Utilizing the integrated difference of two survival functions to quantify the treatment contrast for designing, monitoring, and analyzing a comparative clinical study

Abstract

Consider a comparative, randomized clinical study with a specific event time as the primary end point. In the presence of censoring, standard methods of summarizing the treatment difference are based on Kaplan-Meier curves, the logrank test, and the point and interval estimates via Cox's procedure. Moreover, for designing and monitoring the study, one usually utilizes an event-driven scheme to determine the sample sizes and interim analysis time points. When the proportional hazards (PHs) assumption is violated, the logrank test may not have sufficient power to detect the difference between two event time distributions. The resulting hazard ratio estimate is difficult, if not impossible, to interpret as a treatment contrast. When the event rates are low, the corresponding interval estimate for the 'hazard ratio' can be quite large due to the fact that the interval length depends on the observed numbers of events. This may indicate that there is not enough information for making inferences about the treatment comparison even when there is no difference between two groups. This situation is quite common for a postmarketing safety study. We need an alternative way to quantify the group difference. Instead of quantifying the treatment group difference using the hazard ratio, we consider an easily interpretable and model-free parameter, the integrated survival rate difference over a prespecified time interval, as an alternative. We present the inference procedures for such a treatment contrast. This approach is purely nonparametric and does not need any model assumption such as the PHs. Moreover, when we deal with equivalence or noninferiority studies and the event rates are low, our procedure would provide more information about the treatment difference. We used a cardiovascular trial data set to illustrate our approach. The results using the integrated event rate differences have a heuristic interpretation for the treatment difference even when the PHs assumption is not valid. When the event rates are low, for example, for the cardiovascular study discussed in this article, the procedure for the integrated event rate difference provides tight interval estimates in contrast to those based on the event-driven inference method. The design of a trial with the integrated event rate difference may be more complicated than that using the event-driven procedure. One may use simulation to determine the sample size and the estimated duration of the study. The procedure discussed in this article can be a useful alternative to the standard PHs method in the survival analysis.