1 Journal of Great Lakes Research 37 (2011) Contents lists available at ScienceDirect Journal of Great Lakes Research journal homepage: A Bayesian a...

Author:
Kimberly Stanley

0 downloads 90 Views 1MB Size

No documents

Contents lists available at ScienceDirect

Journal of Great Lakes Research j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j g l r

A Bayesian assessment of the PCB temporal trends in Lake Erie ﬁsh communities Somayeh Sadraddini a, M. Ekram Azim a, Yuko Shimoda a, Satyendra P. Bhavsar b, c, Ken G. Drouillard d, Sean M. Backus e, George B. Arhonditsis a, c,⁎ a

Ecological Modeling Laboratory, Department of Physical and Environmental Sciences, University of Toronto, Toronto, Ontario, Canada M1C 1A4 Ontario Ministry of Environment, Environmental Monitoring and Reporting Branch, Toronto, Ontario, Canada M9P 3V6 Centre for Environment, University of Toronto, Toronto, Ontario, Canada M5S 3E8 d Great Lakes Institute for Environmental Research, University of Windsor, Windsor, Ontario, Canada N9B 3P4 e Water Quality Monitoring & Surveillance Division, Water Science and Technology Directorate, Environment Canada, Burlington, Ontario, Canada L7R 4A6 b c

a r t i c l e

i n f o

Article history: Received 2 October 2010 Accepted 25 May 2011 Available online 19 July 2011 Communicated by Paul Helm Keywords: Polychlorinated biphenyls Bayesian inference Bioaccumulation Dynamic linear modeling Lake Erie Fish contamination

a b s t r a c t The temporal trends of polychlorinated biphenyls (PCBs) in Lake Erie ﬁsh were evaluated using 30 years of ﬁsh contaminant data (1977–2007). The ﬁrst step of our statistical analysis was based on simple exponential decay models parameterized with Bayesian inference techniques to assess the declining rates in four intensively sampled ﬁsh species, i.e., walleye (Stizostedion vitreum), coho salmon (Oncorhynchus kisutch), rainbow trout (Oncorhynchus mykiss) and white bass (Morone chrysops). Because the exponential model postulates monotonic decrease of the PCB levels, we included ﬁrst- or second-order random error terms in our statistical formulations to accommodate non-monotonic patterns in the dataset studied. Generally, our results suggest that the PCBs have been decreasing over the last 30 years with relatively weak rates that vary among the different ﬁsh species examined. Yet, our analysis with the exponential decay model also identiﬁed an increasing trend in the PCB concentrations of walleye skinless–boneless ﬁlet data, which is manifested after the mid-90s. In the second step, we used dynamic linear modeling (DLM) analysis to account for the fact that the ﬁsh length covaries with the PCB concentrations and that different sized ﬁsh may have been sampled over time. Our DLM analysis suggests that the previously reported trend of the walleye ﬁlet data is actually an artifact associated with the bias of the ﬁsh sampling practices followed. The coho salmon and rainbow trout PCB concentrations have been decreasing steadily during the study period but the associated rates were relatively weak. Finally, the PCB trends in white bass appear to have been stabilized over that last decade, although the robustness of this result remains to be conﬁrmed due to the temporal inconsistencies of the information used. We conclude by emphasizing the importance of explicitly accounting for the different covariates (e.g., length, age, lipid content) that can potentially hamper the detection of the actual temporal trends of ﬁsh contaminants. © 2011 International Association for Great Lakes Research. Published by Elsevier B.V. All rights reserved.

Introduction Among the bioaccumulative, toxic and persistent organic pollutants (POPs), polychlorinated biphenyls (PCBs) are of particular concern and historically have restricted the use of valuable commercial and recreational ﬁshery resources in the Great Lakes. PCBs were ﬁrst traced in the Great Lakes in the 1930s and their concentrations peaked in the late 1960s-early 1970s (Tanabe, 1988). As a result, PCB contamination was identiﬁed as a major threat to the integrity of the aquatic biota in

⁎ Corresponding author at: Ecological Modeling Laboratory, Department of Physical and Environmental Sciences, University of Toronto, Toronto, Ontario, Canada M1C 1A4. Tel.:+1 416 208 4858. E-mail addresses: [email protected] (S. Sadraddini), [email protected] (M.E. Azim), [email protected] (Y. Shimoda), [email protected] (S.P. Bhavsar), [email protected] (K.G. Drouillard), [email protected] (S.M. Backus), [email protected] (G.B. Arhonditsis).

the Great Lakes area. There has also been a growing concern that individuals who eat considerable amount of ﬁsh from the Great Lakes have greater exposure to toxic chemicals (Cole et al., 2004; Johnson et al., 1999). Consumption of Great Lakes sport ﬁsh has been one of the signiﬁcant determinants of PCB burden in the human body (Humphrey, 1988; Tee et al., 2003). The reported human health risks from PCBs involve susceptibility to cardiovascular problems including coronary heart disease (Tomasallo et al., 2010), decreased verbal learning and increased depression (Fitzgerald et al., 2008), neurobehavioral alterations, motor immaturity, hyporeﬂexia, and lower psychomotor scores (Faroon et al., 2000), alterations/disruptions of thyroid stimulating hormone, triiodothyronine, thyroxine and sex steroid hormone functions (Turyk et al., 2006) and diabetes (Turyk et al., 2009a, 2009b). Thus, while the health beneﬁts of ﬁsh consumption are widely advocated, the potential human health impacts from the worrisome PCB levels underscore the importance of meticulously updating the ﬁsh consumption advisories typically issued by the Ontario Ministry of the

0380-1330/$ – see front matter © 2011 International Association for Great Lakes Research. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jglr.2011.06.005

508

S. Sadraddini et al. / Journal of Great Lakes Research 37 (2011) 507–520

Environment and different US state governments in the Great Lakes area. Responding to increased public pressure and advocacy for virtual elimination of persistent toxic pollutants from the Great Lakes, various regulatory actions were undertaken at different government levels. PCB production was effectively banned in the 1970s in North America. The Great Lakes Water Quality Agreement (GLWQA) between Canada and USA was signed in 1972 and was subsequently revised in 1978. The agreement included a call for monitoring and research programs to identify the spatiotemporal contaminant trends in the sediments and biota (IJC, 1978, 2006). Implementation of these regulatory actions resulted in decreased levels of most contaminants in the Great Lakes ﬁsh through the 1980s, but the rate of decrease in ﬁsh are reported to have diminished since the early 1990s; especially in Lake Erie (Bhavsar et al., 2007; Carlson et al., 2010). The reasons for these trends are not fully known, but existing mechanistic explanations include the food web alterations induced from invasive species in the Great Lakes (Hogan et al., 2007; Morrison et al., 1998) and also shifts in the trophodynamics associated with global warming (French et al., 2006). The reported contaminant trends could have also been inﬂuenced by many factors such as the type of statistical analysis performed, data pooling across locations, type of samples (whole ﬁsh vs. ﬁlet portions), seasonality, and lack of explicit consideration of important covariates such as the ﬁsh size, age, feeding habits, behavioral patterns, reproductive status, growth and lipid content. Inappropriate data analysis and interpretation of statistical trends derived from incomplete information can conceivably provide misleading results. Therefore, holistic modeling frameworks that explicitly incorporate all the causal factors are essential to delineate the actual contaminant trends and to elucidate the underlying mechanisms. To this end, different modeling strategies have been used to assess the trajectories of the historical ﬁsh contaminant data. Stow et al. (2004) analyzed lake trout PCBs from Lake Michigan with dynamic linear modeling and Bayesian model averaging (BMA) of four individual models: exponential decay model, exponential decay model with a nonzero asymptote, double exponential model and mixed order model. Hickey

St. Clair River

et al. (2006) evaluated the temporal trends of chlorinated organic contaminants in Great Lakes trout and walleye using the single exponential decay model and the double exponential model that distinguishes between fast and slow decay processes, while both approaches explicitly accounted for the presence of a stable contaminant source resulting in an irreducible or baseline concentration. Similarly, a recent analysis by Gewurtz et al. (2010) used exponential decay models to examine the spatiotemporal trends of organochlorines and mercury in ﬁsh species from the St. Clair River/Lake St. Clair corridor. Finally, Bhavsar et al. (2010) used General Linear Model Univariate analysis with Tukey's posthoc multiple comparison along with the non-parametric Mann–Kendal test to detect Hg trends in time and space of lake trout and walleye in Lakes Superior, Georgian Bay, Huron, Erie and Ontario during the 1980–1990 and 2000–2007 periods. In this study, we have undertaken a systematic and rigorous trend analysis based on a 30-yr dataset of PCB concentrations in ﬁsh from Lake Erie. Our aim is to illustrate a Bayesian methodological framework that can objectively detect the temporal trends of ﬁsh contaminant concentrations. In the ﬁrst step, we use single exponential decay models to assess the declining rates in four intensively sampled ﬁsh species, i.e., walleye (Stizostedion vitreum), coho salmon (Oncorhynchus kisutch), rainbow trout (Oncorhynchus mykiss) and white bass (Morone chrysops). Because the exponential decay model postulates monotonic decrease of the PCB levels, we include ﬁrst- or second-order random walk error terms to accommodate the likelihood of non-monotonic patterns in the time series data. In the second step, we use dynamic linear modeling to verify the temporal trends derived from the ﬁrst phase of our statistical framework by explicitly accounting for the covariance between PCB concentrations and ﬁsh length (Stow et al., 2004). Our study also revisits the capacity of several of the hypotheses proposed in the literature to explain the temporal PCB trends in Lake Erie ﬁsh communities. Finally, we conclude by underscoring that all modeling frameworks aiming to impartially identify the actual temporal trends of ﬁsh contaminants should explicitly examine the role of the different covariates (e.g., length, gender, lipid content, weight, season, location) and thus

Long Point Bay

Lake St. Clair Rondeau Harbour Detroit River

Point Pelee SCALE 1: 1,000,000 10

0

10

20

30

40

50

60

70

80

90 100

KILOMETERS

Fig. 1. Map of Lake Erie with the four sampling sites: 1: Western Basin, 2: Central Basin, 3: Long Point Bay, and 4: Eastern Basin. The area around Pelee Island (black circle) represents the federal (Environment Canada) sampling location.

S. Sadraddini et al. / Journal of Great Lakes Research 37 (2011) 507–520 Table 1 Basic statistics of PCB concentrations (ng/g wet weight) in walleye whole ﬁsh data and in skinless–boneless ﬁlet data from walleye, coho salmon, rainbow trout and white bass in Lake Erie. Species

N

Mean

SD

Walleye (WF) Stizostedion vitreum Walleye (SBF) Stizostedion vitreum Coho salmon Oncorhynchus kisutch Rainbow trout Oncorhynchus mykiss White bass Morone chrysops

969

1329

899

Median

Inter Skewness quart.

Kurtosis

987 1100

900

2.42

9.66

114

119 80

105

2.79

1.93

694

463

251 410

311

1.18

2.87

302

399

291 326

350

1.58

5.14

1165

309

245 240

264

1.84

4.37

control the possible bias introduced by the typical sample collection practices and/or changes in other ecological parameters.

