A probabilistic approach to exposure risk assessment

ORIGINAL PAPER

A probabilistic approach to exposure risk assessment

Crispin M. Mutshinda Æ Imoh Antai Æ Robert B. O’Hara

Published online: 20 April 2007

� Springer-Verlag 2007

Abstract The introduction of hazardous substances into

the environment has long been recognized as being a cause

of several diseases in humans, wildlife, and plants. The

damaging character of suspected contaminants is usually

assessed via a ‘‘reject/retain’’ design with no explicit link

between levels of exposure and intensities of the potential

adverse health effects even though this connection may be

important for the development of public health regulations

that limit exposure to hazardous substances. Here, we

propose a probabilistic approach to exposure risk assess-

ment as a way around this typical flaw. We develop a

Bayesian model using proximity to the source of an alleged

contaminant as a surrogate for exposure. Subsequently, we

carry out an experimental study based on simulated data to

illustrate the model implementation with real world data.

We also discuss a possible way of extending the model to

accommodate potential heterogeneity in the spatial distri-

bution of the focal disease.

Keywords Environmental hazards � Inverse square law �Environmental risk assessment � Spatial heterogeneity

1 Introduction

The introduction of hazardous substances into the envi-

ronment has long been recognized as a potential cause of

several diseases in humans, wildlife, and plants. In many

cases, including contact with air pollutants (e.g., Hood

2003; Wilhelm and Ritz 2003), chemical spills (e.g.,

Wilhelm and Ritz 2005), and exposure to electromagnetic

radiation from stationary sources such as overhead power

lines (e.g., Wertheimer and Leeper 1979; Feychting and

Ahlbom 1993; Olsen et al. 1993; Theriault and Li 1997),

the adverse health effects are potentially enhanced by the

proximity of susceptible individuals to a source of the

contaminant and by the duration of the contact. Thus, given

a critical exposure time, distance to the source provides a

sensible proxy for exposure, in consistency with the inverse

square law, which establishes an inverse relationship be-

tween the concentration of a contaminant and the square of

the distance from the source (e.g., Bushong 1993).

The International Commission for Radiological

Protection (ICRP 1991) distinguishes between occupa-

tional, medical, and public exposures: occupational

exposure is one that occurs at workplaces and primarily as

a result of work; medical exposures are those incurred

during medical diagnosis, screening or treatment, usually

from medical equipments, whereas public exposure in-

cludes all exposures other than medical or occupational.

Because of the numerous variables involved in dealing

with medical and occupational exposures, the scope of the

methodology presented in this paper will be limited to

public exposure. We refer to exposure risk as the proba-

bility of disease occurrence in response to environmental

contamination, whereas the process by which the potential

adverse effects of exposure are assessed and characterized

is known as exposure risk assessment. Bates et al. (2003),

C. M. Mutshinda (&) � R. B. O’Hara

Department of Mathematics and Statistics,

University of Helsinki, P.O. Box 68,

Gustaf Hallstromin katu 2b, Helsinki 00014, Finland

e-mail: [email protected]

I. Antai

Department of Marketing, Supply Chain Management

and Corporate Geography, Swedish School of Economics

and Business Administration, Hanken,

P.O. Box 479, 00101 Helsinki, Finland

123

Stoch Environ Res Risk Assess (2008) 22:441–449

DOI 10.1007/s00477-007-0143-0

amongst others, delineate four major activities involved in

the exposure risk assessment process: (1) hazard identi-

fication identifies the potential hazards and the alleged

health defects, (2) exposure assessment identifies popula-

tions, which could be exposed and the possible pathways,

(3) dose–response assessment quantifies the relationship

between levels of exposure and levels of potential adverse

effects and, (4) risk characterization combines the results

of the previous three steps to determine some outcome of

interest.

