Download A framework for assessing the biological risks of increasing salinity

A Framework for Assessing the
Biological Risks of Increasing
Salinity in Victoria
W.J. Dixon
June 2007
Arthur Rylah Institute for Environmental Research
Technical Report Series No. 162
Arthur Rylah Institute for Environmental Research
Technical Report Series No. 162
A framework for assessing the biological risks
of increasing salinity in Victoria
William J. Dixon
June 2007
Published by: Arthur Rylah Institute for Environmental Research
Department of Sustainability and Environment
PO Box 137
Heidelberg, Victoria 3094
Australia
Telephone: (03) 9450 8600
www.dse.vic.gov.au/ari/reports
This publication may be cited as:
Dixon, W.J. (2007) A framework for assessing the biological risks of increasing
salinity in Victoria. Arthur Rylah Institute for Environmental Research, Technical
Report Series No. 162, Department of Sustainability and Environment, Heidelberg,
Victoria.
© The State of Victoria
Department of Sustainability and Environment 2007
This publication is copyright. Apart from any fair dealing for private study, research,
criticism or review allowed under the Copyright Act 1968, no part of this publication
may be reproduced, stored in a retrieval system or transmitted in any forms or by
any means, electronic, photocopying or other, without the prior permission of the
copyright holder.
ISBN 978-1-74208-052-9 (Print)
ISBN 978-1-74208-053-6 (Online)
ISSN 1835-3827 (Print)
ISSN 1835-3835 (Online)
Disclaimer
This publication may be of assistance to you but the State of Victoria and its
employees do not guarantee that the publication is without flaw of any kind or is
wholly appropriate for your particular purposes and therefore disclaims all liability
for any error, loss or other consequence, which may arise from you relying on any
information in this publication.
Authorised by the Victorian Government, Melbourne.
Abbreviations
ANZECC – Australian and New Zealand Environment and Conservation Council
ARI - Arthur Rylah Institute for Environmental Research
ARMCANZ - Agriculture and Resource Management Council of Australia and New
Zealand
CDF – Cumulative Density Function
DPI – Department of Primary Industries (Victoria)
DSE - Department of Sustainability and Environment (Victoria)
EC05 - Effective Concentration 5%: The concentration of a toxicant that results in a
5% effect
ECD - Environmental Concentration Distribution
FFG – The Flora and Fauna Guarantee Act 1988 (Victoria)
GLM - Generalised Linear Model
GLMM - Generalised Linear Mixed Model
LC50 - Lethal Concentration 50%: The concentration of a toxicant that results in a
50% lethal effect
LD50 - Lethal Dose 50%
LOEC - Lowest Observed Effect Concentration
LOEL - Lowest Observed Effect Level
LR - Likelihood Ratio
MCMC – Markov Chain Monte Carlo
ML – Maximum Likelihood
NOEC - No Observed Effect Concentration
PBA – Probability Bounds Analysis
PDF - Probability Density Function
SSD - Species Sensitivity Distribution
I
II
Abstract
Increasing salinity presents a significant threat to a wide range of native plants and
animals in Victoria. In order to ensure adequate conservation of our biodiversity
assets, management action is often required to mitigate or ameliorate the effects of
salinity. In many situations, however, little is known about which species, areas or
ecosystems might be most affected and how. Furthermore, management efforts must
often be prioritised in order to make the best use of limited resources for the
protection, enhancement and restoration of our biodiversity assets. Such decision
making processes can often be better informed through quantitative risk and
scenario assessments. In particular, explicit recognition of the uncertainties involved
in risk assessments can help to avoid erroneous decisions, increase transparency
and highlight critical information gaps. The integration of more objective and
defensible scientific approaches into the decision making process has many other
benefits, including satisfying the aims of various legislative policies and guidelines,
ensuring efficiency in government investment, facilitating a broader landscape
approach to land and water management, and maximising the potential for
biodiversity protection.
This report examines and develops risk assessment methods for predicting the
effects of increasing salinity on biodiversity assets. A detailed investigation of
various approaches to modelling the risks of salinity is presented and the relative
merits and limitations of each are discussed. In summary, techniques that have
previously been applied in this area of risk modelling have focused on the use of
scalar summaries of effect, ‘area under the curve’ formulations of risk and Species
Sensitivity Distributions (SSDs). However, these techniques involve assumptions that
are potentially incorrect and may limit their applications in risk assessment. This
report describes the development of a number of alternative methods for risk
modelling that require fewer statistical assumptions, correctly characterise and
propagate uncertainty and can be used to develop a greater ecological understanding
of
potential
effects.
The
methods
presented
range
from
basic
graphical
representations of tolerance data to complex probabilistic models. In particular,
Probability Bounds Analysis and Bayesian Markov Chain Monte Carlo (MCMC)
techniques are recommended as the basis for improved risk modelling approaches.
The quality and availability of data on which to base risk assessments is also
identified as an important factor contributing to uncertainty in risk management
decisions. It is recommended that these issues be given careful consideration before
deciding on a course of action. Details and examples of suggested risk assessment
techniques are given in the Appendix and also in a series of associated case studies
(Dixon 2007a, b; Dixon & Scroggie 2007).
3
4
Contents
ABBREVIATIONS............................................................................................................ I
ABSTRACT ............................................................................................................... III
ACKNOWLEDGEMENTS ............................................................................................. 1
1 INTRODUCTION AND BACKGROUND ................................................................. 2
AIM OF SALT IMPACT MODELLING .......................................................................................... 3
RECOMMENDATIONS OF STAGE ONE REVIEW ............................................................................. 4
2 REVIEW OF METHODS FOR SALT IMPACT MODELLING. ..................................... 6
OVERVIEW OF CURRENT METHODS FOR MODELLING SPECIES SENSITIVITY DISTRIBUTIONS .................... 6
Parametric Modelling – Probability Distributions ............................................................. 6
Bootstrap Techniques.................................................................................................. 7
Non-parametric Methods ............................................................................................. 8
SPECIES SENSITIVITY DISTRIBUTIONS IN PROBABILISTIC RISKS ASSESSMENTS ................................... 9
3
SOURCES OF UNCERTAINTY IN SPECIES SENSITIVITY DISTRIBUTIONS .......... 11
DATA ASSOCIATED UNCERTAINTY ........................................................................................ 11
PREVIOUS APPROACHES TO THE TREATMENT OF UNCERTAINTY IN TOXICITY BASED RISK ASSESSMENTS. 13
LIMITATIONS OF PREVIOUS APPROACHES TO UNCERTAINTY ANALYSIS AND SSD MODELLING IN
ECOTOXICOLOGY ............................................................................................................. 14
4
REFINED APPROACHES FOR SALT IMPACT MODELLING ....................................18
METHODS FOR UNCERTAINTY PROPAGATION IN RISK ASSESSMENTS .............................................. 18
Interval Arithmetic and Probability Bounds Analysis ..................................................... 18
Bayesian Markov Chain Monte Carlo Estimation Techniques .......................................... 19
UNCERTAINTY CHARACTERISATION AND PROPAGATION IN SSD MODELLING ...................................... 20
5 A GENERALISED LINEAR MODEL APPROACH TO PREDICTIVE EXPOSUREEFFECTS MODELLING ............................................................................................. 23
Probability Bounds Analysis........................................................................................ 24
Bayesian Markov Chain Monte Carlo Methods .............................................................. 26
6 SUMMARY AND CONCLUSIONS. ........................................................................... 29
APPENDIX 1: BASIC METHODS FOR SALT IMPACT MODELLING ............................. 32
A.1 - LOOKUP TABLES AND PLOTS ..................................................................................... 32
A.2 – BURRLIOZ SOFTWARE EXAMPLE ................................................................................ 35
A.3 - PARAMETRIC SSD MODELLING .................................................................................. 37
REFERENCES ........................................................................................................... 39
5
Acknowledgements
Particular thanks are extended to Dr Robin Hale for his editorial and formatting
assistance. A number of other people contributed to the production of this report
and their efforts are greatly appreciated: Prof. Mark Burgman (University of
Melbourne), Dr Scott Ferson (Applied Biomathematics Corp.) Phil Papas (ARI), Tom
Ryan (Environous), Dr Micheal Scroggie (ARI), Dr Charles Todd (ARI), and Dr Jenny
Wilson (DSE). This research was funded by the National Action Plan for Salinity and
Water Quality (NAP) and the National Heritage Trust (NHT).
1
1
Introduction and Background
Salinity presents a significant threat to much of Victoria’s biodiversity, yet little is
known about how effects might occur and what can be done to alleviate them. In
order to ensure suitable protection, enhancement and restoration of our natural
resources, land and water managers are often tasked with deciding how best to
allocate limited resources. It is increasingly recognised that such decisions may be
better informed through quantitative risk and scenario assessments (Thompson &
Graham 1996; Thompson 2002; Burgman 2005). The integration of these more
objective and defensible scientific approaches into the decision making process has
many benefits, including satisfying aims within relevant state strategies, ensuring
efficiency in government investment, using a broader landscape approach to land
and water management, and maximising the potential for biodiversity protection.
The creation of the Salt Sensitivity Database, available through Land and Water
Australia (LWA)1, has provided an important source of collated information on the
salinity tolerance of a range of Australian species (Bailey et al. 2002). However, few
tools are available that can be used to combine this sensitivity data with projections
of increases in salinity in order to determine the risk to species, communities or
ecosystems. In particular, approaches for evaluating uncertainties in these types of
risk assessments have, to date, not been adequately described. Consequently,
decisions must currently be based on risk estimates involving unknown degrees of
confidence.
This document describes the refinement process and outcomes for the Salt Impact
Modelling approach developed as part of Stage 2 of the National Action Plan for
Salinity and Water Quality and the National Heritage Trust funded project: Strategic
Frameworks to Indicate Biodiversity thresholds to Salinity (hereafter ‘The Project’).
Stage 1 of The Project consisted of an initial exploration of different methods for
predicting the effects of salinity on biodiversity, including basic modelling
techniques and exploration of possible vital attributes or aggregation models that
could be used to extrapolate between or provide analogues of different organisms.
The approach sought to utilise data from the LWA database to develop models that
could guide managers in making planning decisions to protect biodiversity assets
from salinity (Cant et al. 2003). The development and use of Salt Impact Models
aimed to “identify which biodiversity assets are most at threat from salinity in order
to target research and management resources to protect those assets” (Cant et al.
2003 p7). An initial trial of the approach, and subsequent review (Webb & Hart 2004),
1
www.rivers.gov.au/Tools_and_Techniques/Salt_Sensitivity_Database/index.aspx
2
revealed that such tools are likely to be useful to managers but that the modelling
techniques
required
further
development.
A
series
of
regional
stakeholder
workshops, consultation with a number of relevant experts and case study work to
test the utility of the refined SIM
approach also contributed to the process
of
refining the modelling approach.
This report describes the outcomes of Stage 2 of The Project on developing ‘refined’
Salt Impact Modelling approaches. A review of methods that are currently applied to
modelling the environmental risks of hazardous agents is provided first. Second, the
sources of uncertainty in ecotoxicological risk assessments are outlined and the
problems and limitations of previous approaches to analysing this uncertainty
discussed. Third, a number of new, alternative techniques for risk modelling are
described and presented as recommended methods for salt impact modelling. These
new approaches focus on the use of Probability Bounds Analysis (PBA) and Bayesian
Markov Chain Monte Carlo (MCMC) methods of uncertainty propagation and are
intended to provide a valuable addition to the suite of assessment and management
tools available for mitigating and ameliorating the risks of salinity to key
biodiversity assets. To aid in understanding the approaches developed, a series of
associated case studies have been performed that describe the methods in further
detail and give examples of their application (Dixon 2007a, b; Dixon & Scroggie
2007)2. Further detail on some basic approaches to modelling salinity effects are also
given in Appendix 1.
Aim of Salt Impact Modelling
The aim of Salt Impact Modelling is to predict the likely effects of changes in levels
of salinity at a particular site for a certain species or group of species. Regional scale
spatial predictions are also possible using the techniques presented here, however
these will invariably involve greater uncertainties due to the lack of suitable
information on the distribution of organisms and the effects of salt at these scales.
The predictions made through Salt Impact Modelling are intended to provide useful
information to decision makers in managing the effects of salinity and can also form
a basis for quantitative risk comparisons between places or situations in order to
assist in the prioritisation and allocation of resources and management works. Due
to the uncertain nature of much of the available data, estimates of confidence and/or
uncertainty in risk estimates will be critical in describing and communicating the
reliability that can be ascribed to predictions, and therefore subsequent management
2
Associated material can be found at www.dse.vic.gov.au/ari/reports
3
decisions. Further refinement of predictions can occur through monitoring, the
integration of more complex ecological models, and additional field research.
The revised Salt Impact Models (SIMs) will also aid in a more strategic approach to
conservation management by providing quantitative information about which
biodiversity assets are most at risk from increasing salinity and the relative
uncertainty surrounding the data on which management decisions can be made. The
SIM approach will enable more informed and objective decisions to be made based
on the likely outcomes of a range of different management scenarios and the likely
impact of any management changes on key biodiversity assets. The SIMs help to
achieve State and Regional conservation goals by providing a rigorous method of
assessing risks, in addition to helping manage threatened species and communities
because specific species can be targeted for assessment.
Recommendations of Stage One Review
A review of Stage 1 of the Project was conducted by Dr Angus Webb & Professor
Barry Hart of the Water Studies Centre, Monash University. This review made a
number of suggestions about refining the modelling process. These included:

