Environ. Sci. Technol. 1998, 32, 2325-2330
A Size-Based Probabilistic
Assessment of PCB Exposure from
Lake Michigan Fish Consumption
C R A I G A . S T O W * ,† A N D S O N G S . Q I A N ‡
Duke University, Nicholas School of the Environment,
Box 90328, Durham, North Carolina 27708, and Portland
State University, Environmental Sciences and Resources,
Portland, Oregon 97207-0751
The state of Wisconsin has recently issued a fish
consumption advisory that includes suggested consumption
rates for Lake Michigan fish, based on fish size and PCB
concentration. To evaluate the size-based exposure risk from
Lake Michigan fish consumption, we estimated PCB
exposure probabilities for five Lake Michigan fish species
using two Bayesian models. The models confirm that
very few individuals of any of the five species are likely
to have PCB concentrations low enough to fall into
the category in which consumption is unrestricted. Among
smaller fish (<50 cm), brown trout have the highest PCB
levels, while lake trout are the most contaminated among
larger fish (>60 cm). Eating meals from multiple individuals
of some species results in a high probability that at
least one of the meals will exceed 1.9 mg/kg, the upper
PCB concentration recommended for consumption in the
advisory.
Introduction
Human exposure to polychlorinated biphenyls (PCBs) from
Great Lakes fish consumption has been a health concern
and an arena of contention for more than 25 years. One
early study reported birth weight anomalies among babies
born to women who consumed large amounts of Lake
Michigan fish, and continuing studies on the same cohort
have recounted enduring problems (1, 2). However, a
subsequent study of a later cohort did not corroborate prior
conclusions (3). Differences between the studies occurred
perhaps, in part, because in the interim, PCB concentrations
in Lake Michigan fish declined substantially (4, 5), resulting
from a production ban and use restrictions imposed in the
1970s. Though fish PCB concentrations have dropped since
the 1970s, declines after the early- to mid-1980s have been
minimal in Lake Michigan, and concentrations remain
relatively high, particularly in Lake Michigan (6) and Lake
Ontario (7) fishes.
One tool to address health concerns has been the issuance
of fish consumption advisories cautioning the public of
possible risks associated with eating contaminated fish.
However, varying standards arising from myriad jurisdictions
have caused a confusing array of warnings. In 1993, an
advisory task force drafted a protocol to standardize consumption advisories (8). In response, the state of Wisconsin
issued an advisory for Lake Michigan fishes based on
* To whom all correspondence should be addressed. E-mail:
[email protected]; phone: 919-613-8105; fax: 919-684-8741.
† Duke University.
‡ Portland State University.
S0013-936X(97)00840-7 CCC: $15.00
Published on Web 06/27/1998
 1998 American Chemical Society
standards established in the protocol (9). The advisory
contains the following five consumption categories based
on fish PCB concentration: (a) at concentrations below 0.05
mg/kg, fish can be eaten without restriction; (b) between
0.05 and 0.2 mg/kg consumption should be restricted to 52
meals/year; (c) from 0.2 to 1.0 mg/kg consumption is limited
to 12 meals/year; (d) concentrations between 1 and 1.9 mg/
kg result in a six meal/year restriction; and (e) people are
advised not to eat any fish with PCB concentrations greater
than 1.9 mg/kg. Because anglers cannot readily know the
PCB concentration of their catch, the advisory translates these
concentration-based consumption categories into fish size
ranges for the important recreational species.
However, PCB concentrations among individual fish are
highly variable, even among similar sized fish of the same
species (10). In this analysis, we use two Bayesian models
to provide a probabilistic assessment of PCB exposure from
consumption of five Lake Michigan salmonids, based on fish
size. A probabilistic approach is essential to capture the
variability among individuals. The Bayesian approach
provides a straightforward probabilistic interpretation of
consumption risk that could lead readily into a broader
decision framework (11).
