A Design for Selecting Among k Treatments with Two Binary Endpoints in Comparison to a Control Treatment

Abstract In this paper, we introduce a method for selecting among $$k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> experimental treatments, each having two Bernoulli endpoints (e.g., efficacy and safety), the subset that contains the treatments whose efficacy and safety rates are superior to those of a control treatment. We then identify the treatment within the selected subset that has the highest efficacy rate. If no treatment meets the criteria of surpassing the control in both efficacy and safety, the control treatment is selected instead. Throughout, we consider the case in which the association between the two binary endpoints is characterized by a common, known odds ratio $$\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> </mml:math> shared by the control and all experimental arms. We employ a quadrinomial distribution to perform the exact calculations and derive the formulas for the proposed procedure. All designs use the exact counts of outcomes rather than the typical normal approximation, allowing for more accurate sample size determination to meet the required probability guarantees.