Area-APC Model Sample Clauses

Area-APC Model. To study the space and time variation for the risk of disease, many general or more heavily parameterized Bayesian approaches in Area-APC (AAPC) models have been proposed to study spatio-temporal mappings of disease rates [Xxxxxx et al., 1997, Xxxxxx and Xxxxx, 2009]. AAPC models have been studied in some recent work in considering spatial correlation in time or important cohort effects [Xxxxxxx, 2006a]. In an analysis of lung cancer rates in Tuscany, 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 and µi are the spatial terms, 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. Two different spatial effects considered in this model can be viewed as random effects where unstructured spatial effects νi represents the spatial heterogeneity and structured spatial effects µi considers the spatial clustering [Xxxxxxx et al., 2003]. A Kronecker product of the structure matrix for the relevant dimensions [Xxxxxxx, 2006a, Xxxxxxx et al., 2003, Xxxxxx and Xxxx, 2004] is used to derive the prior distribution for the interaction terms. The joint distribution for spatio-temporal interactions is modeled as a multinormial distri- bution. For example, the joint spatio-period interactions ϕ = (ϕip, i = 1, ..., N, p = 1, ..., P ) are taken as ϕ ∼ N (0, τϕKµ p). The structure matrix Kµ p is the Kronecker product of Kµ for the spatial effect and Kp for the period effect, such as Kµ p = Kµ ⊗ Kp. Since both Kµ and Kp are symmetric and singular matrices, their Kronecker product Kµ p is symmetric and singular as well. Therefore, the joint density for spatio-temporal effects is improper [Xxxxxx et al., 1997]. Xxxxxx and Xxxxx [2009] pointed out that the proper posterior may not always result, thus extra care must be taken when using the improper priors. As an alternative solution, Xxxxxxx [2006b] introduced the parsimonious product interactions schemes with generic form αiβp, i = 1, ..., N, p = 1, ..., P, where αi are the structured spatial effects, subject to Σi αi = 0, while Σ P−1 βp = exp(ηp)/[1...