Environ Sci Pollut Res (2015) 22:13980–13989
DOI 10.1007/s11356-015-4602-8
RESEARCH ARTICLE
Non-parametric kernel density estimation of species sensitivity
distributions in developing water quality criteria of metals
Ying Wang 1,2 & Fengchang Wu 2 & John P. Giesy 2,3 & Chenglian Feng 2 &
Yuedan Liu 4 & Ning Qin 2 & Yujie Zhao 2
Received: 19 January 2015 / Accepted: 23 April 2015 / Published online: 9 May 2015
# Springer-Verlag Berlin Heidelberg 2015
Abstract Due to use of different parametric models for establishing species sensitivity distributions (SSDs), comparison of
water quality criteria (WQC) for metals of the same group or
period in the periodic table is uncertain and results can be
biased. To address this inadequacy, a new probabilistic model,
based on non-parametric kernel density estimation was developed and optimal bandwidths and testing methods are proposed. Zinc (Zn), cadmium (Cd), and mercury (Hg) of group
IIB of the periodic table are widespread in aquatic environments, mostly at small concentrations, but can exert detrimental effects on aquatic life and human health. With these metals
as target compounds, the non-parametric kernel density estimation method and several conventional parametric density
estimation methods were used to derive acute WQC of metals
for protection of aquatic species in China that were compared
and contrasted with WQC for other jurisdictions. HC5 values
for protection of different types of species were derived for
three metals by use of non-parametric kernel density estimation. The newly developed probabilistic model was superior to
conventional parametric density estimations for constructing
SSDs and for deriving WQC for these metals. HC5 values for
the three metals were inversely proportional to atomic number, which means that the heavier atoms were more potent
toxicants. The proposed method provides a novel alternative
approach for developing SSDs that could have wide application prospects in deriving WQC and use in assessment of risks
to ecosystems.
Keywords SSD . Metals . HC5 . Probabilistic . Taxa .
Hazard
Introduction
Responsible editor: Thomas Braunbeck
Electronic supplementary material The online version of this article
(doi:10.1007/s11356-015-4602-8) contains supplementary material,
which is available to authorized users.
* Fengchang Wu
wufengchang@vip.skleg.cn
1
College of Water Sciences, Beijing Normal University,
Beijing 100875, China
2
State Key Laboratory of Environmental Criteria and Risk
Assessment, Chinese Research Academy of Environmental Science,
Beijing 100012, China
3
Department of Veterinary Biomedical Science and Toxicology
Centre, University of Saskatchewan, 44 Campus Drive,
Saskatoon, SK, Canada
4
The Key Laboratory of Water and Air Pollution Control of
Guangdong Province, South China Institute of Environmental
Sciences, The Ministry of Environment Protection of PRC,
Guangzhou 510065, China
Water quality criteria (WQC) are maximum acceptable threshold values for chemical substances or environmental parameters used to protect wildlife and humans from adverse effects
(US EPA 1976). The species sensitivity distribution (SSD)
which was proposed by Kooijman (Kooijman 1987), is one
method used to derive WQC (Anzecc 2000; CCME 2007) and
conduct assessments of risk to the environment (US EPA
1998), which has been widely used by the European Union
(including the Netherlands), Canada, Australia, New Zealand,
and Hong Kong, among other countries and organizations to
derive WQC. The method is based on the concept that species
have differential sensitivities to stressors, such as chemical
toxicants, that can be described by a probability function. If
information on relative sensitivities of a sufficiently large random sample of species that are expected to occur in a particular community or environment is available, the probability of
observing a more sensitive species can be estimated. Thus,
those selected species are used as surrogates that are assumed
Environ Sci Pollut Res (2015) 22:13980–13989
to represent the community structure of a specific ecosystem
and the toxicity data can be used to describe the SSD curve
(Posthuma et al. 2002). Conventional SSD methodologies assume that toxicity data for pollutants, expressed as the lethal
concentration to affect 50 % of individuals (LC50), effective
concentrations (non-lethal) to affect 50 % of individuals
(EC50), the no observable effect concentration (NOEC), or
lowest observable effect concentration (LOEC) can be accurately described by a parametric distribution, generally the
log-normal probability function, for which the log10 values
follow the normal probability density function. Development
of WQC uses statistical methods, to fit the probability function
describing hazardous concentration (HCp) along with various
measures of uncertainty to predict threshold values and promulgate WQC to protect particular assemblages of species.
Commonly used parametric models used for developing
SSDs, which are then used in development of WQCs, include
among others, log-normal (Van Vlaardingen et al. 2004), loglogistic (Pennington 2003), Burr Type III (Shao 2000),
Weibull (Van Straalen 2002), Gompertz (Newman et al.
2000), Sigmoid (Cao and Wu 2010), Gaussian (Wu et al.
2011a), and exponential growth (Wu et al. 2012a).