509

ﬁsh (WF) and skinless–boneless ﬁlet (SBF) samples for walleye and only SBF samples for the remaining ﬁsh species. The whole-ﬁsh samples were collected from Pelee Island in the western part of Lake Erie, while the ﬁlet samples were pooled from four sites on the Canadian side, i.e., western basin, central basin, Long Point Bay, and eastern basin (Fig. 1). The number of observations for each species is given in Table 1. The analytical procedure is described for the OMOE samples by Bhavsar et al. (2007) and for the EC samples by Borgmann and Whittle (1983). Our modeling framework consists of two steps that aim to detect the presence of statistically signiﬁcant non-monotonic trends associated with the ﬁsh PCB concentrations (step I), and to examine if these temporal trends are actually detected when we explicitly account for the covariance between PCB levels and ﬁsh length (step II). Bayesian inference was used as a means for estimating model parameters due to its ability to include prior information in the modeling analysis and to explicitly handle the model structure and parameter uncertainty (Gelman et al., 2004). Bayesian inference treats each parameter θ as random variable, and uses the likelihood function to express the relative plausibility of obtaining different values of this parameter when particular data have been observed:

Methods The present study is based on the provincial (Ontario Ministry of the Environment; OMOE, Canada) dorsal ﬁlet measurements used for ﬁsh consumption advisories, and the federal (Environment Canada; EC) whole ﬁsh measurements used to assess overall environmental contamination and risk to ﬁsh and ﬁsh-consuming wildlife (Bhavsar et al., 2010). In our analysis, the selection of ﬁsh species was driven by the data availability and their ecological/commercial importance. We examined four intensively sampled species, i.e., walleye (S. vitreum), coho salmon (O. kisutch), rainbow trout (O. mykiss) and white bass (M. chrysops). The examination of the PCB trends was based on whole

πðθ j dataÞ =

πðθÞLðdataj θÞ

ð1Þ

∫ πðθÞLðdataj θÞdθ θ

where π(θ) represents our prior statements regarding the probability distribution that more objectively depicts the existing knowledge on the θ values, L(data|θ) corresponds to the likelihood of observing the data given the different θ values, and π(θ|data) is the posterior probability that expresses our updated beliefs on the θ values after the existing data from the system are considered. The denominator in Eq. (1) is the expected value of the likelihood function, and acts as a

Table 2 Deviance information criterion, posterior mean values, 2.5% and 97.5% percentiles (italicized numbers) of the stochastic nodes of the models used to describe the temporal trends of PCB concentrations in walleye (both WF and SBF portions), coho salmon (SBF), white bass (SBF) and rainbow trout (SBF). Models

Exponential model with 1st order smoothing

Parametersa

DIC PCB0

k

ω

σε

Exponential model with 2nd order smoothing

DIC PCB0

k

ω

σε

Walleye

Walleye

Coho Salmon

White Bass

Rainbow Trout

WF

SBF

SBF

SBF

SBF

1788 982 1054 1131 − 0.003 − 0.001 N−0.001b 0.307 0.424 0.581 0.572 0.599 0.627 1789 994 1062 1137 − 0.002 − 0.001 N−0.001b 0.430 0.617 0.860 0.572 0.599 0.627

2123 87 164 326 − 0.089 − 0.040 − 0.003 0.426 0.592 0.823 0.740 0.776 0.814 2121 87 161 308 −0.086 − 0.039 − 0.002 0.579 0.832 1.181 0.741 0.777 0.814

1132 284 386 570 − 0.040 −0.013 N−0.001b 0.243 0.369 0.559 0.510 0.538 0.568 1128 285 388 558 − 0.034 − 0.010 N−0.001b 0.321 0.495 0.756 0.509 0.537 0.567

2374 237 277 362 − 0.027 − 0.009 N−0.001b 0.331 0.444 0.599 0.635 0.661 0.689 2374 238 274 359 − 0.026 − 0.009 N−0.001b 0.447 0.617 0.844 0.635 0.662 0.689

623 252 325 428 −0.023 − 0.007 N−0.001b 0.300 0.481 0.734 0.602 0.653 0.709 629 241 301 388 − 0.023 − 0.007 N−0.001b 0.154 0.325 0.668 0.612 0.667 0.727

a DIC: Deviance Information Criterion; PCB0: the PCB concentration at t = 0; k: the decay coefﬁcient; ω: the conditional standard deviation of the random error terms δt, representing the annual discrepancies from the trajectory delineated by the common exponential decay model; and σε: the measurement standard error. b Denotes a negative value very close to zero.

Year

400

40

e 2

400

80

g 2

80 40

8

i

2

80 40

8 4

Year

2008

800

2004

2

2000

c

1996

8

1992

4000

Random Walk

Random Walk

40

1988

800

Random Walk

80

1984

800 400

Random Walk

2

1980

800 400

Random Walk

PCB Concentration (ng/g ww)

a

1976

2008

2004

2000

1996

1992

8000 4000

1988

8000

1984

PCB Concentration (ng/g ww) 800

1980

PCB Concentration (ng/g ww) 4000

1976

PCB Concentration (ng/g ww) 4000

PCB Concentration (ng/g ww)

510 S. Sadraddini et al. / Journal of Great Lakes Research 37 (2011) 507–520

b

400 1

0

-1

-2

d

1

0

80 -1

-2

f

1

0

40 -1

-2

h

1

0

-1

-2

j

1

0

-1

-2

S. Sadraddini et al. / Journal of Great Lakes Research 37 (2011) 507–520

scaling constant that normalizes the integral of the area under the posterior probability distribution. Step I — exponential decay models with random walk terms The ﬁrst step of the analysis was based on the exponential decay model (Stow et al., 2004): kt

PCBt = PCB0 e + δt + ε

ð2Þ

where PCBt is the PCB concentration in year t; PCB0 is the PCB concentration at t = 0; k is the decay coefﬁcient. A fundamental weakness of the simple exponential decay model is the postulation of a monotonic decrease of the PCB levels, and therefore its inability to capture systematic deviations from this trend. To accommodate possible non-monotonic patterns in the time series data, we included (zero mean) random error terms δt representing the annual deviations from the trajectory delineated by the common exponential decay model. To reﬂect the prior belief that these annual discrepancies are correlated, we assumed a ﬁrst-order random walk prior speciﬁed as (Arhonditsis et al., 2008a,b; Shaddick and Wakeﬁeld, 2002): 8 2 > N δt + 1 ; ω for t = 1 > > > > > ! < > δ + δt + 1 ω2 2 ; for t = 2; …; T−1 N t−1 p δt jδ−t ; ω e> 2 2 > > > > > > : N δ ; ω2 for t = T

ð3Þ

t−1

where δ−t denotes all elements of δt except from the error associated with a particular year t, ω 2 is the conditional variance and the prior density p(ω 2) was based on a conjugate inverse-gamma (0.001, 0.001) distribution. This statistical approach implies that the ﬁrst-order differences of the annual PCB levels are smooth, and that the probability of sudden jumps between consecutive years is unlikely. Alternatively, we examined a second-order random walk prior for δt representing prior beliefs that the rate of change (gradient) of the PCB concentrations over the study period was smooth: 8 2 > > > N 2δt + 1 −δt + 2 ; ω for t = 1 > > ! > > 2 > > 2δt−1 + 4δt + 1 −δt + 2 ω > > N ; for t = 2 > > 5 5 > > > > > ! < > −δt−2 + 4δt−1 + 4δt + 1 −δt + 2 ω2 2 p δt jδ−t ; ω e N ; for t = 3; …; T−2 ð4Þ > 6 6 > > > > ! > > > −δt−2 + 4δt−1 + 2δt + 1 ω2 > > > ; for t = T−1 N > > 5 5 > > > > > > : N −δ + 2δ ; ω2 for t = T; t−2

t−1

Because of the added complexity, recent work by Azim et al. (2011) showed that the present statistical formulation is prone to poor parameter identiﬁcation when coupled with more complex models (e.g., mixed-order model). In particular, the Azim et al. (2011) study noted that the predicted decay coefﬁcients from the mixedorder model were accompanied by substantial standard deviations

511

(coefﬁcients of variation ≈ 70–114%), which counterbalances the support provided by its higher performance relative to simpler models. Thus, our strategy was to select the simplest possible model structure (single exponential decay model) combined with the random walk term to quantify the decreasing rates as well as to detect deviations from this trend during the study period. Finally, the ε term represents the measurement error and is assumed to follow a Gaussian distribution, N(0, σε2). Contrary to the time variant random error terms δt, the measurement error does not depend on time and the prior density p(σε2) was again based on a conjugate inversegamma (0.001, 0.001) distribution. Sensitivity analysis The Bayesian conﬁguration of the single exponential model was based on non-informative prior distributions for the parameters PCB0 [~N(0,10,000)I(0,)] and k [~N(0,10,000)I(,0)], i.e., normal distributions with mean 0 and variance 10,000 constrained to sample positive and negative values, respectively. To determine the robustness of the results to this assumption, the ﬁrst-order exponential decay model was also run using three different PCB0 priors. Speciﬁcally, we used normal (Prior 1) and lognormal (Prior 2) parameter distributions parameterized such that 95% of the respective values lay within the minimum and maximum PCB concentrations in the ﬁrst year examined, and a multivariate normal prior that accounts for the covariance between the parameters PCB0 and k (Prior 3). [The WinBUGS codes associated with this sensitivity analysis exercise can be found in the Appendix.] Finally, we compared the impact of the speciﬁcation k ∈[−∞,0] relative to the general characterization of k ∈[−∞,+∞], and the results were practically identical. The only difference was that the k marginal posteriors tended to be somewhat ﬂatter due to the inclusion of the random walk term. Evidently, this separation of the space assigned to the k and δt terms alleviates the identiﬁcation problem associated with the more complex structure of our statistical formulation. Step II — dynamic linear modeling In the next phase, dynamic linear modeling analysis was used to examine to what extent the PCB temporal trends detected in the ﬁrst phase are actually manifested, if we explicitly account for the fact that the ﬁsh length covaries with the PCB concentrations and that unequal number of ﬁsh samples of different sizes may have been sampled over time. The main advantage of the DLMs is the explicit recognition of structure in the time series, i.e., the data are sequentially ordered and the level of the response variable at each time step is related to its levels at earlier time steps in the data series (Lamon et al., 1998; Stow et al., 2004). In contrast with regression analysis, parameter estimates are inﬂuenced only by prior and current information, not by subsequent data. Parameter values are dynamic and reﬂect shifts in both the level of the response variable and the underlying ecological processes. Using Bayes' Theorem, the DLM process updates our knowledge regarding the parameters with the likelihood of the data and our prior knowledge. DLMs easily handle missing values/unequally spaced data, and minimize the effect of outliers (Pole et al., 1994). All DLMs consist of an observation equation and a system equation (West and Harrison, 1989). In particular, the DLMs used herein were speciﬁed as follows: Observation equation: ln ½PCBti = levelt + βt ln½lengthti + ψti

ψti e N½0; Ψt

ð5aÞ

Fig. 2. Temporal trends of PCB concentrations (1977–2007) using the exponential decay model with (a, b) walleye skinless–boneless ﬁlet, (c, d) walleye whole ﬁsh, (e, f) white bass skinless–boneless ﬁlet, (g, h) coho salmon skinless–boneless ﬁlet, and (i, j) rainbow trout skinless–boneless ﬁlet data from Lake Erie. The circles indicate the measured values, while the solid and dashed lines correspond to the median and the 95% credible intervals of the posterior predictive distributions, respectively. The ﬁrst-order random walk terms correspond to the δ annual values in Eq. (2) introduced to account for the structural deﬁciencies of the exponential decay model.

