IOl
with the observation i. The proposed model is robust in the sense that it allows the
data to speak and correct prior assumptions when the prior beliefs happen not to be
confirmed by the data. However, the precision of estimates under the proposed model
can be lower than under a partially exchangeable hierarchical regression because in-
ference has to account for the uncertainty in the grouping. This was confirmed in the
simulation study. The proposed model performed better when the prior beliefs were
not matched by the simulation truth.
The proposed model is useful when sample sizes are small in some subpopulations.
With large samples in all subpopulations, a separate model for each subpopulation
could be considered. However, small samples are usually the case in early phase
clinical trials. In particular due to the rare nature of the sarcomas studied in the
motivating phase II clinical trial borrowing strength is essential to obtain precise
estimations.
The proposed model included a novel distribution over random partitions that
gives increased a priori weights to homogeneous partitions. The additional compu-
tational effort compared to a conventional Polya urn (induced by a Dirichlet process
mixture) is minimal. Depending on the application, in principle any sampling model
and any distribution for the cluster-specific effects can be considered. The proposed
model is a particular case of the more general PPMx model introduced in Müller et
al. (2009). Their model uses covariate information to change the prior probability of
clustering. They consider continuous, ordinal and nominal covariates.
In summary, we have proposed an easy to implement model to borrow strength
across non-exchangeable subpopulations. The approach compromises between bor-
rowing too much strength (such as the hierarchical regression model) and no borrow-
ing at all (such as separate models for each subpopulation). The proposed model is
suitable to estimate the success rates in the motivating sarcoma trial presented here
by borrowing strength across the different subtypes of the disease. It considers a pair
of sarcoma subtypes to be a priori likely to be exchangeable when they are related