself assumed to be a random va riate from some specified distribution). An obvious application for this idea in SSD methodology would be to treat multiple test results for a given taxon as random draws from a possible distribution of t est values for that taxon. This would help with concerns about how to summarize multiple test results fo r a given taxon (for example, whether to use the arithmetic mean versus geometric mean versus lowest test result). Furthermore, it would implicitly incorporate the natural statistical notion that mult iple results for a taxon are a data strength, rather than a weakness (as variable results within taxa are co mmonly viewed, Etterson pers. obs.). The above two ideas are reinforced by the observed re sult in this report that SSD analysis with limited data benefits enormously from the imposition of minimum data requirements (see Section 4.2 above and associated sub-sections in Results). Thus diffe rent taxonomic groups, both at the species and genus level, exhibit markedly different sensitivitie s to a given chemical. It is natural to assume that th esePage 57

sensitivities are often phylogenetically conserved and this assumption is reinforced from studies showing similar sensitivities among related taxa for a given mode of action (de Zwart 2002). In this case

, it may be the case that related species share their sensi tivity by sharing an adverse outcome pathway triggered by the chemical. Knowledge of relationship s among species ( http://tolweb.org ) and methods for modeling the distribution of character traits (t oxicity test results in our context) on phylogenetic trees (see for example Huelsenbeck 2003) have increase d greatly since the publication of the 1985 Guidelines (USEPA 1985). Use of phlyogenetic structur e in SSD analyses is likely to be more useful with highly specific modes of action (such as acetylcholine sterase inhibition analyzed in this report) compared to more general modes of action such as narco sis. Development of the methods described in this paragraph would require a substantial commitment of time and research, but with a potentially large benefit to our understanding of the distributi on of toxicity across taxa.



### 5.2. Order statistics Graphical methods for SSDs are an example of a comm on technique in the branch of statistics referred to as the theory of order statistics. Their primary advantages, as reviewed above, are the abilit y to weight the fit in the region of the HC 5 (or any other quantile of interest) and the ability to handle unbounded test results. Their primary disadvantages a re that they are limited to distributions in the location/scale family (in our analyses these are nor mal, logistic, and triangular) and they do not permi t the use of formal model selection techniques. Howeve r, given an ordered set of toxicity test results it i s possible to develop relatively simple expressions for t he likelihood (under any continuous distribution)
for the ordered data that are flexible in the amoun t of weight applied to any given region of the likelihood. This theory is briefly described below. Suppose that n test results are available for a given chemical and let x i represent the i th ordered EC50 (1 ≤ i ≤ n ), where the results are ordered from lowest to highest . Further, let f ( x ) and F ( x ) represent the probability density and cumulative dens ity functions for the particular distribution from which the (unordered) data are assumed to arise. The n the pdf for the ordered data ( f 0 ), weighted
toward the first i order statistics is: 

## ( ) ( ) 

# ( ) 

# ( )

1 

### | 1 i n i
o i j j 

### f F x f x −

= 

### ∝ − 

# ∏ X θ

In particular, suppose n = 100 and it is considered desirable to give particu lar weight to the first ten order statistics corresponding to the lower tenth quant ile of the empirical cdf. These are x 1 , x 2 , x 3 , ... x 10 . The remaining 90 order statistics are assumed to fall i n the upper 90 th quantile of the distribution. The joint probability density, f
o ( x ) , for the order statistics is proportional to: 

## ( ) ( ) 

# ( ) 

# ( ) 10 90

10 1 

### | 1

o j j 

### f F x f x

= 

### ∝ − 

# ∏ X θ

Like the graphical methods described by Erickson and S tephan (1988), the pdf above uses all available data, but weights the fit more strongly to the lower 10% of the data. The upper 90 order statistics are simply assumed to fall within the upper 90 th quantile of the distribution, but their specific val ue is not used. Obviously, any unbounded data that fall withi n this upper quantile may be accommodated by simply counting them in the exponent to 1 − F ( x 10 )

.Page 58



### 5.3. Bayesian methods Several features of Bayesian methods make them attra ctive for use in the analysis of SSDs. First, in attempts to understand the toxicity of a chemical, there may be data available about the toxicity of related chemicals, which could serve as valuable prior information in statistical analyses. Or, data from multiple chemicals could be pooled and analyzed using hierarchical models that include either fixed or random effects of a specific chemical. The latter me thods, especially using some of the complicated hierarchical structures described immediately above, a re much easier to fit using Bayesian methods. Finally, it is often desirable to make some kind of probabilistic statement about the location of the HC 5 . However, under a classical statistical framework, the HC 5 is fixed (but unknown) resulting in cumbersome interpretations of confidence intervals and the like. Under a Bayesian framework the HC 5 would be viewed as a random variable with an estimab le posterior distribution from which probabilistic
statements could be formulated.



## 6. Acknowledgments
This report benefitted greatly from comments by R. Erickson, M. Frankenberry, K. Garber, C. Russom, and C. Stephan. K. Garber conducted and cont ributed the preliminary analysis on MOA-specific parameters for the logistic distribution (Section 8.9 ).



## 7. Literature cited
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Standards Volume 11.06. ASTM International. West Con shohocken, PA, USA. Burnham, K.P. and D.R. Anderson. 2002. Model Selecti on and Multimodel Inference: A Practical Information-Theoretic Approach, 2 nd Ed. Campbell, E., M. Palmer, Q. Shao, and D. Wilson. 200 0. BurrliOZ. CSIRO, Mathematical and Information

Sciences. www.cmis.csiro.au/envir/burrlioz

. Chapman, P.F., M. Reed, A. Hart, et al. (2007). Met hods of Uncertainty Analysis. Work Package 4 in EUFRAM: Concerted action to develop a European fram ework for probabilistic risk assessment of the environmental impacts of pesticides. Version 3, March 2007. http://www.eufram.com/

, accessed 8 August 2011.Page 59

de Zwart, D. 2002. Observed regularities in SSDs for aquatic species. In Posthuma, L. and G.W. Suter (Eds.), The use of Species Sensitivity distributions in E cotoxicology. SETAC Press, Pensacola, FL. Edwards, A.W.F. 1992. Likelihood: Expanded edition. Johns Hopkins University Press. Baltimore, MD,

USA. Efron, B. and R.J. Tibshirani. 1994. An Introduction to the Bootstrap. Chapman and Hall/CRC. Boca Raton,

