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CPS1201 M. Iftakhar Alam et al.
selection based on the probability of success only. For any other choice of ,
the design is expected to allocate the most efficacious doses to the cohorts
and to give precise estimates of the parameters leading to the
recommendation of the best dose for further study.

An Example
To explore the proposed methodology, we introduce an example which is
based on the continuation ratio doseresponse model. Simulation studies,
detailed in Section 2, are conducted to investigate the properties of the design.
For an experimental drug, we use a flexible continuation ratio model
(Agresti, 1990), which is given by

(,ϑ) (,ϑ)
log { 1 } = + and log { 2 } = +
(,ϑ) 1 2 1− (,ϑ) 3 4
2
0

The above equations have the following solutions

1
(, ) = ,
0
2 )(1+
(1+ 1 + 3 +
4 )
2
1 +
(, ) = ,
1
4 )
2 )(1+
(1+ 1 + 3 +
and
3 +
4
(, ) = 1+ 3 + .
2
4

If we are at the kth stage in a trial, then cohorts have been treated with
doses selected from the set of ordered doses . Let be the × 1vector of
doses with components and let R be the × 3 outcome matrix with =

1
( , , ) as the th row, = 1,2, . . . , k. Note that , , = , where is
0
0
1
2
1
2
the number of subjects in a cohort treated with dose The successive

components of are the counts of neutral, successful and toxic responses for

the lth cohort. Thus, the likelihood function is

(|, ) ∝ ∏{ ( , )} 10 { ( , )} 11 { ( , )} 12 .
1
1
0
1
1

2
=1

Since maximum likelihood estimation is unsuitable because of small
sample sizes at the early stages of a trial, we employ a Bayesian approach to
estimate the parameters . The posterior estimate of at the kth stage is
̂
Θ
= ∫ () (|, ) ,

∫ () (|, )
Θ
where p(ϑ) is the joint prior distribution of the parameters. Let us assume that
0 < < , 0 < < , < < , and < < , and that the joint
4
2
1
4
2
1
3
3
1
2
prior distribution is uniform. Then we obtain
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