| Background:There are always controversies about the non-inferiority margin in drug and device non-inferiority trials. According to the ICHE10(International Conference on Harmonization of Technical Requirements for Registration of Pharmaceuticals for Human Use) and CPMP (Committee for Proprietary Medical Products), the effect of the new treatment should not only better than a placebo but also non-inferiority to the standard treatment. But there is still controversy about the non-inferiority margin because of the lacking of a uniform standard. As the drug and device clinical trials usually contain new treatment and standard treatment only, the comparison of the effect between new treatment and placebo must refer to the historical placebo-control trials. So Hung et al. gave a method which defined the non-inferiority margin based on the rate difference after considering the placebo effect. And Then Chow and Shao gave a method which defined the non-inferiority margin based on the rate ratio after considering the placebo effect. Although both of them considered the effect of placebo, there appeared other question:Can both of the two methods give the similar type one error and power? If not, in which situation should people select the first method, and in which situation should people select other? According to this question, I have referred to lots of literatures at home and abroad and found that there was no related research. Objective:To provide a reference on the definition of non-inferiority margin in China by comparing the type one error and power between the RD method and RR method, which used Monte Carlo Simulation and SAS9.2programming that based on binary data, and considered randomized and paired design.Methods:1. Define the non-inferiority marginLet Pc0, Pp0be the event rates for the control treatment and the placebo, respectively, in the historical trial populations. Let Pc and Pt denote the event rates associated with the control treatment, the experimental treatment, respectively, in the patient population targeted by the active-controlled trial. Supposed that the higher the event rates, the better the effect.(1) Non-inferiority margin based on rate differenceThe basic thought is that the effect difference between the experimental treatment and placebo contains at least a certain percent of (f) the difference between the control treatment and placebo. Use formula1to calculate: Δrd=(1-f)(Pc0-Pp0)(Formula1)Δrd in the formula1denotes the non-inferiority margin, Pc0and Pp0are the event rates for the control treatment and the placebo, respectively, in the historical trial populations. The control treatment effect is derived from a random-effects meta-analysis and defined as the lower bound of a95%CI.f denotes the percent that the effect difference between the experimental treatment and placebo contains of the difference between the control treatment and placebo.(2) Non-inferiority margin based on rate ratioThe basic thought is that the effect ratio between the experimental treatment and placebo contains at least a certain percent of (f) the ratio between the control treatment and placebo. Use formula2to calculate: Δrr in the formula2denotes the non-inferiority margin, Pc0and Pp0are the event rates for the control treatment and the placebo, respectively, in the historical trial populations. The control treatment effect is derived from a random-effects meta-analysis and defined as the lower bound of a95%CI.f denotes the percent that the effect ratio between the experimental treatment and placebo contains of the ratio between the control treatment and placebo.2. Simulation with randomized design(1) Simulation of type one errorParameters definition:Let Pc0be the event rate for the control treatment in the historical trial populations, given value of0.6,0.65,0.7,0.75...0.9,0.95;f equals0.5,0.6,0.7,0.8respectively. Pp0is the event rate for the placebo in the historical trial populations, defined as0.35. The calculation of non-inferiority margin is based on formula1and formula2. Let0<Pt<min(Pc0-Δrd, Pc0/Δrr). n denotes the sample size, defines as50、100、150... and500respectively. Considering different combinations of f,n and Pc0, then using SAS9.2to sample from binary data and the simulation will take100000times for each combination. For RD margin, to calculate the ratio of the times that the lower bound of (Pt-Pc) CI greater than-Δrd and the total100000times, namely the type one error. For RR margin, to calculate the ratio of the times that the lower bound of Pc/Pt CI smaller than Δrr and the total100000times, namely the type one error. Then calculate the relative ratio of both based on RD margin.Test method:The CI method.1) CI based on RD:According to one-side100(1-a)%confidence level, to calculate the lower bound CL of the CI of (Pt-Pc), then the non-inferiority conclusion can be established if [CL,∞] is entirely in the range of [-Δrd,∞] or CL>-Δrd.2) CI based on RR: According to one-side100(l-a)%confidence level, to calculate the upper bound C, of the CI of Pc/Pt, then the non-inferiority conclusion can be established if [-∞,Cu] is entirely in the range of [-∞,Δrr] or Cu<Δrr(2) Simulation of powerParameters definition:Let Pc0be the event rate for the control treatment in the historical trial populations, given value of0.6,0.65,0.7,0.75...0.9,0.95;f equals0.5,0.6,0.7,0.8respectively. Pp0is the event rate for the placebo in the historical trial populations, defined as0.35. The calculation of non-inferiority margin is based on formula1and formula2. Pt is defined as (1) Pc-Pt≠0, then Pt> max(Pc0-Δrd, Pc0/Δrr),(2) Pc-Pt;.n denotes the sample size, defines as50、100、150... and500respectively. Considering different combinations of f,n and Pc0, then using SAS9.2to sample from binary data and the simulation will take100000times for each combination. For RD margin, to calculate the ratio of the times that the lower bound of (Pt-Pc) CI greater than-Δn and the total100000times, namely the power. For RR margin, to calculate the ratio of the times that the lower bound of Pc/Pt CI smaller than Δrr and the total100000times, namely the power. Then calculate the relative ratio of both based on RD margin.Test method:The CI method.3. Simulation with paired design(1) Simulation of type one errorParameters definition:Let Pc0be the event rate for the control treatment in the historical trial populations, given value of0.6,0.65,0.7,0.75...0.9,0.95;f equals0.5,0.6,0.7,0.8respectively. Pp0is the event rate for the placebo in the historical trial populations, defined as0.35. The calculation of non-inferiority margin is based on formula1and formula2. LetO<Pt<min(Pc0-Δrd,Pc0/Δrr). n denotes the sample size, defines as50、100、150...and500respectively. Considering different combinations of f,n and Pc0, then using SAS9.2to sample from binary data, and the simulation will take100000times for each combination. For RD margin, to calculate the ratio of the times that the lower bound of [Pt-Pc) CI greater than-Δrd and the total100000times, namely the type one error. For RR margin, to calculate the ratio of the times that the lower bound of Pc|Pt CI smaller than Δrr and the total100000times, namely the type one error. Then calculate the relative ratio of both based on RD margin.Test method:The CI method.