A ratio test in active control non-inferiority trials with a time-to-event endpoint

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.