FL. USA. Elphick, J.R.F., K.D. Bergh, and H.C. Bailey. 2011. Chronic toxicity of chloride to freshwater species: effects of hardness and implications for water quality guidelines. Environmental Toxicology and Chemistry 30:239-246. Erickson, R.J. and C.E. Stephan. 1988. Calculation o f the final acute value for water quality criteria for aquatic organisms. U.S. Environmental Protection Ag ency. Duluth, MN. EPA/600/3-88-018. Etterson, M.A. and L.R. Nagy. 2008. Is mean squared er ror a consistent indicator of model accuracy for spatially structured demographic models. Ecological Modelling 211:202-208. Garber, K. S. Raimondo, and P. TenBrook. 2010. Expl oration of methods for characterizing effects of chemical stressors to aquatic animals. Working paper dr aft 11/2/2010. Harville, D.A. 1977. Maximum likelihood approaches t o variance component estimation and to related problems. Journal of the American Statistical Associat ion 72:320-340. Host, G.E., R.R. Regal, and C.E. Stephan. 1995. Anal yses of acute and chronic data for aquatic life. U.S. Environmental Protection Agency. Huelsenbeck, J.P., R. Nielsen, and J.P. Bollback. 20 03. Stochastic mapping of morphological characters. Systematic Biology 52(2):131–158. Kooijman, S.A.L.M. 1987. A safety factor for LC 50 values allowing for differences in sensitivity among species. Water Research 21:269-276. Manly, B.F.J. 1997. Randomization, Bootstrap, and Mo nte Carlo Methods in Biology, 2 nd Ed. Chapman & Hall/CRC, Boca Raton, FL. Mathworks. 2010. Matlab; the language of technical computing. Mathworks, Inc. Natick, MA. USA. Newman, M.C., D.R. Ownby, L.C.A. Mézin, D.C. Powell , T.R.L Christensen, S.B. Lerberg, and B.-A. Anderson. 2000. Applying species sensitivity distribut ions in ecological risk assessment: assumptions of distribution type and sufficient numbers o f species. Environmental Toxicology and Chemistry 19:508-515. Pennington, D.W. 2003. Extrapolating ecotoxicologic al measures from small data sets. Ecotoxicology and Environmental Safety 56:238-250. Posthuma, L, T.P. Traas, and G.W. Suter. 2002. Gener al introduction to species sensitivity distributions. Pp. 3-10 in L. Posthuma, G.W. Suter, and T.P. Traas (Eds.) Specie s Sensitivity Distributions in Ecotoxicology. Lewis. Boca Raton, FL. USA. Raimondo, S., D.N. Vivian, and M.G. Barron. 2010. W eb-based Interspecies Correlation Estimation (WebICE) for Acute Toxicity: User Manual. Version 3.1. E PA/600/R-10/004. Office of Research and Development, U. S. Environmental Protection Agency. Gulf Breeze, FL. Seber, G.A.F. 2002. The estimation of animal abunda nce and related parameters, 2 nd Ed. Reprint of 1982 edition. Blackburn Press. Caldwell, NJ. USA. Shao, Q. 2000. Estimation for hazardous concentration s based on NOEC toxicity data: an alternative approach. Environmetrics 11:583-595.Page 60

Shao, Q. 2003. Notes on maximum likelihood estimation for the three-parameter Burr XII distribution. Computational Statistics & Data Analysis 45:675-687. U.S. EPA. 1985. Guidelines for deriving numerical na tional water quality criteria for the protection of aquatic organisms and their uses. U.S. Environmental Protection Agency. Duluth, MN PB85- 227049 or EPA 822/R-85-100. Suter, G.W. 1998. Comments on the interpretation of distributions in “Overview of recent developments in ecological risk assessment.” Risk Analysis 18:3-4. Suter, G.W. 2002. North American history of species se nsitivity distributions. Pp. 11-18 in L. Posthuma, G.W. Suter, and T.P. Traas (Eds.) Species Sensitivity Distributions in Ecotoxicology. Lewis. Boca
Raton, FL. USA. Stephan, C.E. 2011. Clarification of the guidance g iven in the 1985 guidelines concerning evaluation of results of toxicity tests with aquatic organisms. Sixt h Draft. 5 October 2011. TenBrook, P. L., R.S. Tjeerdema, P. Hann, and J. Ka rkoski. 2008. Reviews of Environmental Contamination and Toxicology 199, DOI: 10.1007/978-0-387-09808-1. TenBrook PL, Palumbo AJ, Fojut TL, Hann P, Karkoski J, Tjeerdema RS. 2010. The University of California

Davis Methodology for Deriving Aquatic Life Pestici de Water Quality Criteria. Reviews of Environmental Contamination and Toxicology. Volume 209. van Straalen, N.M and C.A.J. Denneman. 1989. Ecotox icological evaluation of soil quality criteria. Ecotoxicology and Environmental Safety 18:241-251. van Straalen, N.M. 2002. Threshold models for species s ensitivity distributions applied to aquatic risk assessment for zinc. Environmental Toxicology and Chem istry 11:167-172. Williams, B.K, J.D. Nichols, and M.J. Conroy. 2002. Analysis and Management of Animal Populations. Academic Press. San Diego, CA. 817 Pp. Zajdlik & Associates. 2005. Statistical analysis of th e SSD approach for development of Canadian Water Quality Guidelines. Canadian Council of Ministers o f the Environment. Project #354-2005.Page 61



## 8. Supplementary results
This section provides full results of SSD analyses not presented in text.



### 8.1. HC 5 

### s estimated using all available GMAVs

Table S8.1.1. Estimated HC 5 s for malathion using all GMAVs for freshwater taxa ( n = 57) Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 3.318 2.554 3.819 4.321 1.556 0.588 2.330 3.639
Moment Estimators 3.165 3.394 2.871 9.527 1.181 Graphical Methods 2.355 2.126 2.453

Table S8.1.2. Estimated HC 5 s for diazinon using all GMAVs for freshwater taxa ( n = 28) Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 0.912 0.780 1.248 0.978 0.546 0.607 0.481 0.733
Moment Estimators 0.820 0.886 0.736 2.777 0.408 Graphical Methods 0.468 0.380 0.527

Table S8.1.3. Estimated HC 5 s for chlorpyrifos using all GMAVs for freshwater taxa ( n = 30) Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 0.045 0.026 0.078 0.091 0.086 0.005 0.090 0.074

Moment Estimators 0.041 0.044 0.037 0.130 0.070 Graphical Methods 0.025 0.021 0.028

Table S8.1.4. Estimated HC 5 s for fenitrothion using all GMAVs for freshwater taxa ( n = 30) Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 0.820 1.015 0.219 0.527 0.034 1.023 0.573 1.016
Moment Estimators 0.743 0.802 0.668 2.475 0.022 Graphical Methods 0.439 0.362 0.491

Table S8.1.5. Estimated HC 5 s for methyl parathion using all GMAVs for freshwater taxa ( n = 29) Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 1.070 0.864 1.548 1.222 0.529 0.824 0.556 0.904
Moment Estimators 0.964 1.043 0.863 3.339 0.389 Graphical Methods 0.552 0.449 0.621Page 62

Table S8.1.6. Estimated HC 5 s for dichlorvos using all GMAVs for freshwater taxa ( n = 22) Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 0.262 0.334 0.273 0.226 0.152 0.076 0.188 0.197

Moment Estimators 0.221 0.244 0.193 1.010 0.105 Graphical Methods 0.096 0.070 0.116

Table S8.1.7. Estimated HC 5 s for chlorpyrifos using all GMAVs for saltwater taxa ( n = 21) Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 0.016 0.012 0.011 0.052 0.057 0.000 0.048 0.051
Moment Estimators 0.014 0.015 0.013 0.045 0.048 Graphical Methods 0.007 0.006 0.009

Table S8.1.8. Estimated HC 5 s for fenthion using all GMAVs for saltwater taxa ( n = 12) Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average Maximum Likelihood 0.123 0.105 0.133 0.055 0.069 0.092 0.051 0.101
Moment Estimators 0.095 0.103 0.084 0.345 0.050

Table S8.1.9. Estimated HC 5 s for carbaryl using all GMAVs for freshwater taxa ( n = 55) Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 9.840 13.966 9.096 7.759 3.311 9.707 8.487 9.707

Moment Estimators 9.442 10.017 8.693 24.025 2.578 Graphical Methods 7.298 6.671 7.572

Table S8.1.10. Estimated HC 5 s for carbaryl using all GMAVs for saltwater taxa ( n = 9) Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 4.767 4.265 8.756 5.410 7.821 3.318 3.028 5.538
Moment Estimators 3.552 3.799 3.233 10.266 6.505 Graphical Methods 1.148 0.775 1.460

Table S8.1.11. Estimated HC 5 s for propoxur using all GMAVs for freshwater taxa ( n = 13) Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 19.953 16.262 27.387 28.107 18.660 3.038 22.657 21.276

Moment Estimators 16.966 17.907 15.732 39.804 15.789 Graphical Methods 8.480 6.605 9.910Page 63

Table S8.1.12. Estimated HC 5 s for methomyl using all GMAVs for freshwater taxa ( n = 16) Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 72.219 69.515 80.708 73.932 67.928 55.055 55.982 66.886

Moment Estimators 67.535 69.425 64.980 104.406 63.608 Graphical Methods 49.831 44.587 53.390



