Olivier Allais
email: olivier.allais@ivry.inra.fr

Jessica Tressou
email: jessica.tressou@agroparistech.fr

Using decomposed household food acquisitions as inputs of a Kinetic Dietary Exposure Model

Keywords: household surveys, individualization, linear mixed model, risk assessment, spline-estimation

Foods naturally contain a number of contaminants that may have di¤erent and long term toxic e¤ects. 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.

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 example, excessive exposure to methylmercury, a contaminant mainly found in …sh and other seafood (mollusks and shell…sh) may have neurotoxic e¤ects such as neuronal loss, ataxia, visual disturbance, impaired hearing, and paralysis [START_REF] Who | Methylmercury, environmental health criteria 101[END_REF]. 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 generally estimated from independent individual food consumption surveys (generally not exceeding 7 days) and food contamination data. Several models have been developed to estimate the distribution of usual dietary intake from short-term measurements (see for example, [START_REF] Nusser | A semiparametric transformation approach to estimating usual intake distributions[END_REF][START_REF] Ho¤mann | Estimating the distribution of usual dietary intake by short-term measurements[END_REF]. The proportion of consumers whose usual weekly intake exceeds the PTWI can then be viewed as a risk indicator (see for example, [START_REF] Tressou | Probabilistic exposure assessment to food chemicals based on extreme value theory. application to heavy metals from …sh and sea products[END_REF]. This kind of risk assessment does not account for the underlying dynamic process, i.e. for the fact that the contaminant 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 …rst used in a decomposition model, similar to the one proposed by [START_REF] Chesher | Diet revealed?: Semiparametric estimation of nutrient intake-age relationships[END_REF][START_REF] Chesher | Individual demands from household aggregates: Time and age variation in the quality of diet[END_REF] in the nutrition …eld, 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 de…ned 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 de…nition 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, [START_REF] Hamilton | Time Series Analysis[END_REF] 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 [START_REF] Engle | Non-parametric estimation of the relationship between weather and electricity demand[END_REF], way. In the present article, the individual contaminant intake is …rstly 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 [START_REF] Ruppert | Semiparametric regression[END_REF]. These choices yield a simple linear mixed model which is estimated by REstricted Maximum Likelihood (REML, [START_REF] Patterson | Recovery of inter-block information when block sizes are unequal[END_REF]. One major extension of the proposed model compared to [START_REF] Chesher | Diet revealed?: Semiparametric estimation of nutrient intake-age relationships[END_REF] 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 application of the methodology to the case of methylmercury exposure in the French population using the 2001 SECODIP panel. It includes an empirical validation of the proposed methodology on individual data as well as the results of our estimation procedure and some tests on the structure of the decomposition model. 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 de…ned (see also [START_REF] Verger | Integration of time as a description parameter in risk characterisation: application to methyl mercury[END_REF], for a di¤erent presentation of the same model). 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 [START_REF] Sempé | Auxologie, méthode et séquences[END_REF]) for individuals under 18. In both cases, gender di¤erentiation is introduced.

Assume that estimations of the individual weekly intakes are available, that is y i;h;t denotes the intake of individual i belonging to household h for the t th week (with i = 1; . . . ; n h;t ; h = 1; . . . ; H and t = 1; . . . ; T ); and D i;h;t denotes the same quantity expressed on a body weight basis. The cumulative exposure up to the t th week of this individual is then given by S i;h;t = exp( ) S i;h;t 1 + D i;h;t ;

(1) where > 0 is the natural dissipation rate of the contaminant in the organism. This dissipation parameter is de…ned 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 l 1=2 ; is estimated to 6 weeks, so that = ln(2)=l 1=2 := ln(2)=6 [START_REF] Smith | Methyl mercury pharmacokinetics in man: A reevaluation[END_REF].

The autoregressive model de…ned by (1) and a given initial state S i;h;0 = D i;h;0 has a stationary solution since exp( ) < 1: As a convention, S i;h;0 is set to the mean of all positive exposures (D i;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 S i;h;0 tends to zero as t increases. We call this autoregressive model "KDEM" for Kinetic Dietary Exposure Model.

The individual cumulative exposure S i;h;t can be considered to be the long term exposure of an individual for su¢ ciently 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 de…ned as the cumulative exposure reached after say 6l 1=2 = 36 weeks.

