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A survival model of the effects of bottom-water hypoxia on the population density of an estuarine clam (Macoma balthica)

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Abstract

The effect of bottom-water hypoxia on the population density of the clam Macoma balthica is estimated using a survival-based approach. We used Bayesian parameter estimation to fit a survival model to times-to-death corresponding to multiple dissolved oxygen (DO) concentrations assessed from scientific experts. We describe guidelines for ensuring the accuracy of such assessments and claim that elicitation of quantities that pertain to measurable variables of interest, rather than unobservable parameters, should improve the use of judgment-based information in Bayesian analyses. When directly relevant data are lacking, predictions based on subjective assessments can serve as the basis for preliminary management decisions and additional data collection efforts. To inform pending water quality controls for the Neuse River estuary, North Carolina, we combined the survival model with a model describing the time dependence of DO. For current conditions, the mean summer survival rate is predicted to be only 11%. However, if sediment oxygen demand (SOD) is reduced as a result of nutrient management, summer survival rates will increase, reaching 23% with a 25% reduction in SOD and 46% with a 50% SOD reduction. Full model predictions are expressed as probabilities to provide a quantitative basis for risk-based decision-making and experimental design.
A survival model of the effects of bottom-water
hypoxia on the population density of an estuarine
clam (Macoma balthica)
Mark E. Borsuk, Sean P. Powers, and Charles H. Peterson
Abstract: The effect of bottom-water hypoxia on the population density of the clam Macoma balthica is estimated
using a survival-based approach. We used Bayesian parameter estimation to fit a survival model to times-to-death
corresponding to multiple dissolved oxygen (DO) concentrations assessed from scientific experts. We describe guide-
lines for ensuring the accuracy of such assessments and claim that elicitation of quantities that pertain to measurable
variables of interest, rather than unobservable parameters, should improve the use of judgment-based information in
Bayesian analyses. When directly relevant data are lacking, predictions based on subjective assessments can serve as
the basis for preliminary management decisions and additional data collection efforts. To inform pending water quality
controls for the Neuse River estuary, North Carolina, we combined the survival model with a model describing the
time dependence of DO. For current conditions, the mean summer survival rate is predicted to be only 11%. However,
if sediment oxygen demand (SOD) is reduced as a result of nutrient management, summer survival rates will increase,
reaching 23% with a 25% reduction in SOD and 46% with a 50% SOD reduction. Full model predictions are
expressed as probabilities to provide a quantitative basis for risk-based decision-making and experimental design.
Résumé : Une méthode basée sur la survie nous a permis d’étudier les effets de l’hypoxie des eaux du fond sur la
densité de la population de la moule Macoma balthica. Nous avons utilisé une estimation des paramètres de type
bayésien pour ajuster un modèle de survie à des données de temps jusqu’à la mort à plusieurs concentrations
d’oxygène dissous (DO) déterminées par des experts. Nous donnons des lignes directrices pour assurer la précision
d’une telle évaluation et indiquons que la mise de l’avant de données quantitatives relatives à des variables pertinentes,
plutôt que de paramètres impossibles à observer, devrait améliorer l’utilisation des informations sujettes à des
jugements dans les analyses bayésiennes. Lorsque les données directement pertinentes font défaut, les prédictions
basées sur des évaluations subjectives peuvent servir de base à des décisions préliminaires de gestion et à la récolte de
données additionnelles. Pour obtenir des informations relatives aux contrôles actuels de qualité de l’eau de l’estuaire de
la Neuse en Caroline du Nord, nous avons combiné le modèle de survie à un modèle qui décrit la dépendance
temporelle de DO. Dans les conditions présentes, la survie prédite en été n’est que de 11 %. Cependant, une réduction
de la demande en oxygène des sédiments (SOD) résultant d’une meilleure gestion des apports de nutriments permettrait
une survie accrue, pouvant atteindre 23 % pour une réduction de 25 % de SOD et 46 % pour une réduction de 50 %
de SOD. Les prédictions complètes du modèle sont exprimées en probabilités pour fournir une base quantitative aux
prises de décision avec analyse des risques et pour établir des plans d’expérience.
[Traduit par la Rédaction] Borsuk et al. 1274
Introduction
Similar to many other intermittently stratified estuaries
(Nixon 1995), the Neuse River estuary, North Carolina, experi-
ences frequent periods of bottom-water hypoxia in the summer,
believed to be worsening because of anthropogenic eutro-
phication (Paerl et al. 1998). Concerns regarding the effects of
low dissolved oxygen (DO) on the state’s fish and shellfish
populations have prompted consideration of watershed nutrient
management strategies (N.C. Division of Water Quality 2001).