In practice, the damaging character of suspected con-

taminants is usually assessed via an all-or-nothing ‘‘reject/

retain’’ design with no explicit link between levels of

exposure and intensities of the adverse health effects,

even though this connection may be important for the

development of public health regulations that limit

exposure to hazardous substances. In this paper we pro-

pose a probabilistic approach to exposure risk assessment

as a way around this typical shortcoming. We develop a

Bayesian model using proximity to the source of an al-

leged contaminant as a surrogate for exposure. Subse-

quently, we conduct an experimental study based on

simulated data to illustrate how the model can be imple-

mented with real world data whose acquisition turns out

to be highly involved, owing to numerous constraints.

Although the proposed model is essentially a log-linear

model, which has been quite extensively studied in the

statistical literature (e.g., McCullagh and Nelder 1989;

Lindsey 1997; Dobson 2002), attempts to tackle exposure

risk from a Bayesian perspective remain so far atypical,

despite the convoluted nature of environmental pollution,

which makes this approach particularly promising. We

devote the next section to describing the basics of the

Bayesian inference and Markov Chain Monte Carlo

(MCMC) methods; we refer readers interested in more

details to the appropriate literature such as Robert (2001)

and Gelman et al. (2003).

2 Theoretical background

Bayesian inference is an approach to statistical inference in

which all forms of uncertainty are expressed in terms of the

probabilities. A Bayesian analysis starts with the formu-

lation of a model, p(y|h), that is assumed to describe the

data conditionally on the unknown parameter of interest, h2Q. Subsequently, a prior distribution, p(h), intended to

embody the analyst’s state of knowledge about the plau-

sible parameter values before seeing the data is formulated.

Finally, as data become available, the prior distribution is

updated to the posterior distribution, p(h|y), by means of

the Bayes’ theorem:

p hjyð Þ ¼ p hð Þ p yjhð ÞR

Hp hð Þ p yjhð Þ dh

/ p hð Þ p yjhð Þ ð1Þ

The posterior distribution takes account of both the prior

uncertainty about the parameters and the variability in the

data, and provides a legitimate tool for inferring model

parameters and outcomes of future observations. This is all

done using probabilistic statements that turn out to be

intuitive by contrast to classical tools such as p values and

conventional confidence intervals, which are often erro-

neously interpreted. Nevertheless, two notes of caution

should be kept in mind when applying a Bayesian analysis.

First, Bayesian priors are subjective in the sense that two

analysts faced with the same problem may have different

states of knowledge about the phenomenon of interest and

consequently start from significantly different priors, which

may result in sensibly different conclusions. Most frequ-

entist statisticians view this dependence on prior specifi-

cation as conferring an arbitrary character to the Bayesian

inference whereas Bayesians consider the prospect of

combining available knowledge with data information as a

real advantage, which makes this approach a learning

process, provided any prior input is duly motivated. A prior

distribution can be based on information available in the

literature, experts’ opinions or information from any other

relevant source. On the other hand, the lack of prior

knowledge leads to the use of so-called non-informative,

‘‘flat’’ or ‘‘vague’’ priors as for example, a uniform dis-

tribution on some large compact region or a centred

Gaussian with large variance. A detailed account of the

prior specification issue is given by Spiegelhalter et al.

(1999) and Gelman (2002). Second, the posterior density is

generally not available in closed-form since the computa-

tion of normalizing constant in the Bayes’ theorem usually

involves a high dimensional integration with no analytic

solution, requiring therefore some form of numerical

approximation. The current prominence of Bayesian

applications in practically all quantitative sciences is

undeniably a consequence of the development of Markov

Chain Monte Carlo (MCMC) techniques (e.g., Gelfand and

Smith 1990; Casella and George 1992; Casella and Robert

1999; Gelman et al. 2003), which enable a direct sampling

from distributions with complex algebraic forms. Although

MCMC has been responsible for the revival in Bayesian

applications, its utility is not restricted to the Bayesian

inference. It should also be noted that computational con-

straints are not exclusive to the Bayesian approach; the

optimization problems involved in the classical maximum

likelihood method estimation may in some instances be

computationally intensive, whereas a Bayesian analysis

based on the so-called conjugate priors where the posterior

has the same algebraic form as the prior (e.g., Binomial

442 Stoch Environ Res Risk Assess (2008) 22:441–449

123

likelihood with Beta prior or Poisson likelihood with

Gamma prior) is always straightforward. All members of

the exponential family have conjugate priors (e.g., Gelman

et al. 2003). The motivation for using conjugate priors

remains essentially computational convenience. Indeed,

non-conjugate priors may be preferable in many circum-

stances, and MCMC methods provide a potential key to

computational hurdles. A sketch of the principles of

MCMC techniques is provided below.

