Increasing the sample size at interim for a two-sample experiment without Type I error inflation

For the case of a one-sample experiment with known variance σ² =1, it has been shown that at interim analysis the sample size (SS) may be increased by any arbitrary amount provided: (1) The conditional power (CP) at interim is ≥ 50% and (2) there can be no decision to decrease the SS (stop the trial early). In this paper we verify this result for the case of a two-sample experiment with proportional SS in the treatment groups and an arbitrary common variance. Numerous authors have presented the formula for the CP at interim for a two-sample test with equal SS in the treatment groups and an arbitrary common variance, for both the one- and two-sided hypothesis tests. In this paper we derive the corresponding formula for the case of unequal, but proportional SS in the treatment groups for both one-sided superiority and two-sided hypothesis tests. Finally, we present an SAS macro for doing this calculation and provide a worked out hypothetical example. In discussion we note that this type of trial design trades the ability to stop early (for lack of efficacy) for the elimination of the Type I error penalty. The loss of early stopping requires that such a design employs a data monitoring committee, blinding of the sponsor to the interim calculations, and pre-planning of how much and under what conditions to increase the SS and that this all be formally written into an interim analysis plan before the start of the study.