Methods
Fish PCB Data. The data we used were from five salmonid
species: brown trout (Salmo trutta), chinook salmon (Oncorhynchus tshawytscha), coho salmon (Oncorhynchus
kisutch), lake trout (Salvelinus namaycush), and rainbow trout
(Oncorhynchus mykiss) collected by the Wisconsin and
Michigan Departments of Natural Resources from 1984 to
1994. We chose data from this period because (a) during
this time PCB concentrations have been relatively stable (12)
and (b) a probabilistic assessment requires many more
samples than are routinely collected in shorter intervals. All
data are skin-on filets from individual fish, approximating
the portion of fish that people eat. Composite samples,
though available for some species, were excluded from the
analysis because they understate PCB variability and introduce problems with predictor variable measurement error
(13). Chemical analysis details have been previously presented (6).
Statistical Methods. We used two Bayesian models to
evaluate PCB exposure risk: a parametric and a nonparametric model. Bayesian inference is based on Bayes’
theorem, which can be expressed as
π(θ|y) )
π(θ) f(y|θ)
∫ π(θ) f(y|θ) dθ
θ
where π(θ|y) is the probability of the parameter vector (θ)
after observing the data (y) (the posterior probability of θ);
π(θ) is the probability of θ before observing y (the prior
probability of θ); and f(y|θ) is the likelihood function. The
probability of future observations (y*) is evaluated from the
predictive distribution, which can be expressed as
π(y*|y) )
∫ f(y*|θ)π(θ|y) dθ
θ
where π(y*|y) is the probability of future y observations given
past observations of y.
Detailed derivations of the parametric model are available
in standard Bayesian texts (11, 14, 15). Briefly, the parametric
model is analogous to the familiar classical linear regression
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model with functional form:
Y ) β0 + β1X + ∼ N(0, σ2)
In this instance, Y ) log of PCB concentration (mg/kg); X )
fish length (cm); the β values are estimated from data; and
, the model disturbance term, is assumed normal with mean
zero and variance σ2. Using a noninformative prior distribution:
π(β0, β1, σ) ∝
1
σ
both β0 and β1 have marginal posterior Student t distributions
with n - 2 degrees of freedom (n ) number of observations).
The predictive distribution for future Y, given observations
of Y and X, is also a Student t distribution with n - 2 degrees
of freedom.
Nonparametric, in this context, should not be confused
with familiar nonparametric order statistics. Nonparametric
means that the model is not a concise algebraic formula
summarized by one or more parameters. The nonparametric
model
Y ) f(X) + can be thought of as a mathematical procedure comparable
to a smoothing method (16) where a series of function values,
fi(Xi), are estimated from data in the vicinity of each Xi.
Familiar examples that employ smoothing methods include
moving averages, local error sums of squares (loess) (17),
and generalized additive models (GAMS) (18). The goal of
nonparametric Bayesian regression is to estimate the joint
posterior distribution of the fi. In this application the fi are
rescaled and modeled as a Dirichlet distribution, the multivariate version of a β-distribution. The Dirichlet distribution
is also used for the joint prior distribution of the fi. The
posterior distribution of the fi cannot be derived analytically,
so it is estimated numerically using Markov chain Monte
Carlo simulation. The probability distribution of is not
assumed to be normal; instead, it is estimated from the data
as a mixture of uniform distributions using the same Markov
chain Monte Carlo algorithm used for the joint posterior of
the fi. A detailed presentation of the nonparametric model
is available in Lavine and Mockus (19), and a general
discussion of nonparametric models is available in Green
and Silverman (20).
The output of interest from both models is a predicted
probability that an individual fish of a particular size and
species will exceed a specified PCB concentration. We
estimated both models using log-transformed PCB concentrations.