The underlying principle of MCMC is to set up a suit-

able Markov chain with the desired posterior density as its

stationary distribution. Then, starting from an arbitrary

state in the parameter space, to simulate the chain until it

converges. One way of assessing convergence is to plot

the trajectories hj(t) against the iteration number t,

for j = 1,...,dim(h) (trace plots), and judge convergence

in an informal visual inspection. It might be easier for each

parameter’s component when multiple chains starting from

different initial states are run simultaneously, to judge

convergence by the mixing between the chains. When the

chain has practically converged, it is then necessary to

ignore the early, pre-convergence, part called the ‘‘burn-

in’’ period in order to avoid the effects of the initial choice

of values. After burn-in removal, one usually simulates the

chains for a number of additional iterations. The most

popular MCMC algorithm is the Metropolis–Hastings

(Metropolis et al. 1953; Hastings 1970), which is briefly

described below.

Let h = (h1, ...,hd) denote the d-dimensional parameter of

interest and p(h|y) the required posterior based on the data

y, and let q(h, h*) = p(h*|h) be a proposal kernel where h is

the current state and h* a (candidate) proposal. The

Metropolis–Hastings (MH) algorithm proceeds as follows:

Algorithm 1: Metropolis–Hastings

1. Pick arbitrarily ( )0θ in the support of ( )|p yθ and set 0i =

2. Generate a proposal *θ from ( ) ( )( )( ),. . | iiq pθ θ=

3. Compute ( )( ) ( )( )( )

( )( )( )( )**

*

*

,|, min , 1

| ,

i

i

i i

qp y

p y q

θ θθα θ θ

θ θ θ

⎧ ⎫⎪ ⎪= ⎨ ⎬⎪ ⎪⎩ ⎭

Generate ( )~ 0,1Unifβ and set ( )1 *iθ θ+ = if ( )( )*,iβ α θ θ< . Otherwise set ( ) ( )1i iθ θ+ =

Repeat 2-3 until “convergence”.

Since a (h(i), h*) only depends on p(h|y) through the ratio

p h�jyð Þp h ið Þjyð Þ ; the normalizing constant of p(h|y) cancels out.

The choice of the proposal distribution is essentially

arbitrary subject only to technical constraints such as the

minimization of the autocorrelation. The original

Metropolis algorithm ‘‘Metropolis chain’’ (Metropolis

et al. 1953) uses a symmetric proposal distribution where

q(h*, h(i)) = q(h(i), h*), as for example, a normal dis-

tribution centered at the current state. A symmetric pro-

posal has the practical advantage that a (h(i), h*) takes the

simpler form a h ið Þ; h��

¼ p h�jyð Þp h ið Þjyð Þ : A typical Metropolis

chain is the random walk Metropolis whose proposal

kernel is h* = h(i) + se where e ~ mvnorm(0,S), S is a

suitably chosen covariance matrix, and s is a constant,

usually tuned over the first iterations to get an acceptance

rate of between 20 and 40% by running the chain for

different values of s and monitoring the acceptance rate.

The Gibbs sampler (e.g., Gelfand and Smith 1990) is a

particular case of the M-H algorithm where the proposal

is always accepted (a = 1). The key to the Gibbs sampler

is that one considers univariate conditional distributions.