Since parametric models often have strong basic assumptions
or requirements for underlying data, there are potential inherent
biases, errors, and uncertainties caused by deviations from the
underlying theoretical models, associated with fitting empirical
data. Several authors have suggested that empirical toxicity data
usually deviates from assumptions of statistical distributions
(Brattin et al. 1996; Liu et al. 2014; Wang et al. 2008), so the
SSD parametric estimation is often unable to obtain satisfactory
results and there is no single, universally applicable distribution
that is appropriate for all toxicity data of pollutants (Forbes and
Forbes 1993; Shao 2000; Smith and Cairns Jr 1993; Warne 2002;
Wu et al. 2011b). Therefore, using a parametric distribution for
fitting the method, based on subjective assumptions, exhibits a
general lack of generality and tends to cause distortion of final
WQC values. Several authors, including Posthuma (Posthuma
et al. 2002), Hayashi (Hayashi and Kashiwagi 2010), and
Newman (Newman et al. 2000) have each proposed use of various non-parametric or distribution-free methods, such as Monte
Carlo, bootstrap, and Bayesian methods, respectively, which can
more accurately describe the empirical toxicity data more objectively without making as many assumptions as parametric
methods. However, Monte Carlo simulation and Bayesian
methods are still based on prior parametric distributions, which
might not satisfied the samples (Fox 2010; Hanna et al. 1998),
and the bootstrap method only obtains HCp distributions and its
confidence interval based on simple statistics (Newman et al.
2000), which are not the same distributions for all species. To
avoid the shortcomings of the basic bootstrap method, other
authors (Liu et al. 2014; Wang et al. 2008) have proposed modified bootstrap and bootstrap regression methods. However, if the
sample data includes dispersion in the form of outliers these
13981
methods can cause distortion of the data (Pan 2011) that result
in uncertainties of the derived SSDs. Since several parametric
models are used, there is uncertainty that can result in biases
when developing WQCs for the same group or period elements.
Therefore, a unified method to derive SSD functions was considered to be desirable.
Non-parametric kernel density estimation, which does not
need a priori information and does not depend on overall distributions and parameters that describe them, can estimate unbiased
distribution characteristics based on sample data (Rosenblatt
1956). This approach, with its natural robustness, has been used
previously, where it was applied to data on structures of molecules to indicate the ability of the chemical bond length, bond
valence angle, and torsion angle distributions (McCabe et al.
2014). Kernel density estimation has been used to simulate a
probability distribution describing speeds of wind, and the results
of that study demonstrated that the non-parametric model gave
more accurate estimates and fit the actual distribution of wind
speeds more accurately than did other more traditional parametric
distributions (Qin et al. 2011). Because it has minimal requirements for data and does not need to assume a specific statistical
distribution, estimated derived by use of kernel density methods
can be used to directly obtain SSDs. This provides a new approach for developing WQC for elements in the same group or
period of elements in the periodic table.
To demonstrate use of the approach, non-parametric kernel
density approach, SSDs for metals were developed for use in
deriving WQC to protect freshwater aquatic organisms in
China. Optimal bandwidth and test methods were developed
and are described herein. SSDs were used to derive HC5 and
WQC for three metals. The results were then verified for accuracy and effectiveness of the non-parametric kernel density
estimation to derive WQC evaluated.
Materials and methods
Toxicity data sets
Data on acute toxicity of the metals zinc (Zn), cadmium (Cd),
and mercury (Hg), used in the present study, were obtained
from USEPA ECOTOX database (http://cfpub.epa.gov/
ecotox/) and China National Knowledge Infrastructure
database (CNKI, http://www.cnki.net/) (Table S1). Accuracy
and reliability of data were evaluated by use of standard
methods (Klimisch et al. 1997), which were based on requirements of WQC guidelines and literature (Zhang et al. 2012).
The toxicity data used in this paper was based on the following screening rules. First, forms of metals used in the analysis
contained oxide, chloride, sulfate, nitrate, acetate, and sulfide,
yet all data were expressed as μg/L of the element of interest.
Second, three types of organisms (fish, zooplankton, and benthic animals) were contained (Barnthouse 2004), and at least
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Environ Sci Pollut Res (2015) 22:13980–13989
ten randomly selected organisms were sufficient to obtain an
effective estimation of aquatic ecosystems (Wheeler et al.
2002). Third, data were subjected to rigorous quality assurance guidelines (Buckler et al. 2003). The selected measurement endpoint was lethality, and corresponding toxicity
threshold values were reported as either the LC50 or EC50.
Results of tests of acute lethality in the USEPA database or
literature included the following results: 96-h LC50 or EC50
for fish; 48-h LC50 or EC50 for invertebrates; the toxicological
endpoints to fish and invertebrates mainly for immobility, respiratory inhibition, and lethality; water hardness was between
30 and 60 mg of CaCO3/L; pH ranged from 6 to 8; temperatures were appropriate per species (e.g., 12–15 μg/L for
Oncorhynchus mykiss; 20 μg/L for Daphnia magna).
Exposures were conducted in flow-through as well as static/
renewal. Finally, taxa used for deriving WQC are representative of those in aquatic ecosystems of China, including local
species and alien species that are now widely distributed in
China. When acute toxicity data for the same species
were available at the same duration of exposure, geometric means were calculated as species mean acute
values (SMAVs) (Stephen et al. 1985) (Table 1). It
was concluded, after a review of all of the available
literature, that there was insufficient information on
chronic effects on sublethal endpoints to derive chronic
SSDs based on sublethal measurement endpoints.
Species sensitivity distribution modeling
SSDs are probability distributions that describe differences in
sensitivities among species, compounds or mixtures of in
complex ecosystem, which is estimated from a sample of toxicity data for various species and visualized as a cumulative
distribution function (CDF) (Posthuma et al. 2002). Currently,
the cumulative probability of SSDs is first developed by
Hazen plotting positions (Eq. 1) (Cunnane 1978).
i − 0:5
p¼
;
n
1≤i≤n
ð1Þ
where p is cumulative probability, i is the sort level of taxa,
and n is the total number of taxa. SSDs were then derived
based on a sample of the given random variable of taxa by
parametric estimation methods or nonparametric estimation
methods. Parametric estimation methods are based on assumed theoretical parametric distributions of a continuous
variable to obtain the CDFs of the population (Cao and Wu
2010; Newman et al. 2000; Pennington 2003; Shao 2000; Van
Straalen 2002; Van Vlaardingen et al. 2004; Wu et al. 2011a,
2012a). SSDs are widely used to derive WQC for single
metals or organic pollutants but seldom for WQC of multiple
metals (Campbell et al. 2000; Giesy et al. 1999; Solomon et al.
