Wang Yong-Cheng, Chen Gang, Chi George Y H
Biostatistics, Centocor, Inc., Malvern, Pennsylvania 19355, USA.
J Biopharm Stat. 2006;16(2):151-64. doi: 10.1080/10543400500508754.
There are essentially two kinds of non-inferiority hypotheses in an active control trial: fixed margin and ratio hypotheses. In a fixed margin hypothesis, the margin is a prespecified constant and the hypothesis is defined in terms of a single parameter that represents the effect of the active treatment relative to the control. The statistical inference for a fixed margin hypothesis is straightforward. The outstanding issue for a fixed margin non-inferiority hypothesis is how to select the margin, a task that may not be as simple as it appears. The selection of a fixed non-inferiority margin has been discussed in a few articles (Chi et al., 2003; Hung et al., 2003; Ng, 1993). In a ratio hypothesis, the control effect is also considered as an unknown parameter, and the noninferiority hypothesis is then formulated as a ratio in terms of these two parameters, the treatment effect and the control effect. This type of non-inferiority hypothesis has also been called the fraction retention hypothesis because the ratio hypothesis can be interpreted as a retention of certain fraction of the control effect. Rothmann et al. (2003) formulated a ratio non-inferiority hypothesis in terms of log hazards in the time-to-event setting. To circumvent the complexity of having to deal with a ratio test statistic, the ratio hypothesis was linearized to an equivalent hypothesis under the assumption that the control effect is positive. An associated test statistic for this linearized hypothesis was developed. However, there are three important issues that are not addressed by this method. First, the retention fraction being defined in terms of log hazard is difficult to interpret. Second, in order to linearize the ratio hypothesis, Rothmann's method has to assume that the true control effect is positive. Third, the test statistic is not powerful and thus requires a huge sample size, which renders the method impractical. In this paper, a ratio hypothesis is defined directly in terms of the hazard. A natural ratio test statistic can be defined and is shown to have the desired asymptotic normality. The demand on sample size is much reduced. In most commonly encountered situations, the sample size required is less than half of those needed by either the fixed margin approach or Rothmann's method.
在活性对照试验中，本质上有两种非劣效性假设：固定界值假设和比率假设。在固定界值假设中，界值是预先设定的常数，该假设是根据一个单一参数来定义的，这个参数表示活性治疗相对于对照的效应。对固定界值假设的统计推断很直接。固定界值非劣效性假设的突出问题是如何选择界值，这项任务可能不像看起来那么简单。一些文章（Chi等人，2003年；Hung等人，2003年；Ng，1993年）已经讨论了固定非劣效界值的选择。在比率假设中，对照效应也被视为一个未知参数，然后非劣效性假设根据这两个参数（治疗效应和对照效应）被表述为一个比率。这种类型的非劣效性假设也被称为效应保留假设，因为比率假设可以解释为对照效应的一定比例的保留。Rothmann等人（2003年）在生存时间设定中根据对数风险制定了一个比率非劣效性假设。为了规避处理比率检验统计量的复杂性，在对照效应为正的假设下，比率假设被线性化为一个等价假设。针对这个线性化假设开发了一个相关的检验统计量。然而，这种方法没有解决三个重要问题。第一，根据对数风险定义的保留比例难以解释。第二，为了使比率假设线性化，Rothmann的方法必须假设真实的对照效应为正。第三，检验统计量的功效不强，因此需要巨大的样本量，这使得该方法不切实际。在本文中，比率假设直接根据风险来定义。可以定义一个自然的比率检验统计量，并且证明它具有所需的渐近正态性。对样本量的要求大大降低。在最常见的情况下，所需样本量不到固定界值方法或Rothmann方法所需样本量的一半。