512

S. Sadraddini et al. / Journal of Great Lakes Research 37 (2011) 507–520

System equations: levelt = levelt−1 + ratet + ωt1

ωt1 e N½0; Ωt1

ratet = ratet−1 + ωt2

ωt2 e N½0; Ωt2 ωt3 e N½0; Ωt3

βt = βt−1 + ωt3 1 = Ω2tj

=ζ

t−1

·1 = Ω21j ; 1 = Ψt2

=ζ

t−1

·1 = Ψ12

level1 ; rate1 ; β1 e Nð0; 10000Þ 2

t > 1 and j = 1 to 3 t=1

deviance is deﬁned as −2log(likelihood) or −2log[p(y|θ)]; pD is a measure of the “effective number of parameters” and corresponds to the trace of the product of Fisher's information and the posterior covariance. It is speciﬁed as the posterior mean deviance of the model DðθÞ minus the point estimate of the model deviance when using the means of the posterior parameter distributions, i.e., pD = DðθÞ−Dðθ Þ. Thus, with this Bayesian model comparison, we ﬁrst assess model ﬁt or model “adequacy”, DðθÞ, and then we penalize complexity, pD. A smaller DIC value indicates a “better” model.

2

1 = Ω1j ; 1 = Ψ1 e gammað0:001; 0:001Þ ð5bÞ where ln[PCB]ti is the observed lnPCB concentration at time t in the individual sample i; levelt is the mean lnPCB concentration at time t when accounting for the covariance with the ﬁsh length; ln[length]ti is the observed (standardized) ﬁsh length at time t in the individual sample i; ratet is the rate of change of the level variable; βt is a length (regression) coefﬁcient; ψt, ωtj are the error terms for year t sampled from normal distributions with zero mean and variances Ψt2, Ωtj2, respectively; the discount factor ζ represents the aging of information with the passage of time; N(0, 10,000) is the normal distribution with mean 0 and variance 10,000; and gamma(0.001, 0.001) is the gamma distribution with shape and scale parameters of 0.001. The prior 2 , distributions for the parameters of the initial year level1, rate1, β1, 1/Ω1j 2 and 1/Ψ1 are considered “non-informative” or vague. In this study, we opted for a parsimonious DLM construct, in which the same discount factor was implemented on all the priors for the ﬁrst year of the study, and therefore non-constant and data-driven variances (with respect to time) were introduced without having to estimate a large number of parameters. Namely, we examined different discounts between 0.9 and 1.0 and the results reported here are based on a discount value of 0.95. Discounts were selected by conducting a model search, in which models with different discount factors were compared on the basis of their difference in loge likelihoods (Lamon et al., 1998). Model computations Sequence of realizations from the model posterior distributions were obtained using Markov Chain Monte Carlo (MCMC) simulations (Gilks et al., 1998). Speciﬁcally, we used the general normal-proposal Metropolis algorithm as implemented in the WinBUGS software; this algorithm is based on a symmetric normal proposal distribution, whose standard deviation is adjusted over the ﬁrst 4000 iterations, such as the acceptance rate ranges between 20% and 40%. We used three chain runs of 80,000 iterations and samples were taken after the MCMC simulation converged to the true posterior distribution. Convergence was assessed using the modiﬁed Gelman–Rubin convergence statistic (Brooks and Gelman, 1998). Generally, we noticed that the sequences converged very rapidly (≈1000 iterations), and the summary statistics reported in this study were based on the last 75,000 draws by keeping every 20th iteration (thin = 20) to avoid serial correlation. The accuracy of the posterior parameter values was inspected by assuring that the Monte Carlo error for all parameters was less than 5% of the sample standard deviation.

Results Table 1 shows the summary statistics of the observed total PCB concentrations in four ﬁsh species. Coho salmon had the highest PCB concentration (mean 463 and median 410 ng/g wet weight) followed by rainbow trout (mean 399 and median 326 ng/g wet weight), white bass (mean 309 and median 240 ng/g wet weight), and walleye (mean 114 and median 80 ng/g wet weight). Yet, the high standard deviation and interquartile range values reﬂect the substantial interand intra-annual variability associated with the PCB levels of the individual ﬁsh species. The positive skewness and kurtosis suggest right skewed and leptokurtic distributions; thus, the natural log transformation was implemented for the subsequent modeling analysis, effectively imposing a log-normal error structure on each model. The posterior estimates for the exponential decay models used to assess the temporal PCB trends in walleye WF and SBF portions are provided in Table 2. The relatively similar DIC values between the models with the ﬁrst and second order temporal smoothing suggest an almost equal support of the two statistical formulations by the observed data. Both approaches predict weakly decreasing trends of the PCB concentrations during the study period, while the decay rates were substantially higher in walleye SBF (k ≈ −0.040 yr −1) than in WF portions (k = − 0.001 yr −1). The predicted mean PCB concentrations in the SBF portions decreased until the mid 1980s, then remained more or less constant through the early 90s, after which increased until the recent years (Fig. 2a). As previously explained, the δ (random walk) terms were used to detect the systematic errors stemming from the structural inadequacies of the single exponential model. In particular, the positive values of the structural error terms during the second half of the survey period represent the inadequacy of the exponential decay model to capture the concurrent increasing trends (Fig. 2b). By contrast, neither the predicted mean PCB patterns show any increasing/decreasing trends throughout the survey period (Fig. 2c), nor the δ terms capture any systematic errors of the model structure for walleye WF data (Fig. 2d). We also examined the robustness of the previous outcomes to the prior distributions assigned to the initial PCB concentrations using Table 3 Sensitivity analysis of the SBF walleye exponential decay model with 1st order random walk using different prior speciﬁcations. Prior 1 and Prior 2 denote normally and lognormally distributed PCB0 priors, parameterized such that 95% of the respective values were lying within the minimum and maximum PCB concentrations measured in 1977; and Prior 3 denotes multivariate normal priors that account for the covariance between the parameters PCB0 and k. Prior 1

Model comparisons The models presented in this analysis were compared using the deviance information criterion (DIC), a Bayesian measure of model ﬁt and complexity (Spiegelhalter et al., 2002). DIC is given by DIC = DðθÞ + pD

ð7Þ

where DðθÞ is the posterior mean of the deviance, a measure of residual variance in data conditional on the parameter vector θ. The

PCB0 k ω σε σk σPCB0 σkPCB0

Prior 2

Prior 3

Mean

SD

Mean

SD

Mean

SD

279 − 0.074 0.592 0.776

112 0.028 0.102 0.019

322 − 0.066 0.599 0.777

363 0.050 0.104 0.019

174 −0.041 0.592 0.777 0.395 211 − 62

89 0.030 0.104 0.019 0.525 330 2142

*σk, σPCB0, and σkPCB0 represent the three elements of the covariance matrix between the initial PCB concentration (PCB0) and the decay coefﬁcient (k).

S. Sadraddini et al. / Journal of Great Lakes Research 37 (2011) 507–520

513

Table 4 Posterior estimates of the length (regression) coefﬁcient (mean values ± standard deviations) for the dynamic linear models used to describe the temporal trends of PCB concentrations in walleye (both WF and SBF portions), coho salmon (SBF), white bass (SBF) and rainbow trout (SBF). Beta coefﬁcient

Walleye WF DIClength 1702

β1977 β1978 β1979 β1980 β1981 β1982 β1983 β1984 β1985 β1986 β1987 β1988 β1989 β1990 β1991 β1992 β1993 β1994 β1995 β1996 β1997 β1998 β1999 β2000 β2001 β2002 β2003 β2004 β2005 β2006 β2007

0.537 ± 0.095 0.454 ± 0.035 0.614 ± 0.087 0.330 ± 0.065 0.247 ± 0.049 0.269 ± 0.043 0.276 ± 0.066 −0.006 ± 0.061 0.357 ± 0.067 0.38 ± 0.076 0.210 ± 0.110 0.113 ± 0.095 0.073 ± 0.144 0.165 ± 0.106 0.209 ± 0.114 0.173 ± 0.108 0.132 ± 0.081 0.168 ± 0.096 0.131 ± 0.132 0.123 ± 0.151 0.257 ± 0.140 0.278 ± 0.220 0.251 ± 0.265 0.194 ± 0.254 0.141 ± 0.136 0.200 ± 0.158 0.122 ± 0.19 0.275 ± 0.128 0.059 ± 0.204

Walleye SBF DICRW 2007

DIClength 2176 0.137 ± 0.108 0.171 ± 0.105 0.186 ± 0.090 0.289 ± 0.059 0.208 ± 0.079 0.168 ± 0.093 0.096 ± 0.086 0.14 ± 0.078 0.132 ± 0.066 0.034 ± 0.079 0.040 ± 0.094 0.165 ± 0.107 0.309 ± 0.125 0.425 ± 0.105 0.376 ± 0.143 0.404 ± 0.165 0.377 ± 0.189 0.361 ± 0.186 0.279 ± 0.153

Coho salmon SBF DICRW 2233

DIClength 1103

DICRW 1129

0.187 ± 0.120 0.155 ± 0.053 0.114 ± 0.086 0.017 ± 0.042 0.066 ± 0.055 0.157 ± 0.059 0.177 ± 0.076 0.177 ± 0.063 0.135 ± 0.069 0.094 ± 0.057 0.136 ± 0.069 0.006 ± 0.072 − 0.035 ± 0.098 − 0.012 ± 0.109 0.186 ± 0.132

0.252 ± 0.121 0.318 ± 0.162 0.251 ± 0.206

0.255 ± 0.153 0.231 ± 0.27 0.182 ± 0.14 0.147 ± 0.133 0.253 ± 0.137 0.345 ± 0.146 0.324 ± 0.13

White bass SBF

0.207 ± 0.341

DIClength 2060 0.432 ± 0.090 0.047 ± 0.063 0.376 ± 0.094 0.473 ± 0.131 0.875 ± 0.057 0.639 ± 0.076 0.358 ± 0.057 0.040 ± 0.157 0.02 ± 0.032 0.411 ± 0.114 0.544 ± 0.073 0.606 ± 0.099 0.411 ± 0.157 0.308 ± 0.146 0.370 ± 0.082 0.242 ± 0.186 0.329 ± 0.168 0.189 ± 0.205 0.329 ± 0.211 0.431 ± 0.18 0.181 ± 0.196 0.155 ± 0.209 0.236 ± 0.084 0.046 ± 0.249 0.064 ± 0.224 −0.064 ± 0.182 0.292 ± 0.171 0.172 ± 0.129 0.643 ± 0.12 0.726 ± 0.118 0.606 ± 0.164

