In this paper, we propose a phase II/III adaptive design for evaluation of drugs efficacy based on continuous endpoints. Under this design structure, the phase II and phase III trials are conducted in the same protocol with the same inclusion/exclusion criteria, the same study design, the same control, the same methods for evaluation, and the same efficacy/safety endpoints. In other words, the data from both the new and original regions are generated within the same study. Another attractive feature is that our phase II/III design is in fact an adaptive phase II/III design that would use the data from patients enrolled from the phase II stage and from the phase III stage in the final analysis. With this approach, the total sample size might be reduced in some cases. That is, shortening the total duration of drug development may be possible.
Doing so can save considerably valuable resource and cost.
Selection of the weighting factors and1 2 might be critical. The investigators should make considerable decision on how they want to spend the type I and type II error probabilities at each stage. If we spend fewer type I error and more power for the phase II stage, the required total sample size for the phase II stage will be larger. Under this condition, if we can reject the null hypothesis at the phase II stage, the possibility of concluding drug efficacy in the final analysis might be increased.
There is another attractive feature in our design. In Section 2, the specification of (i.e., the expected treatment effect for the phase III stage) is based on the linear trend for the phase II stage. That is, cdr d0 . However, the determination of the expected treatment effect for the phase III stage can be estimated from the phase II results. In fact, our phase II/III design can be extended as follows.
First, given and1 , we can determine2 n2 and C2 based on the specification of c (i.e., the undesirable value of slope for the dose response), c (i.e., the expected value of slope for the dose response at the phase II stage), and by (5) and (7).
After the phase II trial succeeds, we can obtain the estimates of and from the phase II stage. Using theses estimates, we can therefore calculate the required total
sample size and the critical value for the phase III stage. Doing so may increase the accuracy of the estimate of the required sample size for the phase III stage, and may consequently improve the possibility of success in the final analysis.
Another point we wish to make is the control of the type I error rate. In traditional approaches, if the type I error rates controlled at phase II and phase III are both 0.05, the actual overall type I error rate is in fact equal to 0.05×0.05=0.0025.
However in our phase II/III design, the actual type I error rate is only equal to 0.05. In other words, the type I error rate of our phase II/III design is 20 times larger than the traditional approaches. In other words, the traditional approach is more conservative than our phase II/III design. Similarly, in traditional approaches, if the values of power for both phase II and phase III are both 0.8, the actual overall power is equal to 0.8×0.8=0.64. On the other hand, in our phase II/III design, the actual power is equal to 0.8 which is 1.25 times larger than the traditional approaches. That is, our phase II/III can gain more power than the traditional approach. This phenomenon can be observed from Figures 2, 3, 4, and 5.
After the success of the phase II stage, the determination of dose level for the phase III stage is also critical. First of all, we need to choose the dose level with the desired response. However, the choice of dose level should not only depend on the effect but also on drug safety. While the dose response increases as the dose level increases, the toxicity might also increase as the dose level increases. In this case, we may choose a lower dose level which has less effect but higher safety. Even if the linear trend of the dose response for the phase II stage is statistically significant, the dose response might increase first and then reach the upper limit for larger dose levels.
In this case, we may select the first dose level reaching the upper limit. If toxicity is also considered, the dose level might be reduced.