The general Probabilistic Risk Assessment (PRA) approach used was highly
supported but it was recommended that some changes be made to the methods
to ensure credibility and suitability of the Project;

Further clarification of the advantages and disadvantages of the approach used
was required, in particular, discussion of relative and absolute risks, as well as
the limitations and uncertainties of the methods used;

Consideration and discussion of the relative merits and inferential consequences
of using different data sources. Specifically, issues associated with using field
and laboratory tolerance information and the ecological relevance of exceeding
particular thresholds or likelihoods of effect;

Validation and statistical testing of assumptions associated with parametric or
other modelling approaches;

Consideration and development of approaches to link together the results of the
SIMs with salinity models such as the Catchment Assessment Tool (CAT); and

Use of a case study approach to test models and validate predictions to develop
credibility and acceptability of the approach.
4
These points formed the starting point for this review and the inclusion of the
alternative modelling approaches described in this report.
5
2
Review of Methods for Salt Impact Modelling
Overview of Current Methods for Modelling Species Sensitivity
Distributions
The designation of ‘safe’ levels of exposure to hazardous agents has become a
fundamental component of the regulation, control and management of both human
and environmental health around the world (Posthuma et al. 2002b). In human and
occupational health situations these (safe) levels are commonly described as
exposure standards, such as maximum allowable dietary or inhalation doses. In
environmental situations such levels are routinely expressed as ‘environmental
quality criteria’, which specify the amount of a hazardous agent below which adverse
effects are considered unlikely to occur (or considered acceptable) (Suter 1993; Rand
et al. 1995; Cairns Jr 1999). Although commonly used to describe synthetic
chemicals, the term ‘hazardous agents’ is used here to refer to a wide range of
potentially toxic natural and anthropogenic phenomena, including salinity.
The most common approach currently used to evaluate the environmental
‘acceptability’ of levels of hazardous agents for environmental applications involves
the construction of a Species Sensitivity Distribution (SSD). These are statistical or
mathematical models that relate an amount of hazardous agent with an effect,
expressed as a proportion of a ‘group of species’ responding (Posthuma et al. 2002a).
This group of species may be defined a number of ways, such as those present at a
specific site or all species of a particular taxonomic type. While empirical models are
probably the most commonly used, a limited number of parametric and nonparametric approaches have been explored, with fewer still of these incorporating
uncertainty analysis (Verdonck et al. 2001b; Posthuma et al. 2002a). In the SSD
approach the risk to the defined group of species is considered to be the overlap of
the SSD (the likelihood of effect from exposure) and Environmental Exposure
Distribution (ECD, the likelihood of exposure to the hazardous agent).
Parametric Modelling – Probability Distributions
Species Sensitivity Distributions, as popularised by Aldenberg and Slob (1993), are
predominately modelled by fitting parametric relationships, usually as a Cumulative
Distribution Function (CDF), to sets of tolerance data. Techniques for performing
parametric distribution fitting are included as standard in many statistical packages
and this method has been widely applied in regulatory situations in Australia and
around the world (Hart et al. 1999; ANZECC & ARMCANZ 2000; van Straalen & van
Leeuwen 2002; Van Sprang et al. 2004).
6
Parametric SSD methods have a number of advantages, including: (1) relatively little
data is required to develop a model, (2) the fitting techniques are included as
standard in many statistical packages and (3) the methods of testing and checking
model adequacy and fit are relatively well understood. Unfortunately, these methods
often require a number of assumptions to be made that may limit their application.
These issues are discussed further in Section 3.
In Australia, parametric methods for SSD modelling were applied by Fox (1999) and
Shao (2000) in developing the modelling approach used in the Australian and New
Zealand Guidelines for Fresh and Marine Water Quality (ANZECC & ARMCANZ 2000).
This method involved fitting a Burr Type III distribution, a three parameter
exponential distribution of which the log-log distribution is a special case (Fox 1999;
ANZECC & ARMCANZ 2000; Shao 2000). This distribution was considered to have
the advantage of being able to encompass a wide range of shapes (although there
may be some problems of colinearity with three parameter distributions). As part of
the ANZECC guidelines, Shao (2000) developed a software
program,
“BurrliOZ”,
which fits the Burr Type III distribution to scalar values and calculates confidence
intervals. Unfortunately, the Burr Type III distribution is not included in many
statistical packages and the BurrliOZ program has limited goodness of fit
applications, making determination of model adequacy difficult. An example using
the BurrliOZ software is given in Appendix 1 (Section A2).
In other Australian examples, Webb and Hart (2004) used log-normal models to
describe the effects of salinity at a number of sites in the Goulburn-Broken
catchment and applied graphical methods for propagating uncertainty, while Kefford
et al. (2005) and Kefford et al. (2006) investigated the use of empirical SSD models
for macro-invertebrates.
Bootstrap Techniques
Bootstrap and other re-sampling techniques are increasingly being applied in risk
modelling because of their ability to cope with multi-modal data sets, lack of
assumptions about distribution type and shape, and the relative ease of computation
of associated statistics (Manly 1997; Frey & Burmaster 1999; Bixio et al. 2002). Jagoe
and Newman (1997) and Newman et al. (2000) have suggested applying bootstrap
techniques to modelling SSDs. These authors propose using bootstrap estimation
methods to overcome the problems and assumptions associated with many
parametric distributional analyses. In an example of the application of boot strap
techniques, 30 toxicity data sets were used to compare between the utility of
lognormal, log-logistic and Gompertz distributions for modelling SSDs (Jagoe &
7
Newman 1997; Newman et al. 2000). The number of values in each data set ranged
from 20 to 91, with an average of 32 and 80% of data sets having less than 40 values.
The authors found that half of the 30 data sets failed the test of lognormality
(alpha=0.05). Goodness of fit tests using Pearson’s X2 statistic indicated that a loglogistic model was a better fit to the data twice as often as the lognormal
distribution, but that the Gompertz distribution was the best fit overall. Most data
sets were found to be multimodal, which was considered to be the result of
similarities between taxonomic groups within the data sets. A strong correlation was
found between the bootstrap and log-normal percentiles (r2 = 0.954). The authors
concluded that assumptions of log-normality were not supported, that the SSD
approach was useful given the limited data available, and that it provided a direct
means of comparing cumulative exposure concentration distribution to cumulative
SSD.
In a response to Newman et al. (2000), Verdonck et al. (2001a) highlight a number of
methodological flaws in the bootstrap approach that was applied. In particular,
because of the small number of observations in each dataset, the sample size for
each bootstrap sample was necessarily larger than those in the original data set. This
meant that all data values were used in the construction of the boot-strap
distribution, not a random sub-sample, and that some values were sampled multiple
times. Although Newman (2000) refers to the use of this approach in the first (1991)
edition of Manly’s work on bootstrap and randomisation methods, this section has
been removed from the second edition as it is not a particularly rigorous practice
(Manly 1997). For many SSD modelling situations the number of data points may be
very few, creating problems with both bootstrap and randomisation approaches.
Newman et al. (2000) further applied this re-sampling approach to calculating
confidence limits for percentiles of the SSD, although the reasoning involved in this
method is again circular, in that the more re-samples taken, the narrower the
confidence intervals were, as also noted by Verdonck et al. (2001a). It is thus evident
that these particular applications of bootstrap and re-sampling methods to SSD
modelling involve inferential and reasoning problems that should be avoided.
Non-parametric Methods
A number of other approaches to modelling SSDs have been proposed. These include
such methods as the ‘Hazen Plotting System’ (Verdonck et al. 2001b), fitting
empirical CDFs, and other non-parametric methods (van der Hoeven 2001; Verdonck
et al. 2001b). While some of
these methods do overcome particular limitations
associated with parametric techniques, they unfortunately tend to require nonstandard statistical packages and have not been widely applied. Further, many of the
8
applications of these methods to date have focused on calculating particular
percentiles of the distribution of effect and not on characterising the whole
distribution or its associated uncertainties. The use of empirical distributions, where
possible, does have the advantage that assumptions of underlying shape are not
required.
Species Sensitivity Distributions in Probabilistic Risks Assessments
The predominant approach to deriving probabilistic measures of the environmental
risk of hazardous agents involves specifying both the Species Sensitivity Distribution
(SSD)
and
Environmental
Concentration
Distribution
(ECD)
as
probability
distributions (Aldenberg et al. 2002; Forbes & Calow 2002; Posthuma et al. 2002a; de
Zwart 2003). In this formulation, the risk is defined as the overlap between these two
distributions, which is also the integral or Area Under the Curve (AUC) of the joint
probability distribution (van Straalen & Denneman 1989; Aldenberg et al. 2002;
Verdonck et al. 2002). Graphical interpretations of the AUC have been described
(Soloman et al. 2000; Soloman & Sibley 2002; Webb & Hart 2004), however these
approaches have been criticised as mathematically incorrect methods for uncertainty
propagation
(Verdonck
et
al.
2003).
A
number
of
different
mathematical
formulations of the AUC have also been described (van Straalen 1990; Cardwell et al.
1993; Aldenberg & Jaworska 2000; Aldenberg et al. 2002; van Straalen 2002a;
Warren-Hicks et al. 2002). Aldenberg et al. (2002) reviewed these different
approaches and found them to be mathematically equivalent expressions of the
integral of the joint probability distribution, which they also describe as the chance
of one distribution exceeding the other, expressed as:

