Using decomposed household food acquisitions as inputs of a
Kinetic Dietary Exposure Model.
Olivier Allais
and Jessica Tressou
Abstract
Foods naturally contain a number of contaminants that may have different and long term
toxic effects. This paper introduces a novel approach for the assessment of such chronic food
risk that integrates the pharmakokinetic properties of a given contaminant. The estimation of
such a Kinetic Dietary Exposure Model (KDEM) should be based on long term consumption
data which, for the moment, can only be provided by Household Budget Surveys such as the
SECODIP panel in France. A semi parametric model is proposed to decompose a series of
household quantities into individual quantities which are then used as inputs of the KDEM.
As an illustration, the risk assessment related to the presence of methylmercury in seafoods is
revisited using this novel approach.
Keywords: household surveys, individualization, linear mixed model, risk assessment,
spline-estimation.
INRA-CORELA, Laboratoire de recherche sur la consommation, Ivry sur Seine, France.
INRA-Mét@risk, Méthodologies d’analyse des risques alimentaires, Paris, France and Hong Kong University of
Science and Technology, Department of Information and Systems Management, Hong Kong.
The second author research is in part supported by Hong Kong RGC Grant #601906.
Corresponding author: Dr Jessica Tressou; Mail address: Hong Kong University of Science and Technology,
ISMT, Clear Water Bay, Kowloon, Hong Kong; Email: tressou@inapg.fr.
http://arxiv.org/abs/0704.0517v1
Introduction
The quantitative assessment of dietary exposure to certain contaminants is of high priority to the
Food and Agricultural Organization and the World Health Organization (FAO/WHO). For exam-
ple, excessive exposure to methylmercury, a contaminant mainly found in fish and other seafood
(mollusks and shellfish) may have neurotoxic effects such as neuronal loss, ataxia, visual disturbance,
impaired hearing, and paralysis (WHO, 1990). Quantitative risk assessments for such chronic risk
require the comparison between a tolerable dose of the contaminant called Provisional Tolerable
Weekly Intake (PTWI) and the population’s usual intake. The usual intake distribution is gener-
ally estimated from independent individual food consumption surveys (generally not exceeding 7
days) and food contamination data. Several models have been developed to estimate the distribu-
tion of usual dietary intake from short-term measurements (see for example, Nusser et al., 1996;
Hoffmann et al., 2002). The proportion of consumers whose usual weekly intake exceeds the PTWI
can then be viewed as a risk indicator (see for example, Tressou et al., 2004). This kind of risk
assessment does not account for the underlying dynamic process, i.e. for the fact that the contami-
nant is ingested over time and naturally eliminated at a certain rate by the human body. Moreover,
longer term measurements of consumption are available through household budget surveys (HBS).
In this paper, we propose to use HBS data to quantify individual long term exposure to a
contaminant. This data provides long time series of household food acquisitions which are first used
in a decomposition model, similar to the one proposed by Chesher (1997, 1998) in the nutrition
field, in order to obtain time series of individual intakes. Then, the pharmacokinetic properties of
the contaminant are integrated into an autoregressive model in which the current body burden is
defined as a fraction of the previous one plus the current intake.
From a toxicological point of view, this approach is, to our knowledge, novel and hence requires
the definition of an ad-hoc long term safe dose as proposed in the next section. We refer to this
autoregressive model as Kinetic Dietary Exposure Model (KDEM).
From a statistical point of view, such autoregressive models are well known in general time series
analysis (see for example, Hamilton, 1994) and most of the paper is devoted to the description of
the decomposition model. This statistical model aims at estimating individual quantities from total
household quantities and structures. This problem is similar to that studied by Engle et al. (1986),
Chesher (1997, 1998), and Vasdekis and Trichopoulou (2000), and is addressed in a slightly different
way. In the present article, the individual contaminant intake is firstly viewed as a nonlinear
function of age within each gender, with time and socioeconomic characteristics being secondly
introduced in a linear way. The nonlinear function is represented by a truncated polynomial
spline of order 1 that admits a mixed model spline representation (section 4.9 in Ruppert et al.,
2003). These choices yield a simple linear mixed model which is estimated by REstricted Maximum
Likelihood (REML, Patterson and Thompson, 1971). One major extension of the proposed model
compared to Chesher (1997) is the introduction of dependence between the individual intakes of a
given household.
In the next section, focusing on the methylmercury example even though the method is much
more general and could be applied to any chronic food risk, SECODIP data are described along
with the construction of a household intake series and the individual cumulative and long term
exposure concepts yielding the KDEM. Section 2 is devoted to the statistical methodology used to
decompose the household intake series into individual intake series, namely the presentation of the
model and its estimation and tests. Section 3 displays the results for the quantification of long term
exposure to methylmercury of the French population using the 2001 SECODIP panel. Finally, a
discussion on the use of household acquisition data, with the focus on the French SECODIP panel,
is conducted in section 4 with respect to the proposed long term risk analysis.
1 Motivating example: risk related to methylmercury in seafoods
in the French population
In this section, the Kinetic Dietary Exposure Model (KDEM) and the concept of long term risk are
defined. Then a brief panorama of consumption data in France is given and the way the SECODIP
HBS data will be used as an input of the KDEM is described.
1.1 Cumulative exposure and long term risk: the Kinetic Dietary Exposure
Model (KDEM)
The main objective of the analysis is to assess individuals’ long term exposure to a contaminant to
deduce whether these individuals are at risk or not. As mentioned in the introduction the only ”safe
dose” reference is the PTWI expressed in terms of body weight (relative intake). Unfortunately,
TNS SECODIP did not record the body weight of the individuals until 2001. The body weights
are thus estimated from independent data sets; namely the French national survey on individual
consumption (INCA, CREDOC-AFSSA-DGAL, 1999) for people older than 18, and the weekly
body weight distribution available from French health records (Sempé et al. (1979)) for individuals
under 18. In both cases, gender differentiation is introduced.
Assume that estimations of the individual weekly intakes are available, that is yi,h,t denotes the
intake of individual i belonging to household h for the tth week (with i = 1, . . . , nh,t; h = 1, . . . ,H
and t = 1, . . . , T ), and Di,h,t denotes the same quantity expressed on a body weight basis. The
cumulative exposure up to the tth week of this individual is then given by
Si,h,t = exp(−η) · Si,h,t−1 +Di,h,t, (1.1)
where η > 0 is the natural dissipation rate of the contaminant in the organism. This dissipation
parameter is defined from the so called half life of the contaminant,which is the time required for
the body burden to decrease by half in the absence of any new intake. For methylmercury, the half
life, denoted by l1/2, is estimated to 6 weeks, so that η = ln(2)/l1/2 := ln(2)/6 (Smith and Farris,
1996).
The autoregressive model defined by (1.1) and a given initial state Si,h,0 = Di,h,0 has a stationary
solution since exp(−η) < 1. As a convention, Si,h,0 is set to the mean of all positive exposures
(Di,h,t)t=1,. . . ,T . However, this convention has little impact on the level of an individual’s long term
exposure since the contribution of the initial state Si,h,0 tends to zero as t increases. We call this
autoregressive model ”KDEM” for Kinetic Dietary Exposure Model.
The individual cumulative exposure Si,h,t can be considered to be the long term exposure of an
individual for sufficiently large values of t. For methylmercury, the long term steady state of the
individual exposure to a contaminant is reached after 5 or 6 half lives according to Dr P. Granjean,
a methylmercury expert. Thus, the long term individual’s exposure to methylmercury is defined
as the cumulative exposure reached after say 6l1/2 = 36 weeks.
The risk assessment usually consists of comparing the exposure with the so called Provisional
Tolerable Weekly Intake (PTWI). This tolerable dose, determined from animal experiments and
extrapolated to humans, refers to the dose an individual can ingest throughout his entire life
without appreciable risk. For methylmercury, the PTWI is set to 1.6 microgram per kilogram of
body weight per week (1.6 µg/kg bw, see FAO/WHO, 2003).
In our dynamic approach, the long term exposure is compared to a reference long term exposure
denoted by Sref , and defined as the cumulative exposure of an individual whose weekly intake is
equal to the PTWI, d, such as
Sref = lim
1− exp(−η) , (1.2)
where
d exp(−η(t− s)) = dexp(−η(t+ 1))− 1
exp(−η)− 1 . (1.3)
For methylmercury, the reference for long term exposure Sref is 14.6 µg/kg bw. An individual
is then assumed to be at risk if his cumulative exposure Si,h,t exceeds the reference S
t for any
t > 6l1/2.
