2.2 Estimation and tests In the rescaled household model (2.3), the error ε$_{h,t}$ ≡ ∑ n$_{h,t}$ i =1 ε$_{i,h,t}$/ √ n$_{h,t}$ is i.i.d. Gaussian with a zero mean and a variance R such that ∀ t, t ′ and ∀ h = h $^{′}$, 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 f$_{S}$ is estimated with the REML technique via the estimation of σ 2 $_{u$_{S}$}$. The estimation was conducted using R ○ SAS MIXED procedure. To get estimators for σ 2 ε 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 σ 2 n is first 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 σ 2 ε and ρ and their standard deviations. The individual intake is then predicted by where ̂ β , ˆ γ , and ˆ α are the estimators of β , γ , and α respectively and ̂ u is the best prediction of the random effect u in the model (2.3). Confidence and prediction intervals can be built for the prediction ̂ y$_{i,h,t}$ 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 σ 2 u$_{M}$ = σ 2 u$_{F}$ = σ 2 u true? 11 ̂ y$_{i,h,t}$ = x$_{i,h,t}$ ̂ β + z$_{i,h,t}$ ̂ u + w$_{h,t}$ ˆ γ + δ$_{t}$ ˆ α, (2.6) V ( ε$_{h,t}$ ) = ρσ 2 $_{ε}$n$_{h,t}$ + (1 − ρ ) σ 2 ε and cov ( ε$_{h,t}$, ε$_{h}$ $^{′}$,t′ ) = 0 . (2.5)