p(ECD SSD)   PDFECD (x).CDFSSD (x)dx
(1)
or

p(ECD SSD)   (1CDFECD (x)).PDFSSD (x)dx
9
(2)
Where:
-
ECD is the Environmental Concentration Distribution,
-
SSD is the Species Sensitivity Distribution,
-
PDF is a Probability Density Function,
-
CDF is a Cumulative Distribution Function, and
-
and x is the level of a hazardous agent.
Aldenberg et al (2002) further describe a simplification of this risk formulation,
termed the ‘Probabilistic Risk Quotient’:
Risk = p(ECD > SSD)
= p(ECD / SSD > 1)
= p(log(ECD) – log(SSD) > 0)
If both the ECD and SSD are assumed to be random (log)normally distributed
variables, then the result is also a normal distribution with mean, µ = µ
standard deviation, σ=√(σ
σ
SDD
) + (σ
2
ECD
) , where µ
2
SDD
ECD
and µ
SDD
ECD
-µ
are the means, and σ
SDD
ECD
and
and
the standard deviations, of the log-transformed ECD and SSD respectively
(Aldenberg et al. 2002; Verdonck et al. 2003). Although this solution provides a
probabilistic measure of risk, it does not allow for propagation of measures of
uncertainty and the assumption of (log)normality is generally untested.
10
3
Sources of Uncertainty in Species Sensitivity
Distributions
The evaluation of uncertainty in risk estimates is of central importance to ensuring
rigor, transparency and success in the risk management process (Morgan & Henrion
1990; Winkler 1996; Suter et al. 2000; Andrews et al. 2004; Burgman 2005). Despite
widespread
acknowledgment
of
the
need
for
uncertainty
analysis
in
risk
assessments (NRC 1993, 1994; Pastorok 2002; Burgman 2005), to date there have
been very few applications of such techniques in toxicological risk assessments
(Aldenberg & Jaworska 2000; Posthuma et al. 2002a; Verdonck et al. 2002; Van
Sprang et al. 2004). Furthermore, it has been recognised that current probabilistic
formulations involve a number of untested assumptions and may not be amenable to
uncertainty analysis (Suter 1998; Forbes & Calow 2002; Newman et al. 2002; Suter et
al. 2002; van Straalen 2002a; van der Hoeven 2004).
Uncertainty in risk assessments arises from a multitude of sources including natural
variability in the sensitivity of organisms to toxicants, confounding and modifying
factors of toxicity, systematic measurement error, lack of knowledge about
mechanisms determining toxicity, and extrapolation across spatial, temporal and
organisational scales (Chapman et al. 1996; Chapman et al. 1998; Cairns Jr 1999). A
distinction is often made between aleatory uncertainty, or the non-reducible inherent
variability in a parameter, and epistemic uncertainty, or incertitude about the
process or situation that should, at least in part, be reducible (Ferson & Ginzburg
1996; Regan et al. 2002a; Ferson et al. 2003).
Data Associated Uncertainty
Perhaps the most significant source of uncertainty in risk assessments involves the
data (or lack there of) on which they are based. Issues surrounding the comparability
of differing spatial and temporal scales of observation and exposure, test endpoints,
taxonomic resolution and a general lack of information on the tolerance of many
species to hazards contribute to uncertainty in the predictions made from risk
models.
Information about the effects of salinity generally comes from either toxicity studies
or field observations. Toxicity studies performed in laboratories, glass houses or
field trials are designed to determine a particular magnitude of effect at different
levels of exposure to a hazardous agent. These studies can also provide
more
detailed information of use in risk assessments, such as the toxic mode of action or
11
the presence of acute responses thresholds. However, because many of these studies
are necessarily performed over short time periods and away from the complicating
factors found in an organism’s natural environment, their applicability to field
situations is often unknown. A multitude of factors may affect an organisms’
sensitivity to a toxicant, and hence the relevance of laboratory/toxicity data to real
world situations. These include:

The length of the exposure period;

The rate at which exposure is increased (i.e. direct transfer vs. gradual change);

The frequency of exposure;

The season or life history stage at which exposure occurs;

The endpoint (i.e. acute vs chronic);

The presence of other stressors, such as predators, chemical pollutants, drought,
food availability; and