Results
Lake and brown trout and coho and chinook salmon exhibit
pronounced PCB concentration increases with size, while
the pattern among rainbow trout is somewhat obscured by
a small cloud of relatively highly contaminated small
individuals (Figure 1). The nonparametric model better
captures nonlinearity among the data, particularly among
lake trout (Figure 1a); however, the 90% credible region of
the model fit is considerably smaller for the parametric model
(Figure 1b). We investigated alternative transformations to
linearize the PCB:size data; however, in all cases the final
probabilities differed little from the result obtained with a
log transformation.
The two models provide qualitatively similar assessments
of consumption risk, though they differ quantitatively (Figure
2). For all species except rainbow trout, the parametric model
indicates a lower probability of exceeding PCB concentrations
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of 1.0 or 1.9 mg/kg in small fish than the nonparametric
model, but a higher probability of exceeding 1.0 or 1.9 mg/kg
in large fish. For rainbow trout, the parametric model
indicates a lower probability of exceeding higher PCB
concentration in all fish sizes. For all species, the parametric
model indicates a probability near 1 of exceeding a 0.05 mg/
kg PCB concentration. The nonparametric model indicates
a slightly lower probability of exceeding 0.05 mg/kg in the
smallest individuals of each species.
Both models indicate that, among smaller fish (<50 cm),
brown trout have the highest probability of exceeding 1.0 or
1.9 mg/kg, while lake trout are the most likely to exceed
these concentrations among larger (>60 cm) fish (Figure 2).
Rainbow trout and coho salmon are the least likely to exceed
1.0 or 1.9 mg/kg though the probability is only slightly higher
in similar sized chinook. Chinook are considerably larger
than brown or lake trout before the probability of exceeding
1.0 or 1.9 mg/kg reaches 50%.
None of the species considered have been placed in the
unrestricted consumption category (Figure 2). The lowest
probability of exceeding 0.05 mg/kg is approximately 62%
for small rainbow trout based on the nonparametric model.
All other species have an 80% or greater probability of
exceeding 0.05 mg/kg based on either model.
Rainbow trout fall into the 52 meals/year category up to
43.2 cm, when the probability of exceeding 0.2 mg/kg is 49%
and 65%, based on the nonparametric (NP) and parametric
(P) models, respectively. Rainbow trout larger than 43.2 cm
are restricted to 12 meals/year with probabilities below 50%
of exceeding 1.0 mg/kg in the largest fish, based on either
model.
The least restrictive category placed on lake trout was 12
meals/year, with even the smallest lake trout exceeding a
60% probability of PCBs above 0.2 mg/kg, based on either
model. Lake trout were placed into the 6 meals/year category
at probabilities of exceeding 1.0 mg/kg of 57% and 76% based
on the nonparametric (NP) and parametric (P) models,
respectively. Lake trout enter the do not eat category at
probabilities of exceeding 1.9 mg/kg of 54% (NP) and 69%
(P). The only other fish with comparable probabilities of
exceeding 1.9 mg/kg are extremely large chinook salmon or
brown trout; sizes that are relatively uncommon.
All coho were placed in the 12 meals/year category, with
the probability of exceeding 1.0 mg/kg never exceeding 60%.
Small chinook are in the 12 meal/year category with
probabilities of exceeding 0.2 mg/kg of at least 50% in the
smallest individuals. At 76.2 cm, chinook enter the 6 meals/
year group with probabilities of exceeding 1.0 mg/kg of 53%
(NP) and 66% (P).
Similarly, small brown trout are in the 12 meals/year class
with probabilities of exceeding 0.2 mg/kg always greater than
60%. Brown trout greater than 55.9 cm are in the 6 meals/
year category, with probabilities of exceeding 1.0 mg/kg of
55% (NP) and 75% (P).
Discussion
All model inference is implicitly conditioned on model
selection. Assessment using two models, in concert, helps
deter faulty decisions resulting from poor model choice. Each
model we used has inherent advantages and disadvantages.
