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subtypes to reach any meaningful conclusion. The disease subtypes are the non-
exchangeable experimental units. In general, I consider problems when experimental
units are grouped according to a categorical covariate, with small sample sizes in some
groups and when the experimental units may not be exchangeable across the values
of the covariate. In the motivating application the covariate is the overall prognosis
for each disease subtype. Borrowing strength across the categories is necessary to
increase the precision of inference in the small-size categories, in our case the disease
subtypes with few patients. A standard procedure to borrow strength across sub-
populations is a hierarchical model when the experimental units are exchangeable,
or a hierarchical regression under partial exchangeability. The hierarchical regres-
sion groups all experimental units with the same categorical covariate together and
borrows strength across them. When the outcome is binary the model becomes a hi-
erarchial logistic regression model (HLRM). Under the hierarchical regression model
the grouping is fixed. Inappropriate grouping induced by the covariate can lead to
poor inference.
I propose a semi-parametric model that is more robust against inappropriate
grouping by considering random grouping, or partitions. The model introduces the
covariate through the prior on the random partition. Inference for the parameter
corresponding to the i—th subpopulation borrows more strength from observations in
subpopulations that are with high probability grouped together with i. The data can
correct a prior guess on the grouping when the prior (grouping) beliefs are inaccurate.
In the same chapter I compare the proposed model with some parametric approaches.
I compare the performance in terms of bias, mean square error and coverage probabil-
ities under different scenarios. I show that the proposed model is more robust against
inappropriate grouping than standard approaches. Unfortunately, we could not show
that the model is any better than the HLRM in terms of stopping probabilities and
average sample sizes when applied in a clinical trial design. That is, the proposed
model does not detect inefficient treatments any faster than the HLRM. The model