To assist in the selection of a nutrient load reduction target, we
have been developing a set of linked ecological response mod-
els (e.g., Borsuk et al. 2001a, 2001b). Integrated as a Bayesian
probability network (Reckhow 1999), the overall model is in-
tended to predict endpoints that reflect ultimate management
concerns, such as fish and shellfish health and abundance
Can. J. Fish. Aquat. Sci. 59: 1266–1274 (2002) DOI: 10.1139/F02-093 © 2002 NRC Canada
1266
Received 25 February 2002. Accepted 13 June 2002. Published on the NRC Research Press Web site at http://cjfas.nrc.ca on
20 August 2002.
J16782
M.E. Borsuk.1,2 Division of Environmental Science and Policy, Nicholas School of the Environment and Earth Sciences,
Duke University, Durham, NC 27708, U.S.A.
S.P. Powers and C.H. Peterson. Institute of Marine Sciences, University of North Carolina at Chapel Hill, Morehead City,
NC 28557, U.S.A.
1Corresponding author (e-mail: mark.borsuk@eawag.ch).
2Present address: Department of Systems Analysis, Integrated Assessment, and Modelling (SIAM), Swiss Federal Institute for
Environmental Science and Technology (EAWAG), P.O. Box 611, 8600 Dübendorf, Switzerland.
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(Borsuk et al. 2001c). In this paper, we describe the model
component linking hypoxia to summer survival of the clam
species Macoma balthica.
The summer survival rate of M. balthica was chosen as a
key management endpoint because summer is the season of
greatest concern regarding DO depletion and because
M. balthica plays a critical role in the Neuse River estuarine
ecosystem. This later-succession bivalve is the major com-
ponent of benthic biomass in the estuary as well as the most
valuable food resource for demersal fish species. The diets
of juvenile demersal fishes, such as spot (Leiostomus
xanthurus), croaker (Micropogonias undulatus), and floun-
der (Paralichthys lethostigma), are dominated by M. balthica
siphons, and blue crabs (Callinectes sapidus) also prey upon
this species, facilitated by its thin shell and shallow burial
depths (Skilleter and Peterson 1994). Field studies have
shown that late-summer patterns of abundance of
M. balthica in the Neuse River estuary closely match the
patterns of extended exposure to summertime hypoxia (S.P.
Powers, unpublished data). However, experimental studies
have not yet been performed under natural field conditions
to directly address the sensitivity of this species to low DO.
Although the direct applicability of existing sample infor-
mation is limited, research scientists studying the effects of
hypoxia on M. balthica possess a great deal of knowledge
gained from laboratory and microcosm experiments (e.g.,
Diaz and Rosenberg 1995; de Zwaan and Babarro 2001) and
literature review (Peterson et al. 2000). In the Bayesian ap-
proach to model building and decision analysis (Punt and
Hilborn 1997), such knowledge can be quantified and used
to generate initial results, which might later be revised as ap-
propriate field data are collected (Winkler 1980). Personal
knowledge, expressed using probabilities, has been shown to
be as admissible as quantitative measurements for providing
the basis for rational decision-making (Savage 1954). There-
fore, the development of our survival model for M. balthica
relies upon the carefully elicited judgment of two marine biol-
ogists. Assessments are performed according to established
protocols (Spetzler and Stael von Holstein 1975) and relate
to measurable quantities that could be easily substituted by
field or laboratory data where applicable. In this way, our
analysis can serve as both an example of the survival analy-
sis approach to ecological assessment and as a demonstra-
tion of the elicitation and application of scientific knowledge
when data are limited. We believe that, given the urgency
with which guidance is often needed for imminent manage-
ment decisions, the reliance on a logical quantitative model
that utilizes the carefully considered judgments of experi-
enced scientists is preferable to a situation in which decision-
makers are forced to rely on their own limited knowledge to
relate water quality changes to expected ecological health.
Additionally, researchers can make use of preliminary model
results that include a description of uncertainty to design fu-
ture data collection activities based on their ability to im-
prove predictive precision.
Materials and methods
Survival analysis approach
Investigators have recognized that characterizing the de-
gree of tolerance to hypoxia by key species is critical to pre-
dicting ecosystem response. However, most have focused on
estimating single-valued indicators of tolerance that do not
provide complete descriptions. A common choice is a mea-
sure of the time at which 50% of exposed individuals die
from exposure to a specified DO concentration (LT50) (e.g.,
Diaz and Rosenberg 1995). Parametric and nonparametric
methods have also been used to estimate mean biomass and
species density for various DO concentrations in a field set-
ting (Ritter and Montagna 1999). Although such approaches
may be statistically reliable and may inform interspecies
comparisons of typical tolerances, they are not useful for
prediction in most natural systems. In estuaries that experi-
ence periodic hypoxia, benthic organisms are exposed to
changing DO concentrations for varying amounts of time
(Modig and Olaffson 1998), rarely of a duration approaching
the LT50. A more complete characterization of tolerance is
required that describes the effects of shorter durations of ex-
posure, as well as the bivariate effects of how tolerance is
influenced by varying DO concentration over varying expo-
sure times.