This KDEM model requires some long surveys of individual intakes which are not monitored
and can only be approximated from available consumption data and contamination data.
1.2 From household acquisition data to household intake series
Two current major consumption data sources in France are the national survey on individual
consumption (INCA, CREDOC-AFSSA-DGAL, 1999) and the SECODIP panel managed by the
company TNS SECODIP. Most quantitative risk assessments conducted by the French agency for
food safety (AFSSA) use the 7 day individual consumption data of the INCA survey jointly with
contamination data collected by several French institutions. Regarding methylmercury, seafood
contamination data have been collected through different analytical surveys (MAAPAR, 1998-2002;
IFREMER, 1994-1998) and were used in Tressou et al. (2004) and Crépet et al. (2005) combined
with the INCA survey. In this paper, a methodology using the SECODIP data is developed (see
Boizot, 2005, for a full description of this database).
The company TNS SECODIP has been collecting the weekly food acquisition data of about five
thousand households since 1989. All participating households register grocery purchases through
the use of EAN bar codes but other grocery purchases are registered differently: the fresh fruit
and vegetable purchases are recorded by the FL sub-panel while fresh meat, fresh fish and wine
purchases are recorded by the VP sub-panel. The households are selected by stratification according
to several socioeconomic variables and stay in the survey for about 4 years. TNS SECODIP provides
weights for each sub-panel and each period of 4 weeks to make sure of the representativeness of
the results in terms of several socioeconomic variables. TNS SECODIP also defines the notion of
household activity which refers to the correct and regular reporting of household purchases over a
year. For each household, the age and gender of each member of the household are retained in our
decomposition model with some socioeconomic variables: the region, the social class (from modest
to well-to-do), the occupation category and level of education of the principal household earner.
For methylmercury risk assessment, the households of the VP panel are considered; in the 2001
data set, there are H = 3229 active households (corresponding to 9288 individuals) and T = 53
weeks during which the households may or may not acquire seafood. The weekly purchases of
seafood are clustered into two categories (”Fish” and ”Mollusks and Shellfish”) for which the mean
contamination levels are calculated from the MAAPAR-IFREMER data and are given in table 1.
Table 1 around here, see page 21
Household intake series ((yh,t)h=1,...,H;t=1,...,T) are computed as the cross product between weekly
purchases of seafoods which are assimilated to weekly consumptions, and mean contamination
levels. They are expressed in micrograms per week (µg/w). The food ”purchase-consumption”
assimilation is of course arguable and will be the main subject of the final discussion (see section
4). An additional assumption concerns the household size, denoted by nh,t for the household h
and the week t. This can indeed vary over time in the case of a birth or death of a household
member. Since a new born baby will not consume fish in his first few months, we assume that
food diversification (and hence consumption of seafoods) starts at one year of age, yielding a total
sample of 8913 individuals for the 2001 panel. These household intake series are then decomposed
into individual intake series using the model described in the next section. These individual intake
series are then used as imputs of the KDEM.
2 Statistical methodology
In this section, the decomposition model is described and compared to similar models described in
the literature, namely Chesher (1997, 1998); Vasdekis and Trichopoulou (2000). Its estimation and
some structure tests are then presented.
2.1 The decomposition model
2.1.1 General principle
Consider a household composed of nh,t members, each member having unobserved weekly intakes
yi,h,t, with i = 1,. . . , nh,t, h = 1,. . . ,H, and t = 1,. . . , T . The week t intake of a household h is
simply the sum across household members of the individual weekly intakes, such as
yh,t =
nh,t∑
yi,h,t. (2.1)
As detailed below, the individual weekly intake yi,h,t is assumed to depend on
• the age and gender of the individual via a function f,
• some socioeconomic characteristics of the household,
• time (seasonal variations).
There are obviously several ways to model the individual intake under these assumptions
and this choice leads to more or less simple estimation procedures. In Chesher (1997, 1998);
Vasdekis and Trichopoulou (2000), a discretization argument on age is used leading to a penalized
least square estimation of a great number of parameters, that is one parameter for each year of age
and gender. We propose to use a truncated polynomial spline of order 1 for each gender, which
admits a mixed model spline representation for f. As far as socioeconomic characteristics are con-
cerned, Chesher (1997) retained a multiplicative specification whereas Vasdekis and Trichopoulou
(2000) chose the additive one. In the multiplicative model, a change in income for example would
proportionally affect all the individual intakes whereas in the additive setting, they would be af-
fected by the same value. Following Vasdekis and Trichopoulou (2000), we retained the additive
specification since the difference between the two specifications may not be notable, and the addi-
tive setting yields to a much simpler estimation procedure (linear model). Finally, time dependency
is only introduced in Chesher (1998) to track changes with age within cohorts: this time depen-
dency is directly introduced into the function f that is bivariately smoothed according to age and
time (cf. Green and Silverman, 1994). Again, we adopt a simpler specification in which time is
introduced as a dummy variable. All these assumptions yield an individual model of the form
yi,h,t = xi,h,tβ + zi,h,tu+ wh,tγ + δtα+ εi,h,t, (2.2)
where the terms xi,h,tβ + zi,h,tu stand for the mixed model spline representation of the function
f, the term wh,tγ denotes the socioeconomic effects, the term δtα the time effect, and εi,h,t is the
individual error term.
Combining (2.1) and (2.2) , we obtain the final rescaled household model given by
Yh,t = Xh,tβ + Zh,tu+
nh,twh,tγ +
nh,tδtα+ εh,t, (2.3)
where Yh,t ≡
∑nh,t
i=1 yi,h,t/
nh,t, Xh,t ≡
∑nh,t
i=1 xi,h,t/
nh,t, Zh,t ≡
∑nh,t
i=1 zi,h,t/
nh,t, and εh,t ≡
∑nh,t
i=1 εi,h,t/
nh,t.
2.1.2 Specification details
Age-gender function specification Let ai,h,t and si,h denote the age and sex of individual i
of household h for the tth week. Individual dietary intake is generally different according to the
gender of individuals, so the function f takes the following form
f(ai,h,t, si,h) = fM (ai,h,t)1l{si,h=M} + fF (ai,h,t)1l{si,h=F},
where fM(.) and fF (.) are age-intake relationships for males (M) and females (F) respectively, and
1l{A} is the indicator function of event A. The function fS(.) is approximated by a spline of order
one with a truncated polynomial basis for either sex, such as
fS(ai,h,t) = β
0 + β
1 ai,h,t +
uSk (ai,h,t − κS,k)+ , (2.4)
where the (κS,k)k=1,. . . ,KS
are nodes chosen from an age list and
(ai,h,t − κS,k)+ ≡ (ai,h,t − κS,k) 1l{ai,h,t−κS,k>0}
denotes the positive part of the difference between the age of the individual ai,h,t and the node κS,k
and the uSk are random effects assumed to be i.i.d. Gaussian with distribution N
0, σ2uS
. This
last assumption allows us to introduce some penalties into the model and to smooth the function
fS yielding a mixed model representation for the spline as shown in Speed (1991); Verbyla (1999);
Brumback et al. (1999); Ruppert et al. (2003). As in Ruppert et al. (2003), page 125, the total
number of nodes KS is set to min
{∣∣aS,d
∣∣ , 35
, where aS,d is the list of distinct ages for individuals
of sex S, and the nodes κS,k are defined as the
percentile of vector aS,d for k = 1,. . . ,KS .
Defining xi,h,t as a line vector
1l{si,h=M} ai,h,t1l{si,h=M} 1l{si,h=F} ai,h,t1l{si,h=F}
, and
zi,h,t as the line vector
(ai,h,t − κS,k)+ 1l{si,h=S}
k=1,. . . ,KS ; S=M,F
, we finally obtain the first
terms of (2.2) , that is f(ai,h,t, si,h) = xi,h,tβ + zi,h,tu.
Socioeconomic characteristics and time dependency In the application, all the socioe-
conomic characterics are categorical variables. Consider the Q categorical variables W
h,t , q =
1, . . . , Q, with mq modalities, and fix the m
q modality as the reference modality, then the socioe-
conomic effect term in (2.2) and (2.3) is
wh,tγ =
mq−1∑
γq,m1l
where γq,m is the effect of the m
th modality of the socioeconomic variable q.
