The name is absent

90

Separate Models: Assume a separate model for each disease subtype. There is
no borrowing of strength or pooling of information across subtypes. That is, for
i = 1,..., n,

Vi I Pi ~ Bin(p_i, N_i)
logit(p_i) ~ N(μ_i,τ_i)
μ_i ~ 2V(0,1/4) and τ_i ~ Gα(1.25,5).

The hyperprior means match the parameters in the NEPPM model (4.1).

Parametric Hierarchical Model: The model borrows strength across subtypes,
but treats a priori all subtypes symmetrically. No prognosis information is considered.

Vi I P '~ Bin(p_i, N_i)
logit(p_i) ~ N(μ, τ)
μ ~ N(0,1/4) and τ ~ Gα(1.25, 5).

The use of hierarchical models with a priori exchangeable subpopulations is a
standard approach for many biomedical inference problems that require borrowing of
strength across subpopulations.

Hierarchical with Logistic Regression Model (HLRM): The HLRM as-
sumes partial exchangeability by fixing the success rates as a logistic regression on x_i,
the overall prognosis of disease subtype i. In other words, ρ is fixed as the grouping
determined by the prognosis covariate. There is no learning about the partition.

Vi I P ~ Bin(p_i, N_i)
logit(p_i) ~ N(β₀ + βμc_i, τ)
(β₀,βι)^t ~MVN(O√₂∕4),                     (4.10)

where the precision τ is fixed to τ — 18 like in (4.1).

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