1996; TenBrook et al. 2010; Vardy et al. 2011).
Non-parametric kernel density estimation
for constructing SSDs
Kernel density estimation is a non-parametric method that
estimates distributions of populations by use of a kernel function K, which is based on sample data without estimating
parameters based on any theoretical distribution (Silverman
1986). By way of example, in this process, let x1,x2,⋯xn
denote a sequence sample of toxicity data of independent
identically distributed random variables with an unknown
probability density function f(x), and the non-parametric kernel estimator is described by Eq. 2.
n
1 X
x − xi
K
f ð xÞ ¼
hn
nhn i¼1
∧
ð2Þ
where K(x) is a Borel function called the kernel function. It
satisfies the conditions described in Eq. 3.
8
Z
x − xi
<
K
ð
u
Þdu
¼
1
u
¼
ℝ
ð3Þ
hn
:
K ðxÞ≥ 0
where hn >0 is the window width and also called the
smoothing parameter or bandwidth, when n→∞,hn →0,
f n ðxÞ ∧ →f ðxÞða:s:Þ (Parzen 1962).
In general, the first step is to select a kernel function. Common
kernel functions are Parzen bandwidth (uniform), triangle,
Gaussian, and Epanechnikov functions (Table S2). When the
data is independently and identically distributed, the functions
have properties such as point-to-point asymptotic unbiasedness,
consistent gradual unbiasedness, and mean square consistency
(Chen and X. 1993). However, the optimal Gaussian kernel
function, the uniform kernel function, and Epanechnikov kernel
function are nearly equal when they all satisfy the conditions of
the kernel function (Rao 1983). Therefore, in this application, the
Gaussian kernel function was estimated by using Eq. 4.
1
u2
K ðuÞ ¼ pffiffiffiffiffiffi e− 2 ; u∈R
2π
ð4Þ
Determining the optimal bandwidth (hn) is more important
than selecting a kernel function, which might affect the accuracy of kernel estimation and needs a large number of tests to
determine hn (Epanechnikov 1969). When the Gaussian kernel function is selected, the computation of hn is given by
Silverman (1986) (Eq. 5).
hAMISE ≈ 1:06 σ ∧ n−1=5
ð5Þ
Goodness-of-fit evaluation of models
The Kolmogorov-Smirnov (K-S) test and a posteriori
tests were used to test the goodness-of-fit of constructed
Environ Sci Pollut Res (2015) 22:13980–13989
Table 1
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Log-transformed toxicity data information for three metals
Metals
Parameters
All species
Plants
Vertebratesf
Fish
Amphibians
Invertebratesg
Crustaceans
Other
invertebrates
Zn
Na
139
0.09
5.14
3.46
35
0.09
4.33
2.99
63
1.02
5.13
3.78
60
1.02
5.13
3.89
3
1.49
1.67
1.55
41
2.04
5.14
3.38
16
2.04
4.80
3.14
25
2.22
5.14
3.53
0.97
63
–1.37
5.48
2.89
1.22
90
–0.50
4.54
2.09
0.94
1.05
19
–1.37
4.11
2.19
1.42
9
1.29
3.10
2.12
0.54
0.94
26
0.49
5.48
3.27
1.10
33
0.48
4.23
2.44
0.70
0.82
25
1.51
5.48
3.38
0.96
26
0.48
3.74
2.41
0.66
0.10
1
0.49
0.49
0.49
/
7
1.57
4.23
2.55
0.86
0.77
18
1.94
4.79
3.09
0.85
48
–0.50
4.54
1.83
1.07
0.83
5
1.94
3.57
2.37
0.68
25
–0.50
4.54
1.55
1.18
0.70
13
2.10
4.79
3.37
0.76
23
0.63
3.88
2.14
0.85
Minb
Maxc
μd
σe
Cd
Hg
a
a
N
Minb
Maxc
μd
σe
Na
Minb
Maxc
μd
σe
Number of species included in model
b
Minimum value of species toxicity data included in model for the metal
c
Maximum value of species toxicity data included in model for the metal
d
Mean of species toxicity data included in model for the metal
e
Variance of species toxicity data included in model for the metal
f
Vertebrates contain fish and amphibians in this research
g
Invertebrates contain crustaceans and other invertebrates in this research
SSDs. The K-S test, established by Kolmogorov
(Kolmogorov 1933) and Smirnov (Smirnoff 1939), is a
non-parametric distribution-free test of goodness-of-fit
based on the maximum difference between an empirical
and a hypothetical cumulative distribution, which might
be superior to the chi-squared test when it is applicable
(Massey Jr 1951). The established SSD model is
deemed sufficient when the Pks value is greater than
0.05, which means that the test fails to reject the null
hypothesis that the empirical data has the given hypothetical cumulative distribution, and the larger the Pks
value of the K-S test, the better the goodness-of-fit for
models (Qin et al. 2011). The a posteriori test was used
to evaluate differences between the SSD model developed and the observed data, of which indexes contain
root mean square errors (RMSE), coefficients of determination (R2), and error sum of squares (SSE). RMSE,
R 2 , and SSE derived from parametric models, and
RMSE and SSE derived from non-parametric models
were used to compare and check the adequacy of the
approaches. The model with minimum RMSE and SSE values
was deemed the best model for building SSDs and deriving HC5 values (Liu et al. 2014). Computational processes for the SSDs modeling were performed by use of
MALAB (2007b version).