### 8.2. HC 5 

### s estimated using all available SMAVs

Table S8.2.1. Estimated HC 5 s for malathion using all SMAVs for freshwater taxa (N = 71) Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 2.652 1.942 2.745 3.611 0.987 0.425 2.345 2.711
Moment Estimators 2.552 2.740 2.312 7.824 0.736 Graphical Methods 1.981 1.823 2.028

Table S8.2.2. Estimated HC 5 s for diazinon using all SMAVs for freshwater taxa (N = 33) Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 0.858 0.729 1.224 0.946 0.544 0.468 0.447 0.661

Moment Estimators 0.784 0.847 0.704 2.644 0.408 Graphical Methods 0.478 0.398 0.529

Table S8.2.3. Estimated HC 5 s for chlorpyrifos using all SMAVs for freshwater taxa (N = 34) Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 0.030 0.017 0.059 0.064 0.046 0.003 0.063 0.055
Moment Estimators 0.028 0.030 0.025 0.089 0.037 Graphical Methods 0.017 0.015 0.019

Table S8.2.4. Estimated HC 5 s for fenitrothion using all SMAVs for freshwater taxa (N = 36) Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 0.725 0.721 0.219 0.552 0.034 0.711 0.417 0.708

Moment Estimators 0.668 0.721 0.601 2.222 0.022 Graphical Methods 0.423 0.357 0.463

Table S8.2.5. Estimated HC 5 s for methyl parathion using all SMAVs for freshwater taxa (N = 35) Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 1.284 1.090 1.742 1.416 0.530 1.056 0.697 1.095
Moment Estimators 1.180 1.274 1.061 3.948 0.389 Graphical Methods 0.738 0.621 0.810Page 64

Table S8.2.6. Estimated HC 5 s for dichlorvos using all SMAVs for freshwater taxa (N = 25) Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 0.090 0.076 0.153 0.116 0.106 0.015 0.043 0.111

Moment Estimators 0.077 0.085 0.066 0.406 0.074 Graphical Methods 0.033 0.024 0.040

Table S8.2.7. Estimated HC 5 s for chlorpyrifos using all SMAVs for saltwater taxa (N = 26) Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 0.023 0.020 0.013 0.000 0.057 0.000 0.066 0.050
Moment Estimators 0.021 0.023 0.019 0.060 0.048 Graphical Methods 0.013 0.010 0.014

Table S8.2.8. Estimated HC 5 s for fenthion using all SMAVs for saltwater taxa (N = 13) Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 0.042 0.030 0.055 0.051 0.035 0.023 0.011 0.035
Moment Estimators 0.032 0.035 0.029 0.132 0.025 Graphical Methods 0.010 0.007 0.013

Table S8.2.9. Estimated HC 5 s for carbaryl using all SMAVs for freshwater taxa (N = 75) Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 7.778 14.610 0.242 3.079 0.020 10.006 8.185 10.006

Moment Estimators 7.535 8.016 6.911 20.005 0.012 Graphical Methods 6.096 5.694 6.202

Table S8.2.10. Estimated HC 5 s for carbaryl using all SMAVs for saltwater taxa (N = 11) Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 9.009 11.896 12.545 7.871 7.971 10.065 7.030 10.412
Moment Estimators 7.203 7.670 6.597 19.427 6.505 Graphical Methods 2.894 2.093 3.538

Table S8.2.11. Estimated HC 5 s for propoxur using all SMAVs for freshwater taxa (N = 13) Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 19.953 16.262 27.387 28.107 18.660 3.038 22.657 21.276

Moment Estimators 16.966 17.907 15.732 39.804 15.789 Graphical Methods 8.480 6.605 9.910Page 65

Table S8.2.12. Estimated HC 5 s for methomyl using all SMAVs for freshwater taxa (N = 20) Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 46.739 51.295 36.482 41.573 19.909 33.557 39.461 37.875

Moment Estimators 43.920 45.354 41.993 72.910 17.567 Graphical Methods 32.543 29.128 34.818



### 8.3. Comparison plots of analyses at n = 8 (ALC MDR s imposed versus not imposed) Notes: Figures in this series for malathion and carbaryl freshwater tests are Figures 3 and 4 in the main text. Panels represent a. maximum likelihood estimate s, b. moment estimates, c. graphical estimates. Note log 10 scale of Y-axes.

Figure S8.3.1. Average ratio of estimated HC 5 s for diazinon from 1,000 draws of empirical data ( n = 8 GMAVs) to the HC 5 estimated from all available freshwater GMAVs (n = 2 8). normal logistic triangular Gumbel Pareto Weibull Burr 0.0001 0.001 0.01 0.1 1 10 100 a. 8 GMAVs chosen at random 8 GMAVs meeting ALC MDRs

normal logistic triangular Gumbel Pareto 0.0001 0.001 0.01 0.1 1 10 100Ratio of HC5at n = 8 to HC5estimated using 28 GMAVs

b.

normal logistic triangular 0.0001 0.001 0.01 0.1 1 10 100

Distribution c.Page 66

Figure S8.3.2. Average ratio of estimated HC 5 s for chlorpyrifos from 1,000 draws of empirical data ( n = 8 GMAVs) to the HC 5 estimated from all available freshwater GMAVs (n =30 ).

Figure S8.3.3. Average ratio of estimated HC 5 s for fenitrothion from 1,000 draws of empirical data ( n = 8 GMAVs) to the HC 5 estimated from all available freshwater GMAVs (n = 3 0). normal logistic triangular Gumbel Pareto Weibull Burr 0.0001 0.001 0.01 0.1 1 10 100 a. 8 GMAVs chosen at random 8 GMAVs meeting ALC MDRs

normal logistic triangular Gumbel Pareto 0.0001 0.001 0.01 0.1 1 10 100Ratio of HC5at n = 8 to HC5estimated using 30 GMAVs

b.

normal logistic triangular 0.0001 0.001 0.01 0.1 1 10 100

Distribution c.

normal logistic triangular Gumbel Pareto Weibull Burr 0.0001 0.001 0.01 0.1 1 10 100 a. 8 GMAVs chosen at random 8 GMAVs meeting ALC MDRs

normal logistic triangular Gumbel Pareto 0.0001 0.001 0.01 0.1 1 10 100Ratio of HC5at n = 8 to HC5estimated using 30 GMAVs

b.

normal logistic triangular 0.0001 0.001 0.01 0.1 1 10 100

Distribution c.Page 67

Figure S8.3.4. Average ratio of estimated HC 5 s for methyl parathion from 1,000 draws of empirical data ( n = 8 GMAVs) to the HC 5 estimated from all available freshwater GMAVs (n = 29 ).

Figure S8.3.5. Average ratio of estimated HC 5 s for dichlorvos from 1,000 draws of empirical data ( n = 8 GMAVs) to the HC 5 estimated from all available freshwater GMAVs (n = 2 2). normal logistic triangular Gumbel Pareto Weibull Burr 0.0001 0.001 0.01 0.1 1 10 100 a. 8 GMAVs chosen at random 8 GMAVs meeting ALC MDRs

normal logistic triangular Gumbel Pareto 0.0001 0.001 0.01 0.1 1 10 100Ratio of HC5at n = 8 to HC5estimated using 29 GMAVs

b.

normal logistic triangular 0.0001 0.001 0.01 0.1 1 10 100

Distribution c.

normal logistic triangular Gumbel Pareto Weibull Burr 0.0001 0.001 0.01 0.1 1 10 100 a. 8 GMAVs chosen at random 8 GMAVs meeting ALC MDRs

normal logistic triangular Gumbel Pareto 0.0001 0.001 0.01 0.1 1 10 100Ratio of HC5at n = 8 to HC5estimated using 22 GMAVs

b.

normal logistic triangular 0.0001 0.001 0.01 0.1 1 10 100

Distribution c.Page 68

Figure S8.3.6. Average ratio of estimated HC 5 s for chlorpyrifos from 1,000 draws of empirical data ( n = 8 GMAVs) to the HC 5 estimated from all available saltwater GMAVs (n = 21 ).