Survival analysis (Hosmer and Lemeshow 1999) can provide
considerably more useful information for predictive model-
ing than standard dose–response techniques. Survival meth-
ods use time-to-death data for all individuals in a study to
characterize the probability of death as it relates to the level
of a stressor and exposure time. The approach has been
widely used in medical and engineering research but has
only recently been employed for ecological studies (e.g.,
Newman and McCloskey 1996).
Of primary interest in survival analysis is estimation of
the survival and hazard functions. Assuming a time course
of exposure to a stressor with individuals dying over period
T, the distribution of times-to-death of the individuals can be
described by probability density f(t) and cumulative distribu-
tion F(t). Estimates of F(t) would be provided by the number
of individuals dead at time t, divided by the total number of
exposed individuals,
(1) Ft t
()=Number dead at time
Total number exposed
The survival function, S(t), expresses the proportion of the
total still alive at time tand is related to F(t)by
(2) S(t)=1–F(t)
The hazard function, h(t), is the probability of an individ-
ual dying during the next small interval given that it has sur-
vived to the beginning of the interval:
(3) h(t)= ft
St
()
()
The cumulative hazard function, H(t), can be calculated as
(4) H(t)= hu u
t
0
d
()
where uis a variable of integration. H(t) is related to the sur-
vival function according to
(5) H(t) = –ln S(t)
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To describe the change in the distribution of times-to-
death resulting from some change in the level of the stressor,
a form of survival model called an accelerated life model is
often used, where
(6) ln t=g(x)+ε
and tis a vector of times-to-death for all individuals, g(x)is
a function relating ln tto a vector of stressor levels, x, and ε
is a probabilistic term describing the variability in time-to-
death among individuals. This variability may arise because
of intrinsic factors, such as genetic and physiological differ-
ences, or extrinsic factors, including variation in actual ex-
posure concentration. Equation 6 resembles the form of a
standard regression equation, and the function g(x) may be
of any form used in regression analysis. Whereas in typical
regression applications the probabilistic term is assumed to
represent a normally distributed error, times-to-death are not
necessarily normally distributed. Common alternatives in-
clude the logistic, exponential, Weibull, log-normal, and log-
logistic distributions, each of which results in a characteris-
tic survival and hazard function according to eqs. 2 and 3.
Certain distributional families have theoretical support or
computational advantages that can help guide the selection
process in particular applications (Newman and Dixon
1996).
Assessment protocol
The use of existing scientific knowledge to form the basis
for Bayesian model building has often involved the direct as-
sessment of distributions representing the uncertainty in the
parameters of the model (e.g., Walters and Ludwig 1994;
McAllister and Kirkwood 1998). However, it is often diffi-
cult for people, even those trained in probability and statis-
tics, to think directly about model parameters and develop
distributions representing their prior beliefs about the possible
values of those parameters (Winkler 1980). This is especially
true with regard to proper assessment of the correlation struc-
ture among parameters (Morgan and Henrion 1990), which
may be a very important determinant of predictive uncer-
tainty. To overcome these difficulties, we used an indirect
method of parameter estimation in which information was
obtained from the scientific experts in the form of probabil-
ity distributions involving the variables of interest (survival
times), rather than model parameters. These distributions,
called predictive distributions in Bayesian terminology, are
univariate rather than multivariate and involve observable
variables instead of unobservable parameters. In this way,
the distributions relate directly to the experts’ knowledge
and experience, thus providing a meaningful assessment task
(Winkler 1980).
It is important to note that, because model parameters do
not play a part in the assessment process, the assessed pre-
dictive distributions are not model-specific. Therefore, a
number of models of differing functional or distributional
forms can be evaluated, based on their fit to the assessed val-
ues. This process is analogous to fitting a model to a set of
observations and makes the survival method we present en-
tirely general to situations in which some directly relevant
data exist. In either case, distributions for model parameters
can be derived from the assessed or observational informa-
tion using methods of Bayesian inference, as described in
the next section.
