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1,... ,n, n = 12. Denote also by p = (pɪ,... ,pn)t the vector of success rates. Let
p= (Si,..., Sχ) denote a partition of {1,..., n} into K clusters St, к = 1,..., K,
i.e., {1,..., n} = U^L1S⅛ and S⅛ ∩ S⅛∕ = 0 for к ≠ k'. Let xn = {τ1,... ,xn} the set
of values of the ordinal covariates. We assume that all disease subtypes in the same
cluster have similar success rates. More specifically, the logits of the success rate with
indices in the cluster к are drawn from the same normal distribution with mean θk.
The partition p is equivalently characterized by cluster membership indicators φi, for
i = 1,..., n with φi = к if i ∈ Sk- We use p or (φ, K) interchangeably.
We assume the following model (NEPPM):
Vi I Pi ~ Bin(pi, Ni)
logit(pi) ≡ θi Й N(θ*k,τp), for i ∈ Sk
0⅛'~N(O, Te), for к = 1,..., K and p~p(p∖xn), (4.1)
The definition of p(p ∣ xn) will be discussed below. Here, N(m, s) denotes the normal
distribution with mean m and precision s. We fix τp = 18 such that the ratio of θll
and θi2 for any two zɪ, ⅛ in the same cluster is between 1/2.5 and 2.5 with probability
0.95. We fix τβ = 1/4. This value allows for sufficiently spread out values of θk,
and thus PiS. The model defines a compromise between an exchangeable hierarchical
model, separate models and a partially exchangeable hierarchical regression model.
To define the model p(p) we resort to the product partition models (PPMs) (Harti-
gan, 1990; Barry and Hartigan, 1993) ) The idea is to construct a probability distribu-
tion p(pn) on the space of partitions of {1,..., n}, by introducing a cohesion function
c(A) ≥ 0 for every Ac {1,..., n}, measuring how tightly grouped the elements in A
are thought to be. A product partition model is defined as
1 κ κ
p(pn = (Si,..., Sk)) = - ∏ c(Sfe), p(yn I pn) = ∏p(yi : i ∈ Sfc), (4.2)
9n fe=l fc=l
Model (4.2) is easily seen to be conjugate. A remarkable connection between PPMs
and the Dirichlet process (DP) (Ferguson, 1973) is pointed out, for example, in Quin-
tana and Iglesias (2003) and Dahl (2003)). The DP is discrete with probability 1 and