The risk assessment usually consists of comparing the exposure with the so called Provisional In our dynamic approach, the long term exposure is compared to a reference long term exposure denoted by S ref , and de…ned as the cumulative exposure of an individual whose weekly intake is equal to the PTWI, d, such as

S ref = lim t!1 S ref t = d 1 exp( ) ; (2)
where (MAAPAR, 1998(MAAPAR, -2002;; with the INCA survey. In this paper, a methodology using the SECODIP data is developed (see [START_REF] Boizot | Présentation du panel de données SECODIP[END_REF], for a full description of this database). Furthermore, as it is commonly admitted in chronic risk assessment (see [START_REF] Kroes | Assessment of intake from the diet[END_REF], for a description of the common practices in food risk assessment), mean contamination levels are used rather than the distributions of contamination of the di¤erent foods.

S ref t = t X s=0 d exp( (t s)) = d exp( (t + 1)) 1 exp( ) 1 : (3)
The company TNS SECODIP has been collecting the weekly food acquisition data of about …ve thousand households since 1989. All participating households register grocery purchases through the use of EAN bar codes but other grocery purchases are registered di¤erently: the fresh fruit and vegetable purchases are recorded by the FL sub-panel while fresh meat, fresh …sh and wine purchases are recorded by the VP sub-panel. The households are selected by strati…cation 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 de…nes 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 Shell…sh") 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 24

Household intake series ((y h;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 ( g=w).

Before turning to the model description, a few remarks are in order.

1. The use of mean contamination levels in the computation of exposures is a simpli…cation of the model. However it is a current practice in food risk assessment related to chemicals. A comparison between this deterministic approach and a distributional one in which contamination levels are sampled according to their empirical distribution or a chosen parametric distribution was conducted in [START_REF] Tressou | Probabilistic exposure assessment to food chemicals based on extreme value theory. application to heavy metals from …sh and sea products[END_REF]. It mainly shows that the distribution of exposure shows less variability when using mean contamination but that the quality/quantity of observed measures of contamination should be carefully checked as well as the parametric distribution.

2. One underlying assumption of most quantitative food risk assessments based on consumption and contamination data is the independence of the two sets of data. This implies that consumption levels are not correlated to the contamination levels, which in turn can not be considered as determinants of the consumers' choices. Concerning the MeHg health risk, some information campaigns were conducted in France but their impact seems limited as shown in [START_REF] Roosen | The e¤ect of product health information on liking and choice[END_REF]. From an economic point of view, we can therefore assume that the contamination levels of …sh and seafood do not enter as inputs of the consumers'utility function (mostly because they are not aware of the risks and also because of the persistance of the consumption habits). This provides some justi…cation to the hypothesis commonly made in food risk assessment.

3. The food "purchase-consumption" assimilation is of course arguable and will be the main topic of the …nal discussion (see section 4).

4. An additional assumption concerns the household size, denoted by n h;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 …sh in his …rst few months, we assume that food diversi…cation (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.

Statistical methodology

In this section, the decomposition model is described and compared to similar models described in the literature, namely [START_REF] Chesher | Diet revealed?: Semiparametric estimation of nutrient intake-age relationships[END_REF][START_REF] Chesher | Individual demands from household aggregates: Time and age variation in the quality of diet[END_REF]; [START_REF] Vasdekis | Non parametric estimation of individual food availability along with bootstrap con…dence intervals in household budget surveys[END_REF]. Its estimation and some structure tests are then presented.

The decomposition model

General principle

Consider a household composed of n h;t members, each member having unobserved weekly intakes y i;h;t , with i = 1;. . . ; n h;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