Water quality associated modifying factors such as carbonate concentration, pH
or temperature.
Despite these limitations, toxicity experiments do provide direct evidence of the
effect of a toxicant and a means by which to test hypotheses about its effects.
Observations of field tolerance, on the other hand, can potentially provide a more
realistic and encompassing indication of likely effects, although the relevance of
such measures of ‘toxic effect’ may also be affected by the factors listed above.
Unfortunately, few field studies are designed to determine the maximum tolerance
of an organism or community to salinity. Instead, much of the data on the field
tolerance of organisms comes from associated research, such as distributional
surveys, where salinity levels have been recorded. Many of these surveys have only
been conducted once over limited areas and thus only provide a snap-shot of the
range of variable environmental conditions that may be experienced over a wider
spatial and temporal extent. Consequently, this information is often uninformative
about where the observed salinities lie within the range of a species’ occurrence and
how long that level can be tolerated.
Issues of which value to use for a species when there are multiple values available
are also currently unresolved. Arguments might be made for using values from the
closest location or the lowest values to ensure that risk assessments are
conservative. These and other approaches to modelling and propagating uncertainty
are discussed in more detail in the next section.
A final and particularly important source of uncertainty in predicting the effects of
salinity relates to the consideration of ‘salinity’ as a toxicant. This is particularly
12
problematic as ‘salinity’ comprises a complex mixture of many different chemical
ions whose relative composition may vary considerably from place to place. Very
little is known about the toxicity of these different ions on native species, let alone
their combined long-term effects, therefore careful consideration should be given to
the relevance and comparability of different mixtures of salts.
Previous Approaches to the Treatment of Uncertainty in Toxicity
Based Risk Assessments
In the risk formulations described above (Section 2.2), only variability in the mean
response is considered and not other sources of knowable uncertainty associated
with the variables,
parameterized
by
such as
probability
inherent
variability that
distributions
or
is greater than that
uncertainty
associated
with
assumptions of model structure. Although the potential effects of other sources of
uncertainty in SSDs have been widely recognised (Suter 1998; Newman et al. 2000;
Forbes & Calow 2002; Newman et al. 2002; Suter et al. 2002), relatively few
approaches have been suggested that include these in risk estimates (Aldenberg &
Jaworska 2000; Aldenberg et al. 2002; Verdonck et al. 2002; Verdonck et al. 2003;
Van Sprang et al. 2004).
Verdonck et al. (2002; 2003) and Van Sprang et al. (2004) describe a Monte Carlo
approach to estimating uncertainty in probabilistic risk quotients, which attempts to
account for uncertainty in both the SSD and ECD. In this approach, successive
samples are drawn from both distributions and used to create a density of values for
a distribution of risk quotients. This density can then be used to obtain the
likelihood that the risk quotient is equal to one. Similar Monte Carlo approaches
have been widely applied in human health risk assessments (Finkel 1995; Brattin et
al. 1996; Thompson 1999; Andrews et al. 2004).
Aldenberg et al. (2000) and Aldenberg et al (2002a) describe a sampling theory
extrapolation approach to
estimating
confidence intervals for percentiles of the
normal distribution. This uses extrapolation factors derived from the non-central tdistribution that relate the number of observations in the sample with estimates of
confidence in the ‘true’ population percentile. Lookup tables of the extrapolation
factors for 90%, 2-sided confidence intervals are provided for various percentiles of
interest (Aldenberg & Jaworska 2000). Although calculations were made for both
deriving the concentration hazardous to a percentile of a population (the ‘forward’
situation) and the fraction affected at a particular concentration (the ‘reverse’
situations), only uncertainty in the SSD was considered.
13
There are also an increasing number of studies that have investigated uncertainties
in larger and more complex models of exposure and ecological risk (see: Ferson et al.
1996; Fogarty et al. 1996; Burmaster & Wilson 1998; Moore et al. 1999; Jager et al.
2001; Linkov et al. 2001; Regan et al. 2002b; Regan et al. 2002c; Bartell et al. 2003;
Bates et al. 2003; Breton et al. 2003; Crane et al. 2003; Landis 2003; Pastorok et al.
2003; Merrick et al. 2005). Many of these studies begin to address the ecological
aspects of the effects of chemicals, however to date there has been little
consideration of uncertainty in concentration-response relationships within these
models or its impact in overall measures of risk. Instead, toxicity information has
generally been represented as a scalar toxicity reference value or dose.
Limitations of Previous Approaches to Uncertainty Analysis and SSD
Modelling in Ecotoxicology
While the approaches outlined above have provided a useful grounding for SSD
based risk assessments (Forbes & Calow 2002; Posthuma et al. 2002a), they involve a
number of assumptions that may be incorrect and limit their application in risk
management (Suter 1998; Suter et al. 2002; van der Hoeven 2004):
First, the proposition that there is a ‘safe level’ of exposure has been the subject of
some criticism. It has been argued that for some hazardous agents there is no safe
level of exposure (Newman et al. 2000). While this criticism is probably valid for
many anthropogenic toxicants, in cases involving salinity it would be reasonable to
assume that there is some safe level of exposure, given that the compounds involved
occur naturally and many are essential to survival.
Second, the aim of SSD modelling is to estimate a certain percentile of species to be
protected, such as the lower 5th percentile (i.e. assumes that 95% of species will be
protected). While this may appear to be an honourable goal, the basic SSD modelling
process is not informative about which species are lost or the ecological
consequences of the loss of those particular species. This would
require
identification of the expected species to be lost and consideration of the ecological
effects of that loss within the community. However, very little is known about the
behaviour of many of the systems of interest, further adding to the uncertainty in
decisions made.
A third limitation of the SSD approach is that models might only be informative for
species for which there is data. If a community contains 100 species but there is only
information for about 50 of these, predictions that describe the proportion
of
species affected will only be based on the 50 species for which there is data (so there
14
is potentially another 50 species that could be affected that were not included in the
assessment). While the process of fitting probability distributions to the data is used
to extrapolate from a sample of species to a larger number, the test species used are
generally not an unbiased random sample from a larger population of potentially
affected species. Additionally, for many situations there is a high degree of
taxonomic uncertainty, both in the description of different species and their
distribution.
Fourth, a number of authors have noted that the treatment of an exposure-response
relationship as a random variable (i.e. as represented as a probability distribution) is
mathematically incorrect (Suter 1998; Forbes & Calow 2002; Suter et al. 2002; van
Straalen 2002b; van der Hoeven 2004). Particular problems with this probabilistic
description of the SSD include:
-
test species are not randomly selected;
-
interpretation of the underlying variability generating mechanisms is
ambiguous, as is the interpretation of the differentiable relationship between
the PDF and CDF of the SSD;
-
different species are treated as equally important and non-dependant on each
other in a community; and
-
interactions between species are not considered.
A fifth problem is the use of scalar summaries of exposure-response distributions.
The most common of these is referred to as the NOEC or No Observed Effect
Concentration, and is often defined as the highest exposure concentration in a
toxicity test that did not produce a significantly different response from the control
(Rand 1995). A number of other similar ‘no-effect’ summary measures are commonly
used, however these values have been found by a number of authors to be statistical
and experimental artefacts, and to lead to misleading interpretations (Laskowski
1995; Chapman et al. 1996; van der Hoeven 1997; Yanagawa & Kikuchi 2001; Suter et
al. 2002; van der Hoeven 2004). Although in practice it has often been necessary to
base risk assessments on scalar summaries due to a lack of previously published
whole datasets or other measures of effect, the use of scalar summaries is
uninformative with regards to levels of uncertainty and variability in the estimates
produced (Hoekstra & van Ewijk 1993; Chapman et al. 1996; van der Hoeven 1997;
Newman et al. 2000). This has the flow-on effect that uncertainty and variability in
the response to a toxicant is ignored in the calculation of risk estimates, even though
it is obviously a fundamentally important consideration if the intention is to predict
the true range of values over which a species or community could be susceptible to a
toxicant. A number of authors have suggested the use of Effect Concentration (EC)
15
values and whole exposure-response distributions as more informative alternatives
(Chapman et al. 1996; de Bruijn & Hof 1997; van der Hoeven 1997; Chapman et al.
1998), although there have been only a few applications of these in risk assessments
to date (Nayak & Kundu 2001; Englehardt 2004).
A sixth limitation of previously applied SSD approaches involves statistical and
mathematical assumptions required in the formulations of risk described above
(Section 2.2). For example, it has been shown that assumptions of one underlying
distributional type, such as the normal or log-normal, rarely hold when applied to
multiple data sets (Newman et al. 2000; van der Hoeven 2001, 2004). Importantly,
discrepancies caused by these assumptions often have the greatest implications for
values in the tails of a distribution, which is generally the focus of risk assessments
(Hattis 1990; Newman et al. 2000). The assumption of ‘independence’ between
parameters in probabilistic calculations is also often untested or invalid and can lead
to an underestimation of risks (Ferson 2002; Ferson & Hajagos 2004). This can cause
significant problems with many applications of Monte Carlo techniques, where
parameters in a model are assumed to be independent of each other and the
inclusion of correlations between them can be difficult (Ferson 1996; Nayak & Kundu
2001; Ades & Lu 2003; Ferson et al. 2003; Berleant & Zhang 2004; Englehardt 2004;
Ferson & Hajagos 2004). Further limitations arise from the formulation of the risk
model as the closed form joint-integral of two probability distributions, as it is
difficult to expand the model to incorporate other factors, such as modifiers of
toxicity or exposure, random effects or population growth rates. Also, exact
solutions to the integral are only available for certain combinations of distributions.
Finally, methods for propagating uncertainty, such as those discussed below, are in
practice mostly limited to arithmetic operations and may not be appropriate for
many graphical and integration based model formulations (Ferson & Ginzburg 1996;
Ferson et al. 2003; Ferson & Hajagos 2004).
Further to these points, it is important to recognise that:
-
Sensitivity Distributions are really a function that describes the expected
‘map’ or effect for increasing levels of exposure. Standard portrayals of an
SSD show its values over the full range of those possible, whereas the risk is
the result of the SSD function for particular levels of exposure;
-
CDFs must be increasing and thus cannot incorporate alternative exposureresponse functions (such as hormesis or avoidance);
16
-
the use of percentile standards inherent in these approaches is based on the
assumption that the loss of some proportion of species is acceptable, and is
non-informative about which species may be lost;
-
the focus on changes in the number of species within a community assumes
that effects on populations are of no consequence.
While it has been argued that the probabilistic interpretation of species sensitivity
has some utility in representing uncertainty in responses across species and that it is
in the interpretation of the results of such processes that problems arise (Forbes &
Calow 2002; van Straalen 2002a), changing the interpretation of the SSD does not