So, at each step, n random variables are generated

sequentially from n univariate conditionals rather than a

single n-dimensional vector. This supposes that the com-

plete conditional posteriors p(hj|hi, i „ j) are available in

closed-forms. The Gibbs algorithm proceeds as follows:

Algorithm 2: Gibbs sampler

1. Pick arbitrarily ( ) ( ) ( )( )0 0 01 , . . . , dθ θ θ= and set 0i =

2. Generate ( ) ( ) ( )( )11 1 2~ | , ..., ,i i i

kp yθ θ θ θ+ , ( ) ( ) ( ) ( )( )1 12 2 1 1~ | , ,..., ,i i i i

kp yθ θ θ θ θ+ + ,

…, ( ) ( ) ( )( )1 1 11 1~ | ,..., ,i i i

k k kp yθ θ θ θ+ + +−

3. Set 1i i= + and repeat steps 2-3 until “convergence”.

Underlying the usual sample-based inference for a scalar

quantity is the assumption of ‘‘independent and identically

distributed’’ (IID) observations, which does not generally

apply to simulated posterior draws. The autocorrelation

function (ACF) provides the serial correlation between

observations that are k iterations apart. If autocorrelations

die out at lag k, say, then thinning the chain at every nth

observation with n > k, yields a roughly IID sample.

Having obtained the joint conditional distribution of the

unknown variables, one usually needs to marginalize over

the nuisance variables (variables that are not of current

interest) to obtain the distribution of just the quantities of

interest. Mathematically, the required marginalization is

achieved by integrating out the nuisance variables. For a

joint posterior simulated via MCMC, a sample from the

marginal distribution of a component of the parameter

vector can be obtained by overlooking the nuisance

variables.

MCMC algorithms are usually easy to implement. In

practice, a wide range of Bayesian models can be fit-

ted using OpenBUGS, a Bayesian software package freely

available at http://www.mathstat.helsinki.fi/openbugs/

(Thomas et al. 2006).

Stoch Environ Res Risk Assess (2008) 22:441–449 443

123

http://www.mathstat.helsinki.fi/openbugs/

3 Materials and methods

A supposedly exposed area is partitioned into a number of

sectors according to the distances, expressed in a suitable

unit, to the source of the suspected contaminant. Here we

restrict our attention to individuals whose exposure times

exceed a given critical limit and assume that a complete

census of this target population is available in order to avoid

a lengthy discussion on sampling issues. We assume that the

number, yi of affected individuals in sector i is Poisson-

distributed with intensity k i = l *f(di) where l is a dis-

tance-independent ‘‘baseline’’ intensity, di is the distance of

the focal area to the source, and f is a decaying function of

the distance so that the more remote from the source an

individual lives the less vulnerable that individual is. For

example, f dið Þ ¼/ 1

dið Þ2; di 6¼ 0 or f(di) = exp (–adi), which

is used here. More specifically, we assume that

yi � Pois k ið Þ; and k i ¼ wi l exp �a dið Þ ð2Þ

where a 2[0, +¥) is a parameter, which reflects the

strength of association between the suspected contaminant

and the alleged health defect, and the ‘‘population

weights’’ wi > 0 are known scaling factors intended to

correct for disparate population sizes in different sectors.

The population weights can be interpreted as follows: if, w

is set to 1 in a reference sector i whose population size is ni,

then the population size, nj of an arbitrary sector j is to be

corrected by a factor wj ¼ ni

nj: This correction would not be

necessary if the model was based on the proportions of

affected individuals in each sector, in which case a multi-

nomial likelihood would be more appropriate, condition-

ally on the total population size in the area under study.

The Poisson distribution adopted here is known to be an

appropriate model for data arising in form of counts (e.g.,

Gillman and Hails 1997; Davison 2003), in particular when

the number of cases is much less than the number of ex-

posed individuals. A significant a (significantly different

from zero) provides some evidence in favor of the asso-

ciation between the presumed contaminant and the alleged

disease (but one must bear in mind that correlation does not

necessarily imply causality). Unlike the classical setting

where evidence against the null hypothesis of no associa-

tion is often investigated, the Bayesian approach enables

estimating the probability of the alleged health defect at

different levels of exposure, which is important for the

development of public health regulations regarding expo-

sure to hazardous substances.