Rainbow trout SBF DICRW 2501

DIClength 573

DICRW 628

0.430 ± 0.179

0.431 ± 0.173

0.400 ± 0.165 0.419 ± 0.156 0.401 ± 0.151 0.465 ± 0.149

0.390 ± 0.142 0.374 ± 0.135 0.367 ± 0.133 0.326 ± 0.129 0.288 ± 0.122 0.248 ± 0.116 0.267 ± 0.106 0.204 ± 0.098 0.211 ± 0.084 0.146 ± 0.072 0.129 ± 0.053 0.113 ± 0.047 0.157 ± 0.083

DIClength and DICRW represent the deviance information criterion values of the dynamic linear models with the ﬁsh length as a covariate and their random walk counterparts, respectively.

the ﬁrst order exponential decay model with the walleye SBF data (Table 3). During this sensitivity analysis, the same ﬂat prior distributions were assigned to the decay rates, i.e., k ~ N(0,10,000)I (,0). Prior 2 resulted in relatively higher posterior PCB0 estimates (322 ± 363 ng PCB/g wet weight) followed by Prior 1 (279 ± 112 ng PCB/g wet weight) and Prior 3 (174 ± 89 ng PCB/g wet weight). Yet, we also note that the use of a log-normally distributed PCB0 prior (Prior 2) led to a poorly determined posterior estimate. The highest posterior decay coefﬁcients were derived from the Prior 1 (k = − 0.074 ±0.028 yr −1) and Prior 2 (k = − 0.066 ± 0.050 yr −1), and the lowest with the Prior 3 (k = − 0.041 ± 0.030 yr −1) which was also very similar to the one estimated from the informative normal PCB0 prior derived from the PCB concentrations measured in 1977 (Table 2). Generally, while the measurement error σε and the conditional variance ω associated with the different model speciﬁcations were very similar and the inference regarding the presence of a distinct declining trend was unaltered, the actual posterior parameter values appear to be somewhat sensitive to the assumptions made about the PCB0 prior. Interestingly, the sensitivity exercise also suggests that the PCB0 and k values tend to covary, i.e., higher PCB0 estimates are associated with lower k values, although the explicit consideration of such term (σkPCB0) with the Prior 3 has not elucidated the strength of this relationship (− 62 ± 2142). The comparison of the models developed for the four ﬁsh species with the SBF data suggests that the PCB concentrations have been decreasing relatively faster in walleye (k = −0.040 and −0.039 yr −1 for the ﬁrst and second order models, respectively), followed by coho salmon (k = −0.013 and −0.010 yr−1), white bass (k = −0.009 yr −1 for both models) and rainbow trout (k = −0.007 yr−1 for both models).

Similar to the aforementioned results for walleye, the DIC values suggest that the ﬁrst and second order temporal smoothing are almost equally supported by the observed data when we consider both model performance and complexity. The predicted average PCB trends in white bass demonstrate a wax and wane pattern throughout the survey period with a net contaminant decrease in recent years (Fig. 2e), while the random walk terms do not reveal any systematic trends unaccounted for by the exponential decay model (Fig. 2f). The PCB levels in coho salmon declined gradually from 1976 until the most recent years in our dataset (Fig. 2g), although the actual magnitude of this decreasing trend is hard to be accurately quantiﬁed due to extensive data gaps in the 1990s and 2000s. Similarly, the rainbow trout exhibited a net decrease in PCB concentrations from 1980s to 2000s, but the inconsistent information from the earlier years of the study period and the lack of data from the mid-90s impedes the precise delineation of the trends followed (Fig. 2i). The δ terms for the latter two species do not suggest any major deviations from the trajectory postulated by the simple exponential decay model (Figs. 2h and j). The structural ﬂexibility of the DLMs allows the model parameters to change over time and also accommodates the covariance between PCB concentrations and other potentially important ﬁsh characteristics. In a preliminary exploratory analysis, we conducted a model search in which models with identical structures, but different covariates (length, weight, lipid content, gender) were compared on the basis of their difference in loge likelihoods. Differences in loge likelihood between identically structured models with different covariates were interpreted as evidence in favor of one covariate over another. In this exercise, the DLM with the ﬁsh length outperformed all the rest models. Further, the same models also outperformed their random walk counterparts that do not consider any covariates (see DIC values in Table 4). In the second

514

S. Sadraddini et al. / Journal of Great Lakes Research 37 (2011) 507–520

8

a Loge(PCB)

Loge(PCB).yr-1

b

7

0.5

0.0

6 5 4

-0.5 3 2 10

c

0.0

8

e

0.4

7

0.2

6

Loge(PCB)

Loge(PCB).yr-1

6

4

-0.5 0.6

d

8

0.5

Loge(PCB)

Loge(PCB).yr-1

1.0

0.0 -0.2

5 4

-0.4

3

-0.6

2 8

g

f

h

0.4

Loge(PCB)

Loge(PCB).yr-1

7

0.0

-0.4

6 5 4 3

-0.8

2 9

i

8

Loge(PCB)

0.0 -0.4 -0.8

j

7 6 5 4

-1.2

Year

Year

2008

2004

2000

1996

1992

1988

1984

1980

2008

2004

2000

1996

1992

1988

1984

1980

3 1976

Loge(PCB).yr-1

0.4

1976

0.8

S. Sadraddini et al. / Journal of Great Lakes Research 37 (2011) 507–520

phase, we examined whether the inclusion of a second covariate (along with the ﬁsh length) can improve the predictive capacity of our DLM analysis. Aside from the case of walleye SBF, the model with the ﬁsh length as a single covariate was proven to be the most parsimonious construct (i.e., lowest DIC values) to detect ﬁsh contaminant trends in Lake Erie. The DLM analysis identiﬁed three distinct patterns regarding the rates of change of the PCB levels in the ﬁsh species examined (Fig. 3). First, the walleye SBF and WF models show no profound temporal shifts in the rates which remain nearly zero throughout the study period (Figs. 3a and c), and therefore the corrections for the ﬁsh length drive the year-to-year variability associated with the level parameter (Figs. 3b and d). Further, the discrepancy between the walleye SBF trends when partialling out the covariance with the ﬁsh length (Fig. 3b) and the trends from the exponential decay model (Fig. 2b) suggest that the ﬁsh sampling practices in Lake Erie may introduce a bias, and therefore a rigorous assessment of contaminant trends in space and time should rather be based on an explicit consideration of the possible covariates (e.g., length, age, gender, lipid content, season). Second, the PCB rates of change of the white bass concentrations have switched from weakly negative to nearly zero during the 2000s (Fig. 3e), and the predicted length-adjusted mean PCB values appear to have been stabilized after a net decrease during the earlier years of the survey period (Fig. 3f). Third, weakly negative rates of change were apparent for coho salmon and rainbow trout during the entire study period (Figs. 3g and i), reﬂecting the moderate decrease of the corresponding concentrations. Yet, we caution that the robustness of the latter pattern remains to be conﬁrmed due to the temporal gaps of the information used during the early 90s (Fig. 3j) or the most recent years (Fig. 3h). Finally, aside from the cases in which the sampling bias introduces discrepancies (e.g., walleye SBF data), we highlight the remarkable consistency between the trends delineated by the random walk terms of the exponential decay models (lines in the right panels of Fig. 2) and the length-corrected DLM predictions (lines in the right panels of Fig. 3). Discussion The major sources of toxic substances in Lake Erie have been point discharges from industrial and municipal facilities along with the nonpoint urban and agricultural runoff. Many of the persistent, bioaccumulative, and toxic organic contaminants are primarily associated with the high organic carbon and ﬁne-grained (silt/clay) fractions of the sediments and thus are mainly transported as part of the sediment load in rivers and tributaries (Marvin et al., 2002). Atmospheric transport and deposition processes can potentially be another signiﬁcant source of contamination, and existing evidence of PCB congener patterns suggests that the contribution of airborne pollutants can dominate the ﬂuxes from other local sources (Datta et al., 1998). In Lake Erie, the lake-wide average PCB concentration in the sediments was 43 ng/g in 1997, approximately representing a three-fold decrease since 1971 (Painter et al., 2001). Yet, the Detroit River still appears to be an active contaminant source, whereby more than 70% of the sediment-bound pollutants accumulate in the relatively small western basin within 20–30 km of the mouth of the river (Carter and Hites, 1992). As a result, the local PCB ﬁsh body burdens are substantially higher, e.g., the average PCB levels in walleye SBF were approximately 50 ng/g higher in the western relative to the eastern basin during the study period, and therefore any modeling exercise aiming to detect temporal trends of ﬁsh contaminants in aggregated datasets should ensure the (reasonably) objective representation of the different locations of the lake. Aside from walleye, the dataset used does not have consistent information

515

for all the sites to rigorously examine the site-speciﬁc temporal trends. In the case of walleye SBF though, the development of local dynamic linear models has revealed a more distinctly decreasing trend in the eastern basin relative to the western part of the lake during the earlier years of the study period. Notably, the samples from the eastern part were also characterized from greater mean length (53.29 ± 10.04 cm) and weight (1704 ± 868 g) relative to those collected from the western basin (44.58 ± 8.44 cm and 973 ± 568 g). Thus, the local sampling practices can potentially introduce a systematic bias, which in turn reinforces the need to explicit consider all the possible covariates (e.g., length, gender, lipid content) that can potentially impede the detection of the actual temporal trends of ﬁsh contaminants and consequently may misinform consumption advisories. Depending on their ethology and trophic position, ﬁsh are likely to receive PCBs through three speciﬁc routes, viz., gills, epithelial/dermal tissues and gastrointestinal tract (Schlenk, 2005). Lower trophic level ﬁsh primarily receive contaminants by the diffusion process through gills and epithelial cells, whereas top predators primarily receive them through dietary uptake of contaminated food. Sediment can be directly ingested by bottom dwelling aquatic organisms which in turn are the food source for higher aquatic animals. In the western basin of Lake Erie, ﬁsh are predicted to accumulate less than half of their contaminant body burden from the sediments, while almost 100% of the ﬁsh contaminants directly (i.e., consumption of the bottom sediment) or indirectly (i.e., consumption of organisms that consume sediment or organisms contaminated with sediment) originate from the sediment in the eastern basin (Morrison et al., 2002). Once contaminants are absorbed in the ﬁsh body, they can be distributed to speciﬁc target organs causing direct biological effects; they may be transported to storage repositories with high lipid content; they may be directly excreted from the body without any interaction with target organs or storage depots; or the lipophilic compounds tend to biotransform to more hydrophilic derivatives in order to enhance polarity and subsequent elimination (Schlenk, 2005). In our study, the walleye whole ﬁsh portions had one order of magnitude higher PCB concentrations than the skinless–boneless ﬁlet data. Because of their lipophilic and non-polar nature, PCBs tend to be deposited in fat tissues and therefore the greater levels in whole ﬁsh data are plausible (Elskus et al., 2005). Yet, the signiﬁcant difference between whole ﬁsh and skinless boneless ﬁlet data in Lake Erie is somewhat unusual relative to what has been reported (or assumed) in the literature (Jackson and Schindler, 1996; Stow and Carpenter, 1994). For example, Amrhein et al. (1999) reported average wholeﬁsh to ﬁlet PCB concentration ratios of 1.70 for coho salmon and 1.47 for rainbow trout in Lake Michigan, but these ratios demonstrated substantial variability among individuals and there were also instances characterized by higher concentration in ﬁlet than in wholeﬁsh portions. The same study also reported an almost linear relationship between ﬁlet and whole ﬁsh PCB concentrations (see their Fig. 2), which was on par with the typical assumption that trends in ﬁlet measurements should also reﬂect trends in the corresponding whole ﬁsh levels (Bhavsar et al., 2007). While our results seem to deviate from both popular notions, we caution that the ﬁlet and whole-ﬁsh concentrations were not derived from concurrent samples and therefore these discrepancies largely stem from the different philosophies (and associated sampling practices) of the two datasets used (Bhavsar et al., 2010). In particular, while the substantial mobility of walleye can conceivably alleviate the biases associated with the various lake locations sampled each year, the whole-ﬁsh samples were almost exclusively collected from Pelee Island in the