HC5 and WQC derivation
In the process of deriving WQCs, a significant reason for developing SSDs is to derive an acceptable concentration of contaminants to protect a certain proportion of species, known as hazardous concentrations (HC). If the proportion of aquatic species
protected set to be 95 %, the acceptable concentration of contaminant is hazardous for the remaining 5 % species, the concentration is thereby defined as HC5 (Van Straalen and Denneman
1989). HC5 is the basis for deriving WQC (US EPA 2005;
Van Straalen and Denneman 1989). Acute WQC were calculated
by use of an assessment (safety) factor of 2.0 to insert some
conservatism to account for possible uncertainties (Eq. 6) (Van
Sprang et al. 2004; Wu et al. 2011a, 2012b).
Acute WQC ¼ acute HC5=2
ð6Þ
Results and discussion
SSDs based on non-parametric kernel density estimation
method modeling
There was a total of 292 species for which there was data for
toxicity of Zn (139), Cd (63), and Hg (90), that was deemed to
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Environ Sci Pollut Res (2015) 22:13980–13989
be valid, including 63 plants, 122 vertebrates (111 fish
and 11 amphibians), and 107 invertebrates (46 crustacean and 61 other invertebrates). The range of toxicity
data for Cd was largest with values ranging from 0.04
to 3.01×105 and with a standard deviation of 3.86×105.
The range of toxicity data for Zn was least with values
ranging from 1.22 to 1.40 × 105 and with a standard
deviation of 2.27×105. Since the range of values of data
was relatively large, the raw data was transformed as
the log10, which reduced the relative difference and
smoothed the data, which made the data more amenable
to do the calculations.
After log10 transformation, the Gaussian kernel function was used to construct the non-parametric kernel
density estimation models of SSDs for the three metals.
First, optimum widths were calculated according to
Eq. 5 (Table 2), which takes both smoothness of the
density curve and goodness-of-fit of the model into consideration. Thus, the non-parametric kernel density estimation
Table 2
Distribution
PDF
Zn
Normal
1
y ¼ pffiffiffiffi
e−
2πσ
Logistic
y¼ e
ðx − μÞ2
2σ2
x−μ
σ
x−μ
σ
σ 1þe
Log-normal
y ¼ xσp1 ffiffiffiffi
e−
2π
Log-logistic
y¼ e
2
ðlnx−μÞ2
2σ2
logðxÞ−μ
σ
logðxÞ−μ
σ
σx 1þe
Non-parametric kernel
Normal
–
Logistic
y¼ 1
y ¼ pffiffiffiffi
e
2πσ
σ
−
x−μ
σ
x−μ
1þe σ
e
Log-normal
y ¼ xσp1 ffiffiffiffi
e−
2π
Log-logistic
y¼ e
Non-parametric kernel
Normal
–
2
ðlnx − μÞ2
2σ2
logðxÞ−μ
σ
logðxÞ−μ
σ
Logistic
y¼ 1
y ¼ pffiffiffiffi
e
2πσ
σ
x−μ
σ
x−μ
1þe σ
e
y ¼ xσp1 ffiffiffiffi
e−
2π
Log-logistic
y¼ e
y¼
Non-parametric kernel
–
2
ðx − μÞ2
−
2σ2
Log-normal
Sigmoid
2
ðx−μÞ2
2σ2
σx 1þe
Hg
1 X 1 − ð0:296Þ
2
pffiffiffiffiffiffi e
f ð xÞ ¼
41:14 i¼1 2π
ð7Þ
1 X 1 − ð0:564Þ
2
pffiffiffiffiffiffi e
35:53 i¼1 2π
ð8Þ
1 X 1 − ð0:368Þ
2
pffiffiffiffiffiffi e
f ð xÞ ¼
33:12 i¼1 2π
ð9Þ
139
∧
∧
f ð xÞ ¼
∧
63
90
x − xi 2
x − xi 2
x − xi 2
Alternatively, after the test for normality, the parametric
models for SSDs were established by use of the maximum
likelihood estimation method, and then compared with the
non-parametric kernel estimation model. All models passed
the K-S test except for the normal distribution model, the
log-normal distribution model, and the log-logistic distribution model of Zn (Table 2). The Pks values of K-S for
Comparison among several good fitting parametric models and non-parametric models for three metals
Metals
Cd
models of SSDs for three metals were determined (Eqs. 7, 8,
and 9).