It is particularly important to extend the study over a

number of similar situations in order to examine the effects

of potential environmental confounders and mitigate the

likelihood of false alarms. Indeed, it might be the case that

the collected data are not sufficient to identify the subtle

effects of exposure while properly adjusting for con-

founders. The model as presented in Eq. 2 can be

straightforwardly extended to a number of, say, K areas as

follows:

yi;k � Pois k i;k

� �;andk i;k ¼wi;k lk exp �a dið Þ; 16 k6 K

ð3Þ

where the parameters keep the same meaning as in Eq. 2,

but lk is now the baseline intensity associated with area k

only, and wi, k corrects for the population size in sector i of

area k.

Spatial homogeneity in the distribution of the focal

disease across different areas can be examined by testing

for equality of the baseline failure rates lk, for example via

some ANOVA design. Significant differences in baseline

failure rates across areas would suggest the relevance of

anonymous locally acting factors and the necessity of

extending the model to accommodate potential confound-

ers. This can be achieved within the flexible hierarchical

Bayesian framework (e.g., Robert 2001; Gelman et al.

2003), which allows the decomposition of a prior distri-

bution into several conditional levels. By assigning for

example Gamma(tk, bk) priors to the baseline intensi-

ties, lk a spatial discrimination can be worked out from

posterior inference on the hyper-parameters tk and bk.

This approach provides a rational way of modeling the risk

of diseases in connection with the spatial distribution of

populations, as a practical way of integrating the areas of

environmental and biomedical sciences.

After suitable priors have been specified for the model

parameters, the joint posterior can be worked out or sim-

ulated numerically via MCMC. The posterior distribution

or the simulated sample from it includes uncertainty about

the parameters, which can be incorporated into subsequent

inferences. For example, conditionally on the observed

data, y, the predictive distribution, p(Yd|y), of affected

individuals in a sector located d units of distance from the

source of a contaminant and which population weight is w

can be simulated from by using the following algorithm:

Algorithm 3: Simulating a posterior predictive distri-

bution

Set b to 0

1. Generate ( ) ( )* *, ~ , |b b p yµ α µ α and compute ( )* * *expb b bwλ µ α δ= −

2. Generate ( )bb Poisy ** ~ λ

b=b+1

Repeat 1 and 2 until b=B


123

y* = (y*1, ..., y*B) is then a sample of size B from the

posterior predictive distribution of interest, formally de-

fined as

p Y djy� �

¼Z

H

p Y djh� �

p hjyð Þ dh ð4Þ

Notice that in Eq. 4 the likelihood of the data is averaged

across the uncertainty contained in the posterior distribu-

tion of the parameter. Hence, mean(y*) provides an

estimate of E Ydjy� �

;F�1y�

12

� �estimates the corresponding

median, while the a2

and 1� a2

� �percentiles of y* provide

approximate cutoff points for the (1 – a)% credibility set

(Bayesian confidence interval) for (Yd|y). Moreover, the

risk of (Yd|y) reaching a critical level C is estimated by

Pr Yd>Cjy

� �¼ 1

B

PBb¼ 1 1 y�b>Cf g where 1 y�b�Cf g denotes

the indicator function of {y*b ‡ C}.

The usual asymptotic Gaussian approximation applies to

posterior samples. Indeed, as the sample size increases, the

joint posterior tends to be multivariate normal with the

posterior mode as approximate mean and the asymptotic

covariance matrix given by the negative inverse of the

Hessian of the log-posterior evaluated at the posterior

mode. Consequently, an approximate (1 – a)% credibility

set for a scalar parameter h can be obtained by the usual

recipe: h� za=2 � Var h� �1

2

where h is an estimate of the

posterior mode of h;Var h is the asymptotic variance of h;and za /2 the a

2-percentile of the standard normal distribu-

tion. Unlike the classical (1 – a)% confidence interval, a

(1 – a)% credibility set has probability (1 – a)% of con-

taining the true value of the parameter.

