  We consider a Bayesian framework based on "probability of decision" for
dose-finding trial designs. The proposed PoD-BIN design evaluates the posterior
predictive probabilities of up-and-down decisions. In PoD-BIN, multiple grades
of toxicity, categorized as the mild toxicity (MT) and dose-limiting toxicity
(DLT), are modeled simultaneously, and the primary outcome of interests is
time-to-toxicity for both MT and DLT. This allows the possibility of enrolling
new patients when previously enrolled patients are still being followed for
toxicity, thus potentially shortening trial length. The Bayesian decision rules
in PoD-BIN utilize the probability of decisions to balance the need to speed up
the trial and the risk of exposing patients to overly toxic doses. We
demonstrate via numerical examples the resulting balance of speed and safety of
PoD-BIN and compare to existing designs.
