The Comparative Study Of Non-inferiority Evaluation On Continuous Variables In Three-arm Clinical Trial

ObjectiveRecently years, with the development of science technology and health care, there are more and more clinical trials for innovative drugs domestic and overseas. Though, there are some effective drugs in clinical areas, they have some disadvantages, such as terrible side effects, low patient compliance, complicated usage, high price and so on. While some innovative drugs, may not improve therapeutic, yet it still worth to promotion for high security, small drug-resistant, high compliance, low cost, easy to use, etc. How to screen and evaluate these drugs effectively? It is difficult to accomplish with conventional null hypothesis of superiority clinical trial design. Thus, non-inferiority design came into being and became more and more assenting.Earlier non-inferiority trial design mainly concentrated in the two-arm non-inferiority trials. Though it is simple, easy to operate and accord with the ethical requirements, with the development of study and deep application, many defects appears in R&D, data analysis and interpretation of results, of which the biggest is short of the placebo group, cannot affirm the effectiveness of the test treatment directly. But with the placebo group and positive control group as the control to test reference drug, the three-arm non-inferiority clinical trials solves the problem of not able to evaluate directly and the difficulty of ensuring constant assumption of the two-arm non-inferiority clinical trials which has only positive control group. ICH E10has recognized the three-arm non-inferiority clinical trials as the gold standard design. Some researchers even suggested that the three-arm non-inferiority clinical trials must be adopted in evaluating the non-inferiority of trial drugs upon the ethical condition.Many researchers have proposed solutions for non-inferiority evaluation on continuous variables in three-arm clinical trial upon different situations, such as, in2003, Pigeot et al. proposed to formulate non-inferiority as a fraction of the trial sensitivity. This resulted in hypotheses based on the ratio of differences in means. For a given threshold0, the alternative hypothesis indicates that the relative efficacy of the experimental drug is more than1000θper cent of the efficacy of the reference compound compared with placebo. For this ratio hypothesis, a t-distributed test statistic was derived, assuming variance homogeneity. In2008, Halser et al have found that the Pigeot test violates the α-level in cases of heterogeneity, Therefore, the adjusted test statistics T test and the related Fieller’s confidence interval is derived,In2009, Munzel proposed using rank to instead of original data, which is based on nonparametric statistics inference of relative efficacy. Bootstrap method adopted Bootstrap re-sampling technology, re-sample many times among original data, get the threshold of confidence interval of the average deviation ratio upon the pre-given level of type Ⅰ error. If the threshold is beyond the pre-given one, it can get the non-inferiority result.This paper will compare and further summarize the existent non-inferiority statistical inference of continuous variables of the three-arm clinical trials. As well as suggest the selection strategy of methodology, in the hope of providing a theoretical support and reference for three-arm non-inferiority clinical trials. MethodThe research will using SAS9.1system, and Monte Carlo simulation technology. To compare the type Ⅰ error of the existent non-inferiority statistical inference of continuous variables of the three-arm clinical trials and the test efficiency through simulation under the pre-given simulated conditions. Below are the detailed steps.The specific steps of the α—simulation are:Step1,setting parameters. Step2, Base on null hypotheses calculate the means of experimental group. Step3, Fixed sample sizes of the reference and the placebo group, changing the sample size of experimental group. Obtain100000random samples submit to pre-given and different distribution conditions severally according to the pre-given mean and standard deviation. To run the Monte Carlo simulation of type Ⅰ error with Pigeot and Halser methods separately changing the sample size of positive control group and placebo group in turn and simulate separately, to get the ratio of type Ⅰ error of simulation. Since the Monte Carlo simulation of Munzel and Bootstrap methods is time-consuming, setting the simulation times as1000. Step4, The estimation of the type Ⅰ error rate. Step5, Calculate the rate value of the type Ⅰ error and compare these methods.When the continuous variables follow normal distribution but heterogeneity, the difference of the basic steps of α—simulation is:when change the sample ratio of experimental group, positive control group and placebo group, change the standard deviation ratio of the three groups to simulate separately and get the rate of the type Ⅰ error of simulation.Specific steps of the power simulation are:Step1, setting parameters. Step2, Base on alternation hypotheses calculate the means of experimental group. Step3, Change the proportion of (μE-μp)/(μR-μp) generate100000random samples for different distribution conditions, in accordance with the pre-given mean and standard deviation. To run Monte Carlo simulation of Pigeot, Halser et al methods separately. And the Monte Carlo simulation times of Bootstrap and Munzel methods are1000. Step4, The estimation of the power. Step5, Calculate the ratio of refuse null hypotheses.ResultWhen continuous variables follow normal distribution