Similarly, time is only measured by weekly counts throughout the year so that the time effect
in (2.2) and (2.3) is simply
δtα =
τ 6=τR
ατ1l{τ=t},
where ατ is the effect of week τ and τR is the reference week.
Error specification The error at the individual level εi,h,t is assumed to be Gaussian with zero
mean, and the variance-covariance structure is such that
• households are independent, i.e. ∀i, i′, t, t′ and ∀h 6= h′
cov(εi,h,t, εi′,h′,t′) = 0,
• members of the same household are dependent, that is for ∀h, t and i 6= i′,
cov(εi,h,t, εi′,h,t) = ρσ
where ρ measures the dependence between individuals within the same household.
• there is no time dependence, that is ∀i, i′ and ∀t 6= t′
cov(εi,h,t, εi′,h,t′) = 0.
In the rescaled household model (2.3), the error εh,t ≡
∑nh,t
i=1 εi,h,t/
nh,t is i.i.d. Gaussian with
a zero mean and a variance R such that ∀t, t′ and ∀h 6= h′,
V(εh,t) = ρσ
εnh,t + (1− ρ)σ2ε and cov(εh,t, εh′,t′) = 0. (2.5)
2.2 Estimation and tests
The model (2.3) is a linear mixed model that can be estimated using restricted maximum likelihood
(REML) techniques, see Ruppert et al. (2003) for details. An attractive consequence of the use of
the mixed model representation of a penalized spline in (2.4) is that mixed model methodology
and software can be used to estimate the parameters and predict the random effect in the resulting
household model. The amount of smoothing of the underlying functions fS is estimated with the
REML technique via the estimation of σ2uS . The estimation was conducted using R©SAS MIXED
procedure. To get estimators for σ2ε and ρ, asymptotic least square techniques combined with the
linear relationship between the variance given in (2.5) and the household size were used. More
precisely, a residual variance σ2n is first estimated for each household size n = 1, . . . , N = maxnh,t
using an option of the MIXED procedure (see the program for the detailed syntax). Then, ordinary
least square regression and the delta method give estimators for σ2ε and ρ and their standard
deviations.
The individual intake is then predicted by
ŷi,h,t = xi,h,tβ̂ + zi,h,tû+ wh,tγ̂ + δtα̂, (2.6)
where β̂, γ̂, and α̂ are the estimators of β, γ, and α respectively and û is the best prediction of the
random effect u in the model (2.3).
Confidence and prediction intervals can be built for the prediction ŷi,h,t as proposed in Ruppert et al.
(2003) and several tests can be conducted in this model:
1. Are the random effects different according to sex? In other words, is the assertion σ2uM =
σ2uF = σ
u true?
2. Another question is the necessity for such random effects. Is the assertion σ2u = 0 (resp.
σ2uM = 0 or σ
= 0) true?
3. More globally, is the function f the same for both sexes? Is the assertion fM = fS true?
These tests can be conducted using classical likelihood (or restricted likelihood) ratio techniques.
The likelihood ratio statistic is asymptotically distributed as a chi square with a degree of freedom
being the number of tested equalities, except for point 2, where the limiting distribution is known to
be a mixture of chi-square (Self and Liang, 1987; Crainiceanu et al., 2003) because the test concerns
the frontier of the parameter definition (σ2u ∈ [0,+∞[).
3 Applying our methodology to the methylmercury risk assess-
In this section, we illustrate our approach on our motivating example. Firstly, several tests are
conducted on the decomposition model, and secondly, individual long term exposure is compared
to the reference long term exposure described in section 1.
3.1 Estimation and tests on the structure of the model
Table 2 shows the REML estimates for all socioeconomic variables (parameter γ) and the p-values
of Student tests in the model (2.3). The socioeconomic variables used are household income, region
of residence, occupation category and level of education of the principal household earner. For each
socioeconomic variable, the reference modality is given in Table 2. We assume here that
• the function f differs according to the gender but the random effect does not (fM 6= fF and
σ2uM = σ
• the maximum household size N is set to 6 for variance-covariance estimation. Indeed, the
dependence between individuals within the same household depends on the household size
nh in (2.5). For each household size, a variance is estimated, and estimates of ρ and σ
are obtained using asymptotic least square techniques as mentioned in section 2.2. Since
large households are not numerous in the database, the estimations are implemented with a
maximum household size, N , set to 6; it is assumed that there is a common variance for all
households with size greater than N .
In this sub-section, we show the results of several tests we carried out to simplify the inter-
pretation of our study. These tests have been implemented in a hierarchical way, starting with
the highest-order interaction terms, combining to the reference modality the modality which does
not differ significantly from the reference. All tests are performed on the 5% level of significance
and each new hypothesis is tested, conditionally on the results of the previous tests. Each null
hypothesis and the p-value resulting from the appropriate F-test are shown in Table 3.
First of all, concerning the occupation category variable, the self-employed modality does not
significantly differ from the reference modality blue collar workers (H1, Pval = 0.771). Refitting the
model with the reference modality ”Blue collar workers and self employed”, all the socioeconomic
variables are significantly different from the reference. Then, F-tests allow us to conclude that the
resulting three groups are significantly different from each other (H2, H3, H4).
Let us now consider the region of residence variable. First, there are some very substantial
differences among the 4 regions of residence (H5, Pval =< 0.001). However, the modality ”North,
Brittany, and Vendee coast” and the modality ”Paris and its suburbs” should be grouped (H6 c,
Pvalc = 0.881). Then, the other tests implemented for the level of education and income variables
suggest that no further simplification is possible (see p-values of null hypotheses H7, H8, H9 in
Table 3). Finally, the overall F-test comparing our resulting final model to the original model (2.3)
shows that no important variable has been left out of the model (Pval = 0.59).
Table 4 shows the parameter estimates and p-values of the Student’s t-tests for all socioeco-
nomic variables of the reduced final model. The income effects on individual exposure are those
expected: the richer the households are, the higher their exposures are because seafoods are ex-
pensive. Furthermore, living in a coastal region or in Paris and its suburbs brings about larger
individual exposure relatively to living in a non coastal region because of the more ready supply of
seafoods in these regions. Moreover, the more educated you are, the larger the individual exposure
is. The occupation category of the principal household earner has an unexpected effect on the in-
dividual exposure. Indeed a higher exposure is expected for white collar workers and retirees whan
compared to blue collar workers but an opposite effect is observed. This may be explained by the
fact that the reference modality for this variable is a very heterogeneous modality also comprising
managers and self-employed persons (farmers and craftsmen). Another explanation could be that
white collars workers have a higher propensity to eat out in restaurants whereas outside the home
consumption is not included in the model.
Table 2 around here, see page 21
Table 3 around here, see page 22
Table 4 around here, see page 22
Likelihood ratio tests are implemented to test the structure of the final model. First, the
dependence of individual exposures to methylmercury within a household is tested. The null
hypothesis ρ = 0 (cf. equation (2.5)) is rejected (null Pval) which confirms that individuals within
the same household have correlated exposures. Then, we test if the function f is the same for both
genders. The null hypothesis fM = fF is rejected (null Pval) but the null hypothesis σ
= σ2uF
is accepted. This means that individual exposure differs with gender but both functions need the
same amount of smoothing.
3.2 The cumulative and the long term individual exposure
The cumulative individual exposure Si,h,t is calculated from the estimated individual weekly intakes
according to equation (1.1) and the resulting values for t > 35 are compared to the reference
cumulative exposure defined by (1.3). Figure 1 shows the cumulative individual exposure over the
53 weeks of the year 2001 for different individuals. Only certain percentiles of the distribution of
the individual cumulative exposures of the last week are displayed. For example, the curve Pmax
represents the cumulative exposure of an individual whose last week’s cumulative exposure is the
highest. This is the cumulative exposure of a girl who turned one year old during the 30th week of
2001, lives in Paris or its suburbs in a well to do household.
Very few individuals have a cumulative individual exposure above the reference long term ex-
posure. We estimate that only 0.186% of individuals are deemed at risk. This risk index should be
compared to the more common one defined as the percentage of weekly intakes Di,h,t exceeding the
PTWI, denoted R1.6, such as R1.6 =
i=1 1l (Di,h,t > 1.6). R1.6 is equal to 0.45%,
and is slightly higher since each occasional deviation above the PTWI increases the risk index
whereas only long term deviations above this PTWI should be taken into account to assess the
risk.
A deeper analysis of at risk individuals shows that all these vulnerable individuals are children
less than three years old. They represent 5.29% of the children aged between 1 and 3 in 2001.