Established elicitation protocols have been developed to
facilitate assessment of probabilistic quantities and avoid
problems associated with bias and overconfidence (Morgan
and Henrion 1990). Among the basic principles that have
been suggested are the following (Spetzler and Stael von
Holstein 1975): (i) Clearly define the quantity to be evalu-
ated as an unambiguous state variable. A good test is to
evaluate whether a clairvoyant could state the value of the
quantity without requesting additional clarification. (ii)De
-
scribe the quantity using a scale that is meaningful to the
person providing the assessment. This avoids the expendi-
ture of unnecessary time and mental energy making unit
conversions. (iii) Ask questions for which the answers can
be represented as points on a cumulative distribution func-
tion. For continuous variables, such quantities are typically
easier to conceive than those related to probability density
functions. (iv) Choose either a fixed-value or a fixed-
probability method. The fixed-value method requires assess-
ments of the probability that the quantity lies in a specified
range of values. In fixed-probability methods, the values of
the quantity that bound specified probability intervals are as-
sessed. (v) Express probabilistic quantities as frequencies to
make the problem easier for people to consider. (vi) Begin
by asking questions regarding values for the uncertain quan-
tity that may be considered somewhat extreme. This practice
helps to avoid overconfidence (i.e., underestimated uncer-
tainty) in the assessed distribution. (vii) Continue by asking
questions regarding intermediate values, working eventually
towards the most likely value. This further helps reduce over-
confidence or bias that may result from anchoring on a
highly likely value. (viii) Verify the assessed distributions by
asking questions stated differently from how they were orig-
inally posed. For example, if the fixed-value technique was
used, ask follow-up questions using the fixed-probability
method.
The quantities necessary for fitting the survival model
given in eq. 6 are predictive distributions for times-to-death,
t, conditional on multiple DO concentrations. The first au-
thor (M.B.) elicited points on the cumulative distribution
function of times-to-death, F(t), from the two other authors
(S.P. and C.H.P.) using the fixed-probability protocol. Their
assessments were based on literature reviews (Diaz and
Rosenberg 1995; Modig and Olaffson 1998), recent studies
(Jahn and Theede 1997; de Zwaan and Babarro 2001; de
Zwaan et al. 2001), their own population surveys in the
Neuse River estuary (Peterson et al. 2002; S.P. Powers, un-
published data), and their experience with benthic ecology
as a discipline. A typical question was “Given a M. balthica
population of 100 individuals and a fixed ambient DO con-
centration of 1.0 mg·L–1, how much time would you expect
to pass before there are xdead individuals?” This question
was asked for DO concentrations of 0.0, 0.5, 1.0, and
1.5 mg·L–1 and for values of xequal to of 5, 95, 25, 75, and
50, consecutively. Special care was taken to follow the sug-
gestions for accurate assessment outlined above. The sub-
jects answered the questions after evaluating the pertinent
literature and data and were allowed to state a range repre-
senting the uncertainty in their assessments. Additional
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questions were asked regarding the basis for their stated
range and to verify their assessed values.
Bayesian parameter estimation
The assessed quantities describing cumulative predictive
probability distributions were used to specify a functional
and distributional form for the general survival model de-
scribed by eq. 6. The fit of the cumulative probabilities to
multiple distributional families was evaluated using a graphi-
cal method based on appropriate transformations of the data
(Table 1). If a particular distribution provides a suitable fit to
the assessments made for a particular DO concentration, the
resulting plot of the transformed variables forms a straight
line (Newman and McCloskey 1996). The relative spacing of
the lines representing different DO concentrations suggests
an appropriate functional form to describe the effect of DO
concentration on survival times. Once the form of the model
given in eq. 6 was fully characterized, model and distribu-
tional parameter values were estimated using Bayesian infer-
ence (Winkler 1972). We chose the Bayesian over the classical
nonlinear regression approach because it generates full pos-
terior parameter distributions, allowing direct comparisons
of the relative plausibility of alternative parameter values
and giving a more complete assessment of predictive uncer-
tainty (Punt and Hilborn 1997). Our choice of non-
informative parameter priors means that the modes of the
posterior parameter distributions equal the estimates that
would result from regression. However, consideration of the
full parameter and predictive distributions, rather than sim-
ply point estimates, often has important consequences for
both theoretical understanding and practical decision-making
(Ludwig 1996). Posterior parameter distributions were gen-
erated using a Markov Chain Monte Carlo method (Gelman
et al. 1995).
Probabilistic prediction
In the bottom water of the Neuse River estuary, DO con-
centration is not constant (e.g., see Lenihan and Peterson
1998). Rather, the concentration continually changes as a
function of the balance between biological and chemical ox-
ygen demand (largely in the sediments) and reaeration from
the surface layer. The effect on the survivorship of
M. balthica can be accounted for by explicitly including a
time-varying stressor in the derived hazard function,
(7) h(t,x(t)) = ftxt
Stxt
(, ())
(, ())
If the functional form for the time dependency of the stressor,
x(t), is known, then the cumulative hazard function can be
calculated as
(8) H(t)= huxu u
t
0
d
(, ())
Borsuk et al. (2001a) derive a model describing the time
dependency of DO concentration in the bottom water of the
middle channel of the Neuse River estuary during an ex-
tended period without vertical mixing,
(9) x(t)=xpk k k k kt
kk kk
u[ ( ) {exp(( ) ) }]
( ) exp((
vd v vd
vd vd
++ +′
++
1
))tE
+
where tis time since the last mixing event (days), xuis the
DO concentration in the surface layer (mg·L–1), pis the ratio
of the DO concentration in the bottom layer to that in the
surface layer immediately following mixing, kvis the vertical
exchange coefficient (day–1), kdis a rate constant for sediment
oxygen demand (day–1) with a temperature dependency ac-
cording to the Arrhenius formulation, and Eis a normally
distributed error term with mean zero and variance σ2.