The parametric model is based on the assumption of
consistent functional and error structures throughout the
data, with all of the data contributing information to the
parameters that summarize these structures. Scant data in
some regions pose no serious problems if the range of
available data is sufficient to describe the functional relationship. If the model is “true”, then extrapolating beyond
the data is no problem in principle although, in practice,
extrapolation should be a cautious exercise.
FIGURE 1. PCB concentration vs size for rainbow trout (n ) 220), lake trout (n ) 396), coho salmon (n ) 269), chinook salmon (n ) 452),
and brown trout (n ) 220). Nonparametric (column a) and parametric (column b) models with 90% credible intervals about the mean are
depicted for each species.
The nonparametric model is more flexible than the
parametric model because the structural assumptions are
less rigid and more influenced by local data rather than data
throughout the range of observations. The nonparametric
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FIGURE 2. Probabilities of exceeding 0.05 mg/kg (solid), 0.2 mg/kg (dotted), 1.0 mg/kg (fine dash), and 1.9 mg/kg (coarse dash) for each
of five species estimated using the nonparametric and parametric models. Vertical lines indicate advisory category size boundaries
established by the state of Wisconsin. The line type of each vertical line corresponds to the line type of the concentration used to establish
that advisory category. All coho are in the 12 meals/year advisory category.
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FIGURE 3. Species comparison of the probability of exceeding a 1.9 mg/kg PCB concentration, based on the parametric model. The solid
horizontal line at a 0.11 probability indicates the upper size boundary that would be required to keep the probability of consuming at least
one meal exceeding 1.9 mg/kg in the six meal/year category below 50% (assuming the six meals are from six individual fish).
model is constrained to be monotonic, a less restrictive
assumption than the linear structure of the parametric model.
The nonparametric error distribution is not constrained to
be normal or even symmetric. If the data do not encompass
a structure that can be parsimoniously described in a simple
equation, the nonparametric model can still be used to extract
useful probability inference. However, paucity of data in a
particular region is disadvantageous, and extrapolation is
not possible. If the data are really linear (or some other
known structure) and the error term is close to normal, then
precision is lost by using the nonparametric model because
less structural information is incorporated.
In the context of environmental risks, fish PCB concentration is a relatively easy-to-measure, discrete, tangible
entity, suggesting that assessing exposure probability should
be unambiguous. However, the probability assessed is highly
dependent on model choice, a subtlely subjective decision
(21). Probability assessments from the parametric model
are approximately 15-20% higher than those from the
nonparametric model at the size thresholds established by
the Wisconsin advisory (Figure 2). Lower size thresholds
would usually be selected, for a fixed exceedence probability,
using the parametric model.
The parametric model may appear more conservative in
that it generally predicts higher PCB concentrations in large
fish, thus suggesting smaller advisory size boundaries than
the nonparametric model. This difference between models
among large fish results principally because the parametric
model is more certain than the nonparametric (Figure 1),
therefore the chance of being further from the mean is greater.
However, if large fish are scarce and small fish are abundant,
the nonparametric model could lead to more cautious
recommendations, depending on the decision criterion used.
An accurate assessment of the number of fish of each species
caught, by size, could expand this analysis into a full
decision-analytic framework (22) incorporating PCB exposure into decisions regarding annual fish stocking.
Evaluation of the effectiveness of the consumption
advisory is also limited by the lack of information regarding
the number and sizes of fish of each species caught. While
the advisory recommends no more than 6 meals/year of lake
trout in the 58.4-68.6 cm. range, if lake trout in that range
are only infrequently caught, then the advisory provides no
extra protection. Alternatively, if this is a common size range
and anglers heed the advisory, then a considerable exposure
reduction may result.
