LIKELIHOOD Sample Clauses

LIKELIHOOD. XX XXXXX --------------------------------------------------------------
LIKELIHOOD. Cancer death counts yiap at area i, age a and period p are modelled as Poisson distribution with parameters rate λiap and population size Niap, i.e., yiap ~ Poisson(Niapλiap), where λiap is the relative risk for the αth age group and the pth calendar period in the ith area, and can be modeled as log(λiap) = νi + µi + θa + γp + δc + ϕip + ϕic. Therefore, we can write the likelihood as , P (y|λ) ∝ Qi,a,p eNiapλiap (Niapλiap)yiap ∝ Qi,a,p eλiap (λiap)yiap where λ = (λiap) includes all i, a, and p and represents all the parameters which need to have prior distributions. Niap is a known quantity which can be ignored here. Priors The joint prior distribution can be written as P (λ) = P (ν, µ, θ, γ, δ, ϕ1, ϕ2), where ν is the joint spatial unstructured effect, µ is the joint spatial structured effect, θ is the age effect, γ is the period effect, δ is the cohort effect, ϕ1 and ϕ2 are the joint spatial- period and spatial-cohort interactions, respectively. Age effects As we introduced before, we apply the carcinogenesis model into the AAPC model by replacing the age effects with hazard functions derived from the carcinogenesis model. For example, the age effect θa at age group a is given a noninformative prior as below: θa ∼ N (θ¯a, τa) where θ¯a = log(h(ta)), and τa ∼ Gamma(0.001, 0.001). Armitage and Doll [1954] assumed multiple transformations happened in stages for a cell to grow into a cancerous tumor. In the Armitage-Doll model, h(t) = cts—1 where s is the number of stages, ta is the age and c is constant. Therefore, we have the following age effects in a nonlinear function of age, θ¯a = c + (s — 1) ∗ log(ta). Hyperpriors are given to c and s at c ∼ N (0, τc) s ∼ N (5, τs) where Gamma hyperpriors are given to two precision terms as below: τc ∼ Gamma(0.001, 0.001) and τs ∼ Gamma(0.001, 0.001). The hazard function derived from the TSCE model is introduced to the AAPC model as well. In contrast to the simple format of the hazard function in the Armitage-Doll model, the hazard function from the TSCE model is more difficult. Four derived parameters will be summarized in the TSCE model, the rate of initiation, ν, the rate of division, α, and death, β, of initial cells, and the rate of malignant conversion, µ. The hazard function in the TSCE model is given by Xxxxxxxxxx [1979, 1981, 1990, 2009] as h(t) = ν e—qt e—pt pq , — α qe—pt — pe—qt where p and q are the roots of a quadratic equation, with p + q = —(α— β — µ) and pq = αµ. Three estimated parameters p, q, and r...


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