Further, no child of a modest households is found to be at risk.
Figure 1 around here, see page 23
4 Discussion
As mentioned in section 1, the use of household acquisition data in a food safety context, and in
our case the use of the SECODIP database for assessing methylmercury dietary intakes, gives rise
to some approximations:
1. Consumption outside of the home is out of the scope of household acquisition data. TNS
SECODIP does not provide any information on the quantities of seafoods consumed out
of the home or bought for outside consumption. Nevertheless, Serra-Majem et al. (2003)
assert that these data are good estimates for the consumption of the whole household.
Vasdekis and Trichopoulou (2000) avoid this question by using the term ”availaibility” in-
stead of intake or consumption. However, as in Chesher (1997), auxiliary information about
outdoor consumption could be introduced in the model as a correction factor accounting for
the propensity to eat outside of the home according to age, sex or socioeconomic variables.
The French INCA survey on individual consumptions gives details about inside / outside the
home consumption for 3003 individuals people aged 3 and older. The mean outside the home
consumption proportion is 20% for seafoods. Applying such a factor to all household intakes
yields a long term risk of 0.226%, and R1.6 = 0.791%. Furthermore, in this case, a small
proportion of consumers older than 3 years old are vulnerable. Nevertheless, children aged
between 1 and 3 in 2001 still represent the most vulnerable consumer group, at 10% of the
corresponding population.
2. The amount of food bought by a household can be different from the amount actually con-
sumed. Indeed, namely for seafoods, a non negligible part is not edible: Favier et al. (1995)
show than on average only 61% of fresh or frozen fish is edible. Besides, Maresca and Poquet
(1994) also demonstrate some part of the purchased food is thrown away, which also reduces
the actual amount of food consumed by a household. However, SECODIP does not specify
whether the quantity of fresh or frozen fish bought is ready to be consumed or as a whole fish
that needs some preparation. Applying such a factor to all household intakes yields a long
term risk of 0.00%, and R1.6 = 0.043%. If both the 20% outside of the home consumption
correction factor and the 61% edible proportion factor are applied to our series, the long term
risk is equal to 0.021%, R1.6 = 0.13%, and 1.06% of the population of children aged between
1 and 3 are vulnerable. These results stress that applying such a correction factor to assess
the actual quantity consumed is probably too strong and is certainly a crude approximation
of the quantity of seafoods ingested. Thus, a more detailed database on fish and seafood is
needed, to realize an accurate assessment of exposure to methylmercury, taking into account
only the edible part of fish and other seafood.
Body weight information is crucial in a food safety context and will be included in the future
SECODIP data since it has now been added to the list of required individual characteristics.
The measurement error afferent to this quantity will remain however, namely for children
whose body weight changes a lot throughout a year. Nevertheless, approximating the weekly
body weight of young children by the median of the weekly body weight distribution available
in French health records is the best approximation possible.
3. The food nomenclature of the SECODIP database is not as detailed as the contamination
database. Unfortunately, fish and seafood species are not well documented so it is not possible
to consider more than two food categories when computing household intakes. This problem
of nomenclature matching is ubiquitous of food risk assessments since contamination analysis
are generally conducted independently from the food nomenclature of consumption data.
These arguments mainly show the disadvantages of the use of household food acquisition data
such as the SECODIP database. Nevertheless, they also present many advantages compared to the
individual food record survey mainly used in France in the food safety context:
• As mentioned before, households respond for a long period of time (the average is 4 years in
the SECODIP panel) which allows us to observe long term behaviors and avoid some well
known biases of individual food record surveys. For example, respondents might over- (under-
) declare certain foods with a good (bad) nutritional value either deliberately or just because
they increased (reduced) their consumption for the short (7 days) period of the survey.
• The individual surveys are expensive and very difficult to conduct. Highly trained interviewers
are required and extraordinary cooperation is required from respondents. Household food
acquisition data can serve many other applications (economics or marketing) and, at least for
the SECODIP data, acquisition recording is simplified by optical scanning of food barcodes.
Conclusion
In this paper, we proposed a methodology to assess chronic risks related to food contamination
using the example of methylmercury exposure through seafood consumption. This methodology
includes the definition of a Kinetic Dietary Exposure Model (KDEM) that integrates the fact that
contaminants are eliminated from the body at different rates, the rate being measured by the half
life of the contaminant. In this paper, the estimation is based on the use of household food acqui-
sition data which are first decomposed into individual intake data through a disaggregation model
accounting for the dependence among household members. Several extensions of this methodology
are currently studied. First, the disaggregation model could be improved by considering a prelim-
inary step in which we determine what member is an actual consumer, in the spirit of the Tobit
model. The KDEM idea is also currently being developed by studying the stability and ergodic
properties of the underlying continuous time piecewise deterministic Markov process (Bertail et al.,
2006). The parameters of this new model are the intake distribution, the inter intake time distri-
bution and the dissipation rate distribution. In this framework, the dissipation parameter η of the
KDEM model is random and the intake and inter-intake distributions can be estimated either from
individual (INCA-type) data or household (SECODIP-type) data.
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Figures and Tables
Table 1: Description of the contamination database (Unit: microgram per kilogram
Mean Min Max Standard Deviation Number of analysis
Fish 0.147 0.003 3.520 0.235 1350
Mollusk and Shellfish 0.014 0.001 0.172 0.011 1293
Table 2: Restricted maximum likelihood estimates (REML) for age and all socioeconomic variables
and the p-value of the Student’s tests (Pval)
Effect Parameter REML Pval
Income (ref: Mean sup)
Well to do γ1 6.027 <0.001
Mean inf γ2 2.686 <0.001
Modest γ3 -1.928 <0.001
Region of residence (ref: Noncoastal regions)
North, Brittany, Vendee coast γ4 0.962 0.003
South West coast γ5 5.232 <0.001
Mediterranean coast γ6 2.303 <0.001
Paris and its suburbs γ7 1.023 0.009
Occupation category of the principal household earner (ref: Blue collar workers)
self-employed persons γ8 -0.122 0.771
white collar workers γ9 -3.733 <0.001
retirees γ10 -5.261 <0.001
no activity γ11 -1.910 0.004
Level of Education of the principal household earner (ref: BAC and higher degree)
student γ12 5.901 <0.001
no or weak diploma γ13 -1.281 <0.001
Table 3: The different steps performed in testing the socioeconomic part of our model. For each
step, the null hypothesis tested and the p-value resulting from the appropriate F-test are shown.
All tests are performed conditionally on the results of the previous tests (Pval)
Null hypothesis Pval
H1 : γ8 = 0 0.771
H2 : γ9 = γ10 0.030
H3 : γ9 = γ11 0.018
H4 : γ10 = γ11 <0.001
H5 : γ4 = γ5 = γ6 = γ7 <0.001
H6 : a : γ4 = γ5 <0.001
b : γ4 = γ6 <0.001
c : γ4 = γ7 0.881
d : γ5 = γ6 <0.001
e : γ5 = γ7 <0.001
f : γ6 = γ7 0.0103
H7 : γ12 = γ13 <0.001
H8 : γ1 = γ2 = γ3 <0.001
H9 : a : γ1 = γ2 <0.001
b : γ1 = γ3 <0.001
c : γ2 = γ3 <0.001
Table 4: Restricted maximum likelihood estimates (REML) for all age and socioeconomic variables
of the reduced final model with all variance components and their standard errors (s.e)
Effect Parameter REML Pval
Income (ref: Mean sup)
Well to do γ1 6.108 <0.001
Mean inf γ2 2.760 <0.001
Modest γ3 -1.915 <0.001
Region of residence (ref: Non coastal regions)
Paris and North, Brittany, Vendee coast γ4= γ7 0.995 <0.001
South west coast γ5 5.156 <0.001
Mediterranean coast γ6 2.250 <0.001
Occupation category of the principal household earner (ref: Blue collar workers and self employed persons)
white collar workers γ9 -3.745 <0.001
retirees γ10 -5.243 <0.001
no activity γ11 -1.871 0.005
Level of education of the principal household earner (ref: BAC and higher degree)
student γ12 5.879 <0.001
no or weak diploma γ13 -1.279 <0.001
REML s.e
Variance of the random effect σu 24.832 6.7316
Variance-covariance structure
variance σ2 1260705 282309
correlation ρ -0.22 0.0434
Figure 1: Cumulative exposure to MeHg (unit: µg per kg of body weight)
	Motivating example: risk related to methylmercury in seafoods in the French population
	Cumulative exposure and long term risk: the Kinetic Dietary Exposure Model (KDEM)
	From household acquisition data to household intake series
	Statistical methodology
	The decomposition model
	General principle
	Specification details
	Estimation and tests
	Applying our methodology to the methylmercury risk assessment
	Estimation and tests on the structure of the model
	The cumulative and the long term individual exposure
	Discussion

  Foods naturally contain a number of contaminants that may have different and
long term toxic effects. This paper introduces a novel approach for the
assessment of such chronic food risk that integrates the pharmacokinetic
properties of a given contaminant. The estimation of such a Kinetic Dietary
Exposure Model (KDEM) should be based on long term consumption data which, for
the moment, can only be provided by Household Budget Surveys such as the
SECODIP panel in France. A semi parametric model is proposed to decompose a
series of household quantities into individual quantities which are then used
as inputs of the KDEM. As an illustration, the risk assessment related to the
presence of methyl mercury in seafood is revisited using this novel approach.