It is worth noting, however, that people who eat 6 meals/
year of some species in the 6 meals/year size category are
very likely to eat at least one meal exceeding the 1.9 mg/kg
limit, which the advisory is intended to prevent. For example,
the minimum probability that a lake trout in the 6 meals/
year category will exceed 1.9 mg/kg occurs for a 58.4 cm fish,
the lower boundary of that category. The nonparametric
model indicates a 38% probability that a 58.4 cm lake trout
will exceed 1.9 mg/kg, and the parametric model indicates
a 34% probability (Figure 2). Using 34%, the lower number,
there is a maximum 66% probability that a single lake trout
will not exceed 1.9 mg/kg. However, assuming independence
(meals from different fish), if one eats two meals the
maximum probability that both will be below 1.9 mg/kg drops
to 44% and continues to drop to 29%, 19%, 13%, and 8% for
three, four, five, and six meals, respectively. In other words,
if one eats six meals from six different lake trout in the 6
meals/year category, there is at least a 92% chance that at
least one of the six will exceed 1.9 mg/kg. This analysis should
not be construed as a toxicological evaluation of the danger
of PCB consumption, only an assessment of the probability
of exceeding the established advisory level.
These models are useful to examine the implications of
setting advisory categories at specific exposure probability
levels. For example, suppose we want to lower the probability
of eating a lake trout meal exceeding 1.9 mg/kg in the 6
meals/year category from 92% to 50%. In other words, the
joint probability that none of the six fish will exceed 1.9 mg/
kg will be set to 50%. For a joint 50% probability, the
individual probabilities of not exceeding 1.9 mg/kg must be
89% (the sixth root of 0.50), indicating that each fish must
have less than an 11% probability (1-0.89) of exceeding 1.9
mg/kg. Based on the parametric model, this restriction would
result in an upper size boundary in the 6 meals/year category
of 49 cm for lake trout (Figure 3). Comparable restrictions
for the other four species would impose upper size boundaries
of 40, 68, 68, and 71 cm for brown trout, chinook, coho, and
rainbow trout, respectively (Figure 3).
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For coho and rainbow trout, such a restriction would not
be overly burdensome, few individuals exceed 1.9 mg/kg.
However, for brown trout, chinook, and lake trout, such a
restriction would severely limit the sizes available for
consumption. One possibility to reduce the exposure
probabilities estimated herein is to spatially subdivide the
data, removing fish from documented “hot spots” (12), and
issuing special advisories for these areas. The tradeoff with
such an approach is that overly complex advisories may be
ignored by the public.
We conducted this analysis by aggregating small annual
data sets from ongoing, long-term monitoring studies and
periodic surveys. Long-term PCB concentration monitoring
is essential for assessing trends. The relative stability of
concentrations since the mid-1980s was postulated early (23)
but has been viewed with some skepticism (24). However,
ascertaining long-term trends and monitoring for exposure
assessment invite different sampling strategies. Variability
obscures trends, increasing the sample size required to lower
uncertainty to acceptable levels. Thus, when trend detection
is the goal, it is advantageous to sample to reduce variability.
Strategies for reducing variability among fish samples include
sampling specific species, sampling at fixed locations, or
sampling specific fish sizes, although this last tactic can be
misleading under some circumstances (25). In contrast,
exposure assessment requires variability documentation.
Monitoring designed to determine trends may bias perceived
exposure. The distinction between these two somewhat
opposing objectives is important. At least four state and two
federal agencies have an interest in both objectives. A
coordinated monitoring program among jurisdictions would
increase the probability that limited resources could be used
effectively to address both in future assessments.
Acknowledgments
Data were provided by the Wisconsin and Michigan Departments of Natural Resources. Mark Borsuk, Conrad Lamon,
Steve Carpenter, and Tara Stow provided helpful comments.
This work was funded by the University of Wisconsin Sea
Grant Institute under grants from the National Sea Grant
College Program, National Oceanic and Atmospheric Administration, U.S. Department of Commerce, and the State
of Wisconsin and by Federal Grant NA90AA-D-SG469, Project
R/MW-41.
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Received for review September 19, 1997. Revised manuscript
received May 5, 1998. Accepted May 13, 1998.
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