y h;t = n h;t X i=1 y i;h;t : (4)
As detailed below, the individual weekly intake y i;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 [START_REF] Chesher | Diet revealed?: Semiparametric estimation of nutrient intake-age relationships[END_REF][START_REF] Chesher | Individual demands from household aggregates: Time and age variation in the quality of diet[END_REF]; [START_REF] Vasdekis | Non parametric estimation of individual food availability along with bootstrap con…dence intervals in household budget surveys[END_REF], 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 concerned, Chesher (1997) retained a multiplicative speci…cation whereas [START_REF] Vasdekis | Non parametric estimation of individual food availability along with bootstrap con…dence intervals in household budget surveys[END_REF] chose the additive one. In the multiplicative model, a change in income for example would proportionally a¤ect all the individual intakes whereas in the additive setting, they would be a¤ected by the same value. Following [START_REF] Vasdekis | Non parametric estimation of individual food availability along with bootstrap con…dence intervals in household budget surveys[END_REF], we retained the additive speci…cation since the di¤erence between the two speci…cations may not be notable, and the additive 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 dependency is directly introduced into the function f that is bivariately smoothed according to age and time (cf. [START_REF] Green | Nonparametric Regression and Generalized Linear Models[END_REF]. Again, we adopt a simpler speci…cation in which time is introduced as a dummy variable. All these assumptions yield an individual model of the form h;t ; q = 1; : : : ; Q; with m q modalities, and …x the m th q modality as the reference modality, then the socioeconomic e¤ect term in (5) and (6) is

y i;h;t =
w h;t = Q X q=1 mq 1 X m=1 q;m 1l n W (q) h;t =m o ;
where q;m is the e¤ect of the m th modality of the socioeconomic variable q:

In section 3.1, tests are conducted to select the most relevant socioeconomic variables and their modalities.

Similarly, time is only measured by weekly counts throughout the year so that the time e¤ect in (5) and ( 6) is simply

t = T X =1 6 = R 1l f =tg ;
where is the e¤ect of week and R is the reference week.

Error speci…cation The error at the individual level " i;h;t is assumed to be Gaussian with zero mean, and the variance-covariance structure is such that households are independent, i.e. 8i; i 0 ; t; t 0 and 8h 6 = h 0 cov(" i;h;t ; " i 0 ;h 0 ;t 0 ) = 0; members of the same household are dependent, that is for 8h; t and i 6 = i 0 ; cov(" i;h;t ; "

i 0 ;h;t ) = 2 " ; (8)
where measures the dependence between individuals within the same household.

there is no time dependence, that is 8i; i 0 and 8t 6 = t 0 cov(" i;h;t ; " i 0 ;h;t 0 ) = 0:

In the rescaled household model (6), the error " h;t P n h;t i=1 " i;h;t = p n h;t is i.i.d. Gaussian with a zero mean and a variance R such that 8t; t 0 and 8h 6 = h 0 ,

V(" h;t ) = 2 " n h;t + (1 ) 2
" and cov(" h;t ; " h 0 ;t 0 ) = 0:

These assumptions are discussed and tested in the application section (see 3.1).

Estimation and tests

The model ( 6) is a linear mixed model that can be estimated using restricted maximum likelihood (REML) techniques, see [START_REF] Ruppert | Semiparametric regression[END_REF] for details. An attractive consequence of the use of the mixed model representation of a penalized spline in (7) is that mixed model methodology and software can be used to estimate the parameters and predict the random e¤ect in the resulting household model. The amount of smoothing of the underlying functions f S is estimated with the REML technique via the estimation of 2 u S . The estimation was conducted using R SAS MIXED procedure. To get estimators for 2 " and ; asymptotic least square techniques combined with the linear relationship between the variance given in (10) and the household size were used. More precisely, a residual variance 2 n is …rst estimated for each household size n = 1; : : : ; N = max n h;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 2 " and and their standard deviations.

The individual intake is then predicted by

d y i;h;t = x i;h;t b + z i;h;t b u + w h;t ^ + t ^ ; (11)
where b , ^ , and ^ are the estimators of , , and respectively and b u is the best prediction of the random e¤ect u in the model (6).

Con…dence and prediction intervals can be built for the prediction d y i;h;t as proposed in [START_REF] Ruppert | Semiparametric regression[END_REF] and several tests can be conducted in this model:

1. Are the random e¤ects di¤erent according to sex? In other words, is the assertion

2 u M = 2 u F = 2 u true?
2. Another question is the necessity for such random e¤ects. Is the assertion 2 u = 0 (resp.

2 u M = 0 or 2 u F = 0) true?
3. More globally, is the function f the same for both sexes? Is the assertion f M = f S 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 [START_REF] Self | Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions[END_REF][START_REF] Crainiceanu | Some properties of likelihood ratio tests in linear mixed models[END_REF] because the test concerns the frontier of the parameter de…nition ( 2 u 2 [0; +1[).

