Strengthening the understanding of the relationship between survival designs and test statistics

In the planning of randomized survival trials, the role of follow-up time of trial participants introduces a level of complexity not encountered in non-survival trials. Of the two commonly used survival designs, one design fixes the follow-up time whereas the other allows it to vary. When the follow-up time is fixed the number of events varies. Conversely, when the number of events is fixed, the follow-up time varies. These two designs influence test statistics in ways that have not been fully explored resulting in a misunderstanding of the design-test statistic relationship. We use examples from the literature to strengthen the understanding of this relationship. Group sequential trials are briefly discussed. When the number of events is fixed, we demonstrate why a two-sample risk difference test statistic reduces to a one-sample test statistic which is nearly equal to the risk ratio test statistic. Some aspects of fixed event designs that need further consideration are also discussed.