2. Another question is the necessity for such random effects. Is the assertion σ 2 u = 0 (resp. σ 2 u$_{M}$ = 0 or σ 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 (Self and Liang, 1987; Crainiceanu et al. , 2003) because the test concerns the frontier of the parameter definition ( σ 2 u ∈ [0 , + ∞ [). 

## 3 Applying our methodology to the methylmercury risk assess- ment 

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 ( f$_{M}$ ̸ = f$_{F}$ and σ 2 u$_{M}$ = σ 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 (2.5). For each household size, a variance is estimated, and estimates of ρ and σ 2 are obtained using asymptotic least square techniques as mentioned in section 2.2. Since 