Fig. 3. Dynamic Linear Modeling analysis. Left panels depict the annual rates of change of PCB concentrations in (a, b) walleye skinless–boneless ﬁlet, (c, d) walleye whole ﬁsh, (e, f) white bass skinless–boneless ﬁlet, (g, h) coho salmon skinless–boneless ﬁlet, and (i, j) rainbow trout skinless–boneless ﬁlet data from Lake Erie. Right panels depict the measured PCB concentrations (gray dots) against the PCB trends when accounting for the covariance with the ﬁsh length (black lines). The solid and dashed lines correspond to the median and the 95% credible intervals of the posterior predictive distributions, respectively.

516

S. Sadraddini et al. / Journal of Great Lakes Research 37 (2011) 507–520

80

a

60

70

2007

2004

1999

1996

1992

1989

1986

1982

20

1977

40

b

50

2006

2002

1999

1996

1992

1989

1986

1983

50

1980

10

1977

30

c

30

80

2007

2004

2001

1998

1995

1992

1989

1986

1983

1980

10

1977

20

d

60

2001

1993

1988

1985

1982

1979

20

1976

40

e 80

60

Year

2007

2004

2001

1998

1994

20

1985

40

1977

Length (cm)

40

western part of Lake Erie as opposed to the ﬁlet data that are based on samples from the entire lake. Consistent with Bhavsar et al.'s (2007) interpretation of the Lake Erie walleye SBF data (see their Fig. 1), our exponential modeling approach suggests that the moderately weak declining rates leveled off since the late 80s and may follow an upward trajectory after the mid-90s. Notably, Bhavsar's study was based on the 45–55 cm size range, which approximately corresponds to the mass of data included within the second and seventh deciles of the dataset used herein. Yet, the dynamic linear modeling analysis appears to negate the previous results, indicating that both the PCB rates of change (Fig. 3a) and the corresponding concentrations when explicitly considering their covariance with the ﬁsh length (Fig. 3b) do not demonstrate any major trends over the time span examined. The discrepancy between the two modeling approaches stems from the systematic increase of the annual median values of the walleye lengths sampled, which was particularly evident with the ﬁlet data (Figs. 4a and b). The same trend was not apparent in the other three species examined in this study (Figs. 4c–e), although the substantial interannual variability of the median length values draws attention to a potential bias if we do not partial out the ﬁsh size effects on PCB bioaccumulation (Amrhein et al., 1999; Carlson and Swackhamer, 2006). Generally, while the image portrayed from our analysis is somewhat inconclusive with regards to the PCB dynamics in walleye over the last three decades, it does certainly cast doubt on the likelihood of achieving the Great Lakes Strategy 2002 objective of 25% decrease in concentrations within a reasonably foreseeable time (Stow et al., 2004; U.S.EPA, 2002). In this regard, our walleye results are again consistent with Bhavsar et al.'s (2007) bootstrap resampling analysis that also ruled out the possibility of compliance with the targeted goals in Lake Erie, although the present analysis does not unequivocally support the same study's projection of an increase in the PCB concentrations. This difference between our and Bhavsar et al.'s (2007) trends after the mid-1990s could be attributed to the consideration of more recent data (for years 2006, 2007) in the present study which were among the lowest measurements since the 1990s as well as to the relatively higher variability in measurements relative to the overall low PCB levels. Among the other ﬁsh species examined, both the exponential decay and the dynamic linear models suggest that the PCB levels in white bass have not undergone any conspicuous changes during the time span examined. Yet, a careful inspection of the observed trends demonstrates an oscillatory pattern which was mainly captured by our exponential model (i.e., Fig. 2e). Similar oscillations are also evident in other Great Lakes time-series for several ﬁsh species and have been mainly attributed to the nature and relative strength of the different prey–predator interactions within the aquatic food webs and/or to the periodicities of the climatic forcing (e.g., Borgmann and Whittle, 1991; French et al., 2006; Scheider et al., 1998). Despite the lack of consistent information for coho salmon, both our modeling approaches suggest negative rates of change and continuous decrease of the PCB levels during the survey period, although these declining trends are signiﬁcantly weaker than those reported for the same species in other systems, e.g., Lake Ontario (French et al., 2006). Similarly, the rainbow trout DLM suggests weakly decrease of the PCB levels, and the odds that the rate parameter has been negative are on average 2.4:1 for coho salmon and 2.3:1 for rainbow trout during the study period. [Note that the odds ratio of the rate parameter being below zero in a particular year is the ratio of the probability mass below zero to the mass above zero.]

Fig. 4. Box plots of the annual sampled length for (a) walleye skinless–boneless ﬁlet, (b) walleye whole ﬁsh, (c) white bass skinless–boneless ﬁlet, (d) coho salmon skinless– boneless ﬁlet, and (e) rainbow trout skinless–boneless ﬁlet in Lake Erie. Extreme values are not included in these plots.

S. Sadraddini et al. / Journal of Great Lakes Research 37 (2011) 507–520

400

a

300

200

100

0 2

120

4

6

8 10 12 14 16 18 20 22 24 26

b 100 80 60 40 20 0

2

6

10

14

18

22

26

30

34 38

Number of observations

400

c

300

200

100

0

2

6

10

14

18

22

26

30

34

38

300

42

d

200

100

0 2

4

6

8

10

12

14

16

120

18

20

e

100 80 60 40 20 0

2

4

6

8

10 12 14 16 18 20 22 24

Lipid concentration (%) Fig. 5. The lipid concentration frequency distributions of the ﬁsh species examined.

517

Many plausible explanations have been proposed to elucidate the limited response of Lake Erie to the various contamination abatement strategies. One popular hypothesis argues that the decrease in external PCB inputs may have altered the contaminant ﬂuxes among the various media (water, atmosphere, sediments), whereby the sediments that historically acted as a net sink for PCBs may have switched into a net source (Pearson et al., 1996). Consequently, the signature of the sediment contributions to the PCB body burdens is predicted to increase and to ultimately reach a chemical equilibrium between sediments and aquatic biota (Morrison et al., 2002). Given that the profound ecological implication of such equilibrium is the resonance of the corresponding concentrations, the plateau-type of pattern (i.e., a decline in the rate of decrease in PCB concentrations in aquatic biota) recently reported in many Great Lakes may partly reﬂect the relatively static character of the contaminants in the sediments (De Vault et al., 1996; Hickey et al., 2006; Huestis et al., 1996). Yet, this mechanistic explanation of the response of the aquatic biota to the external PCB loading reductions may not hold true for the entire Lake Erie due to the signiﬁcant spatial heterogeneity characterizing the sediment contamination in the system. In particular, despite the substantial decrease of the lakewide average PCB concentrations (i.e., from 136 ng/g in 1971 to 43 ng/g in 1997), exceedances of sediment guidelines indicative of contaminated environments still occur in Lake Erie; especially, in the western basin and the southern portion of the central basin (Painter et al., 2001). In eastern Lake Erie, where the fugacity of the various PCB congeners in the sediment is signiﬁcantly higher than their fugacity in the water, the PCB body burden of the local biotic communities primarily stems from sediment-bound chemicals (Morrison et al., 2002). Contrary to the predictions of the proposed mechanism though, the prevailing conditions in the western Lake Erie suggest a smaller chemical disequilibrium between the two phases and thus the PCB burdens mainly originate from the water column (Morrison et al., 2002). If we also consider that the exchanges between atmosphere and water through wet/dry deposition and volatilization can further modulate the PCB ﬂuxes among the various ecosystem components (Jeremiason et al., 1994; Mackay and Bentzen, 1997), the role of the sediments alone may not be sufﬁcient to explain the trajectories delineated by our spatially-integrated models. Another driving factor that has been hypothesized to underlie the PCB temporal trends involves the structural shifts of the Lake Erie food web after the invasion of dreissenids and round gobies in the late 1980s. The invasion of exotic species is hypothesized to have caused a major reconﬁguration of the food web from a pelagic-based to a benthic-based one, which in turn has created new trophodynamics for contaminant transfer to top predators (Hogan et al., 2007). First, the introduction of zebra and quagga mussels has likely induced major changes in the PCB ﬂuxes within the Lake Erie food web, because of their ability to bioaccumulate by ﬁltering contaminated water and scavenging seston; by directly or indirectly inﬂuencing the diet compositions of other biota; and by the selective removal of particulate organic matter from the water column and the subsequent increase of the equilibrium concentrations of the dissolved-phase contaminants which in turn can increase the body burdens of many aquatic organisms (Morrison et al., 1998). Second, round goby invaded the Great Lakes and became extremely abundant in Lake Erie in 1996, causing major shifts in trophic relationships, displacement of native species populations from optimal spawning and feeding habitats, and increased growth rates of top predators (Dubs and Corkum, 1996; Hogan et al., 2007; Ray and Corkum, 1997). As a benthic ﬁsh with diet mainly composed of dreissenids and as component of the diets of many commercially and recreationally important species (e.g., walleye, yellow perch, smallmouth bass), round goby has the potential to accumulate contaminants and then transfer them to the higher trophic levels (Johnson et al., 2005). In particular, the benthivorous ﬁsh species (e.g., yellow perch and

518

S. Sadraddini et al. / Journal of Great Lakes Research 37 (2011) 507–520

0.8

8

a

b

0.4

Loge(PCB)

Loge(PCB).yr-1

7

0.0

6 5 4

-0.4

Year

2008

2004

2000

1996

1992

1988

1984

2

1976

2008

2004

2000

1996

1992

1988

1984

1980

1976

-0.8

1980

3

Year

Fig. 6. Annual rates of change (a) and temporal trends (b) of the PCB concentrations, when accounting for the covariance with the ﬁsh length and the lipid content in walleye skinless–boneless ﬁlet data.