Model parameters were statistically estimated using approxi-
mately 10 years of weekly–biweekly, mid-channel data from
a regular monitoring program, as described in Borsuk et al.
(2001a).
Assuming each extended period without vertical mixing to
be an independent exposure event (i.e., the survivors are not
weakened or sensitized by previous exposure), then tand t
in eqs. 8 and 9 are equivalent. Therefore, once a hazard
function is derived from eq. 7, eqs. 8 and 9 can be used to
calculate the proportion of exposed individuals surviving an
oxygen depletion event of specified duration.
Because of our inability to anticipate precisely when mix-
ing events will occur, we cannot expect to be able to make
precise predictions of population survival at any given time.
This is a problem encountered with any deterministic model.
If we assume, however, that the mechanisms that control
vertical mixing will remain the same into the future, then the
time between events can be described by a probability distri-
bution based on historical data. Monte Carlo simulation can
then be used to generate probabilistic predictions of survival
over some period of interest, in this case the summer season.
The uncertainty in parameter values, as well as model lack-
of-fit, can also be incorporated using the Bayesian posterior
distributions.
Probabilistic predictions of the proportion of the initial
M. balthica population surviving through the summer season
were generated using a two-phase Monte Carlo method. First,
5000 sets of parameter values were randomly selected from
standard parametric distributions matching the Bayesian pos-
terior distributions derived in the present study and by Borsuk
et al. (2001a). For each of these parameter sets, 90 daily val-
ues for the input variables were randomly selected from dis-
tributions derived from historic data for the months of July,
August, and September (Borsuk et al. 2001a). Input variables
© 2002 NRC Canada
Borsuk et al. 1269
Distribution XY
Exponential ln S(t)t
Weibull ln [–ln S(t)] ln t
Normal Probit [F(t)] t
Log-normal Probit [F(t)] ln t
Log-logistic ln [S(t)/F(t))] ln t
Note: The functions S(t) and F(t) represent the
survival and cumulative distribution functions,
respectively, where tis a vector of times-to-death.
Table 1. Linearizing transformations for
selecting among candidate underlying
distributions in survival time models
(Newman and McCloskey 1996).
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consisted of water temperature, percent saturation of oxygen
in the upper layer (to calculate vertical reoxygenation rate),
and time since the last vertical mixing event. Ninety values
for the additive model error term, E, in eq. 9 were also drawn
for each parameter set to account for the variability in DO
concentration not explained by the model. Bottom-water DO
concentrations were then calculated for each of the 90 days
for each of the 5000 parameter sets according to eq. 9. Each
set of 90 values represents a predicted set of daily average
DO concentrations for one summer, and the differences
across the 5000 sets represent uncertainty arising from year-
to-year variability and model uncertainty.
Each predicted DO concentration and time since the last
mixing event (t) was then used to calculate a daily contribu-
tion to the cumulative hazard according to the approximation
(10) H(t+ 1/2) – H(t– 1/2) =
h(t,x(t))[ln(t+ 1/2) – ln(t– 1/2))]
Daily hazard contributions were then summed across each
set of 90 days to calculate a cumulative hazard for the sum-
mer season. Finally, this value was converted to a proportion
of the initial M. balthica population expected to survive
through the summer using eq. 5. The distribution of the 5000
values for this proportion represents year-to-year variability
and model uncertainty. Model predictions were generated for
three scenarios: (i) a baseline condition in which the param-
eters of eq. 9 were assigned the distributions estimated by
Borsuk et al. (2001a), (ii) a hypothetical management condi-
tion in which sediment oxygen demand (represented by the
rate constant kdin eq. 9) is reduced by 25%, and (iii) a man-
agement condition in which sediment oxygen demand is re-
duced by 50%. Monte Carlo simulation was performed using
Analytica, a commercially available software package (Lu-
mina 1997).
Results
Expert assessments
Assessed percentiles of the cumulative predictive distribu-
tion of times-to-death (Table 2) indicate a large amount of
population variability, particularly at higher DO concentra-
tions. Times-to-death at 1.5 mg·L–1 DO range from below 7
to over 63 days. Both the median and spread in these times
is substantially reduced at lower DO concentration, with 95%
of the individuals expected to die within 14 days at complete
bottom-water anoxia (0.0 mg·L–1). The subjects expressed
uncertainty ranges for many of the assessed time points,
which generally corresponded to about 15% in either direc-
tion of the middle value. They felt that these ranges represented
their own 90% confidence intervals.