2
ðlnx−μÞ2
2σ2
logðxÞ − μ
σ
logðxÞ − μ
σx 1þe σ
a
1þe− k ðx − x0 Þ
2
Parameters
pK-S
R2
RMSE
SSE
μ=3.46±0.083
σ=0.97±0.059
μ=3.57±0.078
σ=0.53±0.038
μ=1.16±0.045
σ=0.53±0.032
μ=1.25±0.028
σ=0.20±0.015
0.014
0.9553
0.0610
0.518
0.361
0.9796
0.0412
0.236
7.41E-07
0.7889
0.1326
2.446
0.00898
0.9336
0.0744
0.769
Bandwidth=0.296
μ=2.89±0.15
σ=1.22±0.11
μ=2.98±0.15
σ=0.66±0.069
μ=1.75±0.033
σ=0.26±0.023
μ=1.78±0.026
σ=0.12±0.013
0.907
0.301
–
0.9722
0.0175
0.0481
0.0424
0.146
0.354
0.9764
0.0443
0.124
0.0715
0.9217
0.0807
0.411
0.00898
0.9717
0.0486
0.149
Bandwidth=0.564
μ=2.09±0.099
σ=0.94±0.071
μ=2.12±0.096
σ=0.52±0.046
μ=1.61±0.021
σ=0.20±0.015
μ=1.63±0.020
σ=0.11±0.010
0.654
0.581
–
0.9816
0.0370
0.0391
0.0863
0.138
0.597
0.9887
0.0307
0.0847
0.158
0.9555
0.0609
0.334
0.495
0.9780
0.0428
0.165
a=1.054
x0 =20215
k=1.824
Bandwidth=0.368
0.600
0.9913
0.0266
0.0639
0.897
–
0.0233
0.0488
Parametric models including normal distribution, logistic distribution, log-normal distribution, log-logistic distribution, and sigmoid distribution, and
parameters including the formula of probability density function (PDF), formula parameters of distribution, and goodness-of-fit evaluation results of the
model
Environ Sci Pollut Res (2015) 22:13980–13989
13985
the three non-parametric kernel density estimation models
were PZn =0.907, PCd =0.654, and PHg =0.897, which were
the maximum Pks values among the models of the three
metals, respectively. Values of RMSE and SSE for three
non-parametric kernel density estimation models were the
minimum values among the models of three metals, respectively. Therefore, the non-parametric kernel density estimation
of SSDs of the three metals was the best fitting model. This
result suggested that without the assumption of the sample
data, the kernel density estimation for SSDs built in the present study obtained good fitting capacity, high accuracy, and
the best simulation results.
The kernel density estimation model differed from bootstrap and bootstrap regression methods, because the former
obtained the cumulative density function (CDF) of all taxa
and the latter two only calculated the specific statistic value
estimate and its confidence interval by use of random resampling to construct an empirical distribution function or specific
parameter models (Grist et al. 2002). Moreover, data from
toxicity tests often has outliers and a skewed distribution that
approximated the normal distribution that appeared as the relative Bleptokurtosis and heavy tails.^ Therefore, parametric
models could not fit the sample data well. However, kernel
density estimation can reduce the effects of these outliers and
has a good robustness when used to estimate the SSDs, because it is less constrained by the data and could fit the model
without priori information.
HC5 derivation and comparison
In general, the rationale for using the HC5 for deriving WQC
(US EPA 2005) is that 95 % aquatic taxa will be protected
(Van der Hoeven 2001). Therefore, HC5s were obtained from
parametric models and the kernel density estimation model,
respectively (Table 3). Searching the original data, the species
near 5 % cumulative probability calculated for Zn, Cd, and Hg
were Bufogargarizans, Scenedesmus quadricauda, and
Crustaceans monoculus, of which the acute toxicity values
were 30.8, 4.17, and 2.9 μg/L, respectively. Therefore, the
least deviation between HC5 values and acute toxicity values,
which were calculated by use of the kernel density, and the
normal distribution parametric model were 16.46, 18.71, and
19.31 %, respectively. For Hg, the deviation between the HC5
value calculated by the non-parametric kernel density estimation model and acute toxicity was 26.21 %, which was slightly
larger than that of the normal distribution parametric model.
However, the kernel density estimation model can better fit
the overall trend of sensitivities of taxa and the internal properties of the data (Fig. 1). According to the principle of choosing better fitting models (Wu 2012), the kernel density estimation model was more reliable in obtaining the HC5 value
than the normal distribution parametric model.
The order of magnitude of HC5 values was Hg<Cd<Zn.
HC5 values of metals in group IIB were inversely proportional
to atomic number in the periodic table. Zn, an essential element for organisms, plays an important role as biochemical
enhancer of many enzymes (Friberg et al. 1979), which could
have a negative effect on organism growth when deficiencies
exist. Zn also can be hazardous when its concentrations exceed threshold values. It might cause adverse effects through
combination with biological macromolecules to the organisms, such as reduction of enzyme activity, gene expression
changes, reproduction, and development (Feng et al. 2013;
Poynton et al. 2007). Cd, which is not a required element
and can be a toxic metal, has been defined as a key pollutant
by the United Nations Environment Programme (UNEP) in
1974 (Wu et al. 2011b). In plants, exposed to unusually large
concentrations of Cd can cause adverse effects such as phosphorous deficiency problems with transport of manganese
(Mn) (Godbold and Hüttermann 1985), interference with uptake, transport, and use of essential elements, such as Ca, Mg,
P, and K, and water (Sharma et al. 1985). In aquatic animals,
exposure to Cd can result in anemia, enteropathy, damaging
renal tubules, and osteoporosis for aquatic organisms and
humans (Fox 1979). Its toxicity effects are similar to those
of the required element Zn. Cd has been found to be bound
into a ternary Cd-Zn-protein complex in mammalian kidney
(Friberg et al. 1979), and Zn and Cd act through the same
Table 3 Comparison among HC5 values derived by different models
for three metals
Models
Normal
Logistic
Log-normal
Log-logistic
Sigmoid
Non-parametric kernel
HC5 (μg/L)
Zn
Cd
Hg
72.32
101.94
21.68
88.09
–
35.87
7.59
10.45
5.4
13.14
–
3.39
3.46
3.83
3.98
5.01
3.72
2.14
Fig. 1 Comparison between normal distribution and non-parametric kernel estimate model of probability density function
13986
mechanism of action and exhibit joint toxicity (Guan and
Wang 2004). The mechanism of toxic action of Hg is not
obvious but might be related to the fact that it binds to thiol
groups in proteins and thus affects enzyme activity of organisms (Friberg et al. 1979; Zhang et al. 2012). Mercury is different from Zn and Cd because its affinity for binding to different insoluble cells ligand is very strong, especially to the
nucleus and lysosomes. Hence, toxicity of Hg is the greatest in
group IIB. In addition, metals of group IIB are D-class elements. They are more stable than other D-class elements because their electrons are completely filled orbitals. Toxicity
potencies of metals in group IIB are directly proportional to
periodic number, and sequences are similarly associated with
bulk chemical properties such as ionic radius (Haynes 2012)
and electronegativity (Yu et al. 2009). Results of several studies illustrated that the ionic radius and electronegativity have a
close relationship with toxicity potency (Walker et al. 2003).
