A phase II clinical trial in cancer therapeutics is usually a single-a
rm study to determine whether an experimental treatment (E) holds suff
icient promise to warrant further testing. When the criterion of treat
ment efficacy is a binary endpoint (response/no response) with probabi
lity of response p, we propose a three-stage optimal design for testin
g H-0:p less than or equal to p(0) versus H-1:p greater than or equal
to p(1), where p(1) and p(0) are response rates such that E does or do
es not merit further testing at given levels of statistical significan
ce (alpha) and power (1 - beta). The proposed design is essentially a
combination of earlier proposals by Gehan and Simon. The design stops
with rejection of H-1 at stage 1 when there is an initial moderately l
ong run of consecutive treatment failures; otherwise there is continua
tion to stage 2 and (possibly) stage 3 which have decision rules analo
gous to those in stages 1 and 2 of Simon's design. Thus, rejection of
H-1 is possible at any stage, but acceptance only at the final stage.
The design is optimal in the sense that expected sample size is minimi
zed when p = p(0), subject to the practical constraint that the minimu
m stage 1 sample size is at least 5. The proposed design has greatest
utility when the true response rate of E is small, it is desirable to
stop early if there is a moderately long run of early treatment failur
es, and it is practical to implement a three-stage design. Compared to
Simon's optimal two-stage design, the optimal three-stage design has
the following features: stage 1 is the same size or smaller and has th
e possibility of stopping earlier when 0 successes are observed; the e
xpected sample size under the null hypothesis is smaller; stages 1 and
2 generally have more patients than stage 1 of the two-stage design,
but a higher probability of early termination under H-0; and the total
sample size and criteria for rejection of H-1 at stage 3 are similar
to the corresponding values at the end of stage 2 in the two-stage opt
imal design.