Figure S8.3.7. Average ratio of estimated HC 5 s for fenthion from 1,000 draws of empirical data ( n = 8 GMAVs) to the HC 5 estimated from all available saltwater GMAVs (n = 21 ). normal logistic triangular Gumbel Pareto Weibull Burr 0.0001 0.001 0.01 0.1 1 10 100 a. 8 GMAVs chosen at random 8 GMAVs meeting ALC MDRs

normal logistic triangular Gumbel Pareto 0.0001 0.001 0.01 0.1 1 10 100Ratio of HC5at n = 8 to HC5estimated using 21 GMAVs

b.

normal logistic triangular 0.0001 0.001 0.01 0.1 1 10 100

Distribution c.

normal logistic triangular Gumbel Pareto Weibull Burr 0.0001 0.001 0.01 0.1 1 10 100 a. 8 GMAVs chosen at random 8 GMAVs meeting ALC MDRs

normal logistic triangular Gumbel Pareto 0.0001 0.001 0.01 0.1 1 10 100Ratio of HC5at n = 8 to HC5estimated using 12 GMAVs

b.

normal logistic triangular 0.0001 0.001 0.01 0.1 1 10 100

Distribution c.Page 69

Figure S8.3.8. Average ratio of estimated HC 5 s for carbaryl from 1,000 draws of empirical data ( n = 8 GMAVs) to the HC 5 estimated from all available saltwater GMAVs (n = 9)

. normal logistic triangular Gumbel Pareto Weibull Burr 0.0001 0.001 0.01 0.1 1 10 100 a. 8 GMAVs chosen at random 8 GMAVs meeting ALC MDRs

normal logistic triangular Gumbel Pareto 0.0001 0.001 0.01 0.1 1 10 100Ratio of HC5at n = 8 to HC5estimated using 9 GMAVs

b.

normal logistic triangular 0.0001 0.001 0.01 0.1 1 10 100

Distribution c.Page 70

Figure S8.3.9. Average ratio of estimated HC 5 s for propoxur from 1,000 draws of empirical data ( n = 8 GMAVs) to the HC 5 estimated from all available freshwater GMAVs (n = 1 3). normal logistic triangular Gumbel Pareto Weibull Burr 0.0001 0.001 0.01 0.1 1 10 100 a. 8 GMAVs chosen at random 8 GMAVs meeting ALC MDRs

normal logistic triangular Gumbel Pareto 0.0001 0.001 0.01 0.1 1 10 100Ratio of HC5at n = 8 to HC5estimated using 13 GMAVs

b.

normal logistic triangular 0.0001 0.001 0.01 0.1 1 10 100

Distribution c.Page 71

Figure S3.10. Average ratio of estimated HC 5 s for methomyl from 1,000 draws of empirical data ( n = 8 GMAVs) to the HC 5 estimated from all available freshwater GMAVs (n = 1 6).



### 8.4. Variance plots of analyses at n = 8 (ALC MDRs imposed versus not imposed) Notes: Figure in this series for malathion freshwater tests is Figures 7 in the main text. Panel s represent a. maximum likelihood estimates, b. moment estimates, c . graphical estimates. Note log 10 scale of Yaxes. normal logistic triangular Gumbel Pareto Weibull Burr

0.0001 0.001 0.01 0.1 1 10 100 a. 8 GMAVs chosen at random 8 GMAVs meeting ALC MDRs

normal logistic triangular Gumbel Pareto 0.0001 0.001 0.01 0.1 1 10 100Ratio of HC5at n = 8 to HC5estimated using 16 GMAVs

b.

normal logistic triangular 0.0001 0.001 0.01 0.1 1 10 100

Distribution c.Page 72

Figure S8.4.1. Comparison of variance among HC 5 estimates from 1,000 draws of 8 randomly chosen
freshwater GMAVs for diazinon compared to when the 8 ALC MDRs are imposed. normal logistic triangular Gumbel Pareto Weibull Burr -2 0 2 4 a. 8 GMAVs chosen at random 8 GMAVs meeting ALC MDRs

normal logistic triangular Gumbel Pareto -2 0 2 4Log10(var(HC5)) among 1,000 replicate draws

b.

normal logistic triangular -2 0 2 4

Distribution c.Page 73

Figure S8.4.2. Comparison of variance among HC 5 estimates from 1,000 draws of 8 randomly chosen
freshwater GMAVs for chlorpyrifos compared to when the 8 ALC MDRs are imposed. normal logistic triangular Gumbel Pareto Weibull Burr -4 -3 -2 -1 0 1 a. 8 GMAVs chosen at random 8 GMAVs meeting ALC MDRs

normal logistic triangular Gumbel Pareto -4 -3 -2 -1 0 1Log10(var(HC5)) among 1,000 replicate draws

b.

normal logistic triangular -4 -3 -2 -1 0 1

Distribution c.Page 74

Figure S8.4.3. Comparison of variance among HC 5 estimates from 1,000 draws of 8 randomly chosen
freshwater GMAVs for fenitrothion compared to when the 8 ALC MDRs are imposed. normal logistic triangular Gumbel Pareto Weibull Burr -2 0 2 4 a. 8 GMAVs chosen at random 8 GMAVs meeting ALC MDRs

normal logistic triangular Gumbel Pareto -2 0 2 4Log10(var(HC5)) among 1,000 replicate draws

b.

normal logistic triangular -2 0 2 4

Distribution c.Page 75

Figure S8.4.4. Comparison of variance among HC 5 estimates from 1,000 draws of 8 randomly chosen
freshwater GMAVs for methyl parathion compared to when the 8 ALC MDRs are imposed. normal logistic triangular Gumbel Pareto Weibull Burr -2 0 2 4 6 a. 8 GMAVs chosen at random 8 GMAVs meeting ALC MDRs

normal logistic triangular Gumbel Pareto -2 0 2 4 6Log10(var(HC5)) among 1,000 replicate draws

b.

normal logistic triangular -2 0 2 4 6

Distribution c.Page 76

Figure S8.4.5. Comparison of variance among HC 5 estimates from 1,000 draws of 8 randomly chosen
freshwater GMAVs for dichlorvos compared to when the 8 ALC MDRs are imposed. normal logistic triangular Gumbel Pareto Weibull Burr -2 0 2 4 a.

8 GMAVs chosen at random 8 GMAVs meeting ALC MDRs

normal logistic triangular Gumbel Pareto -2 0 2 4Log10(var(HC5)) among 1,000 replicate draws

b.

normal logistic triangular -2 0 2 4

Distribution c.Page 77

Figure S8.4.6. Comparison of variance among HC 5 estimates from 1,000 draws of 8 randomly chosen
saltwater GMAVs for chlorpyrifos compared to when the 8 ALC MDRs are imposed. normal logistic triangular Gumbel Pareto Weibull Burr -4 -2 0 a.

8 GMAVs chosen at random 8 GMAVs meeting ALC MDRs

normal logistic triangular Gumbel Pareto -4 -2 0Log10(var(HC5)) among 1,000 replicate draws

b.

normal logistic triangular -4 -2 0

Distribution c.Page 78

Figure S8.4.7. Comparison of variance among HC 5 estimates from 1,000 draws of 8 randomly chosen
saltwater GMAVs for fenthion compared to when the 8 ALC MDRs are imposed. normal logistic triangular Gumbel Pareto Weibull Burr -3 -2 -1 0 1 2 a.