The performance of statistical models is often assessed

through simulation studies because the ‘‘true’’ model from

which the data have been generated is known to the ana-

lyst. This enables a judgment of the extent to which the

underlying mechanisms can be revealed by the model-

based analyses. The next section is devoted to a practical

application of the model based upon simulated data, which

is intended to illustrate the model implementation with real

world data on environmental pollution whose acquisition is

particularly complicated, since beyond the marked bud-

getary constraints, it involves dealing with subject privacy

concerns, economic and even political interests. On the

other hand, sufficient data are needed to identify the subtle

effects of exposure while suitably adjusting for potential

confounders and mitigate the likelihood of false alarms.

4 Report on the simulation study

4.1 Parameters and settings

We used computer simulation to generate a dataset ex-

tended to three areas partitioned in 30 equally populated

sectors each, so that all the weight factors wi were set to 1.

The case of unevenly populated sectors could be straight-

forwardly handled by an appropriate scaling of the linear

predictor mi,k defined below. In each area, the distances of

the 30 sectors to the source were assigned values ranging

from 0.1 to 3 in increments of 0.1. In order to ensure that

the results are not obtained by chance, the ‘‘true’’ baseline

intensities,lk, and the parameter a were assigned different

values in the three areas: l = 100; a = 2 in the first

area, l = 20; a = 1 in the second, and l = 10; a = 0.5 in

the third area. The R-script for data generation, a resulting

dataset (in the BUGS format), and the BUGS code for the

model fitting are provided in Appendices 1, 2 and 3.

4.2 Analyses, results and discussion

The log-rescaled intensity derived from Eq. 3 as

mi;k ¼ logki; k

xi; k

� �¼ lmk � a � di; where lmk = log (lk),

provides a linear predictor in d that can be used to fit the

model. We used a Bayesian approach with non-informative

priors: lmj ~ N (0, 0.01) and a ~ Unif(0, 100), where the

normal distribution is parameterized in terms of the mean

and the precision (inverse of the variance). The required

posteriors were simulated using MCMC methods via

OpenBUGS. We ran 30,000 iterations of three chains,

discarding the first 10,000 iterations as burn-in and thin-

ning the remainder to one in every tenth observation.

Convergence of the MCMC was assessed visually by the

mixing of the chains. The sensitivity of the results to the

prior inputs was examined by varying the range of the

priors by orders of magnitude, but the results remained

similar, suggesting non significant influence of the prior

specification on the results obtained.

Table 1 gives the posterior means, standard errors, as

well as lower and upper bounds of the 95% credibility sets

for each parameter, whereas Figs. 1, 2 and 3 display trace

plots of 3,000 posterior draws from the MCMC outputs,

which illustrate the mixing of the chains. We can see that the

MCMC sampler jumps freely around the parameter space,

and that the resulting estimates are close to the true values.

The posterior autocorrelations (Fig. 4) die out at lag 5,

suggesting that the usual IID assumption is reasonable with

a thinning to every kth observation with k > 5. Conse-

quently, we thinned our samples to k = 10. Once the pos-

teriors of the parameters have been estimated, Algorithm 3

can be used to predict the distribution of affected individ-

uals in a specific area, given its distance from a source of

contamination and its population weight.

The right panel in Fig. 5 shows that the variables l and

a are not strongly correlated (the first area has been se-

lected for illustration). A rough IID sample from the

marginal distribution of one parameter can be obtained

from the simulated joint posterior by overlooking the other.


123

The normal QQ-plots in the left and the central panels of

Fig. 5 validate the asymptotic normality of the posteriors

for the two variables, which allows the classical hypothesis

and significance test based on large-sample Gaussian

approximation. This illustrates further the flexibility of the

sample-based posterior analysis, in particular the fact that it

does not completely break with the classical approach.