Environ Sci Pollut Res (2015) 22:13980–13989
Table 4
Comparison of freshwater aquatic criteria
Group
Deriving methods
WQC
(μg/L)
Literatures
Zn
Normal-SSD
Logistic-SSD
36.16
50.97
This research
Log-normal-SSD
10.84
Log-logistic-SSD
Non-parametric kernel-SSD
44.045
17.935
Cd
ExpGro1
48.43
Wu et al. (2012a)
Burr ΙΙΙ-SSD
29.94
Kong et al. (2011)
Percentage of toxicity sorting
Evaluation factors
120
30
US EPA (1996)
CCME (2007)
Normal-SSD
Logistic-SSD
3.795
5.225
This research
Log-normal-SSD
Log-logistic-SSD
2.7
6.57
Non-parametric kernel-SSD
1.695
Derivation and comparison of WQC
Slogistic3
0.4218
Wu et al. (2012a)
Burr ΙΙΙ-SSD
2.265
Kong et al. (2011)
WQC derived in the present study were compared with those
developed by various countries (Table 4). The results indicated that WQC derived in the present study for three metals are
all less than those recommended by USEPA. Comparing with
other values in the literature, WQC derived by for Cd and Hg
during the present study were less different than those recommended by USEPA. The difference for Zn was within an order
of magnitude. Compositions of taxa used in the assessment and their relative sensitivities are important contributors to the SSD and can directly affect the accuracy
of results (Brock et al. 2006). For example, representative fishes to be protected in China belong mostly to the
family Cyprinidae, while in North America, sensitive
valued species to be protected are coldwater fishes of
the family Salmonidae. The species selected in the present study represent the typical species in aquatic environments
of China, which was the reason for the differences from the
WQC recommended by US and Australia.
However, WQC derived in the present study were significantly different from other values in the literature (Kong et al.
2011; Li et al. 2012; Wu et al. 2012a; Zhang et al. 2012) which
indicates that differences exist between WQC derived by use
of the non-parametric kernel density estimation model and
parametric models (Table 4). The number of species used by
Wu et al. (2012a) to derive WQC for Zn and Cd were 45 and
28, respectively, and that used for Hg used by Kong et al.
(2011) was 30. The data set is small therefore has a large
deviation, which might lead to difficulty in obtaining accurate
parametric estimation. The value of SSE derived from the logsigmoid model used by Zhang et al. (2012) was greater than
that in the present study (0.0664>0.0488). The AndersonDarling test used by Li et al. (2012) to test the goodness of
fitting for the log-normal model built by ETX2.0 Software
Percentage of toxicity sorting
Normal-SSD
Logistic-SSD
2
1.73
1.91
US EPA (2001)
This research
Log-normal-SSD
Log-logistic-SSD
Sigmoid-SSD
Non-parametric kernel-SSD
Log-slogistic3-SSD
1.99
2.51
1.86
1.07
1.74
Zhang et al. (2012)
Burr III-SSD
3.33
Kong et al. (2011)
RIVM-SSD
3.42
Li et al. (2012)
Percentage of toxicity sorting
1.4
US EPA (1980)
Hg
(Van Vlaardingen et al. 2004) was not very accurate. The
non-parametric kernel density estimation model was better
for fitting the empirical data and were more accurate than
the above two parametric models. Furthermore, the WQC
derived by the non-parametric kernel density estimation model were closer to the empirical data than WQC recommended
by other jurisdictions, whereas there are no study results about
WQC by use of the non-parametric kernel density estimation
model for SSDs.
Correlation analyses of water quality criteria of metals
When SSDs derived for fewer than ten taxa, using the kernel
density estimation methods were developed (Figs. 2, 3, and 4).
HC5 values of Zn for the pairs of vertebrates and crustaceans,
fish and other invertebrates were similar, which means that
they are same protected at approximately the same concentrations (Table 5; Fig. S1–S3). Similar to results for
Zn, Cd is more harmful for plants, and more hazardous to
vertebrates than invertebrates. However, Hg was different
Environ Sci Pollut Res (2015) 22:13980–13989
Fig. 2 SSDs and HC5 values determination of different creature types for
Zn
from Zn or Cd, in that it was more hazardous to crustaceans,
and more harmful for invertebrates than vertebrates and
plants. Results for these three metals are similar to those of
previous reports (Wu et al. 2011a, b; Zhang et al. 2012). The
reason for this result might because the detoxification mechanism of organisms for the higher trophic level is more completed, so the lower trophic level such as plants and invertebrates is obviously more sensitive than that of vertebrates
(Pavičić et al. 1994).
