Sample sizes for non-inferiority studies based on the difference between two proportions: a unified approach for difference, ratio and odds ratio models


  • Bruno Mario Cesana University of Brescia
  • Paolo Antonelli University of Brescia




Background: It is never easy to make sample size calculation for two-arm, non-inferiority studies with a primary proportion outcome not only because the different parameterisations involved in the difference, ratio, and odds ratio models lead to different non-inferiority margins and different sample size results, but also because of the different efficiency of the respective sample size formulae.

Methods: According to a formal statistical approach, we showed how to express the non-inferiority margins of the three models by keeping the probability (success or failure) of the standard treatment fixed (considered as “known” in the planning phase of a trail), and equal under the null and alternative hypotheses as the statistical basis for sample size calculation.

Results: We have obtained the sample size formulae and their respective power formulae for the three considered models both for success and failure probabilities. A sample size table for non-inferiority success studies is reported for illustrative purposes. In addition, we have compared the sample sizes from the three models by means of graphic and theoretical approaches and we have shown their asymptotic relationships. Furthermore, we have obtained the formulae for switching among the three considered models. Finally, we have correct some previously published formulae for sample size calculations.

Conclusion: The clearly separate approach to the probabilities of success and failure of the three considered models shown in this paper makes it possible to switch among them consistently and equivalently and to choose the probability formulation for the most parsimonious model.