alter the mathematical validity.
17
4
Refined Approaches for Salt Impact Modelling
This section describes three refined approaches for Salt Impact Modelling that
provide probabilistic measures of risk and incorporate uncertainty propagation.
Following on from methods developed in Dixon (2005) the first of these concerns the
application of Probability Bounds Analysis (PBA) to uncertainty characterisation in
SSD models. The other two methods involve the application of PBA and Bayesian
MCMC techniques for propagating uncertainty in generalised linear risk models of
population effects (Dixon 2005). While the two uncertainty propagation approaches
described here are not the only ones that may be appropriate for modelling salinity
risks, they do offer a number of advantages over the standard Monte Carlo based
approaches previously applied to these problems. A general background and
overview of the techniques is first given, before the particular applications of these
methods are outlined in the context of risk modelling. More detail on the application
of these methods is provided in a series of associated case-studies (Dixon 2007a, b;
Dixon & Scroggie 2007).
Methods for Uncertainty Propagation in Risk Assessments
Interval Arithmetic and Probability Bounds Analysis
Probability Bounds Analysis (PBA) is a combination of interval arithmetic and
probability theory that provides a rigorous method for propagating uncertainty
through calculations (Ferson 2002). In particular, PBA provides solutions to problems
involving unknown dependencies between variables and uncertainties in the exact
nature of distributions (Ferson 2002; Ferson & Hajagos 2004). The solving of some
key algorithms used in the computation of probability bounds (Yager 1986;
Williamson & Downs 1990; Ferson et al. 2003) and the recent development of
software that makes use of these techniques (RAMAS RiskCalc) 3, has enabled interval
and PBA approaches to become viable techniques for use in risk assessments (Ferson
2002; Regan et al. 2002b; Regan et al. 2002c; Ferson & Hajagos 2004; Regan et al.
2004).
Conceptually, PBA could be considered an extension of interval arithmetic. If two
uncertain numbers are considered, A and B, defined by the respective intervals [a ,1
a ] and [b , b ], then their sum must lie within the bounds [a +b , a +b ], and their
2
1
2
1
1
2
2
product within the bounds [min(a b , a b , a b , a b ), max(a b , a b , a b , a b )] (Ferson
1
1
1
2
2
1
2
2
1
1
1
2
2
1
2
2
2002). Similar rules exist for other arithmetic operations, and analogous solutions
3
RiskCalc is available from www.ramas.com
18
have been found for convolutions of bounds on whole distributions, referred to as a
probability box (p-box) (Yager 1986; Williamson & Downs 1990; Ferson et al. 2003).
PBA has the particular advantage over other uncertain number techniques that it is
not necessary to assume independence, or to know the dependency, between the
parameters in a model because the bounded solution is guaranteed to encompass
the true value (Ferson & Hajagos 2004). Because p-boxes describe a non-parametric
region of probability space and knowledge of model structure within the box is not
required, various aspects of model and parameter uncertainty can be investigated
simultaneously. The benefit of this is that different types or sources of uncertainty
may be propagated in the same calculation and that the resultant bounds on a
variable are the best possible, given the information available (Ferson et al. 2003;
Ferson & Hajagos 2004). PBA does have the limitation, however, that the relative
likelihood of values occurring inside or outside the bounds specified is not resolved.
To date there have been very few applications of intervals and PBA techniques in risk
assessments of hazardous agents and even fewer in ecotoxicology. Regan et al.
(2002c) applied PBA with an exposure model used to determine soil screening levels
for terrestrial wildlife. Risk was expressed as hazard quotients and the sensitivity
distribution was represented as a single, minimum toxicity reference value. This
study compared PBA with Monte Carlo approaches and showed that significant
discrepancies could result from unjustified assumptions of independence between
parameters (Regan et al. 2002c). In another application, Regan et al. (2002b)
compared two-dimensional Monte Carlo and PBA of a food web exposure model,
again using a minimum value toxicity reference value, and found similar
underestimation of uncertainties even with two-dimensional Monte Carlo techniques
(Regan et al. 2002b).
Bayesian Markov Chain Monte Carlo Estimation Techniques
With the assistance of increased computing power and the refinement of algorithms
used in Markov Chain Monte Carlo (MCMC) estimation procedures, Bayesian
approaches to data analysis and predictive modelling are becoming a powerful
alternative to frequentist or Maximum Likelihood Estimation (MLE) approaches
(Godsill 2001; Gelman et al. 2004; Lee 2004). The particular utility of MCMC methods
in risk assessment applications lies in the relative ease with which correlation
structures between parameters can be captured and propagated through models
(Ades & Lu 2003).
Bayesian approaches to data analysis have been well documented elsewhere
(Congdon 2001, 2003; Gelman et al. 2004; Lee 2004) and so shall only be covered
19
briefly here. Bayes’ theorem states that the posterior probability density p(θ|y) of a
parameter θ, given data y, is related to the prior probability of the parameter
independent of the data p(θ), and the likelihood of the data given the parameter
p(y|θ):
p(θ|y)  p(θ) x p(y|θ)
This general form can be extended to apply to each of the parameters in a model and
to make predictions for y(ỹ):
p(ỹ| y) = ∫p(ỹ|θ)p(θ|y)dθ
Advocates of Bayesian approaches argue that this presents a sensible framework
within which to be explicit about the prior assumptions of the distribution of
parameters within a model and also the structure of the model itself in multiparameter situations (Lee 2004).
Much of the controversy surrounding Bayesian
techniques has stemmed from the use of subjective priors or distributions of p(θ).
However, in many situations, the use of ‘flat’ or non-informative prior distributions
produces results identical to classical methods (Gelman et al. 2004; Lee 2004).
In toxicity risk assessments, a basic Bayesian method has been applied to calculating
confidence intervals for the fraction of species affected at specific exposures
(Aldenberg & Jaworska 2000). A similar approach has also been applied to the
determination of compliance with regulatory percentile standards (McBride & Ellis
2001). Nayak and Kundu (2001) discuss the use of Bayesian methodology to
propagate uncertainty in risk assessments and apply these methods to calculating
hazard quotients for ingestion exposure in humans. In another example Englehardt
(2004) described a Bayesian dose-response assessment for human health risk
assessment of viruses.
Uncertainty characterisation and propagation in SSD modelling
The first of the refined methods presented here involves the application of
Probability Bounds Analysis (PBA) to SSD modelling. In this method, PBA is used to
construct bounds on a SSD from commonly available Effect Concentration (EC)
estimates and their confidence intervals. In a similar manner to fitting probability
distributions to groups of scalar tolerance values, in order to construct a standard
parametric SSD, PBA can be used to combine the upper and lower confidence
intervals into a probability box that describes the uncertainty bounds on the SSD.
Specifically, the ‘histogram’ and ‘mixture’ functions in RAMAS RiskCalc enable p-
20
boxes to be constructed from groups of intervals (min-max) data. This operation is
performed empirically, without assumptions of underlying distributional shapes
(although parametric distributions may also be used), and can be used in
calculations of the (bounded) likelihood that various values are exceeded for a given
exposure distribution.
Using this PBA method, bounds on the SDD can also be constructed using other data
sources, such as the highest and lowest observed field tolerance of a species. In this
way, taxonomic uncertainty can also be incorporated into the model by including the
highest and lowest observed tolerance for the lowest level of taxonomic resolution
available. For example, if an SSD is to be constructed for the salinity tolerance of a
group of organisms that occur in a particular area of interest, except that many of
them have only been identified to Family level, bounds on the SSD could be defined
using the minimum and maximum reported tolerance values for that family. This
carries with it the consequence that the bounds would generally be much wider than
they would for a particular species within that family, encapsulating the increased
uncertainty caused by low taxonomic resolution data. As further research is
conducted and taxonomic resolution increases, the bounds on the SSD model will
generally narrow, reducing uncertainty in the SSD. Where a number of data points
are available for a family or species, the SSD model can be constructed using the
median of all the reported tolerance values for that species (i.e. the medial response)
in order to provide a less biased estimator of effects.
An example of a bounded SSD is given in Figure 1, which compares a normal
distribution fitted to log-transformed tolerance data (blue line), a non-parametric
distribution (ECDF) of the median tolerance values (black line) and a p-box of a
mixture distribution of the corresponding 95% confidence intervals (red lines) (Dixon
2005). The application of this approach is described in more detail in the associated
case study, Dixon (2007a).
21
Proportion of Species Affected
Lower
Bound
1
Upper Bound
0.5
Log-normal
Mean
Tolerance
0
0
1
2
3
4
5
6
Concentration (mg/L)
Figure 1 Example of a bounded Species Sensitivity Distribution model. The blue
line is a normal distribution fitted to log-transformed tolerance data, the black line is
an empirical CDF of the median tolerance values (medial response), and the red
lines are a p-box (mixture distribution) of the corresponding 95% confidence
intervals.
22
5
A Generalised Linear Model Approach to Predictive
Exposure-Effects Modelling
While the uncertain SSD approach described above improves on previous methods, it
still relies on a probabilistic interpretation of species’ response. In order to avoid
this assumption the second and third refined methods presented for Salt Impact
Modelling involve the construction of parametric functions to model the response of
each species’ population. Generalised Linear Models (GLM) and Generalised Linear
Mixed Models (GLMM) form the basis for these parametric risk models and are a
versatile, informative and natural approach to describing exposure effects data.
These approaches are also amenable to uncertainty propagation
using
the
techniques outlined below.
A binomially distributed logit model is the basis for the population level risk model
described here (although other formulations may be appropriate, depending on the
situation):
logit() = B + B *X
0
(3)
1
where: logit = log(/1-),  is the proportion of organisms responding and X is the
concentration of a hazardous agent. This model describes a wide range of sigmoid
shaped curves commonly used to describe toxicological responses where a binomial
outcome (i.e. survival) is concerned. In this case, a logit model is fit to tolerance data
for each species. This is then used to either consider the impact of changes on each
species separately or the individual logit model might be combined into a larger
meta-model of community response (discussed later).
In both the PBA and Bayesian techniques, uncertainty in the model is estimated
directly from the experimental data and described as distributions on the
parameters B and B . However, the methods used to estimate and propagate this
0
1
uncertainty differ. Also, dependencies between the parameters are accounted for in
the PBA approach by effectively including all possible dependencies between the
parameters, while in the Bayesian method correlation structures are estimated
directly from the data and preserved through the calculation of predictions.
Unlike previous ‘joint probability’ approaches to calculating ecotoxicological risk, the
method described here involves arithmetic operations of probabilistic terms within a
parametric model. This approach has four main advantages: First, techniques for
propagating uncertainty are more amenable to arithmetic formulation and therefore
are more easily applied within a parametric function. Second, and perhaps more
importantly, this approach models whole exposure-response distributions, rather
23
than scalar summary values, and is easily expanded to include more terms such as
other sources of uncertainty or modifying factors. Third, it provides an alternative to
the SSD approach and does not require the associated assumptions. Fourth, this
general formulation of risk model can handle a wide range of parameterisations and
