Bayesian AAPC Model. Xxxxxxx et al. [2003] introduced a full area-age-period-cohort (AAPC) model to study the spatio-temporal pattern of disease risk. The model incorporates the main effect of area, age, period and cohort, and interaction terms such as the area-cohort and area-period interactions. The model is as follows: log(λiap) = νi + µi + θa + γp + δc + ϕip + ϕic, where λiap is the relative risk for the αth age group and the pth calendar period in the ith area, νi is the unstructured spatial term for the spatial heterogeneity effects, µi is the structured spatial term to incorporate spatial clustering effect, θa, γp and δc are the age, period, and cohort main effects, ϕip is the space-period interaction and ϕic is the space- cohort interaction. In the prior assumptions, νi is an unstructured area effect, and µi follows an intrinsic conditional Gaussian spatial autoregressive model (ICAR). For the unstructured spatial effect νi, we usually assign a homoscedastic distribution to them such as νi ∼ N (0, σ2) where σ can be further defined as a hyperprior with an inverse-gamma distribution, i.e., σ ∼ IG(αν, βν). For the structured spatial effect µi, Xxxxxxx [2006a] illustrated the joint distribution for spatial effects µ = (µ1, ..., µn) derived from their pairwise differences and a variance term κ as follows: P (µ1, ..., µn) ∝ exp[−0.5κ—1 Σ Σ cij(µi − µj)2], where the cij are contiguity measures based on spatial adjacency between areas i and j. cij = ,, 1, if areas i and j are first-order neighbours; ,, 0, otherwise. Xxxxxxx [2006a] further demonstrated that the conditional prior of µi given the remaining spatial effects µj where j /= i follows a normal distribution. P (µi|µ[i]) ~ N (ωi, τi—1), where µ[i] refers to remaining spatial effects µj where j ΣΣ computed as
i. The weighted average ωi is ω = j cijµj j cij = ωijµj Σ and Σ j cij are conditional variances. This is recognized as the intrinsic conditional autoregressive (ICAR) prior since the conditional distribution involves row-standardised weights [Xxxxxxx, 2006a]. The effects γp and δc are modeled as Gaussian RW1 and RW2 [Xxxxxxxx and Xxxxxxx, 1994]. The Kronecker product of these structure matrices Kµ p = Kµ ⊗ Kp(or Kµ c = Kµ ⊗ Kc) defines the structure matrix for the joint prior and provides a prior for the interaction terms ϕip(or ϕic) [Xxxxxxx, 2006a]. For example, the joint spatio-period interactions ϕ = (ϕip, i = 1, ..., N, p = 1, ..., P ) are taken as ϕ ~ N (0, τϕKµ p). Usually, the hyperprior τϕ is set as a non...