smallmouth bass) have been projected to experience larger increases in their PCB burdens due to the effects of round gobies, whereas the impact on pelagic ﬁsh species (e.g., alewife and walleye) was relatively smaller (Johnson et al., 2005; Morrison et al., 2000). Our results do not refute the existing predictions regarding the relatively minor change of the walleye PCB levels following the invasion of exotic species, and a careful review of the pertinent literature does not provide evidence of strong walleye reliance on round goby (Bur et al., 2008; Jude et al., 2010). Yet, we cannot rule out the existence of indirect trophic paths (e.g., through the impact of round gobies on other prey ﬁshes) that can presumably shape the contaminant patterns in top pelagic predators. In this regard, we also highlight that a concurrent analysis of the total Hg variability in the same species does reveal an increasing trend after the mid-90s (Azim et al., 2011). Aside from the relationship between PCB concentrations and ﬁsh length, the ﬁsh lipid content stands out as one of the possible covariates that have received considerable attention in the literature (Amrhein et al., 1999; Ewald and Larsson, 1994; Rowan and Rasmussen, 1992; Stow, 1995; Voiland et al., 1991). Generally, there are contradictory results regarding the strength of the causal link between ﬁsh lipid content and organochlorine contaminant levels (Amrhein et al., 1999; Larsson et al., 1996). In the Lake Erie dataset, if we compare the species-speciﬁc PCB statistics (Table 1) with their corresponding lipid levels (Fig. 5), we can infer that the lipid content certainly plays a role in contaminant accumulation differences among species. Yet, a preliminary examination (not presented here) of the strength of the PCB:lipid relationship among individuals revealed weak covariance for all the ﬁsh species studied in our analysis. The only exception was the case of the walleye skinless boneless ﬁlet data, in which the most parsimonious dynamic linear model (DIC = 2013) considers both ﬁsh length and lipid content as covariates. Yet, the inference drawn from this model remained practically unaltered relative to the model with the ﬁsh length as the sole covariate (Fig. 6). The same result was also true when we considered other potentially important covariates, such as the ﬁsh age and gender. Stow et al. (1997) reported very similar patterns for ﬁve species of Lake Michigan salmonids, but also identiﬁed a stronger PCB: lipid association when focusing on samples collected during the spawning period (see their Fig. 5). Likewise, Amrhein et al. (1999) found that lipid normalization does not efﬁciently control withinspecies variability, but can accentuate among-species differences. Thus, although the reconciliation of the interplay between ﬁsh contaminant levels and their lipid content warrants consideration, we underscore that the ﬁsh length was the single best covariate for detecting PCB temporal trends in Lake Erie.

In conclusion, our analysis shows that the PCBs have been decreasing over the last 30 years with relatively weak rates that vary among the different ﬁsh species examined. The walleye skinless– boneless ﬁlet data are characterized by an increasing trend in the PCB concentrations after the mid-90s, which however disappears when explicitly considering the ﬁsh length as a covariate. Our DLM analysis also suggests that the walleye WF trends show no profound changes suggesting relatively stable levels throughout the study period. The coho salmon and rainbow trout PCB concentrations have been decreasing steadily during the study period but the associated rates were relatively weak, while the same trends in white bass appear to have been stabilized over that last decade after a weak decrease during the 70s and 80s. The robustness of the latter results remains to be veriﬁed due to the temporal gaps and inconsistencies of the information used. The different trends demonstrated by the various ﬁsh species herein stress the importance of considering more than one ﬁsh species for proper spatial/temporal trend assessments. Finally, we emphasize that the differences between the two phases of our statistical analysis pinpoint the potential bias introduced by all the time-series analysis strategies (single and double exponential decay, mixed order models, simple regression analysis) that do not consider the role of important covariates. Yet, aside from the studies that screened the datasets prior to the analysis and subsequently focused on a speciﬁc (narrow) ﬁsh length or lipid content range, much of the contemporary literature draws inference on statistical trends that fail to explicitly account for potentially important covariates (e.g., length, age, lipid content, location). One of the take-home messages from our study is that the likelihood of a systematic sampling bias can impede the detection of the actual temporal trends of ﬁsh contaminants, and thus may misleadingly guide ﬁsh consumption advisories. Acknowledgments This project has received funding support from the Ontario Ministry of the Environment (Best in Science Research ProgramGrant Funding Agreement 89002). Such support does not indicate endorsement by the Ministry of the contents of the study. All the model codes pertinent to this analysis are available upon request from the corresponding author. Appendix A. Supplementary data Supplementary data to this article can be found online at doi:10. 1016/j.jglr.2011.06.005.

S. Sadraddini et al. / Journal of Great Lakes Research 37 (2011) 507–520

References Amrhein, J.F., Stow, C.A., Wible, C., 1999. Whole-ﬁsh versus ﬁllet polychlorinatedbiphenyl concentrations: an analysis using classiﬁcation and regression tree models. Environ. Toxicol. Chem. 18, 1817–1823. Arhonditsis, G.B., Papantou, D., Zhang, W.T., Perhar, G., Massos, E., Shi, M.L., 2008a. Bayesian calibration of mechanistic aquatic biogeochemical models and beneﬁts for environmental management. J. Mar. Syst. 73, 8–30. Arhonditsis, G.B., Perhar, G., Zhang, W., Massos, E., Shi, M., Das, A., 2008b. Addressing equiﬁnality and uncertainty in eutrophication models. Water Resour. Res. 44 (W01420, 19 pp.). Azim, M.E., Kumarappah, A., Bhavsar, S., Backus, S., Arhonditsis, G.B., 2011. Detection of the spatiotemporal trends of mercury contamination in Lake Erie ﬁsh communities: a Bayesian approach. Environ. Sci. Technol. 45, 2217–2226. Bhavsar, S.P., Jackson, D.A., Hayton, A., Reiner, E.J., Chen, T., Bodnar, J., 2007. Are PCB levels in ﬁsh from the Canadian Great Lakes still declining? J. Great Lakes Res. 33, 592–605. Bhavsar, S.P., Gewurtz, S.B., McGoldrick, D.J., Keir, M.J., Backus, S.M., 2010. Changes in mercury levels in Great Lakes ﬁsh between 1970s and 2007. Environ. Sci. Technol. 44, 3273–3279. Borgmann, U., Whittle, D.M., 1983. Particle-size-conversion efﬁciency and contaminant concentrations in Lake Ontario biota. Can. J. Fish. Aquat. Sci. 40, 328–336. Borgmann, U., Whittle, D.M., 1991. Contaminant concentration trends in LakeOntario lake trout (Salvelinus nmaycush) — 1977 to 1988. J. Great Lakes Res. 17, 368–381. Brooks, S.P., Gelman, A., 1998. General methods for monitoring convergence of iterative simulations. J. Comput. Graph. Stat. 7, 434–455. Bur, M.T., Stapanian, M.A., Bernhardt, G., Turner, M.W., 2008. Fall diets of red-breasted merganser (Mergus serrator) and walleye (Sander vitreus) in Sandusky Bay and adjacent waters of Western Lake Erie. Am. Midl. Nat. 159, 147–161. Carlson, D.L., Swackhamer, D.L., 2006. Results from the US Great Lakes ﬁsh monitoring program and effects of lake processes on bioaccumulative contaminant concentrations. J. Great Lakes Res. 32, 370–385. Carlson, D.L., De Vault, D., Swackhamer, D., 2010. On the rate of decline of persistent organic contaminants in lake trout (Salvelinus namaycush) from the Great Lakes, 1970–2003. Environ. Sci. Technol. 44, 2004–2010. Carter, D.S., Hites, R.A., 1992. Fate and transport of Detroit River drived pollutants throughout Lake Erie. Environ. Sci. Technol. 26, 1333–1341. Cole, D.C., Kearney, J., Sanin, L.H., Leblanc, A., Weber, J.P., 2004. Blood mercury levels among Ontario anglers and sport-ﬁsh eaters. Environ. Res. 95, 305–314. Datta, S., McConnell, L.L., Baker, J.E., Lenoir, J., Seiber, J.N., 1998. Evidence for atmospheric transport and deposition of polychlorinated biphenyls to the Lake Tahoe basin, California — Nevada. Environ. Sci. Technol. 32, 1378–1385. De Vault, D.S., Hesselberg, R., Rodgers, P.W., Feist, T.J., 1996. Contaminant trends in lake tout and walleye from the Laurentian Great Lakes. J. Great Lakes Res. 22, 884–895. Dubs, D.O.L., Corkum, L.D., 1996. Behavioural interactions between round gobies (Neogobius melanostomus) and mottled sculpins (Cottus bairdi). J. Great Lakes Res. 22, 838–844. Elskus, A.A., Coller, T.K., Monosson, E., 2005. Interactions between lipids and persistent organic pollutants in ﬁsh. In: Mommsen, T.P., Moon, T.W. (Eds.), Biochemistry and Molecular Biology of Fishes, vol. 6. Elsevier BV, pp. 119–152. Ewald, G., Larsson, P., 1994. Partitioning of 14C-labelled 2,2′,4,4′-tetrachlorobiphenyl between water and ﬁsh lipids. Environ. Toxicol. Chem. 13, 1577–1580. Faroon, O., Jones, D., De Rosa, C., 2000. Effects of polychlorinated biphenyls on the nervous system. Toxicol. Ind. Health 16, 307–333. Fitzgerald, E.F., Belanger, E.E., Gomez, M.I., Cayo, M., McCaffrey, R.J., Seegal, R.F., Jansing, R.L., Hwang, S.A., Hicks, H.E., 2008. Polychlorinated biphenyl exposure and neuropsychological status among older residents of upper Hudson River communities. Environ. Health Perspect. 116, 209–215. French, T.D., Campbell, L.M., Jackson, D.A., Casselman, J.M., Scheider, W.A., Hayton, A., 2006. Long-term changes in legacy trace organic contaminants and mercury in Lake Ontario salmon in relation to source controls, trophodynamics, and climatic variability. Limnol. Oceanogr. 51, 2794–2807. Gelman, A., Carlin, J.B., Stern, H.S., Rubin, D.B., 2004. Bayesian Data Analysis, Second ed. Chapman & Hall, New York. Gewurtz, S.B., Bhavsar, S.P., Jackson, D.A., Fletcher, R., Awad, E., Moody, R., Reiner, E.A., 2010. Temporal and spatial trends of organochlorines and mercury in ﬁshes from the St. Clair River/Lake St. Clair corridor, Canada. J. Great Lakes Res 36, 100–112. Gilks, W., Roberts, G.O., Sahu, S.K., 1998. Adaptive Markov Chain Monte Carlo through regeneration. J. Am. Stat. Assoc. 93, 1045–1054. Hickey, J.P., Batterman, S.A., Chernyak, S.M., 2006. Trends of chlorinated organic contaminants in Great Lakes trout and walleye from 1970 to 1998. Arch. Environ. Contam. Toxicol. 50, 97–110. Hogan, L.S., Marschall, E., Folt, C., Stein, R.A., 2007. How non-native species in Lake Erie inﬂuence trophic transfer of mercury and lead to top predators. J. Great Lakes Res. 33, 46–61. Huestis, S.Y., Servos, M.R., Whittle, D.M., Dixon, D.G., 1996. Temporal and age-related trends in levels of polychlorinated biphenyl congeners and organochlorine contaminants in Lake Ontario lake trout (Salvelinus namaycush). J. Great Lakes Res. 22, 310–330. Humphrey, H.E.B., 1988. Chemical contaminants in the Great Lakes: the human health aspects. In: Evans, M.S. (Ed.), Toxic Contaminants and Ecosystem Health: A Great Lakes Focus. Wiley Inter Science, New York, pp. 153–168.