underlying error-distributions, including higher-order, non-linear effects and models
with mixtures of error distributions.
Probability Bounds Analysis
In the PBA approach to uncertainty characterisation in GLMs, a logit or other model
is fit to experimental data using maximum likelihood techniques found as standard
in many statistical packages (such as SAS, S-Plus or R). Uncertainty in the fit of the
model to the data is determined using the standard error (SE) values for the
parameter estimates (i.e. 95% intervals on the mean parameter estimates are ±
1.96*SE) and these are used to describe lower and upper bounds on the parameters,
B and B . Model predictions are then computed with RiskCalc, Version 4.0 (Ferson
0
1
2002), using the logistic equation:
~
y 
1
1 exp(B0 B1X )
(4)
Or more specifically with bounds of the parameters:
~
y 
1
1 exp
([B0(MIN) , B0(MAX) ] [B1(MIN) , B1(MAX) ] *(dist(X) ~ (mean[MIN,MAX] , SD[MIN,MAX] )))
(5)
where dist(X) (the Environmental Concentration Distribution) can be any probability
distribution, scalar, interval or p-box. In this example the distribution is shown as a
p-box with bounds on its parameters, shown as minimum and maximum values for
the mean and standard deviation. A mean predicted response representing
the
results of a Monte Carlo simulation can also be calculated to enable comparisons to
be made with the results of PBA. An example of the output from this type of
PBA/GLM risk assessment is given in Figure 2.
24
1
Lab
Pr
0.5
Field
Both
Mean
0
0
1
Proportion Affected
Figure 2 Example of risk distribution produced from Probability Bounds Analysis
uncertainty propagation in a GLM risk model showing the relative contribution of
various sources of uncertainty. The y-axis gives the likelihood that an exposure level
has been exceeded, while the x-axis gives the corresponding proportion of the
population expected to be affected. The mean response is given in black the
uncertainty in the exposure distribution in blue, the uncertainty in the tolerance
relationship in red and the combined uncertainties in green.
25
Bayesian Markov Chain Monte Carlo Methods
In
the
Bayesian
approach,
model
fitting
and
generating
predictions
occur
simultaneously. Although computationally intensive, this has the advantage that
correlation structures between parameters are preserved and incorporated into the
predictions (Ades & Lu 2003). The Bayesian techniques also produce a continuous
density of likelihoods, so that the probability of observing any effect size can be
determined, not just the min-max bounds. However, this comes at the cost
of
limiting the range of dependencies that could occur. Binomial logit models can be fit
using MCMC techniques in programs such as WinBugs or OpenBugs (Spiegelhalter et
al. 2003). The algorithms applied in MCMC estimation, such as the Gipps sampling
method,
are
iterative
and
involve
sequential
sampling
from
approximate
distributions for the parameters in a model, θ , θ ,…θ , in order to find the best fitting
1
2
n
distribution for the parameters, conditional on the data. At each successive iteration,
the ‘random walk’ sampling chain is corrected such that the next sample draw
depends only on the previous value, with the aim that the chain of samples
eventually converges on the target distribution, p(θ|y). Recall that this is the posterior
probability density of a parameter given the data, conditional on the other
parameters in the model.
The use of Bayesian techniques to estimate logistic models for standard bioassay
data is discussed in Gelman et al. (2004) and Goodman (2004).
The general risk model is of the form of Equation 3:
logit (θ ) = B + B *X
i
0
1
i
where θ is the probability of an effect (y |n ), given concentration X , and logit(θ ) =
i
i
i
i
i
log(θ |1- θ ). For each replicate, the number of organisms responding y out of the total
i
i
i
n , is binomially distributed (Bin), such that:
i
y |θ  Bin(n , θ )
i
i
i
i
which gives the posterior distribution:
p(B , B |y, n, X )  p(B , B |n, X ) x p(y |B , B , n, X )
0
1
0
1
0
1
And the likelihood function:
l(B0 , B1 | y)  i
yi
(1 i )
ni yi
i.e. the likelihood l, of the parameters B and B , conditional on the data y, is given by
0
1
the product () of the binomial probabilities (Gelman et al. 2004; Lee 2004).
26
In the Bayesian MCMC fitting procedure, the parameters to be estimated are defined
as nodes within an overall model that specifies the full joint probability distribution
of all quantities. Estimation then seeks to find the joint posterior density of the
parameters, conditional on the observed quantities (y, the data), the ‘prior’
assumptions and the constraints of the model. As with the PBA approach,
predictions can be made to find the likely effects for different exposure distributions
or to determine which exposure regimes may have minimal effect on the population.
An example of the output from a Bayesian risk assessment is given in Figure 3 which
compares the risks of exposure to two different populations of a species. Further
details and examples of the application of these techniques to population level risk
assessment are given in an associated case study, Dixon (2007b).
27
1
Cumulative Probability (Exccedence)
0.9
0.8
W o o rine n
0.7
R o und
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0 .1
0.2
0.3
0 .4
0 .5
0.6
0 .7
0 .8
0 .9
P ropo rtion A ffe c te d
Figure 3 Example of uncertainty risk distributions produced using Bayesian MCMC techniques. Compares the effects of a hazardous agent
(salinity) on two different populations of a species. The x-axis gives the proportion of the population affected, while the y-axis is the
probability of exceeding these values, given the exposure distributions. The spread of points describes the uncertainty in each distribution.
28
1
6
Summary and Conclusions
While risk assessment tools, such as probabilistic modelling, can provide the basis
for a more objective and transparent approach to decision making, it should be
noted that predicting the ecological changes from increasing salinity is a complex
and difficult task. The tolerance of a species to a hazardous agent may not be
constant and can vary between locations and times. The effects of a hazardous agent
on other species and ecological processes can lead to intricate indirect effects that
may not be predictable or well understood. Additionally, the limited understanding
of the behaviour of most ecosystems restricts the ability to predict their response to
disturbance.
For these reasons, risk assessment approaches are focused on using the information
available to inform decision making in the best way possible. Many models used in
risk assessment may not be correct or accurate, yet still provide adequate
comparisons of the likely outcomes of different management actions. In risk
assessments of the effects of salinity, no particular set of models or data will be
suitable for all situations and thus risk assessments must have some element of
localisation or specificity.
In refining the approach for Salt Impact Modelling, an investigation into methods for
uncertainty analysis in risk models was conducted. This found that one of the major
limitations of previous approaches has been describing uncertainty as a lack of fit of
scalar values to an assumed model, rather than the propagation of uncertainty
estimated from the original data or additional information.
Unknowable sources of incertitude, such as extrapolation from laboratory to field
scales of exposure and effect can introduce considerable uncertainty into risk
assessments. While accurate prediction of in situ effects may not be the goal of
many risk assessments, quantification of uncertainty is just as important when
comparing risks under management scenarios that involve tradeoffs and relative
risks.
Species Sensitivity Distribution based approaches to risk assessment have many
attractive properties (e.g. being able to extrapolate to community level effects) and in
the past have provided a useful framework within which to consider the
environmental risk of hazardous agents (Forbes & Calow 2002). However, these and
other current approaches used in ecotoxicological risk assessments are limited by
their underlying assumptions and involve unknown uncertainties. In particular, the
notion that the loss of a proportion of species is acceptable, without specific regard
to either the community structure, which species are expected to be lost or the effect
29
on species’ populations, obviously has limited ecological relevance (Suter et al. 2002;
van der Hoeven 2004).
An approach was described for characterising uncertainty in SSDs using Probability
Bounds Analysis to construct bounds on the model, based on Effect Concentration
(EC%) data and associated confidence intervals. This uncertain SSD method can be
easily implemented in the calculation of both the forward and inverse situations
(predicting effects and setting exposure standards) using currently available
software, although it should be noted that some of the assumptions involved in this
approach may be incorrect (as discussed above).
In comparison to the SSD approach, a more detailed method for risk modelling was
described that focused on each species’ response. These species level responses can
then be built into a larger model of community effects that incorporates the whole
response and uncertainties of each population and may also include interactions and
complexities between species. Single species risk assessments are also important in
the management of threatened or target species and are focused on predicting
effects on populations, rather than communities (although the methods used may be
expanded to apply to communities). The application of Bayesian MCMC and PBA
techniques to these models enables quantification of distributions of risk and
associated uncertainty, as well as direct quantitative comparisons between species,
situations or scenarios.
In extending these methods to multi-species or community level risk assessments, a
number of approaches are possible depending on the information available, the
outcomes required and the assumptions that can be justified. If the aim of a risk
assessment is to minimise effects on a particular species and/or those it depends on
then separate modelling of the risks to the individual species would provide
information relevant to the management of the situation. If a larger model of the
expected community sensitivities to exposure was required, a hierarchical approach
that formed a (community) distribution from underlying (species) population
tolerance models could be undertaken (e.g. Dixon & Scroggie 2007). Bayesian MCMC
techniques are highly amenable to hierarchical modelling and it would be possible to
construct a model of species within a community that might be affected, where each
underlying population has a cut off for inclusion (e.g. 95% chance of 50% of the
population affected). Mixture models of different underlying distributions might also
be suitable for community analysis and treatment of overdispersion in larger
datasets. Finally, perhaps the preferred approach would be to include the ‘uncertain’
risk distributions as a parameter in a larger model of exposure and ecological effect,
where the interdependencies between species and their populations is considered.
30
The methods described in this report enable quantification of risk distributions and
associated uncertainties from the original data, propagation of this uncertainty
through the risk models and quantitative comparisons between risk distributions. In
developing these alternative risk models few assumptions were required about the
underlying distribution of parameters and those assumptions that were made are
easily
tested.
The
approaches
developed
are
also
amenable
to
alternative
distributional structures, re-parameterisation, inclusion of other estimates of
uncertainty and do not assume independence between the parameters. Although the
examples provided are focused on single species populations, the approaches used
can be extended to apply to communities or to feed into larger models of ecosystem
behaviour and risk.
The integration of more objective and defensible scientific approaches into the
decision making process helps achieve regional and State conservation goals by
providing a rigorous and transparent method of assessing risks and prioritising
resources. This ensures efficiency in government investment, facilitates a landscape
level approach to land and water management, and helps to maximise the potential
for biodiversity protection.
31
Appendix 1: Basic Methods for Salt Impact Modelling
This section describes a number of basic methods for predicting the effects of
salinity on biodiversity. The first approach described is to use tolerance or look-up
tables to compare between measures of sensitivity for different species of interest.
Although simplistic, this approach has the advantage that it does not make the
statistical assumptions of SSD based techniques.
The other two methods described here are for SSD approaches to effects modelling.
