2.1.2 Specification details Age-gender function specification 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 different according to the gender of individuals, so the function f takes the following form where f$_{M}$ ( . ) and f$_{F}$ ( . ) are age-intake relationships for males (M) and females (F) respectively, and 1l$_{{}$$_{A}$$_{}}$ 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 where the ( κ$_{S,k}$ )$_{k}$$_{=1}$$_{,}$ . . . ,K$_{S}$ are nodes chosen from an age list and denotes the positive part of the difference between the age of the individual a$_{i,h,t}$ and the node κ$_{S,k}$ and the u S k are random effects assumed to be i.i.d. Gaussian with distribution N ( 0 , σ 2 u$_{S}$ ) . 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 κ$_{S,k}$ are defined as the ( k +1 K$_{S}$ +2 ) th percentile of vector a$_{S,d}$ for k = 1 , . . . , K$_{S}$ . Defining x$_{i,h,t}$ as a line vector ( $^{1l}${ s$_{i,h}$ = M } a$_{i,h,t}$ $^{1l}${ s$_{i,h}$ = M } $^{1l}${ s$_{i,h}$ = F } a$_{i,h,t}$ $^{1l}${ s$_{i,h}$ = F } ) , and z$_{i,h,t}$ as the line vector { ( a$_{i,h,t}$ − κ$_{S,k}$ )$_{+}$ $^{1l}${ s$_{i,h}$ = S $_{}}$} k =1 , . . . ,K$_{S}$ ; S = M,F , we finally obtain the first terms of (2.2) , that is f ( a$_{i,h,t}$, s$_{i,h}$ ) = x$_{i,h,t}$β + z$_{i,h,t}$u. 9 ( a$_{i,h,t}$ − κ$_{S,k}$ )$_{+}$ ≡ ( a$_{i,h,t}$ − κ$_{S,k}$ $^{) 1l}${ a$_{i,h,t}$ − κ$_{S,k}$> 0 } f$_{S}$ ( a$_{i,h,t}$ ) = β S 0 + β S $_{1}$a$_{i,h,t}$ + K$_{S}$ ∑ k =1 u S k ( a$_{i,h,t}$ − κ$_{S,k}$ )$_{+}$ , (2.4) f ( a$_{i,h,t}$, s$_{i,h}$ ) = f$_{M}$ ( a$_{i,h,t}$ $^{)1l}${ s$_{i,h}$ = M } + f$_{F}$ ( a$_{i,h,t}$ $^{)1l}${ s$_{i,h}$ = F } ,