Introduction
The quantitative assessment of dietary exposure to certain contaminants is of high priority to the
Food and Agricultural Organization and the World Health Organization (FAO/WHO). For exam-
ple, excessive exposure to methylmercury, a contaminant mainly found in fish and other seafood
(mollusks and shellfish) may have neurotoxic effects such as neuronal loss, ataxia, visual disturbance,
impaired hearing, and paralysis (WHO, 1990). Quantitative risk assessments for such chronic risk
require the comparison between a tolerable dose of the contaminant called Provisional Tolerable
Weekly Intake (PTWI) and the population’s usual intake. The usual intake distribution is gener-
ally estimated from independent individual food consumption surveys (generally not exceeding 7
days) and food contamination data. Several models have been developed to estimate the distribu-
tion of usual dietary intake from short-term measurements (see for example, Nusser et al., 1996;
Hoffmann et al., 2002). The proportion of consumers whose usual weekly intake exceeds the PTWI
can then be viewed as a risk indicator (see for example, Tressou et al., 2004). This kind of risk
assessment does not account for the underlying dynamic process, i.e. for the fact that the contami-
nant is ingested over time and naturally eliminated at a certain rate by the human body. Moreover,
longer term measurements of consumption are available through household budget surveys (HBS).
In this paper, we propose to use HBS data to quantify individual long term exposure to a
contaminant. This data provides long time series of household food acquisitions which are first used
in a decomposition model, similar to the one proposed by Chesher (1997, 1998) in the nutrition
field, in order to obtain time series of individual intakes. Then, the pharmacokinetic properties of
the contaminant are integrated into an autoregressive model in which the current body burden is
defined as a fraction of the previous one plus the current intake.
From a toxicological point of view, this approach is, to our knowledge, novel and hence requires
the definition of an ad-hoc long term safe dose as proposed in the next section. We refer to this
autoregressive model as Kinetic Dietary Exposure Model (KDEM).
From a statistical point of view, such autoregressive models are well known in general time series
analysis (see for example, Hamilton, 1994) and most of the paper is devoted to the description of
the decomposition model. This statistical model aims at estimating individual quantities from total
household quantities and structures. This problem is similar to that studied by Engle et al. (1986),
Chesher (1997, 1998), and Vasdekis and Trichopoulou (2000), and is addressed in a slightly different
way. In the present article, the individual contaminant intake is firstly viewed as a nonlinear
function of age within each gender, with time and socioeconomic characteristics being secondly
introduced in a linear way. The nonlinear function is represented by a truncated polynomial
spline of order 1 that admits a mixed model spline representation (section 4.9 in Ruppert et al.,
2003). These choices yield a simple linear mixed model which is estimated by REstricted Maximum
Likelihood (REML, Patterson and Thompson, 1971). One major extension of the proposed model
compared to Chesher (1997) is the introduction of dependence between the individual intakes of a
given household.
In the next section, focusing on the methylmercury example even though the method is much
more general and could be applied to any chronic food risk, SECODIP data are described along
with the construction of a household intake series and the individual cumulative and long term
exposure concepts yielding the KDEM. Section 2 is devoted to the statistical methodology used to
decompose the household intake series into individual intake series, namely the presentation of the
model and its estimation and tests. Section 3 displays the results for the quantification of long term
exposure to methylmercury of the French population using the 2001 SECODIP panel. Finally, a
discussion on the use of household acquisition data, with the focus on the French SECODIP panel,
is conducted in section 4 with respect to the proposed long term risk analysis.
1 Motivating example: risk related to methylmercury in seafoods
in the French population
In this section, the Kinetic Dietary Exposure Model (KDEM) and the concept of long term risk are
defined. Then a brief panorama of consumption data in France is given and the way the SECODIP
HBS data will be used as an input of the KDEM is described.
1.1 Cumulative exposure and long term risk: the Kinetic Dietary Exposure
Model (KDEM)
The main objective of the analysis is to assess individuals’ long term exposure to a contaminant to
deduce whether these individuals are at risk or not. As mentioned in the introduction the only ”safe
dose” reference is the PTWI expressed in terms of body weight (relative intake). Unfortunately,
TNS SECODIP did not record the body weight of the individuals until 2001. The body weights
are thus estimated from independent data sets; namely the French national survey on individual
consumption (INCA, CREDOC-AFSSA-DGAL, 1999) for people older than 18, and the weekly
body weight distribution available from French health records (Sempé et al. (1979)) for individuals
under 18. In both cases, gender differentiation is introduced.
Assume that estimations of the individual weekly intakes are available, that is yi,h,t denotes the
intake of individual i belonging to household h for the tth week (with i = 1, . . . , nh,t; h = 1, . . . ,H
and t = 1, . . . , T ), and Di,h,t denotes the same quantity expressed on a body weight basis. The
cumulative exposure up to the tth week of this individual is then given by
Si,h,t = exp(−η) · Si,h,t−1 +Di,h,t, (1.1)
where η > 0 is the natural dissipation rate of the contaminant in the organism. This dissipation
parameter is defined from the so called half life of the contaminant,which is the time required for
the body burden to decrease by half in the absence of any new intake. For methylmercury, the half
life, denoted by l1/2, is estimated to 6 weeks, so that η = ln(2)/l1/2 := ln(2)/6 (Smith and Farris,
1996).
The autoregressive model defined by (1.1) and a given initial state Si,h,0 = Di,h,0 has a stationary
solution since exp(−η) < 1. As a convention, Si,h,0 is set to the mean of all positive exposures
(Di,h,t)t=1,. . . ,T . However, this convention has little impact on the level of an individual’s long term
exposure since the contribution of the initial state Si,h,0 tends to zero as t increases. We call this
autoregressive model ”KDEM” for Kinetic Dietary Exposure Model.
The individual cumulative exposure Si,h,t can be considered to be the long term exposure of an
individual for sufficiently large values of t. For methylmercury, the long term steady state of the
individual exposure to a contaminant is reached after 5 or 6 half lives according to Dr P. Granjean,
a methylmercury expert. Thus, the long term individual’s exposure to methylmercury is defined
as the cumulative exposure reached after say 6l1/2 = 36 weeks.
The risk assessment usually consists of comparing the exposure with the so called Provisional
Tolerable Weekly Intake (PTWI). This tolerable dose, determined from animal experiments and
extrapolated to humans, refers to the dose an individual can ingest throughout his entire life
without appreciable risk. For methylmercury, the PTWI is set to 1.6 microgram per kilogram of
body weight per week (1.6 µg/kg bw, see FAO/WHO, 2003).
In our dynamic approach, the long term exposure is compared to a reference long term exposure
denoted by Sref , and defined as the cumulative exposure of an individual whose weekly intake is
equal to the PTWI, d, such as
Sref = lim
1− exp(−η) , (1.2)
where
d exp(−η(t− s)) = dexp(−η(t+ 1))− 1
exp(−η)− 1 . (1.3)
For methylmercury, the reference for long term exposure Sref is 14.6 µg/kg bw. An individual
is then assumed to be at risk if his cumulative exposure Si,h,t exceeds the reference S
t for any
t > 6l1/2.
This KDEM model requires some long surveys of individual intakes which are not monitored
and can only be approximated from available consumption data and contamination data.