Bayesian analyses based on vague priors are known to

yield similar results to the classical maximum likelihood

estimation, and frequentist statisticians often think that the

two approaches are equivalent. Indeed, even though the

results from the two approaches may seem identical at first

glance, their interpretation is not always the same (e.g.,

confidence intervals are intervals for statistics that would

be calculated from replicate data sets whilst credibility

intervals are intervals for possible values of parameters). In

addition, under the Bayesian framework, the posterior from

a previous analysis can serve as prior input as new data are

obtained, which renders the Bayesian approach more

fruitful when non-trivial prior information is available as is

often the case. More importantly, if prediction about future

observations is of concern, the two approaches may lead to

significantly different results. Indeed, suppose that one has

gotten data y = (y1, ...,yn) and wants to infer about a future

observation, say Pr (yn+1 2 A|y). A solution under classical

statistics, based on the ‘‘plug-in’’ principle, is to compute

Pr ynþ1 2 Ajh ¼ h� �

which ignores the uncertainty of the

estimate h by conditioning on an event that is known to be

only approximately true. On the other hand, the Bayesian

Table 1 Posterior summaries; each parameter is indexed by the

corresponding area

Parameter True value Posterior mean SE 2.5 pc 97.5 pc

l1 100 103.6 7.31 89.75 118.70

l2 50 50.2 3.87 43.0 58.12

l3 10 10.11 1.42 7.50 13.09

a1 2.0 2.046 0.09 1.86 2.24

a2 1 0.99 0.06 0.87 1.12

a3 0.5 0.48 0.002 0.30 0.67

mu[1]

iteration26999 28000 29000

80.0

100.0

120.0

140.0

alpha[1]

iteration26999 28000 29000

1.6

1.8

2.0

2.2

2.4

2.6

Fig. 1 3,000 MCMC steps of

the posteriors of l (top) and

a (bottom) for the first area

plotted against the iteration

number. The true parameter

values are l = 100 and a = 2.

This figure, as well as Figs. 2

and 3, illustrates the mixing of

the three chains

mu[2]

iteration26999 28000 29000

30.0

40.0

50.0

60.0

70.0

alpha[2]

iteration26999 28000 29000

0.6

0.8

1.0

1.2

1.4


the posteriors of l (top) and

a (bottom) for the second area



values are l = 50 and a = 1


123

solution consists of averaging the likelihood of possible

outcomes over the posterior distribution of the parameter.

That is, calculating Pr ynþ1 2 Ajyð Þ ¼R

A

RH p ynþ1; hjyð Þ

dh dynþ1: It turns out that the posterior predictive distri-

bution includes uncertainties inherent to the parameter

estimate and to the fact that any future value is itself a

random event, whereas only the second source is taken into

account by the classical counterpart. Consequently, pre-

diction intervals based on classical statistics tend to be too

short. However, when the number of observations gets

infinitely large compared to the number of parameters, the

two approaches often agree (e.g., Robert 2001).

5 Concluding remarks

In this paper we have been concerned with the typical flaw

of the missing link between levels of exposure to a con-

taminant and intensities of the alleged health defect. We

proposed a probabilistic approach as a way around this

mu[3]

iteration26999 28000 29000

5.0

10.0

15.0

20.0

alpha[3]

iteration26999 28000 29000

0.0

0.2

0.4

0.6

0.8

1.0


the posteriors of l (top) and a(bottom) for the third are,



values are l = 10 and a = 0.5

mu[1]

lag0 20 40

-1.0-0.50.00.51.0

-1.0-0.50.00.51.0

-1.0-0.50.00.51.0

mu[2]

lag0 20 40

mu[3]

lag0 20 40

alpha[1]

lag0 20 40

-1.0 -0.5 0.0 0.5 1.0

alpha[2]

lag0 20 40

-1.0 -0.5 0.0 0.5 1.0

alpha[3]

lag0 20 40

-1.0 -0.5 0.0 0.5 1.0

Fig. 4 Estimated

autocorrelation functions for all

six parameters. In all cases, the

autocorrelation vanishes

practically at lag 5


123

shortcoming. We developed a Bayesian model dealing with

a situation where the distance from the source of an alleged

contaminant was used as a proxy for exposure, and con-

ducted an experimental study based upon simulated data to

illustrate the model implementation with real world data.