(2) Simulation of powerParameters definition:Let Pc0be the event rate for the control treatment in the historical trial populations, given value of0.6,0.65,0.7,0.75...0.9,0.95;f equals0.5,0.6,0.7,0.8respectively. Pt0is the event rate for the placebo in the historical trial populations, defined as0.35. The calculation of non-inferiority margin is based on formula1and formula2. Pt is defined as (1) Pc-Pt≠0, then Pt=max(Pc0-Δrd,Pc0/Δrr),(2) Pt-Pc.n denotes the sample size, defines as50/100n150...and500respectively. Considering different combinations of,n and Pc0, then using SAS9.2to sample from binary data and the simulation will take100000times for each combination. For RD margin, to calculate the ratio of the times that the lower bound of (Pt-Pc) CI greater than-Δrd and the total100000times, namely the power (1-β). For RR margin, to calculate the ratio of the times that the lower bound of Pc/Pt CI smaller than Δrr and the total100000times, namely the power. Then calculate the relative ratio of both based on RD margin.Test method:The CI method. Result:(1) Randomized designFrom the perspective of type one error, the type one error of RD margin is always below the0.025test level and decreases with the increase of Pc in different combinations of f and Pc0. When n=50, and Pc0<0.8, the type one error of RR margin fluctuates near the0.025level. When n=50, and PcQ>0.8, the type one error of RR margin is always above the0.025test level and increase s with the increases of Pc. When n>50, the type one error of RR margin fluctuates near the0.025level. The comparison of the two error showed that the type one error of RD margin is always smaller than that of RD margin no matter different values of n and f,and the difference between them increases with the increases of Pc. The comparison of different combinations of n and/showed that no matter the value of f,when n=50, the type one error of RR margin fluctuates near the0.025level, and the type one error then decreases with the increases of the sample size. The type one error of RD margin is always smaller than that of RR margin and decreases with the increases of sample size.From the perspective of power, when Pc-Pt≠0, the power of is always smaller than that of the RR margin and both of them increase with the increases of Pc. When Pc<0.75, the difference between them is quiet small, then increases with the increases of Pc. When Pt=Pc, the power of is always smaller than that of the RR margin and both of them increase with the increases of Pc. We found an interesting phenomenon that the difference of the power appears increasing then decreasing with the increases of Pc and the smaller the difference, the greater the n combines with the smaller the f.(2) Paired designFrom the perspective of type one error, the type one error of RD margin is always below the0.025test level and decreases with the increase of Pc in different combinations of f and Pc0. When n=50,f<0.7and Pc<0.75, the type one error of RR margin fluctuates near the0.025level. When n=50,f>0.7and Pc>0.75, the type one error of RR margin is always above the0.025test level and increase s with the increases of Pc. When n>50and Pc<0.75, the difference between them is quite small. When Pc>0.75, the difference increases slowly with the increases of Pc. The comparison of the two error showed that the type one error of RD margin is always smaller than that of RD margin no matter different values of n and f,and the difference between them increases with the increases of Pc. The comparison of different combinations of n and f showed that no matter the value of f,when n=50, the type one error of RR margin fluctuates near the0.025level, and the type one error then decreases with the increases of the sample size. The type one error of RD margin is always smaller than that of RR margin and decreases with the increases of sample size.From the perspective of power, when Pc-Pt≠0, the power of is always smaller than that of the RR margin and both of them increase with the increases of Pc. When Pc<0.75, the difference between them is quiet small, and then increases with the increases of Pc. When Pt=Pc, the power of is always smaller than that of the RR margin and both of them increase with the increases of Pc. We found an interesting phenomenon that the difference of the power appears increasing then decreasing with the increases of Pc and the smaller the difference, the greater the n combines with the smaller the f.Conclusion:From both of type one error and power perspective no matter randomized or paired design, we got the following two conclusions:(1) From the perspective of statistical property, the nearer the type one error to singnificance level (0.025), the higher the power, and the effective the method. This study showed that statistical property of RR margin method was much better than that of RD method, it also meaned that it was better to determine margin by RR method, especially under the situation of≥8or n<50.(2) From the perspective of practical application, the smaller the type one error the beeter. Form this way, it will be much easier to identify medicines which cannot satisfy the standard, then the benefit and health of patients will be guaranteed. Our study showed that the type one error of RD method was always smaller than that of RR method. The improvement of the power of latter was at the cost of type one error. So the RD margin is recommended to guarantee the benefit of the patients. |
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