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(pi, Ni)
logit(pi) ~ N(μi,τi)
μi ~ 2V(0,1/4) and τi ~ 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(pi, Ni)
logit(pi) ~ 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 xi,
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(pi, Ni)
logit(pi) ~ N(β0 + βμci, τ)
(β0,βι)t ~MVN(O√2∕4), (4.10)
where the precision τ is fixed to τ — 18 like in (4.1).