519

International Joint Commission (IJC), 1978. Great Lakes Water Quality Agreement as Amended by Protocol. Windsor, ON, Canada. International Joint Commission (IJC), 2006. A Guide to the Great Lakes Water Quality Agreement: Background for the 2006 Governmental Review. 1-894280-53-9. Canada and the United States. Jackson, L.J., Schindler, D.E., 1996. Field estimates of net trophic transfer of PCBs from prey ﬁshes to Lake Michigan salmonids. Environ. Sci. Technol. 30, 1861–1865. Jeremiason, J.D., Hornbuckle, K.C., Eisenreich, S.J., 1994. PCBS in Lake-Superior, 1978– 1992 — decreases in water concentrations reﬂect loss by volatilization. Environ. Sci. Technol. 28, 903–914. Johnson, B.L., Hicks, H.E., De Rosa, C.T., 1999. Key environmental human health issues in the Great Lakes and St. Lawrence River basins. Environ. Res. 80, S2–S12. Johnson, T.B., Bunnell, D.B., Knight, C.T., 2005. A potential new energy pathway in central Lake Erie: the round goby connection. J. Great Lakes Res. 31 (Suppl. 2), 238–251. Jude, D.J., Rediske, R., O'Keefe, J., Hensler, S., Giesy, J.P., 2010. PCB concentrations in walleyes and their prey from the Saginaw River, Lake Huron: a comparison between 1990 and 2007. J. Great Lakes Res. 36, 267–276. Lamon, E.C., Carpenter, S.R., Stow, C.A., 1998. Forecasting PCB concentrations in Lake Michigan salmonids: a dynamic linear model approach. Ecol. Appl. 8, 659–668. Larsson, P., Backe, C., Bremle, G., Eklöv, A., Okla, L., 1996. Persistent pollutants in a salmon population (Salmo salar) of the southern Baltic Sea. Can. J. Fish. Aquat. Sci. 53, 62–69. Mackay, D., Bentzen, E., 1997. The role of the atmosphere in Great Lakes contamination. Atmos. Environ. 31, 4045–4047. Marvin, C., Alaee, M., Painter, S., Charlton, M., Kauss, P., Kolic, T., MacPherson, K., Takeuchi, D., Reiner, E., 2002. Persistent organic pollutants in Detroit River suspended sediments: polychlorinated dibenzo-p-dioxins and dibenzofurans, dioxin-like polychlorinated biphenyls and polychlorinated naphthalenes. Chemosphere 49, 111–120. Morrison, H.A., Gobas, F., Lazar, R., Whittle, D.M., Haffner, G.D., 1998. Projected changes to the trophodynamics of PCBs in the western Lake Erie ecosystem attributed to the presence of zebra mussels (Dreissena polymorpha). Environ. Sci. Technol. 32, 3862–3867. Morrison, H.A., Whittle, D.M., Haffner, G.D., 2000. The relative importance of species invasions and sediment disturbance in regulating chemical dynamics in western Lake Erie. Ecol. Model. 125, 279–294. Morrison, H.A., Whittle, D.M., Haffner, G.D., 2002. A comparison of the transport and fate of polychlorinated biphenyl congeners in three Great Lakes food webs. Environ. Toxicol. Chem. 21, 683–692. Painter, S., Marvin, C., Rosa, F., Reynoldson, T.B., Charlton, M.N., Fox, M., Thiessen, L., Estenik, J.F., 2001. Sediment contamination in Lake Erie: a 25-year retrospective analysis. J. Great Lakes Res. 27, 434–448. Pearson, R.F., Hornbuckle, K.C., Eisenreich, S.J., Swackhamer, D.L., 1996. PCBs in Lake Michigan water revisited. Environ. Sci. Technol. 30, 1429–1436. Pole, A., West, M., Harrison, J., 1994. Applied Bayesian Forecastingand Time Series Analysis. Chapman & Hall, New York. Ray, W.J., Corkum, L.D., 1997. Predation of zebra mussels by round gobies, Neogobius melanostomus. Environ. Biol. Fish. 50, 267–273. Rowan, D.J., Rasmussen, J.B., 1992. Why don't Great Lakes ﬁsh reﬂect environmental concentrations of organic contaminants — an analysis of between-lake variability in the ecological partitioning of PCBs and DDT. J. Great Lakes Res. 18, 724–741. Scheider, W.A., Cox, C., Hayton, A., Hitchin, G., 1998. Current status and temporal trends in concentrations of persistent toxic substances in sport ﬁsh and juvenile forage ﬁsh in the Canadian waters of the Great Lakes. Environ. Monit. Assess. 53, 57–76. Schlenk, D., 2005. Pesticide biotransformation in ﬁsh. In: Mommsen, T.P., Moon, T.W. (Eds.), Biochemistry and Molecular Biology of Fishes, vol. 6. Elsevier BV, pp. 171–190. Shaddick, G., Wakeﬁeld, J., 2002. Modelling daily multivariate pollutant data at multiple sites. J. R. Stat. Soc. Ser. C App. Stat. 51, 351–372. Spiegelhalter, D.J., Best, N.G., Carlin, B.R., van der Linde, A., 2002. Bayesian measures of model complexity and ﬁt. J. R. Stat. Soc. Ser. B Stat. Methodol. 64, 583–616. Stow, C.A., 1995. Factors associated with PCB concentrations in Lake-Michigan salmonids. Environ. Sci. Technol. 29, 522–527. Stow, C.A., Carpenter, S.R., 1994. PCB accumulation in Lake-Michigan coho and chinook salmon — individual-based models using allometric relationships. Environ. Sci. Technol. 28, 1543–1549. Stow, C.A., Jackson, L.J., Amrhein, J.F., 1997. An examination of the PCB:lipid relationship among individual ﬁsh. Can. J. Fish. Aquat. Sci. 54, 1031–1038. Stow, C.A., Lamon, E.C., Qian, S.S., Schrank, C.S., 2004. Will Lake Michigan lake trout meet the Great Lakes strategy 2002 PCB reduction goal? Environ. Sci. Technol. 38, 359–363. Tanabe, S., 1988. PCB problems in the future-foresight from current knowledge. Environ. Pollut. 50, 5–28. Tee, P.G., Sweeney, A.M., Symanski, E., Gardiner, J.C., Gasior, D.M., Schantz, S.L., 2003. A longitudinal examination of factors related to changes in serum polychlorinated biphenyl levels. Environ. Health Perspect. 111, 702–707. Tomasallo, C., Anderson, H., Haughwout, M., Imm, P., Knobeloch, L., 2010. Mortality among frequent consumers of Great Lakes sport ﬁsh. Environ. Res. 110, 62–69.

520

S. Sadraddini et al. / Journal of Great Lakes Research 37 (2011) 507–520

Turyk, M.E., Anderson, H.A., Freels, S., Chatterton, R., Needham, L.L., Patterson, D.G., Steenport, D.N., Knobeloch, L., Imm, P., Persky, V.W., 2006. Associations of organochlorines with endogenous hormones in male Great Lakes ﬁsh consumers and non-consumers. Environ. Res. 102, 299–307. Turyk, M., Anderson, H., Knobeloch, L., Imm, P., Persky, V., 2009a. Organochlorine exposure and incidence of diabetes in a cohort of Great Lakes sport ﬁsh consumers. Environ. Health Perspect. 117, 1076–1082. Turyk, M., Anderson, H.A., Knobeloch, L., Imm, P., Persky, V.W., 2009b. Prevalence of diabetes and body burdens of polychlorinated biphenyls, polybrominated diphenyl

ethers, and p, p′-diphenyldichloroethene in Great Lakes sport ﬁsh consumers. Chemosphere 75, 674–679. U.S. EPA, 2002. Great Lakes Strategy 2002 available at http://www.epa.gov/glnpo/gls/ glstoc.html Ofﬁce of Water, U.S.EPA., Washington, DC. Voiland Jr., M.P., Gall, K.L., Lisk, D.J., MacNeill, D.B., 1991. Effectiveness of recommended fat-trimming procedures on the reduction of PCB and Mirex levels in brown trout (Salmo trutta) from Lake Ontario. J. Great Lakes Res. 17, 454–460. West, M., Harrison, P.J., 1989. Bayesian Forecasting and Dynamic Models. SpringerVerlag, New York.

A BAYESIAN ASSESSMENT OF THE PCB TEMPORAL TRENDS IN LAKE ERIE FISH COMMUNITIES (Electronic Supplementary Material)

Somayeh Sadraddini1, M. Ekram Azim1, Yuko Shimoda1, Satyendra P. Bhavsar2,3,4, Ken G. Drouillard4, Sean M. Backus5, and George B. Arhonditsis1,3*

1

Ecological Modeling Laboratory

Department of Physical and Environmental Sciences University of Toronto, Toronto, Ontario, M1C 1A4, Canada 2

Ontario Ministry of Environment, Environmental Monitoring and Reporting Branch Toronto, Ontario, M9P 3V6, Canada 3

Centre for Environment

University of Toronto, Toronto, Ontario M5S 3E8

4

Great Lakes Institute for Environmental Research

University of Windsor, Windsor, Ontario, N9B 3P4, Canada 5

Water Quality Monitoring & Surveillance Division, Water Science and Technology Directorate Environment Canada, Burlington, Ontario L7R 4A6

*Corresponding author E-mail: [email protected]; Tel: +1 416-208-4858; Fax: +1 416-287-7279