8 GMAVs chosen at random 8 GMAVs meeting ALC MDRs

normal logistic triangular Gumbel Pareto -3 -2 -1 0 1 2Log10(var(HC5)) among 1,000 replicate draws

b.

normal logistic triangular -3 -2 -1 0 1 2

Distribution c.Page 79

Figure S8.4.8. Comparison of variance among HC 5 estimates from 1,000 draws of 8 randomly chosen
freshwater GMAVs for carbaryl compared to when the 8 ALC MDRs are imposed. normal logistic triangular Gumbel Pareto Weibull Burr 0 2 4 6 a. 8 GMAVs chosen at random 8 GMAVs meeting ALC MDRs

normal logistic triangular Gumbel Pareto 0 2 4 6Log10(var(HC5)) among 1,000 replicate draws

b.

normal logistic triangular 0 2 4 6

Distribution c.Page 80

Figure S8.4.9. Comparison of variance among HC 5 estimates from 1,000 draws of 8 randomly chosen
saltwater GMAVs for carbaryl compared to when the 8 ALC MDRs are imposed. normal logistic triangular Gumbel Pareto Weibull Burr -4 -2 0 2 a.

8 GMAVs chosen at random 8 GMAVs meeting ALC MDRs

normal logistic triangular Gumbel Pareto -10 -5 0 5Log10(var(HC5)) among 1,000 replicate draws

b.

normal logistic triangular -6 -4 -2 0 2

Distribution c.Page 81

Figure S8.4.10. Comparison of variance among HC 5 estimates from 1,000 draws of 8 randomly chosen
freshwater GMAVs for propoxur compared to when the 8 ALC MDRs are imposed. normal logistic triangular Gumbel Pareto Weibull Burr -2 0 2 4 a. 8 GMAVs chosen at random 8 GMAVs meeting ALC MDRs

normal logistic triangular Gumbel Pareto 0 1 2 3 4Log10(var(HC5)) among 1,000 replicate draws

b.

normal logistic triangular 0 1 2 3

Distribution c.Page 82

Figure S8.4.11. Comparison of variance among HC 5 estimates from 1,000 draws of 8 randomly chosen
saltwater GMAVs for methomyl compared to when the 8 ALC MDRs are imposed.



### 8.5. Comparison plots of analyses at n = 3 (random subset versus 3 typical OPP species) Notes: Figures in this series for malathion and propoxur freshwater tests are Figures 5 and 6 in the main text. Panels represent a. maximum likelihood estimate s, b. moment estimates, c. graphical estimates. Note log 10 scale of Y-axes. normal logistic triangular Gumbel Pareto Weibull Burr 0 1 2 3 4 a. 8 GMAVs chosen at random 8 GMAVs meeting ALC MDRs

normal logistic triangular Gumbel Pareto 0 1 2 3 4Log10(var(HC5)) among 1,000 replicate draws

b.

normal logistic triangular 0 1 2 3 4

Distribution c.Page 83

Figure S8.5.1. Average ratio of estimated HC 5 s for diazinon from 1,000 draws of empirical data ( n = 3 SMAVs) to the HC 5 estimated from all available freshwater SMAVs (n = 33). normal logistic triangular Gumbel Pareto Weibull Burr 0.0001 0.001 0.01 0.1 1 10 100 a. 3 SMAVs chosen at random Typical OPP species

normal logistic triangular Gumbel Pareto 0.0001 0.001 0.01 0.1 1 10 100Ratio of HC5at n = 3 to HC5estimated using 33 SMAVs

b.

normal logistic triangular 0.0001 0.001 0.01 0.1 1 10 100

Distribution c.Page 84

Figure S8.5.2. Average ratio of estimated HC 5 s for chlorpyrifos from 1,000 draws of empirical data ( n = 3 SMAVs) to the HC 5 estimated from all available freshwater SMAVs (n = 3 4).

Figure S8.5.3. Average ratio of estimated HC 5 s for fenitrothion from 1,000 draws of empirical data ( n = 3 SMAVs) to the HC 5 estimated from all available freshwater SMAVs (n = 36). normal logistic triangular Gumbel Pareto Weibull Burr 0.0001 0.001 0.01 0.1 1 10 100 a. 3 SMAVs chosen at random Typical OPP species

normal logistic triangular Gumbel Pareto 0.0001 0.001 0.01 0.1 1 10 100Ratio of HC5at n = 3 to HC5estimated using 34 SMAVs

b.

normal logistic triangular 0.0001 0.001 0.01 0.1 1 10 100

Distribution c.

normal logistic triangular Gumbel Pareto Weibull Burr 0.0001 0.001 0.01 0.1 1 10 100 a. 3 SMAVs chosen at random Typical OPP species

normal logistic triangular Gumbel Pareto 0.0001 0.001 0.01 0.1 1 10 100Ratio of HC5at n = 3 to HC5estimated using 36 SMAVs

b.

normal logistic triangular 0.0001 0.001 0.01 0.1 1 10 100

Distribution c.Page 85

Figure S8.5.4. Average ratio of estimated HC 5 s for methyl parathion from 1,000 draws of empirical data ( n = 3 SMAVs) to the HC 5 estimated from all available freshwater SMAVs (n = 3 5). normal logistic triangular Gumbel Pareto Weibull Burr 0.0001 0.001 0.01 0.1 1 10 100 a. 3 SMAVs chosen at random Typical OPP species

normal logistic triangular Gumbel Pareto 0.0001 0.001 0.01 0.1 1 10 100Ratio of HC5at n = 3 to HC5estimated using 35 SMAVs

b.

normal logistic triangular 0.0001 0.001 0.01 0.1 1 10 100

Distribution c.Page 86

Figure S8.5.5. Average ratio of estimated HC 5 s for dichlorvos from 1,000 draws of empirical data ( n = 3 SMAVs) to the HC 5 estimated from all available freshwater SMAVs (n = 25).

Figure S8.5.6. Average ratio of estimated HC 5 s for chlorpyrifos from 1,000 draws of empirical data ( n = 3 SMAVs) to the HC 5 estimated from all available saltwater SMAVs (n = 26 ). normal logistic triangular Gumbel Pareto Weibull Burr 0.0001 0.001 0.01 0.1 1 10 100 a. 3 SMAVs chosen at random Typical OPP species

normal logistic triangular Gumbel Pareto 0.0001 0.001 0.01 0.1 1 10 100Ratio of HC5at n = 3 to HC5estimated using 25 SMAVs

b.

normal logistic triangular 0.0001 0.001 0.01 0.1 1 10 100

Distribution c.

normal logistic triangular Gumbel Pareto Weibull Burr 0.0001 0.001 0.01 0.1 1 10 100 a. 3 SMAVs chosen at random Typical OPP species

normal logistic triangular Gumbel Pareto 0.0001 0.001 0.01 0.1 1 10 100Ratio of HC5at n = 3 to HC5estimated using 26 SMAVs

b.

normal logistic triangular 0.0001 0.001 0.01 0.1 1 10 100

Distribution c.Page 87

Figure S8.5.7. Average ratio of estimated HC 5 s for fenthion from 1,000 draws of empirical data ( n = 3 SMAVs) to the HC 5 estimated from all available saltwater SMAVs (n = 13 ).