Bayesian parameter estimation
Plots of the appropriate transformations of the data re-
vealed that the log-normal distribution provides the best fit
to the assessments based on the average R2values for the
four lines corresponding to the four DO concentrations (av-
erage R2values were as follows: log-normal, 0.970; log-
logistic, 0.964; exponential, 0.939; normal, 0.918; Weibull,
0.817). However, the log-logistic distribution was also found
to be suitable and was chosen based on the ability of its cu-
mulative distribution function to be expressed parametri-
cally. This characteristic greatly simplifies the process of
fitting a model to the assessed data. The four lines in the
log-logistic plot are approximately parallel (Fig. 1), indicat-
ing that, after a log-transformation, the population distribu-
tion of times-to-death has a variance that is constant across
all assessed oxygen concentrations. The fact that the lines
are equally spaced indicates that the relationship between
DO concentration and the log-transformed survival time is
linear. Based on these observations, eq. 6 can be written as
(11) ln t=a+bx +ε
where εhas a logistic distribution with mean zero and con-
stant variance, σ2=π2c2/3, where cis a scale parameter
(Evans et al. 2000). Alternatively, the distribution of log-
transformed times-to-death at a particular DO concentration
can be viewed as having a logistic distribution with mean
(a+bx) and variance π2c2/3. In this case, the cumulative
distribution function of times-to-death becomes
(12) F(t|x) = {1 + exp[–(t–(a+bx))/c]}–1
(Evans et al. 2000).
Because our assessment followed a fixed-probability ap-
proach, in which assessments were made of elapsed time for
a specified cumulative mortality, the expression to be fit to
the data is the inverse distribution function,
(13) t=a+bx +clog[α/(1 α)]
where αis the specified mortality.
Bayesian posterior parameter distributions (Fig. 2) indi-
cate that each of the three model parameters is well deter-
mined. Coefficients of variation are all approximately 5%.
The posterior distribution of the standard deviation of the
lack-of-fit term, s, corresponds well with the average 15%
uncertainty claimed by the scientists. Together with the un-
certainty in the parameter estimates, this term can be inter-
preted as the contribution of knowledge uncertainty to the
predictive distribution, whereas the logistic distribution indi-
cated by the term εin eq. 11 represents the natural variabil-
ity across individuals in the M. balthica population.
The model fit to assessed data (Fig. 3) shows some amount
of mismatch at exposure times greater than 20 days. This
may reflect lack-of-fit of the model or, perhaps, lack of ex-
perience by the scientists with these relatively rare long-term
depletion events. However, for the more common exposure
times of less than 20 days, the model shows a close fit to the
assessed data.
© 2002 NRC Canada
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Dissolved oxygen concentration (mg·L–1)
Percent dead 1.5 1.0 0.5 0.0
5 7 3–4 2 1–2
25 14 6–7 3–4 2–3
50 14–21 14 7 4–5
75 35–49 21–28 7–14 6–9
95 49–63 28 21 10–14
Table 2. Assessed number of days corresponding to each specified
cumulative mortality and ambient dissolved oxygen concentration.
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Probabilistic prediction
The proportion of the M. balthica population expected to
survive through the summer season is currently predicted to
be low (Fig. 4) with a mean value of 0.11 (Table 3). For
comparison, during the summer of 1997, the first year of ex-
tensive benthic surveying, the M. balthica clam population
in the Neuse was estimated to be reduced to less than 20%
of its spring population (Peterson et al. 2002). However, if
sediment oxygen demand should be reduced as a result of a
nutrient management and resultant diminished algal growth,
the survival model predicts that a greater proportion of the
M. balthica population will survive (Fig. 4). The mean sur-
vival rate approximately doubles to 0.23 with a 25% reduc-
tion in sediment oxygen demand and then doubles again to
0.46 at a sediment oxygen demand reduction of 50% (Ta-
ble 3). However, the uncertainty resulting from natural vari-
ability and model uncertainty also increases, widening the
predictive distribution and greatly increasing the 90% credi-
ble interval (Fig. 4 and Table 3).
Discussion
Survival models are more useful than conventional dose–
response methods for predicting the effects of a stressor on
field populations. Both exposure concentration and duration
can be incorporated into the analysis, as well as any other
covariates believed to be important in determining an organ-
ism’s response (e.g., size, age). Unfortunately, data applica-
ble to such analyses are not always available. The formal
elicitation of scientific judgment is one solution to this prob-
lem and is a distinguishing feature of the survival model pre-
sented here.
Although the use of assessed distributions for model
building may appear to be a subjective process, it must be
kept in mind that professional judgment is already implicit
in all scientific modeling. Whether it is involved in deciding
what processes to consider, what mathematical form appro-
priately characterizes those processes, what experimental re-
sults are relevant, or how to extrapolate experimental results
© 2002 NRC Canada
Borsuk et al. 1271
Fig. 1. Plot of assessment results transformed to evaluate the fit
of a log-logistic distribution (see Table 1). Symbols represent
assessed values corresponding to dissolved oxygen concentrations
of 1.5 mg·L–1 (circles), 1.0 mg·L–1 (squares), 0.5 mg·L–1
(triangles), and 0.0 mg·L–1 (diamonds). Vertical error bars
indicate average assessed uncertainty of 15% in either direction.
