We give some examples for the purpose of illustration. Suppose the test drug has dose levels of 10, 20, and 30 respectively. Also assume that the placebo group has dose level of 0. Given =10, ,= (0.05, 0.20), c =0, and c=0.1, Tables 1, 2, 3, and 4 illustrate the phase II/III designs for different combinations of design parameters with =0.6 and 0.8, and1 =1 and 2, respectively. For each , we1 consider various combinations of values for . The tabulated results include the2 required sample size (n ) at the phase II stage, the required sample size (2 n ) at the3 phase III stage, the critical value for the observed value of slope that would reject the test drug at the phase II stage (C ), the critical value for the observed mean difference2 that would reject the test drug at the phase III stage (C ), numbers of sample sizes3 required for the traditional phase II and phase III trials (n,nB, and n respectively), and the ratios of the total sample size for our phase II/III design vs. the total sample size for the traditional designs (r ands r ).c
For instances, the first line in Table 1 considers the case of =10, ( cc, ')= (0, 0.1), (,')= (0, 1), and ,= (0.05, 0.2). In this case, the phase II stage needs to recruit 100 patients for each group (i.e. 4×100=400 for total). When the study is completed at the phase II stage, if the observed value of slope ˆ does not exceed 0.0079, the trial is terminated after the phase II stage and considers the test drug is concluded as lack of efficacy. Oppositely, if the observed value of the estimator of slope ˆ is greater than 0.0079, the trial continues to phase III stage and assume that the dose level of 10 (i.e. =1) is selected. We need to enroll additional 1022 patients for each group of the d0 and d . After the recruitment of the patients at phase III1 stage is completed, if the overall observed absolute value of mean difference, ˆ, based on the cumulative data n2n3 obtained at the end of the trial does not exceed 0.6369, we will reject the test drug. On the other hand, if the observed absolute value of ˆ is greater than 0.6369, we can conclude that the effect of test drug is different from that of the control group. In addition, the numbers of required sample sizes for
the traditional phase II trial and phase III trial are 1237 (and 1764 for Bonferroni p-value adjustment) and 1570 per group respectively. In this case, the required total sample size for our phase II/III design can be reduced by around 33% and 76%
respectively compared with two traditional approaches. From all tables, since all of values of r ands r are less than 1, the required total sample size for our phase II/IIIc design is possibly smaller than those required by the traditional methods.
From the tables, as decreases, the required sample size per group for the2 phase III stage decreases but the required sample size per treatment group for the phase II stage increases. This makes sense, since the phase II stage will spend more power than the phase III as decreases. In addition, the critical value at the final2 analysis also increases as decreases. This phenomenon can be observed from (8).2 On the other hand, it is notable that the sample size at the phase II stage increases as
increases. This fact is because that the larger the1 is, the fewer type I error rate1 will be spent by the phase II stage. In other words, the investigators should make considerable decision on how they want to spend the type I and type II error probabilities at each stage.
Note that in the phase II/III design, when is sufficiently large, the sample2 size required for the phase III stage might be greater than that required for the traditional phase III trial. Larger indicates that more power will be spent at the2 phase III stage. In this case, the contribution of the patients from the phase II stage strongly decreases indicating that we need to recruit more patients for the phase III stage. Also it should be noted that when is large enough and1 is small, it is2 difficult to find n and2 C to satisfy constraints (5) and (7). This makes intuitive2 sense since we spend fewer type I error and more power for the phase II stage. In this case, n2 might be extremely large.
A simulation study was conducted to compare our proposed phase II/III design with the traditional design in terms of success rate. Suppose the test drug has dose levels of 10, 20, and 30 respectively. Also assume that the placebo group has dose
level of 0. Figure 2 displays simulation results for the case of =10, ( cc , )= (0, 0.1),
= 1, and ,= (0.05, 0.2), =0.6 with various values of1 . For instance,2 given =0.2, we can derive that2 n =75,2 C =0.0092,2 n =1137,3 C =0.6209,3 n=1237, and n =1570. We assume that the true values of are respectively 0, 0.02, 0.04,…, 0.30. Assuming the linear trend, =10 . For each (and thus), the success rate was derived from simulations of 10,000 replicates. From Figure 2, our phase II/III design can reach the desired power as assumed when =0.1. Also our phase II/III design performs better or at least the same than the traditional designs. In Figures 3, we show the simulation results when =0.8 and1 =1 with =0.3, 0.5,2 0.7, and 0.9. Figures 4 and 5 display the simulation results when =0.6 and1 =2 with =0.2, 0.4, 0.6, and 0.8, and when2 =0.8 and1 =2 with =0.3, 0.5, 0.7,2 and 0.9, respectively. All figures exhibit the same phenomenon as Figure 2.