References
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List of Tables
Table 1. Designs with =10, c, c' =(0, 0.1), =1, =0.6, k=3,1 ,=(0.05, 0.2), and d0,d1,d2,d3 0,10,20,30
2 n2 n3 C2 C3 n nB n rs† rc‡
0.1 100 1022 0.0079 0.6369 1237 1764 1570 0.3022 0.2397 0.2 75 1137 0.0092 0.6209 1237 1764 1570 0.3182 0.2525 0.3 60 1239 0.0102 0.6036 1237 1764 1570 0.3361 0.2666 0.4 51 1346 0.0112 0.5851 1237 1764 1570 0.3581 0.2840 0.5 43 1465 0.0121 0.5650 1237 1764 1570 0.3835 0.3042 0.6 37 1606 0.0131 0.5427 1237 1764 1570 0.4154 0.3295 0.7 32 1785 0.0140 0.5172 1237 1764 1570 0.4572 0.3627 0.8 28 2032 0.0151 0.4866 1237 1764 1570 0.5163 0.4096 0.9 24 2448 0.0162 0.4449 1237 1764 1570 0.6172 0.4896
†: rs 4n22n3 / 4n2n
‡: rc 4n22n3/
4nB2n
Table 2. Designs with =10, c, c' =(0, 0.1), =1, =0.8, k=3,1 ,=(0.05, 0.2), and d0,d1,d2,d3 0,10,20,30
2 n2 n3 C2 C3 n nB n rs† rc‡
0.3 103 845 0.0312 0.5524 1237 1764 1570 0.2599 0.2062 0.4 90 948 0.0335 0.5331 1237 1764 1570 0.2789 0.2213 0.5 80 1060 0.0355 0.5122 1237 1764 1570 0.3017 0.2393 0.6 71 1191 0.0375 0.4894 1237 1764 1570 0.3296 0.2615 0.7 64 1356 0.0395 0.4637 1237 1764 1570 0.3670 0.2911 0.8 58 1582 0.0415 0.4331 1237 1764 1570 0.4199 0.3331 0.9 53 1964 0.0436 0.3921 1237 1764 1570 0.5119 0.4060
†: rs 4n22n3 / 4n2n
‡: rc 4n22n3/
4nB2n
Table 3. Designs with =10, c, c' =(0, 0.1), =2, =0.6, k=3,1 ,=(0.05, 0.2), and d0,d1,d2,d3 0,10,20,30
2 n2 n3 C2 C3 n nB n rs† rc‡
0.2 100 124 0.0079 1.2974 310 441 393 0.3198 0.2541
0.3 75 188 0.0092 1.2646 310 441 393 0.3337 0.2651
0.4 60 231 0.0102 1.2296 310 441 393 0.3465 0.2753
0.5 51 269 0.0112 1.1915 310 441 393 0.3662 0.2910
0.6 43 307 0.0121 1.1498 310 441 393 0.3880 0.3082
0.7 37 349 0.0131 1.1033 310 441 393 0.4176 0.3318
0.8 32 399 0.0140 1.0502 310 441 393 0.4571 0.3631
0.9 28 465 0.0151 0.9864 310 441 393 0.5143 0.4086
†: rs 4n22n3 / 4n2n
‡: rc 4n22n3/
4nB2n
Table 4. Designs with =10, c, c' =(0, 0.1), =2, =0.8, k=3,1 ,=(0.05, 0.2), and d0,d1,d2,d3 0,10,20,30
2 n2 n3 C2 C3 n nB n rs† rc‡
0.5 103 92 0.0312 1.2138 310 441 393 0.2942 0.2337
0.6 90 130 0.0335 1.1547 310 441 393 0.3060 0.2431
0.7 71 207 0.0375 1.0398 310 441 393 0.3445 0.2737
0.8 64 254 0.0395 0.9772 310 441 393 0.3771 0.2996
0.9 58 316 0.0415 0.9055 310 441 393 0.4265 0.3388
†: rs 4n22n3 / 4n2n
‡: rc 4n22n3/
4nB2n
List of Figures
(a) (b)
Figure 1. Schma of our phase II/III design and traditional approach. (a) our phase II/III design; (b) the traditional approach.
Phase II stage
Pairwise comparison between each dose and placebo
Is any dose superior to
placebo? Stop
Phase III stage (1) Recruit subjects
for the selected dose and placebo group.
(2) Evaluate the treatment effect only based on the data from phase III stage.
Phase II stage
Establish the dose-response by the simple linear regression
Is the slope larger than a pre-specified
value?
Stop
Phase III stage
(1) Recruit subjects for the best dose and placebo group.
(2) Evaluate the treatment effect based on the accumulated data from both phase II and III stages.
=0.22 =0.42
=0.62 =0.82
our phase II/III design; traditional phase II and phase III trials Figure 2. Simulated success rates for the case of =10, c, c' =(0, 0.1), =1,
=0.6, k=3, and1 d0,d1,d2,d3 0,10,20,30.
=0.32 =0.52
=0.72 =0.92
our phase II/III design; traditional phase II and phase III trials Figure 3. Simulated success rates for the case of =10, c, c' =(0, 0.1), =1,
=0.8, k=3, and1 d0,d1,d2,d3 0,10,20,30.
=0.22 =0.42
=0.62 =0.82
our phase II/III design; traditional phase II and phase III trials Figure 4. Simulated success rates for the case of =10, c, c' =(0, 0.1), =2,
=0.6, k=3, and1 d0,d1,d2,d3 0,10,20,30.
=0.32 =0.52
=0.72 =0.92
our phase II/III design; traditional phase II and phase III trials Figure 5. Simulated success rates for the case of =10, c, c' =(0, 0.1), =2,
=0.8, k=3, and1 d0,d1,d2,d3 0,10,20,30.