1.2 From household acquisition data to household intake series
Two current major consumption data sources in France are the national survey on individual
consumption (INCA, CREDOC-AFSSA-DGAL, 1999) and the SECODIP panel managed by the
company TNS SECODIP. Most quantitative risk assessments conducted by the French agency for
food safety (AFSSA) use the 7 day individual consumption data of the INCA survey jointly with
contamination data collected by several French institutions. Regarding methylmercury, seafood
contamination data have been collected through different analytical surveys (MAAPAR, 1998-2002;
IFREMER, 1994-1998) and were used in Tressou et al. (2004) and Crépet et al. (2005) combined
with the INCA survey. In this paper, a methodology using the SECODIP data is developed (see
Boizot, 2005, for a full description of this database).
The company TNS SECODIP has been collecting the weekly food acquisition data of about five
thousand households since 1989. All participating households register grocery purchases through
the use of EAN bar codes but other grocery purchases are registered differently: the fresh fruit
and vegetable purchases are recorded by the FL sub-panel while fresh meat, fresh fish and wine
purchases are recorded by the VP sub-panel. The households are selected by stratification according
to several socioeconomic variables and stay in the survey for about 4 years. TNS SECODIP provides
weights for each sub-panel and each period of 4 weeks to make sure of the representativeness of
the results in terms of several socioeconomic variables. TNS SECODIP also defines the notion of
household activity which refers to the correct and regular reporting of household purchases over a
year. For each household, the age and gender of each member of the household are retained in our
decomposition model with some socioeconomic variables: the region, the social class (from modest
to well-to-do), the occupation category and level of education of the principal household earner.
For methylmercury risk assessment, the households of the VP panel are considered; in the 2001
data set, there are H = 3229 active households (corresponding to 9288 individuals) and T = 53
weeks during which the households may or may not acquire seafood. The weekly purchases of
seafood are clustered into two categories (”Fish” and ”Mollusks and Shellfish”) for which the mean
contamination levels are calculated from the MAAPAR-IFREMER data and are given in table 1.
Table 1 around here, see page 21
Household intake series ((yh,t)h=1,...,H;t=1,...,T) are computed as the cross product between weekly
purchases of seafoods which are assimilated to weekly consumptions, and mean contamination
levels. They are expressed in micrograms per week (µg/w). The food ”purchase-consumption”
assimilation is of course arguable and will be the main subject of the final discussion (see section
4). An additional assumption concerns the household size, denoted by nh,t for the household h
and the week t. This can indeed vary over time in the case of a birth or death of a household
member. Since a new born baby will not consume fish in his first few months, we assume that
food diversification (and hence consumption of seafoods) starts at one year of age, yielding a total
sample of 8913 individuals for the 2001 panel. These household intake series are then decomposed
into individual intake series using the model described in the next section. These individual intake
series are then used as imputs of the KDEM.
2 Statistical methodology
In this section, the decomposition model is described and compared to similar models described in
the literature, namely Chesher (1997, 1998); Vasdekis and Trichopoulou (2000). Its estimation and
some structure tests are then presented.
2.1 The decomposition model
2.1.1 General principle
Consider a household composed of nh,t members, each member having unobserved weekly intakes
yi,h,t, with i = 1,. . . , nh,t, h = 1,. . . ,H, and t = 1,. . . , T . The week t intake of a household h is
simply the sum across household members of the individual weekly intakes, such as
yh,t =
nh,t∑
yi,h,t. (2.1)
As detailed below, the individual weekly intake yi,h,t is assumed to depend on
• the age and gender of the individual via a function f,
• some socioeconomic characteristics of the household,
• time (seasonal variations).
There are obviously several ways to model the individual intake under these assumptions
and this choice leads to more or less simple estimation procedures. In Chesher (1997, 1998);
Vasdekis and Trichopoulou (2000), a discretization argument on age is used leading to a penalized
least square estimation of a great number of parameters, that is one parameter for each year of age
and gender. We propose to use a truncated polynomial spline of order 1 for each gender, which
admits a mixed model spline representation for f. As far as socioeconomic characteristics are con-
cerned, Chesher (1997) retained a multiplicative specification whereas Vasdekis and Trichopoulou
(2000) chose the additive one. In the multiplicative model, a change in income for example would
proportionally affect all the individual intakes whereas in the additive setting, they would be af-
fected by the same value. Following Vasdekis and Trichopoulou (2000), we retained the additive
specification since the difference between the two specifications may not be notable, and the addi-
tive setting yields to a much simpler estimation procedure (linear model). Finally, time dependency
is only introduced in Chesher (1998) to track changes with age within cohorts: this time depen-
dency is directly introduced into the function f that is bivariately smoothed according to age and
time (cf. Green and Silverman, 1994). Again, we adopt a simpler specification in which time is
introduced as a dummy variable. All these assumptions yield an individual model of the form
yi,h,t = xi,h,tβ + zi,h,tu+ wh,tγ + δtα+ εi,h,t, (2.2)
where the terms xi,h,tβ + zi,h,tu stand for the mixed model spline representation of the function
f, the term wh,tγ denotes the socioeconomic effects, the term δtα the time effect, and εi,h,t is the
individual error term.
Combining (2.1) and (2.2) , we obtain the final rescaled household model given by
Yh,t = Xh,tβ + Zh,tu+
nh,twh,tγ +
nh,tδtα+ εh,t, (2.3)
where Yh,t ≡
∑nh,t
i=1 yi,h,t/
nh,t, Xh,t ≡
∑nh,t
i=1 xi,h,t/
nh,t, Zh,t ≡
∑nh,t
i=1 zi,h,t/
nh,t, and εh,t ≡
∑nh,t
i=1 εi,h,t/
nh,t.
2.1.2 Specification details
Age-gender function specification Let ai,h,t and si,h denote the age and sex of individual i
of household h for the tth week. Individual dietary intake is generally different according to the
gender of individuals, so the function f takes the following form
f(ai,h,t, si,h) = fM (ai,h,t)1l{si,h=M} + fF (ai,h,t)1l{si,h=F},
where fM(.) and fF (.) are age-intake relationships for males (M) and females (F) respectively, and
1l{A} is the indicator function of event A. The function fS(.) is approximated by a spline of order
one with a truncated polynomial basis for either sex, such as
fS(ai,h,t) = β
0 + β
1 ai,h,t +
uSk (ai,h,t − κS,k)+ , (2.4)
where the (κS,k)k=1,. . . ,KS
are nodes chosen from an age list and
(ai,h,t − κS,k)+ ≡ (ai,h,t − κS,k) 1l{ai,h,t−κS,k>0}
denotes the positive part of the difference between the age of the individual ai,h,t and the node κS,k
and the uSk are random effects assumed to be i.i.d. Gaussian with distribution N
0, σ2uS
. This
last assumption allows us to introduce some penalties into the model and to smooth the function
fS yielding a mixed model representation for the spline as shown in Speed (1991); Verbyla (1999);
Brumback et al. (1999); Ruppert et al. (2003). As in Ruppert et al. (2003), page 125, the total
number of nodes KS is set to min
{∣∣aS,d
∣∣ , 35
, where aS,d is the list of distinct ages for individuals
of sex S, and the nodes κS,k are defined as the
percentile of vector aS,d for k = 1,. . . ,KS .
Defining xi,h,t as a line vector
1l{si,h=M} ai,h,t1l{si,h=M} 1l{si,h=F} ai,h,t1l{si,h=F}
, and
zi,h,t as the line vector
(ai,h,t − κS,k)+ 1l{si,h=S}
k=1,. . . ,KS ; S=M,F
, we finally obtain the first
terms of (2.2) , that is f(ai,h,t, si,h) = xi,h,tβ + zi,h,tu.
Socioeconomic characteristics and time dependency In the application, all the socioe-
conomic characterics are categorical variables. Consider the Q categorical variables W
h,t , q =
1, . . . , Q, with mq modalities, and fix the m
q modality as the reference modality, then the socioe-
conomic effect term in (2.2) and (2.3) is
wh,tγ =
mq−1∑
γq,m1l
where γq,m is the effect of the m
th modality of the socioeconomic variable q.
Similarly, time is only measured by weekly counts throughout the year so that the time effect
in (2.2) and (2.3) is simply
δtα =
τ 6=τR
ατ1l{τ=t},
where ατ is the effect of week τ and τR is the reference week.
Error specification The error at the individual level εi,h,t is assumed to be Gaussian with zero
mean, and the variance-covariance structure is such that
• households are independent, i.e. ∀i, i′, t, t′ and ∀h 6= h′
cov(εi,h,t, εi′,h′,t′) = 0,
• members of the same household are dependent, that is for ∀h, t and i 6= i′,
cov(εi,h,t, εi′,h,t) = ρσ
where ρ measures the dependence between individuals within the same household.