Figure S8.5.8. Average ratio of estimated HC 5 s for carbaryl from 1,000 draws of empirical data ( n = 3 SMAVs) to the HC 5 estimated from all available freshwater SMAVs (n = 7 5). normal logistic triangular Gumbel Pareto Weibull Burr 0.0001 0.001 0.01 0.1 1 10 100 a. 3 SMAVs chosen at random Typical OPP species

normal logistic triangular Gumbel Pareto 0.0001 0.001 0.01 0.1 1 10 100Ratio of HC5at n = 3 to HC5estimated using 13 SMAVs

b.

normal logistic triangular 0.0001 0.001 0.01 0.1 1 10 100

Distribution c.

normal logistic triangular Gumbel Pareto Weibull Burr 0.0001 0.001 0.01 0.1 1 10 100 a. 3 SMAVs chosen at random Typical OPP species

normal logistic triangular Gumbel Pareto 0.0001 0.001 0.01 0.1 1 10 100Ratio of HC5at n = 3 to HC5estimated using 75 SMAVs

b.

normal logistic triangular 0.0001 0.001 0.01 0.1 1 10 100

Distribution c.Page 88

Figure S8.5.9. Average ratio of estimated HC 5 s for carbaryl from 1,000 draws of empirical data ( n = 3 SMAVs) to the HC 5 estimated from all available saltwater SMAVs (n = 11 ). normal logistic triangular Gumbel Pareto Weibull Burr 0.0001 0.001 0.01 0.1 1 10 100 a. 3 SMAVs chosen at random Typical OPP species

normal logistic triangular Gumbel Pareto 0.0001 0.001 0.01 0.1 1 10 100Ratio of HC5at n = 3 to HC5estimated using 11 SMAVs

b.

normal logistic triangular 0.0001 0.001 0.01 0.1 1 10 100

Distribution c.Page 89

Figure S8.5.10. Average ratio of estimated HC 5 s for methomyl from 1,000 draws of empirical data ( n = 3 SMAVs) to the HC 5 estimated from all available freshwater SMAVs (n = 20). normal logistic triangular Gumbel Pareto Weibull Burr 0.0001 0.001 0.01 0.1 1 10 100 a. 3 SMAVs chosen at random Typical OPP species

normal logistic triangular Gumbel Pareto 0.0001 0.001 0.01 0.1 1 10 100Ratio of HC5at n = 3 to HC5estimated using 20 SMAVs

b.

normal logistic triangular 0.0001 0.001 0.01 0.1 1 10 100

Distribution c.Page 90



### 8.6. Variance plots of analyses at n = 3 using rand om SMAVs Notes: Figures in this series for malathion and propoxur freshwater tests are Figures 8 and 9 in the main text. Panels represent a. maximum likelihood estimate s, b. moment estimates, c. graphical estimates. Note log 10 scale of Y-axes.

Figure S8.6.1. Variance in HC 5 estimates from 1,000 draws of 3 randomly chosen freshwa ter SMAVs for
diazinon

. normal logistic triangular Gumbel Pareto Weibull Burr

0 2 4 6 a.

normal logistic triangular Gumbel Pareto 0 2 4 6Log10(var(HC5)) among 1,000 replicate draws

b.

normal logistic triangular 0 2 4 6

Distribution c.Page 91

Figure S8.6.2. Variance in HC 5 estimates from 1,000 draws of 3 randomly chosen freshwa ter SMAVs for chlorpyrifos

.

Figure S8.6.3. Variance in HC 5 estimates from 1,000 draws of 3 randomly chosen freshwa ter SMAVs for
fenitrothion

. normal logistic triangular Gumbel Pareto Weibull Burr

0 1 2 3 4 a.

normal logistic triangular Gumbel Pareto 0 1 2 3 4Log10(var(HC5)) among 1,000 replicate draws

b.

normal logistic triangular 0 1 2 3 4

Distribution c.

normal logistic triangular Gumbel Pareto Weibull Burr 0 2 4 6 a.

normal logistic triangular Gumbel Pareto 0 2 4 6Log10(var(HC5)) among 1,000 replicate draws

b.

normal logistic triangular 0 2 4 6

Distribution c.Page 92

Figure S8.6.4. Variance in HC 5 estimates from 1,000 draws of 3 randomly chosen freshwa ter SMAVs for methyl parathion

. normal logistic triangular Gumbel Pareto Weibull Burr

0 2 4 6 a.

normal logistic triangular Gumbel Pareto 0 2 4 6Log10(var(HC5)) among 1,000 replicate draws

b.

normal logistic triangular 0 2 4 6

Distribution c.Page 93

Figure S8.6.5. Variance in HC 5 estimates from 1,000 draws of 3 randomly chosen freshwa ter SMAVs for
dichlorvos

. normal logistic triangular Gumbel Pareto Weibull Burr

0 2 4 6 8 a.

normal logistic triangular Gumbel Pareto 0 2 4 6 8Log10(var(HC5)) among 1,000 replicate draws

b.

normal logistic triangular 0 2 4 6 8

Distribution c.Page 94

Figure S8.6.6. Variance in HC 5 estimates from 1,000 draws of 3 randomly chosen saltwat er SMAVs for chlorpyrifos

. normal logistic triangular Gumbel Pareto Weibull Burr

-1 0 1 2 3 a.

normal logistic triangular Gumbel Pareto -1 0 1 2 3Log10(var(HC5)) among 1,000 replicate draws

b.

normal logistic triangular -1 0 1 2 3

Distribution c.Page 95

Figure S8.6.7. Variance in HC 5 estimates from 1,000 draws of 3 randomly chosen saltwat er SMAVs for
fenthion

. normal logistic triangular Gumbel Pareto Weibull Burr

0 1 2 3 4 5 a.

normal logistic triangular Gumbel Pareto 0 1 2 3 4 5Log10(var(HC5)) among 1,000 replicate draws

b.

normal logistic triangular 0 1 2 3 4 5

Distribution c.Page 96

Figure S8.6.8. Variance in HC 5 estimates from 1,000 draws of 3 randomly chosen freshwa ter SMAVs for carbaryl

. normal logistic triangular Gumbel Pareto Weibull Burr

0 2 4 6 8 a.

normal logistic triangular Gumbel Pareto 0 2 4 6 8Log10(var(HC5)) among 1,000 replicate draws

b.

normal logistic triangular 0 2 4 6

Distribution c.Page 97

Figure S8.6.9. Variance in HC 5 estimates from 1,000 draws of 3 randomly chosen saltwat er SMAVs for carbaryl

. normal logistic triangular Gumbel Pareto Weibull Burr

0 2 4 6 8 a.

normal logistic triangular Gumbel Pareto 0 2 4 6Log10(var(HC5)) among 1,000 replicate draws

b.

normal logistic triangular 0 2 4 6

Distribution c.

normal logistic triangular Gumbel Pareto Weibull Burr 0 2 4 6 a.

normal logistic triangular Gumbel Pareto 0 2 4 6Log10(var(HC5)) among 1,000 replicate draws

b.

normal logistic triangular 0 2 4 6

Distribution c.Page 98

Figure S8.6.10. Variance in HC 5 estimates from 1,000 draws of 3 randomly chosen freshw ater SMAVs for methomyl

.Page 99



### 8.7. Goodness-of-fit results for n = 20 and n = 50. Table S8.7.1. Power of parametric bootstrap goodness- of-fit tests for maximum likelihood at n = 20.

normal logistic triangular Gumbel Pareto Weibull Burr

normal 0.020 0.067 0.080 0.087 0.733 0.167 0.033 logistic 0.080 0.073 0.193 0.173 0.773 0.133 0.093 triangular 0.080 0.107 0.047 0.127 0.727 0.120 0.080
Gumbel 0.167 0.113 0.427 0.040 0.433 0.373 0.047

Pareto 0.720 0.587 0.913 0.113 0.040 0.213 0.473
Weibull 0.273 0.187 0.467 0.247 0.940 0.040 0.353

Burr 0.067 0.067 0.247 0.093 0.733 0.240 0.033

Table S8.7.2. Power of parametric bootstrap goodness- of-fit tests for moment estimates at n = 20.

normal logistic triangular Gumbel Pareto

normal 0.040 0.047 0.040 0.187 1.000 logistic 0.093 0.067 0.133 0.267 1.000 triangular 0.073 0.087 0.053 0.240 1.000
Gumbel 0.240 0.193 0.300 0.047 1.000

Pareto 0.747 0.693 0.813 0.233 1.000
Weibull 0.307 0.247 0.360 0.733 1.000

Burr 0.093 0.067 0.107 0.133 1.000

Table S8.7.3. Power of parametric bootstrap goodness- of-fit tests for graphical estimates at n = 20.

normal logistic triangular

normal 0.033 0.007 0.053 logistic 0.127 0.053 0.180 triangular 0.053 0.027 0.067
Gumbel 0.260 0.160 0.313

Pareto 0.793 0.660 0.820
Weibull 0.327 0.207 0.347

Burr 0.113 0.027 0.180

Table S8.7.4. Power of parametric bootstrap goodness- of-fit tests for maximum likelihood at n = 50.