and homogeneity of variance, whether the t hypothesis test statistical of Pigeot method under small samples or large samples, The type Ⅰ error rate always maintain the pre-given level, and in certain sample sizes the power researched pre-given level and increased with the increasing of the ratio of (μE-μp)/(μR-μp) and the sample size, show good statistical performance. When estimate the big sample, the fluctuate of type Ⅰ error of its Fieller’s confidence interval is within10%, which shows good robustness. While for the small sample, few fluctuate of the type Ⅰ error is beyond20%, show poor robustness. Whether with small sample or big sample, the type Ⅰ error rate of Halser t test statistical is beyond the pre-given level, the fluctuate is within10%and in certain sample sizes the power researched pre-given level and increased with the increasing of the ratio of c and the sample size, show steady statistical performance. When estimate the big sample, the fluctuate of type Ⅰ error of its Fieller’s confidence interval is within10%, which shows good robustness. While for the small sample, the type Ⅰ error deviate from the pre-given level, the fluctuate is beyond20%, shows poor robustness. Under the big sample, few Fieller’s confidence interval statistical type Ⅰ error fluctuate of Munzel method is beyond20%, in certain sample sizes the power lower than pre-given level, show poor statistical performance. While the type I error of t test statistical is far beyond the pre-given level, whose fluctuate is beyond20%, whether with small or big sample. Under big sample, the type I error stain the pre-given level, only few fluctuate beyond20%, in certain sample sizes the power researched pre-given level and increased with the increasing of the ratio of (μE-μp)/(μR-μp) and the sample size, show good statistical performance. While with small sample, the type Ⅰ error rate fluctuate is beyond20%, show poor robustness.When continuous variables follow normal distribution and heterogeneity of variance, regardless of large samples or small samples, the rate of the type Ⅰ error of Pigeot method always deviated from the pre-given level. And it is very conservative when extremely heterogeneity of variance of each group, even cannot improving as increasing each group’s sample size, show poor robustness. The rate of the type Ⅰ error of Halser method always maintain in pre-given level, which fluctuate is within10%, regardless of small sample or large sample, even extremely heterogeneity of variance, in certain sample sizes the power researched pre-given level and increased with the increasing of the ratio of (μE-μp)/(μR-μp) and the sample size, show very steady statistical performance. Regardless of large samples or small samples, the error rate of Munzel method is always deviate from the pre-given level, the fluctuation is beyond20%, in certain sample sizes the power better than pre-given level.show poor statistical performance. In a large sample, few value of Bootstrap method is beyond20%. While in a certain sample, the power can achieve the pre-given level and increased with the increasing of the ratio of (μE-μp)/(μR-μp) and the sample size, shows steady statistical performance. Yet, in a small sample, the rate of the type Ⅰ error will deviate from pre-given level, show poor robustness.To study the statistical performance of each method under abnormal distribution, we choose two classical abnormal distributions in simulate experiment, they are gamma distribution and the logarithmic normal distribution. When continuous variables obey the gamma distribution, the type Ⅰ error rate of method of Pigeot, Halse method and Munzel method are all violated from the pre-given level, Pigeot method performs particularly radical, but Halser method performs particularly conservative. The statistical performance of these three methods are poor. In the case of the large sample, few type Ⅰ error rate of Bootstrap method’s fluctuated is beyond20%, its power increased with the increasing of the ratio of (μE-μp)/(μR-μp) and sample size, shows relative steady statistical performance, while poor statistical performance for small sample. When continuous variable obey lognormal normal distribution, the rate of type Ⅰ error of Pigeot method and Munzel method are deviate from the pre-given level, which fluctuate is beyond20%, show the poor robustness and particular radical. The fluctuate of type Ⅰ error rate of Halser method is beyond20%with small sample, while it always maintain in the pre-given level with big sample, whose fluctuate is within20%, shows good robustness. The fluctuate of type Ⅰ error rate of Bootstrap method is within20%, and its power increased with the increasing of the ratio of (μE-μp)/(μR-μp) and sample size.show steady statistical performance.ConclusionWhen continuous variables in three-arm clinical trials follow normal distribution and homogeneity of variance, the best choice of judgment of non-inferiority is Pigeot method. Halser method is the alternative, Munzel method and Bootstrap method can be used as a secondary reference. When continuous variables in three-arm clinical trials follow normal distribution and heterogeneity of variance, the best judge method is Halser method. In case of big sample, Bootstrap can be adopted as a secondary reference. When continuous variables in three-arm clinical trials don’t obey normal distribution, the Bootstrap method is the best method for three-arm clinical trials, upon the condition of big sample. Therefore, the judgment of non-inferiority of the three-arm clinical trials on continuous variables need to check the distribution, depending on the circumstances and make the right Choice.