The first uses a program called BurrliOZ, which was developed for the ANZECC
guidelines, while the second applies more standard approaches using commonly
available software packages.
A.1 - Lookup Tables and Plots
This basic approach to describing tolerance data provides a way to visualise the
information and indicate which species are likely to be affected at various levels of
salinity. The method involves constructing a lookup table or ranked-plot of the
tolerances of the different species of concern. The limitations of this approach are
that it does not enable extrapolations or estimates of confidence/uncertainty to be
made, and thus the results should be considered cautiously.
As with the other methods described here, the initial step is to define the subject
area, the species of interest contained within it, and to collate the relevant data on
sensitivity. This data should then be entered into a spreadsheet, such as in Microsoft
Excel, with species name in one column and the corresponding tolerance value/s in
the next. The next stage is to ‘sort’ the data in ascending order by tolerance. This is
done by highlighting both columns, clicking ‘tools’ and then ‘sort’ tabs, selecting the
relevant column for the tolerance values and checking the ‘ascending’ box. A plot
can then be then created (i.e. in Excel select the graph wizard and choose column
graph) with species on the x-axis and tolerance on the y-axis. The salinity level of
concern is then read across the graph from the y-axis. Any bars that fall below this
line represent species that are likely to be affected at this level and subsequent
attention can then be given to their conservation status or other specialist concerns.
Visual comparisons may also be made between groups within the plot, such as
between native and introduced species.
This method can only provide a basic indication of likely effects of salinity on
species of concern, and does not provide estimates of confidence or uncertainty.
32
Careful consideration should be given to the relevance and inferential consequences
of using different data sources. An example of the tolerance plot approach is given
in Figure A1.
33
70000
Salinity (us/cm)
60000
50000
Native
Introduced
40000
30000
20000
10000
0
Figure A1 Compares the salt tolerance of native and introduced fishes. It can be seen here that at a level of 20,000 EC (red line) at least 5 native
species would be expected to be affected, a number of which are listed and threatened or endangered. It can also been seen that this level is close to
the tolerance of three other native species. The effect on introduced species can also be compared.
34
A.2 – BurrliOZ Software Example
BurrliOZ is a free program that performs parametric modelling of SSDs. Developed
by Shao (2000), BurrliOZ fits a Burr Type III distribution to sets of tolerance data and
computes confidence intervals around a specified percentile. It has the advantage of
being easy to use and has been used in the ANZECC water quality guidelines (Fox
1999; ANZECC & ARMCANZ 2000; Shao 2000). However, the limitations of this
method are that goodness of fit and model checking statistics are limited and the
distribution itself is not characterised in a way that can be used in risk assessments.
The program is available from CSIRO4 and includes instructional documentation.
Once the species list and tolerance data have been obtained (from sources such as
the Land and Water database), the individual tolerance points are entered into the
program,
along
with
the
effect-percentile
of
interest.
Resultant
hazardous
concentrations and confidence intervals are then computed.
This is an easy way to perform parametric SSD modelling and compute confidence
intervals. However, care should be given to the influence of the data used and the
lack of statistical checking diagnostics. An example of this approach is given in
Figure A2.
4
www.cmis.csiro.au/Envir/burrlioz/
35
Figure A2 Display of typical output from the BurrliOZ program. The solid blue line represents
the Burr Type III distribution that is used to calculate percentiles. A log-log (orange line) and
log-normal (green line) fit are shown for comparison. Red dots represent data points and the
purple line indicates the percentile of choice.
36
A.3 - Parametric SSD Modelling
While a number of statistical packages can fit parametric probability distributions to
data, only a few estimate confidence intervals for percentiles. In particular, recent
versions of Minitab (i.e. 13 and 14) have included this function as standard. The
more specific and powerful packages such as SAS and S-plus are also able to produce
confidence intervals, although they are more expensive and difficult to use.
Many of the published studies involving SSDs assume that data is log-normally
distributed and relatively few provide tests of normality or comparisons with other
parametric models. A few studies have compared the fit of different parametric
models, generally finding that the assumption of (log)normality was not supported
in over half of cases and that log-logistic was a better fit (Newman et al. 2000). There
seems to be little justification for the a priori assumption that species sensitivities
are normally or log-normally distributed (Newman et al. 2000). However, statistical
tests of normality (such as Shapiro-Wilks) should provide some basis for the choice
of distribution. Comparison between the fit of different distributions should be
made using a goodness of fit statistic such as Anderson-Darling, KolmogorovSmirnov or the various likelihood information criterion (such as
Akaike’s
Information Criterion) (Buckland et al. 1997; Burnham & Anderson 2001), and may
be aided by examining plots of the fitted distribution. The choice of distribution is a
much debated issue and one particular model will not be advocated over another
here, except to note that in published studies the log-logistic has been found to be a
better fit more often than other common models (Newman et al. 2000; Posthuma et
al. 2002a).
To fit the model, tolerance or sensitivity data sets are entered into the packages
spread-sheet and the distributions fit (Minitab has a menu command for this
whereas SAS and other programs require command line instructions). Minitab
automatically produces a plot and table of confidence intervals and an Anderson
Darling statistic. These percentiles, confidence intervals and plots can then be used
to assess the fit of the model, choose better fitting models and make inferences
about the likely effects of salinity on species or groups of species. A Minitab example
is given in Figure A3.
37
Figure A3 Compares the fit of a log-logistic (parametric) distribution to data for two different
lakes (shown in red and black). The straight, middle, lines depict the mean response, with the
outer lines representing the upper and lower 95% confidence intervals (a table of these is
also produced by the software). The solid dots are the data points. The parameters for each
distribution are given as ‘loc’ (location) and ‘scale’ in the key, as are the number of data points
(N), Anderson-Darling statistic (AD) and p-value.
38
References
Ades A. E. & Lu G. (2003) Correlations between parameters in risk models: estimation
and propagation of uncertainty by Markov Chain Monte Carlo. Risk Analysis 23:
1165-1172.
Aldenberg T. & Jaworska J. S. (2000) Uncertainty of the hazardous concentration and
fraction affected for normal species sensitivity distributions. Ecotoxicology and
Environmental Safety 46: 1-18.
Aldenberg T., Jaworska J. S. & Traas T. P. (2002) Normal species sensitivity
distributions and probabilistic ecological risk assessment. In: Species Sensitivity
Distributions in Ecotoxicology (eds. L. Posthuma, G. W. Suter II & T. P. Traas) pp. 49102. Lewis Publishers, Boca Raton.
Aldenberg T. & Slob W. (1993) Confidence limits for hazardous concentrations based
on logistically distributed NOEC toxicity data. Ecotoxicology and Environmental
Safety 25: 48-63.
Andrews C. J., Hassenzahl D. M. & Johnson B. B. (2004) Accommodating uncertainty
in comparative risk. Risk Analysis 24: 1323-1335.
ANZECC & ARMCANZ (2000) National Water Quality Management Strategy Australian and New Zealand Guidelines for Fresh and Marine Water Quality.
Australian and New Zealand Environment and Conservation Council (ANZECC) &
Agriculture and Resource Management Council of Australia and New Zealand
(ARMCANZ), Canberra, Australia.
Bailey
P.,
Boon
P.
&
Morris
K.
(2002)
Salt
Sensitivity
Database.
www.rivers.gov.au/Tools_and_Techniques/Salt_Sensitivity_Database/index.aspx.
Land and Water Austalia.
Bartell S. M., Pastorok R. A., Akçakaya H. R., Regan H., Ferson S. & Mackay C. (2003)
Realism and relevance of ecological models used in chemical risk assessment.
Human and Ecological Risk Assessment 9: 907-938.
Bates S. C., Cullen A. & Raftery A. E. (2003) Bayesian uncertainty assessment in
multicompartment deterministic simulation models for environmental risk
assessment. Environmetrics 14: 35-371.
Berleant D. & Zhang J. (2004) Using pearson correlation to improve envelopes around
the distributions of functions. Reliable Computing 10: 139-161.
Bixio D., Parmentier G., Rousseau D., Verdonck F. A. M., Meirlaen J., Vanrolleghem P.
A. & Thoeye C. (2002) A quantitative risk analysis tool for design/simulation of
wastewater treatment plants. Water Science and Technology 46: 301-307.
Brattin W. J., Barry T. M. & Chiu N. (1996) Monte Carlo modeling with uncertain
probability density functions. Human and Ecological Risk Assessment 2: 820-840.
Breton R. L., Teed R. S. & Mooreb D. R. J. (2003) An ecological risk assessment of
phenol in the aquatic environment. Human and Ecological Risk Assessment 9: 549568.
Buckland S. T., Burnham K. P. & Augustin N. H. (1997) Model selection: An
intergrated part of inference. Biometrics 53: 603-618.
Burgman M. A. (2005) Risks and Decisions for Conservation and Environmental
Management. Cambridge University Press, London, UK.
Burmaster D. E. & Wilson J. C. (1998) Risk assessment for chemicals in the
environment. In: The Encyclopedia of Biostatistics, First Edition. John Wiley & Sons,
Inc.
Burnham K. P. & Anderson D. R. (2001) Kullback–Leibler information as a basis for
strong inference in ecological studies. Wildlife Research 28: 111-119.
39
Cairns Jr J. (1999) Absence of certainty is not synonymous with absence of risk.
Environmental Health Perspectives 107: A56-A66.
Cant B., James K. R. & Ryan T. (2003) Salt impact model for strategic framework and
information to indicate biodiversity thresholds to salinity pp. 82. Arthur Rylah
Institute, Department of Sustainability & Environment, Heidelberg, Victoria.
Cardwell R. D., Parkhurst B. R., Warren-Hicks W. & Volosin J. S. (1993) Aquatic
ecological risk. Water Environment and Technology 4: 47–51.
Chapman P. M., Caldwell R. S. & Chapman P. F. (1996) A warning: NOECs are
inappropriate for regulatory use. Environmental Toxicology and Chemistry 15: 77-79.
Chapman P. M., Fairbrother A. & Brown D. (1998) A critical evaluation of safety
(uncertainty) factors for ecological risk assessment. Environmental Toxicology and
Chemistry 17: 99-108.
Congdon P. (2001) Bayesian Statistical Modelling. John Wiley & Sons., Chichester, UK.
Congdon P. (2003) Applied Bayesian Modelling. John Wiley & Sons, Ltd., Chichester,
UK.
Crane M., Whitehouse P., Comber S., Watts C., Giddings J. M., Moore D. R. J. & Grist E.
(2003) Evaluation of probabilistic risk assessment of pesticides in the UK:
chlorpyrifos use on top fruit. Pest Management Science 59: 512-526.
de Bruijn J. H. M. & Hof M. (1997) How to measure no effect. Part IV: How acceptable
is the ECx from an environmental policy point of view? Environmetrics 8: 263-267.
de Zwart D. (2003) Ecological effects of pesticide use in the Netherlands. RIVM
Report 500002003/2003, Bilthoven.
Dixon W. J. (2005) Uncertainty in aquatic toxicological exposure-effect models: the
toxicity of 2,4-Dichlorophenoxyacetic acid and 4-Chlorophenol to Daphnia carinata.
PhD Thesis. RMIT University, Melbourne, Australia.
Dixon W. J. (2007a) The use of probability bounds analysis for characterising and
propagating uncertainty in species sensitivity distributions. Arthur Rylah Institute
for Environmental Research, Technical Report Series No. 163. Department of
Sustainability and Environment, Heidelberg, Victoria.
Dixon W. J. (2007b) Uncertainty propagation in population level salinity risk models.
Arthur Rylah Institute for Environmental Research, Technical Report Series No. 164.
Department of Sustainability and Environment, Heidelberg, Victoria.
Dixon W. J. & Scroggie M. P. (2007) A Bayesian approach to multi-species risk
assessments using whole exposure-response distributions. Arthur Rylah Institute for
Environmental Research, Technical Report Series No. 165. Department of
Sustainability and Environment, Heidelberg, Victoria.
Englehardt J. D. (2004) Predictive Bayesian dose-response assessment for appraising
absolute health risk from available information. Human and Ecological Risk
Assessment 10: 69 - 78.