Broken lines are those estimated by linear least-squares
regression for each oxygen concentration.
Fig. 2. Bayesian posterior density functions for each parameter
in the equation, t=a+bx +clog[α/(1 – α)], where tis the
vector of assessed times-to-death, a,b, and care model parame-
ters, where cis the scale parameter of the log-logistic distribu-
tion, and αis the vector of specified cumulative mortalities. The
parameter srepresents the standard deviation of the model lack-
of-fit term.
Fig. 3. Cumulative distribution plot showing model fit to
assessed data for each of four dissolved oxygen (DO) concentra-
tions. Symbols represent assessed values corresponding to
dissolved oxygen concentrations of 1.5 mg·L–1 (circles),
1.0 mg·L–1 (squares), 0.5 mg·L–1 (triangles), and 0.0 mg·L–1
(diamonds). Horizontal error bars indicate the average assessed
uncertainty of 15% in either direction. Solid curves indicate the
mean values of the predictive distribution resulting from the
survival model.
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to the natural system, judgment is used in every step of the
modeling process. Therefore, when directly relevant data are
limited and yet policy needs require model construction, the
use of carefully elicited judgment for developing relation-
ships is an acceptable alternative. In fact, by including esti-
mates of uncertainty, a scientist is able to account for many
factors driving variability that may be difficult or impossible
to address empirically, as well as communicate the degree of
confidence they have in their assessments. Rather than disre-
garding uncertainty in modeling, a formally elicited assess-
ment makes the expression of uncertainty explicit. Results
expressed as probabilities give decision-makers and stake-
holders a realistic appraisal of the chances of achieving de-
sired outcomes and can provide the basis for a formal
decision analysis (Punt and Hilborn 1997) or risk assessment
(Hilborn et al. 1993).
Of course, we do not advocate relying exclusively on
judgment-based quantities when relevant “hard data” exist.
However, even under such circumstances, incorporation of
existing knowledge and experience into initial model formu-
lation “makes good sense” (Walters and Ludwig 1994). In
the Bayesian framework, model results based on literature
review and scientific judgment provide a prior distribution
for model parameters and predictions (Winkler 1980) that
can then be updated using site-specific data to obtain revised
posterior distributions (Winkler 1972). These can then serve
as the prior for subsequent updates. Such a stepwise process
provides a rational framework for quantifying advancements
in scientific knowledge and predictive ability. The related
exercise of Bayesian preposterior analysis (Winkler 1972)
involves calculating the potential posterior distributions re-
sulting from proposed data collection efforts that have not
yet been undertaken. Such an assessment provides a basis
for designing appropriate scientific studies based on their
ability to reduce uncertainty. This feature should make sub-
jective Bayesian analysis a method of interest even to inves-
tigators whose primary objective is to understand system
behavior rather than to directly support management decision-
making.
Probability elicitations that express questions in terms of
predictive, rather than parameter, distributions provide a more
meaningful and manageable assessment task for scientists.
Subjects can focus their attention on measurable quantities
that are analogous to the results of hypothetical experiments
rather than on unobservable parameters specific to a given
model. Such a practice should help to avoid the problems of
overconfidence or inaccuracy that have plagued subjective
assessments of parameter distributions in previous Bayesian
analyses (Walters and Ludwig 1994). The predictive elicita-
tion procedure also allows for more flexibility in model
building and avoids the need for new assessments if the
mathematical form of the model should change. Finally, it
makes the comparison between assessed values and ob-
served data explicit, facilitating interpretation of results.
The model described here represents the impact on the
M. balthica population resulting exclusively from cumula-
tive mortality from time-varying exposures to hypoxia. The
effects of year-to-year changes in recruitment, harvesting, or
predation are not included. Therefore, model predictions
should be interpreted as an indication of the expected rela-
tive change in summer survival of M. balthica owing to
changes in the severity of hypoxia, rather than as a predic-
© 2002 NRC Canada
1272 Can. J. Fish. Aquat. Sci. Vol. 59, 2002
Fig. 4. Histogram of predicted summer survival rate for Macoma
balthica for three scenarios: (a) a baseline condition with current
sediment oxygen demand; (b) a hypothetical management condi-
tion in which sediment oxygen demand is reduced by 25%; and
(c) a management condition in which sediment oxygen demand
is reduced by 50%. Reduction in SOD (%) Mean Median 90% credible
interval
0 0.11 0.07 0.01–0.35
25 0.23 0.19 0.03–0.60
50 0.47 0.45 0.12–0.86
Note: Results were generated for three scenarios corresponding to
varying reductions in sediment oxygen demand (SOD) relative to current
conditions.