3 Applying our methodology to the methylmercury risk assessment

In this section, we illustrate our approach on the methylmercury risk assessment. Firstly, several tests are conducted on the decomposition model yielding a …nal estimation of the individual intake series based on the SECODIP data. An empirical validation on individual consumption data is then proposed. Finally, individual long term exposure obtained from our model is compared to the reference long term exposure described in section 1.

Estimation and tests on the structure of the model

Table 2 shows a preliminary REML estimation of our model, de…ned in (6); under the following assumptions:

the socioeconomic variables are household income, region of residence, occupation category and level of education of the principal household earner, their modalities (inc. the reference) are given in Table 2.

the function f di¤ers according to the gender but the random e¤ect does not (f M 6 = f F and

2 u M = 2 u F );
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 n h in (10). For each household size, a variance is estimated, and estimates of and 2 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 .

Table 2 around here, see page 24

Selection of the sociodemographic covariates/modalities In this sub-section, we show the results of several tests we carried out to simplify the interpretation 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 di¤er signi…cantly from the reference. All tests are performed on the 5% level of signi…cance 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 signi…cantly di¤er from the reference modality blue collar workers (H1; P val = 0:771). Re…tting the model with the reference modality "Blue collar workers and self employed", all the socioeconomic variables are signi…cantly di¤erent from the reference. Then, F-tests allow us to conclude that the resulting three groups are signi…cantly di¤erent from each other (H2; H3; H4).

Let us now consider the region of residence variable. First, there are some very substantial di¤erences among the 4 regions of residence (H5; P val 0:001). However, the modality "North, Brittany, and Vendee coast" and the modality "Paris and its suburbs" should be grouped (H6 c, P val c = 0:881). Then, the other tests implemented for the level of education and income variables suggest that no further simpli…cation is possible (see p-values of null hypotheses H7, H8, H9 in Table 3). Finally, the overall F-test comparing our resulting …nal model to the original model (6)

shows that no important variable has been left out of the model (P val = 0:59).

Other tests on the …nal model Likelihood ratio tests are implemented to test the structure of the …nal model, that is the one resulting from the previous tests and simpli…cations regarding the socioeconomic variables.

First, the dependence of individual exposures to methylmercury within a household is tested.

The null hypothesis = 0 (cf. equation (10)) is rejected (P val < 0:001) which con…rms that individuals within the same household have correlated exposures. Moreover, a graphical comparison of the exposure curves from the two models shows that accounting for the dependence within the household reduces the mean individual exposure in the adult population and increases the children's [START_REF] Tressou | Méthodes statistiques pour l'évaluation du risque alimentaire[END_REF]. Time dependence in the errors (cf. (9)) was also investigated in [START_REF] Tressou | Méthodes statistiques pour l'évaluation du risque alimentaire[END_REF] (see pages 139 and 147) where two models were confronted: in the …rst one, time independence and within household dependence were assumed, and in the second one, autoregressive correlation of order 1 and within household independence were assumed. A comparison of the Akaike criteria favoured the …rst model. Unfortunately, limitations in the parametrization of the R variance structure unabled the test of joint time and within household dependences but this will be investigated in future work.

Finally, we test if the function f is the same for both genders. The null hypothesis f M = f F is rejected (P val < 0:001) but the null hypothesis 2 u M = 2 u F is accepted. This means that individual exposure di¤ers with gender but both functions need the same amount of smoothing.

Regarding the error structure, basic model checks are quite desappointing. Indeed, normality is rejected by usual tests (Kolmogorov-Smirnov, Cramer-von Mises, Anderson-Darling tests have p-values lower than 1%). This is mainly due to the important disymetry exhibited by the residuals as shown in Figure 1. Moreover, since the dependent variable are positive, the residual plot is truncated by below, see Figure 2. For standard regressions, a (log) transformation is usually applied to the data but in our case this kind of (nonlinear) transformation would annihilate the possibility of prediction for individual quantity, namely log i y i is not equal to i log y i . Other solutions could be studied to account for the disymetry: choice of some disymetric parametric distribution such as the (shifted) lognormal distribution or adopt generalized linear mixed modelling approach.