normal logistic triangular gumbel Pareto Weibull Burr

normal 0.080 0.067 0.247 0.293 1.000 0.513 0.073 logistic 0.073 0.053 0.520 0.320 1.000 0.553 0.067 triangular 0.073 0.113 0.087 0.260 1.000 0.407 0.107
Gumbel 0.460 0.227 0.840 0.040 0.960 0.647 0.073

Pareto 1.000 0.973 1.000 0.107 0.060 0.253 0.860
Weibull 0.500 0.307 0.880 0.400 1.000 0.047 0.607

Burr 0.113 0.067 0.507 0.340 1.000 0.533 0.080Page 100

Table S8.7.5. Power of parametric bootstrap goodness- of-fit tests for moment estimates at n = 50.

normal logistic triangular Gumbel Pareto

normal 0.093 0.087 0.120 0.460 1.000 logistic 0.100 0.067 0.200 0.447 1.000 triangular 0.067 0.140 0.087 0.473 1.000
Gumbel 0.500 0.427 0.580 0.060 1.000

Pareto 1.000 1.000 1.000 0.553 1.000
Weibull 0.547 0.440 0.633 0.973 1.000

Burr 0.133 0.073 0.227 0.387 1.000

Table S8.7.6. Power of parametric bootstrap goodness- of-fit tests for graphical estimates at n = 50.

normal logistic triangular

normal 0.100 0.027 0.153 logistic 0.173 0.060 0.247 triangular 0.040 0.047 0.067
Gumbel 0.507 0.380 0.607

Pareto 0.993 0.993 1.000
Weibull 0.607 0.400 0.680

Burr 0.193 0.073 0.300



### 8.8. Model selection results for SSDs fit to GMAVs when N = 8 and ALC MDRs

are met Notes: The tables in this series pertaining to malathion is Tables 44 in the main text. Average is the average of model-averaged estimates using Akaike wei ghts, calculated following methods described by Burnham and Anderson (2002).

Table S8.8.1. Mean estimated diazinon HC 5 s when n = 8 freshwater GMAVs meeting the ALC MDRs Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 0.33 0.19 0.57 0.51 0.85 0.10 0.41 0.39

Moment Estimators 0.23 0.24 0.21 0.78 0.69 Graphical Methods 0.05 0.03 0.07

Table S8.8.2. Mean estimated chlorpyrifos HC 5 s when n = 8 freshwater GMAVs meeting the ALC MDRs Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 0.05 0.04 0.09 0.10 0.14 0.01 0.07 0.07
Moment Estimators 0.04 0.04 0.04 0.11 0.12 Graphical Methods 0.01 0.01 0.02Page 101

Table S8.8.3. Mean estimated fenitrothion HC 5 s when n = 8 freshwater GMAVs meeting the ALC MDRs Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 1.64 1.45 2.29 1.77 2.43 1.41 1.30 1.14
Moment Estimators 1.16 1.24 1.03 3.25 2.06 Graphical Methods 0.34 0.23 0.45

Table S8.8.4. Mean estimated methyl parathion HC 5 s when n = 8 freshwater GMAVs meeting the ALC

MDRs Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 0.76 0.53 1.53 1.22 2.25 0.39 0.91 0.89
Moment Estimators 0.53 0.59 0.50 1.92 1.89 Graphical Methods 0.12 0.08 0.17

Table S8.8.5. Mean estimated dichlorvos HC 5 s when n = 8 freshwater GMAVs meeting the ALC MDRs Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 0.44 0.64 0.33 0.33 0.21 0.23 0.37 0.20
Moment Estimators 0.29 0.33 0.24 1.27 0.14 Graphical Methods 0.05 0.03 0.07

Table S8.8.6. Mean estimated chlorpyrifos HC 5 s when n = 8 saltwater GMAVs meeting the ALC MDRs Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 0.013 0.009 0.020 0.056 0.090 0.001 0.046 0.040

Moment Estimators 0.008 0.010 0.008 0.030 0.076 Graphical Methods 0.002 0.001 0.003

Table S8.8.7. Mean estimated fenthion HC 5 s when n = 8 saltwater GMAVs meeting the ALC MDRs Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 0.039 0.023 0.061 0.070 0.065 0.015 0.008 0.041

Moment Estimators 0.025 0.028 0.023 0.101 0.050 Graphical Methods 0.005 0.003 0.007

Table S8.8.8. Mean estimated carbaryl HC 5 s when n = 8 freshwater GMAVs meeting the ALC MDRs Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 6.97 8.88 7.17 5.50 6.16 5.44 6.31 2.96
Moment Estimators 4.84 5.43 4.43 12.19 5.06 Graphical Methods 1.70 1.10 1.93Page 102

Table S8.8.9. Mean estimated carbaryl HC 5 s when n = 8 saltwater GMAVs meeting the ALC MDRs Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 3.24 2.38 6.78 4.36 7.74 1.79 3.50 4.61

Moment Estimators 2.30 2.47 2.09 7.00 6.50 Graphical Methods 0.64 0.41 0.83

Table S8.8.10. Mean estimated propoxur HC 5 s when n = 8 freshwater GMAVs meeting the ALC MDRs Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 18.0 13.1 31.6 34.5 38.6 2.0 33.3 27.7
Moment Estimators 13.5 14.4 12.4 34.1 33.9 Graphical Methods 4.7 3.3 5.9

Table S8.8.11. Mean estimated methomyl HC 5 s when n = 8 freshwater GMAVs meeting the ALC MDRs Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 45.6 37.1 59.1 56.0 71.1 27.1 55.2 53.5
Moment Estimators 39.3 40.7 37.4 63.4 67.9 Graphical Methods 22.8 18.7 25.6



### 8.9. Model selection results for SSDs fit to GMAVs when N = 8 regardless of ALC
MDRs Notes: The tables in this series pertaining to malathion and fenitrothion are Tables 43 and 45 in the main text. Average is the average of model-averaged estimates using Akaike weights, calculated following methods described by Burnham and Anderson ( 2002).

Table S8.9.1. Mean estimated diazinon HC5s when n = 8 freshwater GMAVs regardless of MDRs Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 14.3 19.1 12.4 17.7 6.5 17.1 14.1 0.6
Moment Estimators 11.1 11.7 13.0 30.4 7.6 Graphical Methods 4.5 4.0 5.0

Table S8.9.2. Mean estimated chlorpyrifos HC5s when n = 8 freshwater GMAVs regardless of MDRs Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 0.16 0.12 0.21 0.21 0.29 0.06 0.19 0.08

Moment Estimators 0.12 0.11 0.12 0.30 0.24 Graphical Methods 0.04 0.02 0.05Page 103

Table S8.9.3. Mean estimated methyl parathion HC 5 s when n = 8 freshwater GMAVs regardless of MDRs Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 43.6 49.2 17.8 24.1 21.1 33.5 50.4 0.9

Moment Estimators 29.0 17.5 28.1 28.5 32.0 Graphical Methods 22.6 8.5 15.4

Table S8.9.4. Mean estimated dichlorvos HC 5 s when n = 8 freshwater GMAVs regardless of MDRs Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 7.7 4.9 9.3 9.8 15.1 1.9 12.5 0.6
Moment Estimators 6.6 5.6 4.9 12.1 9.5 Graphical Methods 2.5 1.5 2.7

Table S8.9.5. Mean estimated chlorpyrifos HC 5 s when n = 8 saltwater GMAVs regardless of MDRs Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 0.044 0.031 0.059 0.093 0.130 0.008 0.091 0.055
Moment Estimators 0.031 0.031 0.029 0.077 0.125 Graphical Methods 0.011 0.008 0.013

Table S6.8.6. Mean estimated fenthion HC 5 s when n = 8 saltwater GMAVs regardless of MDRs Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 0.49 0.89 0.63 0.38 0.39 0.89 0.57 0.05