Table 3. Summary statistics for predictive distributions on the
proportion of the M. balthica population surviving to the end of
the summer season.
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tion of absolute population density. The intrinsic assumption
is that other pressures on the M. balthica population are in-
dependent of hypoxia and will stay the same into the future.
Although we consider this to be a reasonable assumption for
the Neuse River estuary, it should be noted that there is evi-
dence in the literature to suggest that there may be interact-
ing effects between hypoxia and predation by fish and crabs.
The initiation of moderate hypoxia has been shown to pro-
vide clams with a refuge from predators because of their
greater tolerance to low oxygen conditions and deeper burial
depths (Taylor and Eggleston 2000). However, clams sur-
viving severe hypoxia require a recovery period on the order
of hours to days to resume normal activity, whereas the abil-
ity of blue crabs and fish to avoid hypoxic regions by emi-
gration eliminates their need for a recovery time (Nestlerode
and Diaz 1998). Predators may thus be able to enter affected
areas quickly and forage on susceptible clams and other ben-
thic invertebrates before they have time to recover (Taylor
and Eggleston 2000). Predatory fish and crabs also can use
refuge areas to escape low DO elsewhere and deplete local
benthic prey such as M. balthica (Lenihan et al. 2001). The
net effect of these processes is unclear and may affect the
accuracy of the survival model developed here. Future moni-
toring and experimental work will allow for the appropriate
inclusion of these interaction effects.
Macoma balthica is an important prey resource for demersal
fish and shellfish in the Neuse River estuary and, therefore,
is a key indicator of estuarine ecological health and eco-
nomic value. Ours is the first study to quantitatively predict
the response of the M. balthica population to changes in
benthic oxygen consumption rates. We have shown that the
potential for improved survival as a result of eutrophication
management is substantial, but so is the predictive uncer-
tainty arising from natural variation and knowledge uncer-
tainty. This is an important fact for the public and water
quality managers to realize so that they can maintain appro-
priate expectations. It is also a compelling argument for
adaptive ecosystem management, in which predictive preci-
sion is improved through the process of learning from the re-
sults of initial management actions (Walters and Holling
1990). Adaptive management is particularly useful when sys-
tem response is expected to be slow, thus allowing time for
management actions to be adjusted to account for new find-
ings. It has been estimated that, because of the substantial
sediment repository of organic carbon in the Neuse estuary,
it may take years of diminished algal production before
reductions in oxygen demand are realized (Alperin et al.
2000). Rather than lose patience with the recovery process,
scientists and water quality managers can view this situation
as an opportunity to gain important information for improved
understanding and decision-making.
Acknowledgements
We would like to thank Donald Stanley, Robert Christian,
and the members of the Neuse River Modeling and Moni-
toring (ModMon) project for providing the data used to de-
velop the oxygen dynamics model. We would also like to
thank David Higdon for helpful discussion on statistical is-
sues and Michael Newman for suggesting important refer-
ences. The comments of two anonymous reviewers on an
earlier version of the manuscript helped to focus our
description of the motivation and methods regarding the as-
sessment procedure. M. Borsuk was supported by an EPA
STAR graduate fellowship. Additional support was provided
by the Water Resources Research Institute of the University
of North Carolina, the State of North Carolina General As-
sembly, and the North Carolina Department of Environment
and Natural Resources.
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The contrasting impacts of externally supplied (runoff) and internally generated (nutrient-stimulated phytoplankton blooms) organic matter on oxygen (02) depletion were examined and evalu-ated in the eutrophic, salinity-stratified Neuse River Estuary, North Carolina, USA. This nitrogen (N)-limited estuary is experiencing increasing anthropogenic N loading from expanding urban, agricultural and industrial development in its watershed. Resultant algal blooms, which provided organic matter loads capable of causing extensive low O2 (hypoxic) and depleted O2 (anoxic) condi-tions, have induced widespread mortality of resident fin-and shellfish. Phytoplankton blooms followed periods of elevated N loading, except during extremely high runoff periods (e.g. hurricanes), when high rates of flushing and reduced water residence times did not allow sufficient time for bloom devel-opment. During these periods, hypoxia and anoxia were dominated by watershed-derived organic mat-ter loading. Externally vs internally generated organic matter loading scenarios were examined in sequential years (1994 to 1996) to compare the differential impacts of an average discharge year (l 0 yr mean hydrological conditions) (1994), N-stimulated summer algal blooms [1995), and a major hurricane (Fran; September 1996). The responses of primary production, hypoxia, and anoxia to these hydrolog-ically contrasting years and resultant organic matter loadings help distinguish watershed from internal forcing of 0, dynamics and fish kills.
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