Based on HC5 values, except for amphibians, only crustaceans exhibited differences from previously reported WQC
and consistent with the sequence of binding of the metals
in vitro to protein: Hg2+ >Cd2+ >Zn2+(Amiard et al. 2006).
Therefore, toxicity of the three metals is directly proportional
to atomic number. The sensitivity of crustaceans to the three
metals is Hg>Zn>Cd. It appeared in the marine invertebrate
species such as Mytilusgalloprovincialis as well as other mollusks and crustaceans, which might because crustaceans have
13987
Fig. 4 SSDs and HC5 values determination of different creature types for
Hg
an effective defense mechanism to hazard of some toxic
metals (Pavičić et al. 1994).
Uncertainty analysis of the model
Accuracy and consistency of empirical data on toxicity of
these three metals to various taxa is critical to the model, so
collection and validation of data is important. Data employed
in the present study did not consider effects of bioaccumulation of the three metals and ranges of values observed in the
literatures. In addition, toxicity of pollutants can be affected
by various environmental factors, such as hardness of water,
which was not considered, because there was insufficient data
across the range of possible values in the environment to
quantitatively describe effects of these environmental factors
on bioavailability and or toxic potencies of these metals.
Since width estimation by use of the kernel function in the
model depends on the toxicity data of the species selected,
width determination need to be careful. If hn is too small, then
when using a kernel function in the model to describe the
SSD, the model would be biased and not accurately reflect
Table 5 Comparison among HC5 values derived by non-parametric
kernel models of different taxa for three metals
Groups
Fig. 3 SSDs and HC5 values determination of different creature types for
Cd
Plant
Vertebrates
Fish
Amphibians
Invertebrates
Crustaceans
Other invertebrates
HC5 (μg/L)
Zn
Cd
Hg
3.01
30.93
143.49
–
102.37
34.07
160.31
0.11
14.36
40.03
–
29.28
57.08
92.64
–
11.4
7.67
16.12
0.79
0.38
3.16
13988
the internal characters of the data. In contrast, if hn is too large,
the structural characteristics of the data might not be shown
(Chen and X. 1993; Silverman 1986). Although the present
study obtained good results by the Gaussian kernel function
and its optimal bandwidth, it still could do appropriate adjustments to hn according to the fitting and smoothness of data
curve. Moreover, the non-parametric kernel density estimation
is not very suitable to use with small sample size (<30) (Chen
and X. 1993). Therefore, the supplement and improvement
should be considered for the non-parametric kernel density
estimation model for SSDs in future studies.
Conclusions
Based on the comparison with other approaches, the nonparametric kernel density estimation for SSDs is more simple,
flexible, accurate, and effective to sample data. The case study
of three metals verified the robustness and adaptability of the
method in derivation of WQC. Although the model presented
can reasonably develop SSDs, it does need to be further developed for use with small sample sizes (<30). The proposed
method has expanded the methodological foundation for use
of SSDs in development of WQC and provided solid support
for protection of aquatic organisms, which could be considered wide use in deriving WQC and assessing risk of metals.
Acknowledgments The present study was supported by the Environmental Public Welfare Program (201409037) and the National Natural
Science Foundation of China (Nos. 41130743 and 41473109).
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Table 1 Log-transformed toxicity data information for three metals.
Other
Invertebrates
a
N
139
35
63
60
3
41
16
25
b
min
0.09
0.09
1.02
1.02
1.49
2.04
2.04
2.22
c
max
5.14
4.33
5.13
5.13
1.67
5.14
4.80
5.14
Zn
d
μ
3.46
2.99
3.78
3.89
1.55
3.38
3.14
3.53
e
σ
0.97
1.05
0.94
0.82
0.10
0.77
0.83
0.70
a
N
63
19
26
25
1
18
5
13
b
min
--1.37
--1.37
0.49
1.51
0.49
1.94
1.94
2.10
c
max
5.48
4.11
5.48
5.48
0.49
4.79
3.57
4.79
Cd
d
μ
2.89
2.19
3.27
3.38
0.49
3.09
2.37
3.37
e
σ
1.22
1.42
1.10
0.96
/
0.85
0.68
0.76
a
N
90
9
33
26
7
48
25
23
b
min
--0.50
1.29
0.48
0.48
1.57
--0.50
--0.50
0.63
c
Hg
max
4.54
3.10
4.23
3.74
4.23
4.54
4.54
3.88
d
μ
2.09
2.12
2.44
2.41
2.55
1.83
1.55
2.14
e
σ
0.94
0.54
0.70
0.66
0.86
1.07
1.18
0.85
a
b
c
number of species included in model; minimum value of species toxicity data included in model for the metal; maximum value of
species toxicity data included in model for the metal; d mean of species toxicity data included in model for the metal; e variance of
species toxicity data included in model for the metal; f Vertebrates contain fish and amphibians in this research; g Invertebrates contain
crustaceans and other Invertebrates in this research.