Bootstrap techniques can also be adapted. However, in this premier model, we chose not to engage in such re…nements, convinced that other limitations of the model would need to be addressed …rst, namely the failure of the proposed model to determine non-consumers (see the discussion section). unexpected e¤ect on the individual exposure. Indeed a higher exposure is expected for white collar workers and retirees whan compared to blue collar workers but an opposite e¤ect 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. Finally, the individual exposure is plotted as a function of age in Figure 3 for the female and male populations respectively. Children exposure is quite similar for both gender but the adult female subpopulation tends to be more exposed than the adult male subpopulation. This is not a body weight e¤ect since it remains true when plotting the exposures expressed in g=w instead of g=kgbw=w (graphics not shown). Besides, when considering the 95%-con…dence intervals (CI's)

for the curves, also given in Figure 3 for both subpopulations, we observe that this di¤erence is signi…cant at least for adults between roughly 55 and 75. These CI's also illustrate that our estimation procedure yields more uncertainty and variability at the edge of the graph, that is for the younger and the older individuals. This is common to most spline estimation and also results

here from an extrapolation bias in the case of children. Indeed, the model parameters are estimated from the household model and there is no household exclusively composed of children. However as we shall see in the next section, the error certainly remains within the computed con…dence intervals. Very few individuals have a cumulative individual exposure above the reference long term exposure. We estimate that only 0:186% of individuals are deemed at risk. This risk index should be compared to the more common one de…ned as the percentage of weekly intakes D i;h;t exceeding the PTWI, denoted R 1:6 , such as R 1:6 = 1 nT P T t=1 P H h=1 P n h i=1 1l (D i;h;t > 1:6). R 1:6 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.

Discussion

In our opinion, two main topics need to be discussed after this study: …rst the limitations of the decomposition model and then the use of household acquisition data in a food safety context.

Limitations of the decomposition model As already mentioned in section 3.1, the model does not …t very well yielding nonnormal residuals with an important disymetry so that predicted individual exposures most often sum to a quantity smaller than the observed. This is certainly due to the fact that null consumptions (and thus null exposures) are not accounted for in our model.

This could be improved considering a Tobit type model at the individual level (instead of (5)).

In term of risk, this is important since null consumption of one household member automatically increases the exposures of others. This should be the subject for future investigations.

Regarding the conclusion of the long term risk assessment stating that children between 1 and 3 are the more vulnerable population, the main limitation of the decomposition model is obviously the extrapolation bias concerning the children population. However, the empirical validation proposed in section 3.2 partly restores con…dence in our model although the data used in the validation only concerns individuals over 3 years old. Moreover, this kind of bias is inherent to decomposition model and can also be underlined in other models of this kind [START_REF] Chesher | Diet revealed?: Semiparametric estimation of nutrient intake-age relationships[END_REF][START_REF] Vasdekis | Non parametric estimation of individual food availability along with bootstrap con…dence intervals in household budget surveys[END_REF]. A comparison of the performances of these models was conducted in [START_REF] Tressou | Méthodes statistiques pour l'évaluation du risque alimentaire[END_REF] and illustrated that the mean squared error was lower in our model.

Use of household acquisition data 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, [START_REF] Serra-Majem | Comparative analysis of nutrition data from national, household, and individual levels: results from a WHO-CINDI collaborative project in Canada, Finland, Poland, and Spain[END_REF] assert that these data are good estimates for the consumption of the whole household. [START_REF] Vasdekis | Non parametric estimation of individual food availability along with bootstrap con…dence intervals in household budget surveys[END_REF] avoid this question by using the term "availaibility" instead of intake or consumption. However, as in [START_REF] Chesher | Diet revealed?: Semiparametric estimation of nutrient intake-age relationships[END_REF], 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 2. The amount of food bought by a household can be di¤erent from the amount actually consumed. Indeed, namely for seafoods, a non negligible part is not edible: [START_REF] Favier | Répertoire Général des Aliments, Table de composition[END_REF] show than on average only 61% of fresh or frozen …sh is edible. Besides, [START_REF] Maresca | Collectes sélectives des déchets et comportements des ménages[END_REF] 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 …sh bought is ready to be consumed or as a whole …sh that needs some preparation. Applying such a factor to all household intakes yields a long term risk of 0.00%, and R 1:6 = 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%, R 1:6 = 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 …sh and seafood is needed, to realize an accurate assessment of exposure to methylmercury, taking into account only the edible part of …sh and other seafood.