Ferson S. (1996) What Monte Carlo methods cannot do. Human and Ecological Risk
Assessment 2: 990-1007.
Ferson S. (2002) RAMAS Risk Calc 4.0 Software: Risk Assessment with Uncertain
Numbers. Lewis Publishers,, Boca Raton, Florida.
Ferson S. & Ginzburg L. R. (1996) Different methods are needed to propagate
ignorance and variability. Reliability Engineering & System Safety 54: 133-144.
Ferson S., Ginzburg L. R. & Goldstein R. A. (1996) Inferring ecological risk from
toxicity bioassays. Water, Air and Soil Pollution 90: 71-82.
Ferson S. & Hajagos J. G. (2004) Arithmetic with uncertain numbers: rigorous and
(often) best possible answers. Reliability Engineering & System Safety 85: 135-152.
40
Ferson S., Kreinovich V., Ginzburg L. R., Myers D. S. & Sentz K. (2003) Constructing
Probability Boxes and Dempster-Shafer Structures. Report Number: SAND2002-4015.
Sandia National Laboratories, Albuquerque, New Mexico.
Finkel A. M. (1995) Toward less misleading comparisons of uncertain risks: the
example of Aflatoxin and Alar. Environmental Health Perspectives 103: 376-385.
Fogarty M. J., Mayo R. K., O'Brien L., Serchuk F. M. & Rosenberg A. A. (1996) Assessing
uncertainty and risk in exploited marine populations. Reliability Engineering &
System Safety 54: 183-195.
Forbes V. E. & Calow P. (2002) Species sensitivity distributions revisited: a critical
appraisal. Human and Ecological Risk Assessment 8: 473-492.
Fox D. R. (1999) Setting water quality guidelines - A statistician's perspective. SETAC
News May: 17-18.
Frey H. C. & Burmaster D. E. (1999) Methods for characterizing variability and
uncertainty: comparison of bootstrap simulation and likelihood-based approaches.
Risk Analysis 19: 109-130.
Gelman A., Carlin J. B., Stern H. S. & Rubin D. B. (2004) Bayesian Data Analysis.
Second Edition. Chapman & Hall/CRC, Boca Ranton, Florida, USA.
Godsill S. J. (2001) On the relationship between markov chain monte carlo methods
for model uncertainty. Journal of Computational and Graphical Statistics 10: 1-19.
Hart B. T., Maher B. & Lawrence I. (1999) New generation water quality guidelines for
ecosystem protection. Freshwater Biology 41: 347-359.
Hattis D. (1990) Three candidate "laws" of uncertainty analysis. Risk Analysis 10: 11.
Hoekstra J. A. & van Ewijk P. H. (1993) Alternatives for the no-observed-effect level.
Environmental Toxicology and Chemistry 12: 187-194.
Jager T., den Hollander H. A., van der Poel P., Rikken M. G. J. & Vermeire T. (2001)
Probabilistic environmental risk assessment for Dibutylphthalate (DBP). Human and
Ecological Risk Assessment 7: 1681-1697.
Jagoe R. H. & Newman M. C. (1997) Bootstrap estimation of community NOEC values.
Ecotoxicology 6: 293-306.
Kefford B. J., Nugegoda D., Metzeling L. & Fields E. J. (2006) Validating species
sensitivity distributions using salinity tolerance of riverine macroinvertebrates in the
southern Murray-Darling Basin (Victoria, Australia). Canadian Journal of Fisheries
and Aquatic Sciences 63: 1865-1877.
Kefford B. J., Palmer C. G., Warne M. S. & Nugegoda D. T. (2005) What is meant by
"95% of species"? An argument for the inclusion of rapid tolerance testing. Human
and Ecological Risk Assessment 11: 1025-1046.
Landis W. G. (2003) The frontiers in ecological risk assessment at expanding spatial
and temporal scales. Human and Ecological Risk Assessment 9: 1415 - 1424.
Laskowski R. (1995) Some good reasons to ban the use of NOEC, LOEC and related
concepts in ecology. Oikos 73: 140-144.
Lee P. M. (2004) Bayesian Statistics. Third Edition. Hodder Arnold, London, UK.
Linkov I., von Stackelberg K. E., Burmistrov D. & Bridges T. S. (2001) Uncertainty and
variability in risk from trophic transfer of contaminants in dredged sediments. The
Science of the total environment 274: 255-269.
Manly B. F. J. (1997) Randomization, Bootstrap and Monte Carlo Methods in Biology,
2nd Edition. Chapman & Hall, London, UK.
McBride G. B. & Ellis J. C. (2001) Confidence of compliance: a bayesian approach for
percentiles standards. Water Research 35: 1117-1124.
41
Merrick J. R. W., van Dorp J. R. & Dinesh V. (2005) Assessing uncertainty in
simulation-based maritime risk assessment. Risk Analysis 25: 731-743.
Moore D. R. J., Sample B. E., Suter G. W., Parkhurst B. R. & Teed R. S. (1999) A
probabilistic risk assessment of the effects of methylmercury and PCBs on mink and
kingfishers along east fork Poplar Creek, Oak Ridge, Tennessee, USA. Environmental
Toxicology and Chemistry 18: 2941-2953.
Morgan M. G. & Henrion M. eds. (1990) Uncertainty: A Guide to Dealing with
Uncertainty in Quantitative Risk and Policy Analysis. Cambridge University Press,
New York, NY USA.
Nayak T. K. & Kundu S. (2001) Calculating and describing uncertainty in risk
assessment: the Bayesian approach. Human and Ecological Risk Assessment 7: 307328.
Newman M. C., Ownby D. R., Mezin L. C. A., Powell D. C., Chiristensen T. R. L., Lerberg
S. B. & Anderson B. A. (2000) Applying species-sensitivity distributions in ecological
risk assessment: assumptions of distribution type and sufficient numbers of species.
Environ. Toxicol. Chem. 19: 508-515.
Newman M. C., Ownby D. R., Mézin L. C. A., Powell D. C., Christensen T. R. L., Lerberg
S. B., Anderson B.-A. & Padma T. V. (2002) Species sensitivity distributions in
ecological risk assessment: distributional assumptions, alternate bootstrap
techniques, and estimation of adequate number of species. In: Species Sensitivity
Distributions in Ecotoxicology (eds. L. Posthuma, G. W. Suter II & T. P. Traas) pp. 109132. Lewis Publishers, Boca Raton.
NRC (1993) Issues in Risk Assessment: a Paradigm for Ecological Risk Assessment.
NRC (National Research Council), Committee on Risk Assessment Methodology,
Board on Environmental Studies and Toxicology, Commission on Life Sciences, and
National Research Council, National Academy Press, Washington D.C. USA.
NRC (1994) Science and Judgment in Risk Assessment. NRC (National Research
Council), Committee on Risk Assessment of Hazardous Air Pollutants, Board on
Environmental Studies and Toxicology, Commission on Life Sciences, and National
Research Council, National Academy Press, Washington D.C. USA.
Pastorok R. A., Akçakaya H. R., Regan H., Ferson S. & Bartell S. M. (2003) Role of
ecological modeling in risk assessment. Human and Ecological Risk Assessment 9:
939-972.
Pastorok R. A., Bartell, S.M., Ferson, S., Ginzburg, L.R. ed. (2002) Ecological Modeling
in Risk Assessment: Chemical Effects on Populations, Ecosystems and Landscapes.
Lewis Publishers, Boca Raton, FL.
Posthuma L., Suter II G. W. & Traas T. P. eds. (2002a) Species Sensitivity Distributions
in Ecotoxicology. Lewis Publishers, Boca Raton.
Posthuma L., Traas T. P., de Zwart D. & Suter II G. W. (2002b) Conceptual and
technical outlook on species sensitivity distributions. In: Species Sensitivity
Distributions in Ecotoxicology (eds. L. Posthuma, G. W. Suter II & T. P. Traas) pp. 475508. Lewis Publishers, Boca Raton.
Rand G. M. ed. (1995) Fundamentals of Aquatic Toxicology, Second Edition. Taylor and
Francis, Washington, D.C. USA.
Regan H. M., Colyvan M. & Burgman M. A. (2002a) A taxonomy and treatment of
uncertainty for ecology and conservation biology. Ecological Applications 12: 618628.
Regan H. M., Ferson S. & Berleant D. (2004) Equivalence of methods for uncertainty
propagation of real-valued random variables. International Journal of Approximate
Reasoning 36: 1-30.
42
Regan H. M., Hope B. K. & Ferson S. (2002b) Analysis and portrayal of uncertainty in a
food-web exposure model. Human and Ecological Risk Assessment 8: 1757 - 1777.
Regan H. M., Sample B. E. & Ferson S. (2002c) Comparison of deterministic and
probabilistic calculation of ecological soil screening levels. Environmental Toxicology
and Chemistry 21: 882-890.
Shao Q. (2000) Estimation for hazardous concentrations based on NOEC toxicity
data: an alternative approach. Environmetrics 11: 583-595.
Soloman K. R., Giesy J. & Jones P. (2000) Probabilistic risk assessment of
agrochemicals in the environment. Crop Protection 19: 649-655.
Soloman K. R. & Sibley P. (2002) New concepts in ecological risk assessment: where
do we go from here? Marine Pollution Bulletin 44: 279-285.
Spiegelhalter D. J., Thomas A., Best N. G. & Gilks W. R. (2003) BUGS: Bayesian
inference Using Gibbs Sampling, Version 1.4. MRC Biostatistics Unit, Cambridge.
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., II, Efroymson R. A., Sample B. E. & Jones D. S. (2000) Ecological Risk
Assessment for Contaminated Sites. Lewis Pub., Boca Raton, USA.
Suter G. W., II, Traas T. P. & Posthuma L. (2002) Issues and practices in the derivation
and use of species sensitivity distributions. In: Species Sensitivity Distributions in
Ecotoxicology (eds. L. Posthuma, G. W. Suter II & T. P. Traas) pp. 437-474. Lewis
Publishers, Boca Raton.
Thompson K. M. (1999) Developing univariate distributions from data for risk
analysis. Human and Ecological Risk Assessment 5: 755-783.
Thompson K. M. (2002) Variability and uncertainty meet risk management and risk
communication. Risk Analysis 22: 647-654.
Thompson K. M. & Graham J. D. (1996) Going beyond the single number: using
probabilistic risk assessment to improve risk management. Human and Ecological
Risk Assessment 2: 1008-1034.
van der Hoeven N. (1997) How to measure no effect. Part III: Statistical aspects of
NOEC, ECx and NEC estimates. Environmetrics 8: 255-261.
van der Hoeven N. (2001) Estimating the 5-percentile of the species sensitivity
distribution without any assumptions about the distribution. Ecotoxicology 10: 25-34.
van der Hoeven N. (2004) Current Issues in Statistics and Models for Ecotoxicological
Risk Assessment. Acta Biotheoretica 52: 201–217.
Van Sprang P. A., Verdonck F. A. M., Vanrolleghem P. A., Vangheluwe M. L. & Janssen
C. R. (2004) Probabilistic environmental risk assessment of Zinc in Dutch surface
water. Environmental Toxicology and Chemistry 23: 2993-3002.
van Straalen N. M. (1990) New methodologies for estimating the ecological risk of
chemicals in the environment. In: Proceedings of the 6th Congress of the International
Association of Engineering Geology (ed. D. G. Price) pp. pp. 165–173. A.A. Balkema,
Rotterdam, the Netherlands.
van Straalen N. M. (2002a) Theory of ecological risk assessment based on species
sensitivity distributions. In: Species Sensitivity Distributions in Ecotoxicology (eds. L.
Posthuma, G. W. Suter II & T. P. Traas) pp. 37-48. Lewis Publishers, Boca Raton.
van Straalen N. M. (2002b) Threshold models for species sensitivity distributions
applied to aquatic risk assessment for zinc. Environmental Toxicology and
Pharmacology 11: 167-172.
van Straalen N. M. & Denneman C. A. J. (1989) Ecotoxicological evaluation of soil
quality criteria. Ecotoxicology and Environmental Safety 18: 241-251.
43
van Straalen N. M. & van Leeuwen C. J. (2002) European history of species sensitivity
distributions. In: Species Sensitivity Distributions in Ecotoxicology (eds. L. Posthuma,
G. W. Suter II & T. P. Traas) pp. 19-34. Lewis Publishers, Boca Raton.
Verdonck F., Jaworska J. & Vanrolleghem P. A. (2001a) Comments on species
sensitivity distributions sample size determination based on Newman et al. (2001).
SETAC Globe, Learned Discourse 2: 22-24.
Verdonck F. A. M., Aldenberg T., Jaworska J. S. & Vanrolleghem P. A. (2003)
Limitations of current risk characterization methods in probablistic environmental
risk assessment. Environmental Toxicology and Chemistry 22: 2209-2213.
Verdonck F. A. M., Jaworska J., Janssen C. R. & Vanrolleghem P. A. (2002)
Probabilistic ecological risk assessment framework for chemical substances. In:
Integrated Assessment and Decision Support, Proceedings of the First Biennial Meeting
of the International Environmental Modelling and Software Society (IEMSS), Volume 1
(eds. A. E. Rizzoli & A. J. Jakeman) pp. 144- 149. IEMSS.
Verdonck F. A. M., Jaworska J., Thas O. & Vanrolleghem P. A. (2001b) Determining
environmental standards using bootstrapping, Bayesian and maximum likelihood
techniques: a comparative study. Analytica Chimica Acta 446: 429-438.
Warren-Hicks W. J., Parkhurst B. R. & Butcher J. B. (2002) Methodology for Aquatic
Ecological Risk Assessment. In: Species Sensitivity Distributions in Ecotoxicology (eds.
L. Posthuma, G. W. Suter II & T. P. Traas) pp. 345-382. Lewis Publishers, Boca Raton.
Webb J. A. & Hart B. T. (2004) Environmental Risks from Salinity Increases in the
Goulburn-Broken Catchment - Draft. Water Studies Centre, Monash University,
Melbourne, Australia.
Williamson R. C. & Downs T. (1990) Probabilistic arithmetic I: numerical methods for
calculating convolutions and dependency bounds. International Journal of
Approximate Reasoning 4: 89–158.
Winkler R. L. (1996) Uncertainty in probabilistic risk assessment. Reliability
Engineering & System Safety 54: 127-132.
Yager R. R. (1986) Arithmetic and other operations on Dempster-Shafer structures.
International Journal of Man-Machine Studies 25: 357-366.
Yanagawa T. & Kikuchi Y. (2001) Statistical issues on the determination of the noobserved-adverse-effect levels in toxicology. Environmetrics 12: 319-325.
44