Metals Parameters All Species
Plants
Vertebratesf
Fish
Amphibians Invertebratesg Crustaceans
Table 2 Comparison among several good fitting parametric models and
non-parametric models for three metals. Parametric models including normal
distribution, logistic distribution, log-normal distribution, log-logistic distribution and
sigmoid distribution, and parameters including the formula of probability density
function (PDF), formula parameters of distribution, and goodness-of-fit evaluation
results of the model.
metals Distribution
PDF
Parameters
pK-S
R2
RMSE SSE
μ= 3.46±0.083
1
y=
e
Normal
0.014 0.9553 0.0610 0.518
2πσ
σ= 0.97±0.059
μ= 3.57±0.078
e
y=
Logistic
0.361 0.9796 0.0412 0.236
σ (1 + e )
σ= 0.53±0.038
μ= 1.16±0.045
1
y=
e
log-normal
7.41E-07 0.7889 0.1326 2.446
Zn
xσ 2π
σ= 0.53±0.032
−
( x − µ )2
2σ 2
x−µ
σ
x−µ
σ
2
−
(ln x − µ )2
2σ 2
log( x ) − µ
log-logistic
σ
e
y=
log( x ) − µ


σ x 1 + e σ 


Non- parametric Kernel
2
/
Normal
y=
Logistic
y=
log-normal
y=
−
1
e
2πσ
Bandwidth= 0.296
μ= 2.89±0.15
σ= 1.22±0.11
μ= 2.98±0.15
σ= 0.66±0.069
( x − µ )2
2σ 2
x−µ
Cd
σ
e
σ (1 + e
x−µ
σ
)2
−
1
e
xσ 2π
(ln x − µ )2
2σ 2
log( x ) − µ
log-logistic
σ
e
y=
log( x ) − µ


σ x 1 + e σ 


Non- parametric Kernel
2
/
Normal
y=
Logistic
y=
log-normal
y=
−
1
e
2πσ
2σ 2
σ
σ (1 + e
x−µ
σ
)2
−
1
xσ 2π
Hg
(ln x − µ )2
e
2σ 2
log( x ) − µ
log-logistic
Sigmoid
Non- parametric Kernel
y=
y=
e
σ
log( x ) − µ


σ x 1 + e σ 


a
1 + e − k ( x − x0 )
/
2
0.00898 0.9336 0.0744
0.907
/
0.769
0.0175 0.0424
0.301 0.9722 0.0481
0.146
0.354 0.9764 0.0443
0.124
μ= 1.75±0.033
σ= 0.26±0.023
0.0715 0.9217 0.0807
0.411
μ= 1.78±0.026
σ= 0.12±0.013
0.00898 0.9717 0.0486
0.149
Bandwidth= 0.564
μ= 2.09±0.099
σ= 0.94±0.071
μ= 2.12±0.096
σ= 0.52±0.046
( x − µ )2
x−µ
e
μ= 1.25±0.028
σ= 0.20±0.015
0.654
/
0.0370 0.0863
0.581 0.9816 0.0391
0.138
0.597 0.9887 0.0307 0.0847
μ= 1.61±0.021
σ= 0.20±0.015
0.158 0.9555 0.0609
0.334
μ= 1.63±0.020
σ= 0.11±0.010
0.495 0.9780 0.0428
0.165
a=1.054
x0=20215
k=1.824
Bandwidth= 0.368
0.600 0.9913 0.0266 0.0639
0.897
/
0.0233 0.0488
Table 3 Comparison among HC5 values derived by different models for three metals.
models
Normal
Logistic
log-normal
log-logistic
Sigmoid
Non- parametric
Kernel
Zn
72.32
101.94
21.68
88.09
/
35.87
HC5/(μg·L-1)
Cd
7.59
10.45
5.4
13.14
/
3.39
Hg
3.46
3.83
3.98
5.01
3.72
2.14
Table 4 Comparison of freshwater aquatic criteria.
Group
Zn
Cd
Hg
Deriving methods
WQC(μg·L-1)
Normal-SSD
Logistic-SSD
log-normal-SSD
log-logistic-SSD
Non- parametric
Kernel-SSD
ExpGro1
36.16
50.97
10.84
44.045
Burr Ⅲ
-SSD
29.94
Literatures
This research
17.935
48.43
Percentage of toxicity
sorting
Evaluation factors
Normal-SSD
Logistic-SSD
log-normal-SSD
log-logistic-SSD
Non- parametric
Kernel-SSD
Slogistic3
0.4218
BurrⅢ-SSD
2.265
Wu et al.(Wu et al. 2012)
Kong et al.(Kong et al.
2011)
120
US EPA (U.S.EPA 1996)
30
3.795
5.225
2.7
6.57
CCME(CCME 2007)
This research
1.695
Wu et al.(Wu et al. 2012)
Kong et al.(Kong et al.
2011)
Percentage of toxicity
sorting
Normal-SSD
Logistic-SSD
log-normal-SSD
log-logistic-SSD
Sigmoid-SSD
Non- parametric
Kernel-SSD
1.73
1.91
1.99
2.51
1.86
log-Slogistic3-SSD
1.74
BurrⅢ-SSD
3.33
RIVM-SSD
Percentage of toxicity
sorting
3.42
Zhang et al.(Zhang et al.
2012)
Kong et al.(Kong et al.
2011)
Li et al.(Li et al. 2012)
1.4
US EPA(U.S.EPA 1980)
2
US EPA (U.S.EPA 2001)
This research
1.07
Table 5 Comparison among HC5 values derived by Non-parametric Kernel models of
different taxa for three metals.
Groups
Plant
Vertebrates
Fish
Amphibians
Invertebrates
Crustaceans
Other Invertebrates
Zn
3.01
30.93
143.49
/
102.37
34.07
160.31
HC5/(μg·L-1)
Cd
0.11
14.36
40.03
/
29.28
57.08
92.64
Hg
/
11.4
7.67
16.12
0.79
0.38
3.16