• there is no time dependence, that is ∀i, i′ and ∀t 6= t′
cov(εi,h,t, εi′,h,t′) = 0.
In the rescaled household model (2.3), the error εh,t ≡
∑nh,t
i=1 εi,h,t/
nh,t is i.i.d. Gaussian with
a zero mean and a variance R such that ∀t, t′ and ∀h 6= h′,
V(εh,t) = ρσ
εnh,t + (1− ρ)σ2ε and cov(εh,t, εh′,t′) = 0. (2.5)
2.2 Estimation and tests
The model (2.3) is a linear mixed model that can be estimated using restricted maximum likelihood
(REML) techniques, see Ruppert et al. (2003) for details. An attractive consequence of the use of
the mixed model representation of a penalized spline in (2.4) is that mixed model methodology
and software can be used to estimate the parameters and predict the random effect in the resulting
household model. The amount of smoothing of the underlying functions fS is estimated with the
REML technique via the estimation of σ2uS . The estimation was conducted using R©SAS MIXED
procedure. To get estimators for σ2ε and ρ, asymptotic least square techniques combined with the
linear relationship between the variance given in (2.5) and the household size were used. More
precisely, a residual variance σ2n is first estimated for each household size n = 1, . . . , N = maxnh,t
using an option of the MIXED procedure (see the program for the detailed syntax). Then, ordinary
least square regression and the delta method give estimators for σ2ε and ρ and their standard
deviations.
The individual intake is then predicted by
ŷi,h,t = xi,h,tβ̂ + zi,h,tû+ wh,tγ̂ + δtα̂, (2.6)
where β̂, γ̂, and α̂ are the estimators of β, γ, and α respectively and û is the best prediction of the
random effect u in the model (2.3).
Confidence and prediction intervals can be built for the prediction ŷi,h,t as proposed in Ruppert et al.
(2003) and several tests can be conducted in this model:
1. Are the random effects different according to sex? In other words, is the assertion σ2uM =
σ2uF = σ
u true?
2. Another question is the necessity for such random effects. Is the assertion σ2u = 0 (resp.
σ2uM = 0 or σ
= 0) true?
3. More globally, is the function f the same for both sexes? Is the assertion fM = fS true?
These tests can be conducted using classical likelihood (or restricted likelihood) ratio techniques.
The likelihood ratio statistic is asymptotically distributed as a chi square with a degree of freedom
being the number of tested equalities, except for point 2, where the limiting distribution is known to
be a mixture of chi-square (Self and Liang, 1987; Crainiceanu et al., 2003) because the test concerns
the frontier of the parameter definition (σ2u ∈ [0,+∞[).
3 Applying our methodology to the methylmercury risk assess-
In this section, we illustrate our approach on our motivating example. Firstly, several tests are
conducted on the decomposition model, and secondly, individual long term exposure is compared
to the reference long term exposure described in section 1.
3.1 Estimation and tests on the structure of the model
Table 2 shows the REML estimates for all socioeconomic variables (parameter γ) and the p-values
of Student tests in the model (2.3). The socioeconomic variables used are household income, region
of residence, occupation category and level of education of the principal household earner. For each
socioeconomic variable, the reference modality is given in Table 2. We assume here that
• the function f differs according to the gender but the random effect does not (fM 6= fF and
σ2uM = σ
• the maximum household size N is set to 6 for variance-covariance estimation. Indeed, the
dependence between individuals within the same household depends on the household size
nh in (2.5). For each household size, a variance is estimated, and estimates of ρ and σ
are obtained using asymptotic least square techniques as mentioned in section 2.2. Since
large households are not numerous in the database, the estimations are implemented with a
maximum household size, N , set to 6; it is assumed that there is a common variance for all
households with size greater than N .
In this sub-section, we show the results of several tests we carried out to simplify the inter-
pretation of our study. These tests have been implemented in a hierarchical way, starting with
the highest-order interaction terms, combining to the reference modality the modality which does
not differ significantly from the reference. All tests are performed on the 5% level of significance
and each new hypothesis is tested, conditionally on the results of the previous tests. Each null
hypothesis and the p-value resulting from the appropriate F-test are shown in Table 3.
First of all, concerning the occupation category variable, the self-employed modality does not
significantly differ from the reference modality blue collar workers (H1, Pval = 0.771). Refitting the
model with the reference modality ”Blue collar workers and self employed”, all the socioeconomic
variables are significantly different from the reference. Then, F-tests allow us to conclude that the
resulting three groups are significantly different from each other (H2, H3, H4).
Let us now consider the region of residence variable. First, there are some very substantial
differences among the 4 regions of residence (H5, Pval =< 0.001). However, the modality ”North,
Brittany, and Vendee coast” and the modality ”Paris and its suburbs” should be grouped (H6 c,
Pvalc = 0.881). Then, the other tests implemented for the level of education and income variables
suggest that no further simplification is possible (see p-values of null hypotheses H7, H8, H9 in
Table 3). Finally, the overall F-test comparing our resulting final model to the original model (2.3)
shows that no important variable has been left out of the model (Pval = 0.59).
Table 4 shows the parameter estimates and p-values of the Student’s t-tests for all socioeco-
nomic variables of the reduced final model. The income effects on individual exposure are those
expected: the richer the households are, the higher their exposures are because seafoods are ex-
pensive. Furthermore, living in a coastal region or in Paris and its suburbs brings about larger
individual exposure relatively to living in a non coastal region because of the more ready supply of
seafoods in these regions. Moreover, the more educated you are, the larger the individual exposure
is. The occupation category of the principal household earner has an unexpected effect on the in-
dividual exposure. Indeed a higher exposure is expected for white collar workers and retirees whan
compared to blue collar workers but an opposite effect is observed. This may be explained by the
fact that the reference modality for this variable is a very heterogeneous modality also comprising
managers and self-employed persons (farmers and craftsmen). Another explanation could be that
white collars workers have a higher propensity to eat out in restaurants whereas outside the home
consumption is not included in the model.
Table 2 around here, see page 21
Table 3 around here, see page 22
Table 4 around here, see page 22
Likelihood ratio tests are implemented to test the structure of the final model. First, the
dependence of individual exposures to methylmercury within a household is tested. The null
hypothesis ρ = 0 (cf. equation (2.5)) is rejected (null Pval) which confirms that individuals within
the same household have correlated exposures. Then, we test if the function f is the same for both
genders. The null hypothesis fM = fF is rejected (null Pval) but the null hypothesis σ
= σ2uF
is accepted. This means that individual exposure differs with gender but both functions need the
same amount of smoothing.
3.2 The cumulative and the long term individual exposure
The cumulative individual exposure Si,h,t is calculated from the estimated individual weekly intakes
according to equation (1.1) and the resulting values for t > 35 are compared to the reference
cumulative exposure defined by (1.3). Figure 1 shows the cumulative individual exposure over the
53 weeks of the year 2001 for different individuals. Only certain percentiles of the distribution of
the individual cumulative exposures of the last week are displayed. For example, the curve Pmax
represents the cumulative exposure of an individual whose last week’s cumulative exposure is the
highest. This is the cumulative exposure of a girl who turned one year old during the 30th week of
2001, lives in Paris or its suburbs in a well to do household.
Very few individuals have a cumulative individual exposure above the reference long term ex-
posure. We estimate that only 0.186% of individuals are deemed at risk. This risk index should be
compared to the more common one defined as the percentage of weekly intakes Di,h,t exceeding the
PTWI, denoted R1.6, such as R1.6 =
i=1 1l (Di,h,t > 1.6). R1.6 is equal to 0.45%,
and is slightly higher since each occasional deviation above the PTWI increases the risk index
whereas only long term deviations above this PTWI should be taken into account to assess the
risk.
A deeper analysis of at risk individuals shows that all these vulnerable individuals are children
less than three years old. They represent 5.29% of the children aged between 1 and 3 in 2001.
Further, no child of a modest households is found to be at risk.
Figure 1 around here, see page 23
4 Discussion
As mentioned in section 1, the use of household acquisition data in a food safety context, and in
our case the use of the SECODIP database for assessing methylmercury dietary intakes, gives rise
to some approximations:
1. Consumption outside of the home is out of the scope of household acquisition data. TNS
SECODIP does not provide any information on the quantities of seafoods consumed out
of the home or bought for outside consumption. Nevertheless, Serra-Majem et al. (2003)
assert that these data are good estimates for the consumption of the whole household.
