  The present paper deals with the problem of allocating patients to two
competing treatments in the presence of covariates or prognostic factors in
order to achieve a good trade-off among ethical concerns, inferential precision
and randomness in the treatment allocations. In particular we suggest a
multipurpose design methodology that combines efficiency and ethical gain when
the linear homoscedastic model with both treatment/covariate interactions and
interactions among covariates is adopted. The ensuing compound optimal
allocations of the treatments depend on the covariates and their distribution
on the population of interest, as well as on the unknown parameters of the
model. Therefore, we introduce the reinforced doubly adaptive biased coin
design, namely a general class of covariate-adjusted response-adaptive
procedures that includes both continuous and discontinuous randomization
functions, aimed to target any desired allocation proportion. The properties of
this proposal are described both theoretically and through simulations.