The model was fitted to the data using Markov chain Monte

Carlo methods via the OpenBUGS software. We pointed

out some difficulties connected with the acquisition of

actual data on environmental pollution and emphasized the

necessity for analysts to ensure the adequateness of the data

to identify the effects of the presumed contaminant (if any)

while properly adjusting for confounders, in order to mit-

igate the likelihood of false alerts. We dealt with time by

restricting the target population to susceptible individuals

whose exposure time exceeded a given critical level. To

avoid a lengthy discussion on sampling issues, we assumed

that a census of the target population was available. We

maintain, however, that more insight can be gained by

extending the study to all susceptible individuals and

treating the exposure time as a covariate. The model as

presented here can be tailored to situations with different

exposure proxies such as the concentration of chemicals in

drinking water.

Bayesian inference provides a coherent framework for

learning from evidence as it accumulates, with the

attractive feature that once a model is defined, all answers

follow directly from probability theory, often with an

insightful meaning. Recent advances in computational

algorithms exemplified by the advent of MCMC have

stimulated a tremendous increase in the use of Bayesian

methods in most quantitative sciences over the last dec-

ade. Researchers in different fields would be well advised,

as a matter of practical necessity, to familiarize with the

basics of the Bayesian methodology, to ensure at least

that they are in a position to understand and discuss the

increasing number of Bayesian reports in the literature

(e.g., Tan 2001).

Acknowledgments We are indebted to the OpenBUGS develop-

ment team for making this software package freely available.

Appendix 1

R-code for data generation

d<-seq(0.1,3,0.1);

lbda<-matrix(0,nrow=30,ncol=3)

y<-matrix(0,nrow=30,ncol=3)

mu<-c(100,50,10)

alpha<-c(2,1,0.5)

for(j in 1:3){

for(i in 1:30){

lbda[i,j]<-mu[j]*exp(-alpha[j]*d[i])

y[i,j]<-rpois(1,lbda[i,j])} }

Appendix 2

Data in the BUGS format

list (d = c( 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9,

1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7,1.8,1.9, 2.0, 2.1,

2.2, 2.3, 2.4,2.5, 2.6, 2.7, 2.8, 2.9, 3.0), y = structure

(.Data= c (80, 66, 61, 44, 39, 26, 27, 26, 14, 14, 14, 5,

10, 7, 6, 4, 3, 0, 0, 1, 3, 1, 2, 1, 0, 0, 1, 0, 0, 0, 34,

47, 41, 37, 21, 29, 32, 35, 22, 13, 14, 11, 12, 13, 12, 10,

7, 11, 6, 6, 4, 4, 8, 6, 3, 5, 1, 3, 5, 3, 11, 9, 7, 7, 7,

3, 9, 8, 14, 5, 6, 5, 3, 4, 7, 4, 8, 2, 2, 4, 4, 4, 4, 0,

5, 6, 3, 0, 2, 2), .Dim = c(3, 30)))

-3 -2 -1 0 1 2 3

9010

011

012

0

Normal Q-Q Plot Normal Q-Q Plot

mu 1

Sam

ple

Qua

ntile

s

-3 -2 -1 0 1 2 3

.18

1.9

2.0

2.1

.22

2.3

2.4

alpha 1

Sma

lpQ e

autnil

s e

1.8 1.9 2.0 2.1 2.2 2.3 2.4

0910

011

02 10

alpha 1

m1 u

Fig. 5 From left to right: normal QQ-plots of the posteriors of l1, QQ-plots of the posteriors of a1, and cloud of the pairs (a1, l1). The first area

has been arbitrarily selected for illustration


123

Appendix 3

BUGS-code for the model fitting

model{

for(i in 1:3){

for(j in 1:30){

lbda[i,j]<-exp(moy[i,j])

moy[i,j]<-lm[i]-alpha[i]*d[j]

y[i,j]~dpois(lbda[i,j])}

mu[i]<-exp(lm[i])}

for (i in 1:3){

lm[i]~dnorm(0,0.01)

alpha[i]~dunif(0,100)}}

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A probabilistic approach to exposure risk assessment