A) The WinBUGS codes associated with all the single exponential decay models of the walleye skinless boneless fillet data are as follows: 1) Original approach: Non-informative (flat) prior distributions for the parameters PCB0 and k. a) First-order temporal smoothing model { for (i in 1:N) { LogPCBmod[i]<-log(PCB0*exp(k*time[i])) LogPCBm[i]<-LogPCBmod[i]+delta[time[i]+1] LogPCB[i]~dnorm(LogPCBm[i],mtau) LogPredPCB[i]~dnorm(LogPCBm[i],mtau) PredPCB[i]<-exp(LogPredPCB[i])} delta[1:31]~car.normal(adj[],weights[],num[],tau) for (i in 1:1) { weights[i]<-1; adj[i]<-i+1 num[i]<-1} for (i in 2:30) { weights[2+(i-2)*2]<-1; adj[2+(i-2)*2]<-i-1; weights[3+(i-2)*2]<-1; adj[3+(i-2)*2]<-i+1; num[i]<-2} for (i in 31:31) { weights[(i-2)*2+2]<-1; adj[(i-2)*2+2]<-i-1; num[i]<-1} mtau~dgamma(0.001,0.001) msigma<-sqrt(1/mtau) k~dnorm(0,0.0001)I(,0) PCB0~dnorm(0,0.0001)I(0,) tau~dgamma(0.001,0.001) sigma<-sqrt(1/tau) } b) Second-order temporal smoothing model { for (i in 1:N) { LogPCBmod[i]<-log(PCB0*exp(k*time[i])) LogPCBm[i]<-LogPCBmod[i]+delta[time[i]+1] LogPCB[i]~dnorm(LogPCBm[i],mtau) LogPredPCB[i]~dnorm(LogPCBm[i],mtau) PredPCB[i]<-exp(LogPredPCB[i])} delta[1:31]~car.normal(adj[],weights[],num[],tau) for (i in 1:1) { weights[i] <- 2; adj[i] <- i+1 weights[i+1] <- -1; adj[i+1] <- i+2; num[i] <- 2} for (i in 2:2) { weights[i+1] <- 2; adj[i+1] <- i-1 1

weights[i+2] <- 4; adj[i+2] <- i+1 weights[i+3] <- -1; adj[i+3] <- i+2; num[i] <- 3} for (i in 3:29) { weights[6+(i-3)*4] <- -1; adj[6+(i-3)*4] <- i-2 weights[7+(i-3)*4] <- 4; adj[7+(i-3)*4] <- i-1 weights[8+(i-3)*4] <- 4; adj[8+(i-3)*4] <- i+1 weights[9+(i-3)*4] <- -1; adj[9+(i-3)*4] <- i+2; num[i] <- 4} for (i in 30:30) { weights[(31-4)*4 + 6] <- 2; adj[(31-4)*4 + 6] <- i+1 weights[(31-4)*4 + 7] <- 4; adj[(31-4)*4 + 7] <- i-1 weights[(31-4)*4 + 8] <- -1; adj[(31-4)*4 + 8] <- i-2; num[i] <- 3} for (i in 31:31) { weights[(31-4)*4 + 9] <- 2; adj[(31-4)*4 + 9] <- i-1 weights[(31-4)*4 + 10] <- -1; adj[(31-4)*4 + 10] <- i-2; num[i] <- 2} mtau~dgamma(0.001,0.001) msigma<-sqrt(1/mtau) k~dnorm(0,0.0001)I(,0) PCB0~dnorm(0,0.0001)I(0,) tau~dgamma(0.001,0.001) sigma<-sqrt(1/tau) } 2) Prior 1: Normal PCB0 prior distribution parameterized such that 95% of the respective values lay within the minimum and maximum PCB concentrations in the first year examined. model { for (i in 1:N) { LogPCBmod[i]<-log(PCB0*exp(k*time[i])) LogPCBm[i]<-LogPCBmod[i]+delta[time[i]+1] LogPCB[i]~dnorm(LogPCBm[i],mtau) LogPredPCB[i]~dnorm(LogPCBm[i],mtau) PredPCB[i]<-exp(LogPredPCB[i])} delta[1:31]~car.normal(adj[],weights[],num[],tau) for (i in 1:1) { weights[i]<-1; adj[i]<-i+1 num[i]<-1} for (i in 2:30) { weights[2+(i-2)*2]<-1; adj[2+(i-2)*2]<-i-1; weights[3+(i-2)*2]<-1; adj[3+(i-2)*2]<-i+1; num[i]<-2} for (i in 31:31) { weights[(i-2)*2+2]<-1; adj[(i-2)*2+2]<-i-1; num[i]<-1} mtau~dgamma(0.001,0.001) msigma<-sqrt(1/mtau) k~dnorm(0,0.0001)I(,0) PCB0~dnorm(233,0.0000816)I(0,) tau~dgamma(0.001,0.001) sigma<-sqrt(1/tau) 2

} 3) Prior 2: Lognormal PCB0 prior distribution parameterized such that 95% of the respective values lay within the minimum and maximum PCB concentrations in the first year examined. model { for (i in 1:N) { LogPCBmod[i]<-log(PCB0*exp(k*time[i])) LogPCBm[i]<-LogPCBmod[i]+delta[time[i]+1] LogPCB[i]~dnorm(LogPCBm[i],mtau) LogPredPCB[i]~dnorm(LogPCBm[i],mtau) PredPCB[i]<-exp(LogPredPCB[i])} delta[1:31]~car.normal(adj[],weights[],num[],tau) for (i in 1:1) { weights[i]<-1; adj[i]<-i+1 num[i]<-1} for (i in 2:30) { weights[2+(i-2)*2]<-1; adj[2+(i-2)*2]<-i-1; weights[3+(i-2)*2]<-1; adj[3+(i-2)*2]<-i+1; num[i]<-2} for (i in 31:31) { weights[(i-2)*2+2]<-1; adj[(i-2)*2+2]<-i-1; num[i]<-1} mtau~dgamma(0.001,0.001) msigma<-sqrt(1/mtau) k~dnorm(0,0.0001)I(,0) PCB0<-exp(LnPCB0) LnPCB0~dnorm(4.441,1.3801699) tau~dgamma(0.001,0.001) sigma<-sqrt(1/tau) } 4) Prior 3: Multivariate normal prior accounting for the covariance between the parameters PCB0 and k. model { for (i in 1:N) { LogPCBmod[i]<-log(PCB0*exp(k*time[i])) LogPCBm[i]<-LogPCBmod[i]+delta[time[i]+1] LogPCB[i]~dnorm(LogPCBm[i],mtau) LogPredPCB[i]~dnorm(LogPCBm[i],mtau) PredPCB[i]<-exp(LogPredPCB[i])} delta[1:31]~car.normal(adj[],weights[],num[],tau) for (i in 1:1) { weights[i]<-1; adj[i]<-i+1 num[i]<-1} for (i in 2:30) { weights[2+(i-2)*2]<-1; adj[2+(i-2)*2]<-i-1; weights[3+(i-2)*2]<-1; adj[3+(i-2)*2]<-i+1; num[i]<-2} for (i in 31:31) { 3

weights[(i-2)*2+2]<-1; adj[(i-2)*2+2]<-i-1; num[i]<-1} mtau~dgamma(0.001,0.001) msigma<-sqrt(1/mtau) theta[1:2] ~ dmnorm(dmu[1:2], dtau[1:2, 1:2])I(P[],Q[]) dtau[1:2, 1:2] ~ dwish(R[1:2, 1:2], 2) dsigma2[1:2, 1:2] <- inverse(dtau[1:2, 1:2]) for (i in 1:2) {dsigma[i] <- sqrt(dsigma2[i, i])} k<-theta[1] PCB0<-theta[2] tau~dgamma(0.001,0.001) sigma<-sqrt(1/tau) } B) The WinBUGS code associated with the dynamic linear model for the PCB concentrations in walleye skinless boneless fillet data is as follows: model { # Specification of the observation equation for (i in 1:N) { LogPCBm[i]<-level[time[i]+1]+beta[time[i]+1]*length[i] LogPCB[i]~dnorm(LogPCBm[i],mtau[time[i]+1]) LogPredPCB[i]~dnorm(LogPCBm[i],mtau[time[i]+1]) PredPCB[i]<-exp(LogPredPCB[i])} # Specification of the system equations for the second year until the end of the study period for (t in 2:24) { beta[year[t]]~dnorm(beta[year[t-1]],btau[year[t]]) rate[year[t]]~dnorm(rate[year[t-1]],gtau[year[t]]) levelm[year[t]]<-level[year[t-1]]+rate[year[t]] level[year[t]]~dnorm(levelm[year[t]],ltau[year[t]]) # Specification of the discount factors for the second year until the end of the study period ltau[year[t]]<-ltau.in*pow(0.95,year[t]-1) lsigma[year[t]]<-sqrt(1/ltau[year[t]]) btau[year[t]]<-btau.in*pow(0.95,year[t]-1) bsigma[year[t]]<-sqrt(1/btau[year[t]]) gtau[year[t]]<-gtau.in*pow(0.95,year[t]-1) gsigma[year[t]]<-sqrt(1/gtau[year[t]]) mtau[year[t]]<-mtau.in*pow(0.95,year[t]-1) msigma[year[t]]<-sqrt(1/mtau[year[t]]) } # Specification of the system equations for the first year beta[year[1]]~dnorm(beta[1],btau[year[1]]) rate[year[1]]~dnorm(rate[1],gtau[year[1]]) levelm[year[1]]<-level[1]+growth[year[1]] level[year[1]]~dnorm(levelm[year[1]],ltau[year[1]]) # Specification of the discount factors for the first year ltau[year[1]]<-ltau.in*pow(0.95,year[1]-1) 4

lsigma[year[1]]<-sqrt(1/ltau[year[1]]) btau[year[1]]<-btau.in*pow(0.95,year[1]-1) bsigma[year[1]]<-sqrt(1/btau[year[1]]) gtau[year[1]]<-gtau.in*pow(0.95,year[1]-1) gsigma[year[1]]<-sqrt(1/gtau[year[1]]) mtau[year[1]]<-mtau.in*pow(0.95,year[1]-1) msigma[year[1]]<-sqrt(1/mtau[year[1]]) # Prior distributions for the parameters of the first year beta[1]~dnorm(0,0.0001) rate[1]~dnorm(0,0.0001) level[1]~dnorm(0,0.0001) ltau.in~dgamma(0.001,0.001) ltau[1]<-ltau.in btau.in~dgamma(0.001,0.001) btau[1]<-btau.in gtau.in~dgamma(0.001,0.001) gtau[1]<-gtau.in mtau.in~dgamma(0.001,0.001) mtau[1]<-mtau.in } Inference Data list(N=899, year=c(3,5,6,8,9,10,11,12,13,14,15,16,17,18,20,21,22,23,25,27,28,29,30,31), time=c(paste time.dat here), LogPCB=c(paste walleyePCB.dat here), length=c(paste length.dat here), Initial values 1 list(beta=c(0,NA,0,NA,0,0,NA,0,0,0,0,0,0,0,0,0,0,0,NA,0,0,0,0,NA,0,NA,0,0,0,0,0), rate=c(0,NA,0,NA,0,0,NA,0,0,0,0,0,0,0,0,0,0,0,NA,0,0,0,0,NA,0,NA,0,0,0,0,0), level=c(0,NA,0,NA,0,0,NA,0,0,0,0,0,0,0,0,0,0,0,NA,0,0,0,0,NA,0,NA,0,0,0,0,0), mtau.in=0.2, ltau.in=1, btau.in=1, gtau.in=1, LogPredPCB=c(paste walleyePCB.dat here)) Initial values 2 list(beta=c(1,NA,1,NA,1,1,NA,1,1,1,1,1,1,1,1,1,1,1,NA,1,1,1,1,NA,1,NA,1,1,1,1,1), rate=c(1,NA,1,NA,1,1,NA,1,1,1,1,1,1,1,1,1,1,1,NA,1,1,1,1,NA,1,NA,1,1,1,1,1), level=c(1,NA,1,NA,1,1,NA,1,1,1,1,1,1,1,1,1,1,1,NA,1,1,1,1,NA,1,NA,1,1,1,1,1), mtau.in=0.32, ltau.in=0.32, btau.in=0.32, gtau.in=0.32, LogPredPCB=c(paste walleyePCB.dat here)

5

AZDOC.SITE | To ensure the functioning of the site, we use **cookies**. We share information about your activities on the site with our partners and Google partners: social networks and companies engaged in advertising and web analytics. For more information, see the Privacy Policy and Google Privacy & Terms. Your consent to our cookies if you continue to use this website. Accept