3. 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 a¤erent 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. Moreover, the empirical validation of section 3.2 clearly states that this body weight has a minor impact at least on the estimation of exposure for people older than 3.

4. The food nomenclature of the SECODIP database is not as detailed as the contamination database. Unfortunately, …sh and seafood species are not well documented so it is not possible to consider more than two food categories when computing household intakes. Namely, in the case of …shes, it is clear that the contamination highly depend on the species (tuna being one the most contaminated …sh) and when averaging all …sh contaminations in the computation of the household intake, we reduce the actual variability of the exposure. However, 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 di¢ cult to conduct. Highly trained interviewers are required as well as an extraordinary cooperation from respondents. Household food acquisition data can serve many other applications (economics or marketing) and, at least for the SECODIP data, acquisition recording is simpli…ed 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 de…nition of a Kinetic Dietary Exposure Model (KDEM) that integrates the fact that contaminants are eliminated from the body at di¤erent 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 acquisition data which are …rst decomposed into individual intake data through a disaggregation model accounting for the dependence among household members. One important feature of this model is that it simpli…es into a linear mixed model which can be estimated using standard routines even if the parametrization of the variance components may be di¢ cult. Several extensions of this methodology are currently studied. First, the disaggregation model proposed here presents many limitations and would be considerably improved by considering a preliminary 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 [START_REF] Bertail | A storage model with random release rate for modeling exposure to food contaminants[END_REF]. The parameters of this new model are the intake distribution, the inter-intake time distribution and the dissipation rate distribution. In this framework, the dissipation parameter of the KDEM model is random and the intake and inter-intake time distributions can be estimated either from individual (INCAtype) data or household (SECODIP-type) data, see [START_REF] Verger | Integration of time as a description parameter in risk characterisation: application to methyl mercury[END_REF] for an illustration with the INCA data.

Figures and Tables

Figure 1

 1 Figure 1 around here, see page 25 Figure 2 around here, see page 26

Figure 3

 3 Figure 3 around here, see page 28

Figures 4

 4 Figures 4 and 5 around here, see page 28 and 29

Figure 6

 6 Figure 6 around here, see page 29

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 R 1:6 = 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.

Figure 1 :Figure 2 :

 12 Figure 1: Model check: histogram and normal density adjustment (…nal model, Secodip data).

Figure 3 :Figure 4 :

 34 Figure 3: Individual exposure as a function of age according to gender (unit: g per kilogram of body weight per week).

Figure 5 :

 5 Figure 5: Empirical validation of the decomposition model using individual consumption data (Female population, unit: g per kilogram of body weight per week).

Figure 6 :

 6 Figure 6: Cumulative exposure to MeHg (unit: g per kilogram of body weight)

  the positive part of the di¤erence between the age of the individual a i;h;t and the node S;k and the u S k are random e¤ects assumed to be i.i.d. Gaussian with distribution N 0; 2 u

			K S X	u S k (a i;h;t	S;k ) + ;	(7)
			k=1
	where the ( S;k ) k=1;. . . ;K S are nodes chosen from an age list and
	(a i;h;t	S;k ) + (a i;h;t	S;k ) 1l fai;h;t S;k >0g

x i;h;t + z i;h;t u + w h;t + t + " i;h;t ;

(5) where the terms x i;h;t + z i;h;t u stand for the mixed model spline representation of the function f; the term w h;t denotes the socioeconomic e¤ects, the term t the time e¤ect, and " i;h;t is the individual error term.

Combining (4) and (5) ; we obtain the …nal rescaled household model given by

Y h;t = X h;t + Z h;t u + p n h;