Vasdekis and Trichopoulou (2000) avoid this question by using the term ”availaibility” in-
stead of intake or consumption. However, as in Chesher (1997), auxiliary information about
outdoor consumption could be introduced in the model as a correction factor accounting for
the propensity to eat outside of the home according to age, sex or socioeconomic variables.
The French INCA survey on individual consumptions gives details about inside / outside the
home consumption for 3003 individuals people aged 3 and older. The mean outside the home
consumption proportion is 20% for seafoods. Applying such a factor to all household intakes
yields a long term risk of 0.226%, and R1.6 = 0.791%. Furthermore, in this case, a small
proportion of consumers older than 3 years old are vulnerable. Nevertheless, children aged
between 1 and 3 in 2001 still represent the most vulnerable consumer group, at 10% of the
corresponding population.
2. The amount of food bought by a household can be different from the amount actually con-
sumed. Indeed, namely for seafoods, a non negligible part is not edible: Favier et al. (1995)
show than on average only 61% of fresh or frozen fish is edible. Besides, Maresca and Poquet
(1994) also demonstrate some part of the purchased food is thrown away, which also reduces
the actual amount of food consumed by a household. However, SECODIP does not specify
whether the quantity of fresh or frozen fish bought is ready to be consumed or as a whole fish
that needs some preparation. Applying such a factor to all household intakes yields a long
term risk of 0.00%, and R1.6 = 0.043%. If both the 20% outside of the home consumption
correction factor and the 61% edible proportion factor are applied to our series, the long term
risk is equal to 0.021%, R1.6 = 0.13%, and 1.06% of the population of children aged between
1 and 3 are vulnerable. These results stress that applying such a correction factor to assess
the actual quantity consumed is probably too strong and is certainly a crude approximation
of the quantity of seafoods ingested. Thus, a more detailed database on fish and seafood is
needed, to realize an accurate assessment of exposure to methylmercury, taking into account
only the edible part of fish and other seafood.
Body weight information is crucial in a food safety context and will be included in the future
SECODIP data since it has now been added to the list of required individual characteristics.
The measurement error afferent to this quantity will remain however, namely for children
whose body weight changes a lot throughout a year. Nevertheless, approximating the weekly
body weight of young children by the median of the weekly body weight distribution available
in French health records is the best approximation possible.
3. The food nomenclature of the SECODIP database is not as detailed as the contamination
database. Unfortunately, fish and seafood species are not well documented so it is not possible
to consider more than two food categories when computing household intakes. This problem
of nomenclature matching is ubiquitous of food risk assessments since contamination analysis
are generally conducted independently from the food nomenclature of consumption data.
These arguments mainly show the disadvantages of the use of household food acquisition data
such as the SECODIP database. Nevertheless, they also present many advantages compared to the
individual food record survey mainly used in France in the food safety context:
• As mentioned before, households respond for a long period of time (the average is 4 years in
the SECODIP panel) which allows us to observe long term behaviors and avoid some well
known biases of individual food record surveys. For example, respondents might over- (under-
) declare certain foods with a good (bad) nutritional value either deliberately or just because
they increased (reduced) their consumption for the short (7 days) period of the survey.
• The individual surveys are expensive and very difficult to conduct. Highly trained interviewers
are required and extraordinary cooperation is required from respondents. Household food
acquisition data can serve many other applications (economics or marketing) and, at least for
the SECODIP data, acquisition recording is simplified by optical scanning of food barcodes.
Conclusion
In this paper, we proposed a methodology to assess chronic risks related to food contamination
using the example of methylmercury exposure through seafood consumption. This methodology
includes the definition of a Kinetic Dietary Exposure Model (KDEM) that integrates the fact that
contaminants are eliminated from the body at different rates, the rate being measured by the half
life of the contaminant. In this paper, the estimation is based on the use of household food acqui-
sition data which are first decomposed into individual intake data through a disaggregation model
accounting for the dependence among household members. Several extensions of this methodology
are currently studied. First, the disaggregation model could be improved by considering a prelim-
inary step in which we determine what member is an actual consumer, in the spirit of the Tobit
model. The KDEM idea is also currently being developed by studying the stability and ergodic
properties of the underlying continuous time piecewise deterministic Markov process (Bertail et al.,
2006). The parameters of this new model are the intake distribution, the inter intake time distri-
bution and the dissipation rate distribution. In this framework, the dissipation parameter η of the
KDEM model is random and the intake and inter-intake distributions can be estimated either from
individual (INCA-type) data or household (SECODIP-type) data.
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Figures and Tables
Table 1: Description of the contamination database (Unit: microgram per kilogram
Mean Min Max Standard Deviation Number of analysis
Fish 0.147 0.003 3.520 0.235 1350
Mollusk and Shellfish 0.014 0.001 0.172 0.011 1293
Table 2: Restricted maximum likelihood estimates (REML) for age and all socioeconomic variables
and the p-value of the Student’s tests (Pval)
Effect Parameter REML Pval
Income (ref: Mean sup)
Well to do γ1 6.027 <0.001
Mean inf γ2 2.686 <0.001
Modest γ3 -1.928 <0.001
Region of residence (ref: Noncoastal regions)
North, Brittany, Vendee coast γ4 0.962 0.003
South West coast γ5 5.232 <0.001
Mediterranean coast γ6 2.303 <0.001
Paris and its suburbs γ7 1.023 0.009
Occupation category of the principal household earner (ref: Blue collar workers)
self-employed persons γ8 -0.122 0.771
white collar workers γ9 -3.733 <0.001
retirees γ10 -5.261 <0.001
no activity γ11 -1.910 0.004
Level of Education of the principal household earner (ref: BAC and higher degree)
student γ12 5.901 <0.001
no or weak diploma γ13 -1.281 <0.001
Table 3: The different steps performed in testing the socioeconomic part of our model. For each
step, the null hypothesis tested and the p-value resulting from the appropriate F-test are shown.
All tests are performed conditionally on the results of the previous tests (Pval)
Null hypothesis Pval
H1 : γ8 = 0 0.771
H2 : γ9 = γ10 0.030
H3 : γ9 = γ11 0.018
H4 : γ10 = γ11 <0.001
H5 : γ4 = γ5 = γ6 = γ7 <0.001
H6 : a : γ4 = γ5 <0.001
b : γ4 = γ6 <0.001
c : γ4 = γ7 0.881
d : γ5 = γ6 <0.001
e : γ5 = γ7 <0.001
f : γ6 = γ7 0.0103
H7 : γ12 = γ13 <0.001
H8 : γ1 = γ2 = γ3 <0.001
H9 : a : γ1 = γ2 <0.001
b : γ1 = γ3 <0.001
c : γ2 = γ3 <0.001
Table 4: Restricted maximum likelihood estimates (REML) for all age and socioeconomic variables
of the reduced final model with all variance components and their standard errors (s.e)
Effect Parameter REML Pval
Income (ref: Mean sup)
Well to do γ1 6.108 <0.001
Mean inf γ2 2.760 <0.001
Modest γ3 -1.915 <0.001
Region of residence (ref: Non coastal regions)
Paris and North, Brittany, Vendee coast γ4= γ7 0.995 <0.001
South west coast γ5 5.156 <0.001
Mediterranean coast γ6 2.250 <0.001
Occupation category of the principal household earner (ref: Blue collar workers and self employed persons)
white collar workers γ9 -3.745 <0.001
retirees γ10 -5.243 <0.001
no activity γ11 -1.871 0.005
Level of education of the principal household earner (ref: BAC and higher degree)
student γ12 5.879 <0.001
no or weak diploma γ13 -1.279 <0.001
REML s.e
Variance of the random effect σu 24.832 6.7316
Variance-covariance structure
variance σ2 1260705 282309
correlation ρ -0.22 0.0434
Figure 1: Cumulative exposure to MeHg (unit: µg per kg of body weight)
	Motivating example: risk related to methylmercury in seafoods in the French population
	Cumulative exposure and long term risk: the Kinetic Dietary Exposure Model (KDEM)
	From household acquisition data to household intake series
	Statistical methodology
	The decomposition model
	General principle
	Specification details
	Estimation and tests
	Applying our methodology to the methylmercury risk assessment
	Estimation and tests on the structure of the model
	The cumulative and the long term individual exposure
	Discussion