Moment Estimators 0.37 0.40 0.31 0.93 0.34 Graphical Methods 0.14 0.06 0.19

Table S8.9.7. Mean estimated carbaryl HC 5 s when n = 8 freshwater GMAVs regardless of MDRs Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 83.9 95.5 103.8 82.6 82.4 75.7 95.0 15.0
Moment Estimators 84.0 74.5 72.4 142.5 95.9 Graphical Methods 53.4 38.9 57.3

Table S8.9.8. Mean estimated carbaryl HC 5 s when n = 8 saltwater GMAVs regardless of MDRs Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 6.00 7.50 9.16 6.02 8.21 5.66 5.46 4.86

Moment Estimators 4.37 4.68 3.91 12.05 6.84 Graphical Methods 1.31 0.89 1.72Page 104

Table S8.9.9. Mean estimated propoxur HC 5 s when n = 8 freshwater GMAVs regardless of MDRs Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 31.4 28.3 43.2 44.0 50.7 11.5 40.4 22.6

Moment Estimators 25.7 27.0 26.1 54.3 47.1 Graphical Methods 12.2 8.8 13.1

Table S8.9.10. Mean estimated methomyl HC 5 s when n = 8 freshwater GMAVs regardless of MDRs Fitting Method normal logistic triangular Gumbel Pareto Weibull Burr average
Maximum Likelihood 91.3 98.4 98.1 96.4 92.6 77.3 93.0 68.3
Moment Estimators 85.7 87.2 82.5 125.6 86.5 Graphical Methods 51.7 49.0 57.0



### 8.9. Preliminary analysis of the use of MOA-specifi c parameters for the logistic
distribution

Kristina Garber US Environmental Protection Agency Office of Pesticide Programs Washington, DC

de Zwart (2002) proposed that knowledge of mode of a ction (MOA) could be used to improve the ability to fit SSDs at smaller sample sizes. He h ypothesized that within an MOA, the average test result among species would vary, but that the variance among responses would remain relatively constant. Thus, given a set of pre-existing data conc erning chemicals with a given MOA, the expected variation in response to a new chemical with the same MOA could be specified a priori, and test results for the new chemical could be used only to estimate t he mean response. To illustrate this idea, de Zwart used the logistic distribution and hypothesized that the location parameter (α) would vary among chemicals, but that the scale parameter (β) would rem ain relatively constant. As a preliminary assessment of de Zwart’s method, I est imated β for 23 chemicals (Table ?)and estimated mean values of the estimated β by mode of ac tion. These were compared to the mean β values reported by de Zwart (2002). In total, it was possible to compare β values from 5 modes of action, including nonpolar narcosis, polar narcosis, AChE in hibition by carbamate insecticides, AChE inhibition by organophosphate insecticides and neurotoxicity cause d by organochlorine insecticides. Of these modes of action, the average β calculated for neurot oxicity for this analysis is closest to the average β reported by de Zwart (2002), although the sample si ze used by de Zwart for this mode of action was very small ( i.e. , N=2). The average β calculated in this analysis fo r the nonpolar and polar narcosis modes of action falls within the standard deviation of the av erage β reported by de Zwart. For both AChE inhibition modes of action, the average β calculated in this analysis falls outside of the standard deviation of the average β reported by de Zwart. Th is indicates that the average β reported by de Zwart may be applicable to narcosis modes of action and ne urotoxicity of organochlorine insecticides, but may be limited for AChE inhibitors.Page 105

Table 1. FAVs derived according to 85 guideline met hodology and HC 5 derived using de Zwart (2002) methodology.
Chemical MOA FAV C5 # acute GMAVs β Carbaryl AChE inhibition (carbamate) 4.4 96 55 0.50 Methomyl AChE inhibition (carbamate) 50 61 16 0.50 Propoxur AChE inhibition (carbamate) 9.6 116 13 0.50 Chlorpyrifos AChE inhibition (organophosphate) 0.074 0.47 30 0.71
Diazinon AChE inhibition (organophosphate) 0.54 13 28 0.71 Dichlorvos AChE inhibition (organophosphate) 0.083 15 22 0.71
Fenitrothion AChE inhibition (organophosphate) 0.11 11 30 0.71
Malathion AChE inhibition (organophosphate) 2.1 29 57 0.71 Methyl parathion AChE inhibition (organophosphate) 7.3 17 29 0.71 DDT neurotoxicity 0.85 1.9 49 0.50
Dieldrin neurotoxicity 0.54 2.9 18 0.50 Endosulfan neurotoxicity 0.21 0.67 32 0.50 Endrin neurotoxicity 0.21 0.26 37 0.50 Lindane neurotoxicity 1.9 6.3 23 0.50 Benzenamine* Narcosis (polar) 45 24000 39 0.31 Nonylphenol* Narcosis (polar) 41 50 21 0.31 Ammonia Narcosis (nonpolar) 4800 7100 20 0.39 Benzene* Narcosis (nonpolar) 6300 12000 9 0.39 Chloroform* Narcosis (nonpolar) 26000 24000 8 0.39 Phenol Narcosis (nonpolar) 8600 8300 42 0.39 Toluene* Narcosis (nonpolar) 10000 9900 11 0.39 Pentachlorophenol Uncoupling oxidative phosphorylation 20 24 40 0.38 1,3 Dichloropropene Reactivity 630 ??? 10 ??? *Not a pesticide

Table 2. Comparison of mean β values reported by de Zwart 2002 and mean β values of example chemicals.

Mode of Action De Zwart 2002 Current analysis Average β Standard

deviation of β * N** Average β Standard

deviation of β N**

Nonpolar narcosis 0.39 0.17 34 0.22 0.07 5

Polar narcosis 0.31 0.11 13 0.36 0.26 2 AChE inhibition (carbamates) 0.71 0.099 11 0.52 0.1 9 3 AChE inhibition (organophosphates) 0.50 0.26 27 0.87 0.10 6 Neurotoxicity (organochlorines) 0.50 0.18 2 0.51 0.1 0 5

*Calculated as: 

### N SEM of stdev *

= 

### β
** N = number of chemicals used to determine averag e β

In comparing the results of this analysis to those of de Zwart, (2002) it is important to note three major differences in the two methods. First, this anal ysis had more strict definitions of acute toxicity for aquatic animals (i.e., 48-h duration for crustaceans, 96-h for fish and other invertebrates). Second, thi s analysis developed SSDs based on acute toxicity data f or genera, while de Zwart (2002) developed SSDs for species. Third, this analysis developed SSDs using only aquatic animals, while de Zwart (2002) included effects data to derive SSDs for aquatic ani mals, plants and bacteria. This limitation arises because the modes of action of the different chemical s may be different for animals, plants andPage 106

bacteria. The latter difference in methods represents a significant limitation of the application of mean β values to estimation of C5 values intended to approxim ate FAVs, which are based on effects data for aquatic animals only. This may explain why the AChE inhibitors, which primarily act upon animals, but not plants, have average β values for only animals wh ich differ from the average β reported by de Zwart, which include plants. Because Pennington’s (2003) work depends upon that of de Zwart (2002), it should be noted that the uncertainties and limitations of the β value s recommended by de Zwart (2002)would also extend to the extrapolation factors reported by Penn ington (2003). This is especially relevant to the use of the extrapolation factors for AChE inhibitors, wh ich are based on β values that may be affected by inclusion of non-animal toxicity data in de Zwart’s (2002) analysis.
HC 5 values were generated for each example chemical usin g de Zwart’s (2002)approach (logistic distribution with MOA specific β value) along with th e full data set available for that chemical. For 20 of the 23 chemicals considered, the de Zwart (2002)meth od generated HC 5 values that overpredict the FAVs, with factors ranging 1.2x to 530x. Three of the 23 chemicals considered had HC 5 values that underpredicted the FAV, but were still on the same order of magnitude as the FAV. In these cases, all three chemicals (chloroform, phenol and toluene) are described by the nonpolar narcosis MOA.