t w h;t + p n h;t t + " h;t ; (6) where Y h;t P n h;t i=1 y i;h;t = p n h;t ; X h;t P n h;t i=1 x i;h;t = p n h;t ; Z h;t P n h;t i=1 z i;h;t = p n h;t ; and " h;t P n h;t i=1 " i;h;t = p n h;t : 2.1.2 Speci…cation details Age-gender function speci…cation Let a i;h;t and s i;h denote the age and sex of individual i of household h for the t th week. Individual dietary intake is generally di¤erent according to the gender of individuals, so the function f takes the following form f (a i;h;t ; s i;h ) = f M (a i;h;t )1l fsi;h=Mg + f F (a i;h;t )1l fsi;h=F g ; where f M (:) and f F (:) are age-intake relationships for males (M) and females (F) respectively, and 1l fAg is the indicator function of event A. The function f S (:) is approximated by a spline of order one with a truncated polynomial basis for either sex, such as f S (a i;h;t ) = S 0 + S 1 a i;h;t + denotes S . This last assumption allows us to introduce some penalties into the model and to smooth the function f S 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 K S is set to min a S;d 4 ; 35 , where a S;d is the list of distinct ages for individuals of sex S, and the nodes S;k are de…ned as the k+1 K S +2 th percentile of vector a S;d for k = 1;. . . ; K S . De…ning x i;h;t as a line vector 1l fsi;h=Mg a i;h;t 1l fsi;h=Mg 1l fsi;h=F g a i;h;t 1l fsi;h=F g ; and z i;h;t as the line vector n (a i;h;t S;k ) + 1l fsi;h=Sg o k=1;. . . ;K S ; S=M;F ; we …nally obtain the …rst terms of (5) ; that is f (a i;h;t ; s i;h ) = x i;h;t + z i;h;t u: Socioeconomic characteristics and time dependency In the application, all the socioeconomic characterics are categorical variables. Consider the Q categorical variables W (q)

Table 3

 3

	around here, see page 27
	Table 4 around here, see page 27

Table 1 :

 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 Shell…sh	0.014	0.001	0.172	0.011	1293

Table 2 :

 2 Restricted maximum likelihood estimates (REML) for age and all socioeconomic variables and the p-value of the Student's tests(Pval)

	E¤ect	Parameter	REML	Pval
	Income	(ref: Mean sup)
	Well to do	1	6.027	<0.001
	Mean inf	2	2.686	<0.001
	Modest	3	-1.928	<0.001
	Region of residence	(ref: Noncoastal regions)
	North, Brittany, Vendee coast	4	0.962	0.003
	South West coast	5	5.232	<0.001
	Mediterranean coast	6	2.303	<0.001
	Paris and its suburbs	7	1.023	0.009
	Occupation category of the principal household earner	(ref: Blue collar workers)
	self-employed persons	8	-0.122	0.771
	white collar workers	9	-3.733	<0.001
	retirees	10	-5.261	<0.001
	no activity	11	-1.910	0.004
	Level of Education of the principal household earner	(ref: BAC and higher degree)
	student	12	5.901	<0.001
	no or weak diploma	13	-1.281	<0.001

Table 4 :

 4 Restricted maximum likelihood estimates (REML) for all age and socioeconomic variables of the reduced …nal model with all variance components and their standard errors (s.e)

	E¤ect	Parameter	REML	Pval
	Income	(ref: Mean inf)
	Well to do	1	6.062	<0.001
	Mean inf	2	2.708	<0.001
	Modest	3	-1.931	<0.001
	Region of residence	(ref: Non coastal regions)
	Paris and North, Brittany, Vendee coast	4 = 7	0.984	<0.001
	South west coast	5	5.232	<0.001
	Mediterranean coast	6	2.297	<0.001
	Occupation category of the principal household earner	(ref: Blue collar workers and self employed persons)
	white collar workers	9	-3.704	<0.001
	retirees	10	-5.242	<0.001
	no activity	11	-1.877	0.005
	Level of education of the principal household earner	(ref: BAC and higher degree)
	student	12	5.901	<0.001
	no or weak diploma	13	-1.275	<0.001
			REML	s.e
	Variance of the random e¤ect	u	24.832	6.7316
	Variance-covariance structure
	variance	2	1260705	282309
	correlation		-0.22	0.0434

In this application of the decomposition model, assumptions similar to the previous section are used regarding the basic structure of the model and the introduced socioeconomic variables are the region of residence and the occupation category of the principal household earner. There is no time e¤ect since the INCA data only records the consumption over one week.


Badreddine Douzi

Sébastien Alphonse

Herve Darbon

Alain Filloux

Christian Cambillau

Cedric Bernard

Mariella Tegoni

Romé Voulhoux

Xcpv The

The XcpV/GspI pseudopilin has a central role in the assembly of a quaternary complex within the T2SS pseudopilus

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