A geometrically intuitive framework for the synthesis method in non-inferiority trials

The classical approach to non-inferiority (NI), known as the fixed margin method (FMM), uses a pre-defined NI margin (M). Non-inferiority is concluded if the lower bound of the confidence interval (CI) for the difference between the experimental and control treatments is greater than -M. Typically, M is set using a conservative estimate of the historical effect of the control therapy versus placebo. In contrast, the synthesis method (SM) indirectly evaluates the efficacy of the experimental treatment versus placebo by combining the observed active-controlled effect from the NI trial (Δ_NI) with the historical effect of the active control compared to placebo (Δ_H). Although the SM is known to be more efficient than the FMM, its use remains uncommon. Two perceived limitations of the SM are the inability to (a) pre-specify an explicit margin in the protocol and (b) present the final results as a straightforward CI compared against a fixed threshold. Both limitations are largely overcome by framing the estimates in the bivariate space (Δ_NI, Δ_H). Using this parameterization, the NI margin is defined in the study protocol by the line Δ_NI + (1-δ)Δ_H = 0 where δ represents the minimum acceptable efficacy retention fraction. During the analysis, we can intuitively visualize both FMM and SM in this plane, with FMM depicted as a horizontal confidence segment and SM as a confidence ellipse. We formally demonstrate that comparing these regions to the line Δ_NI + (1-δ)Δ_H = 0 is mathematically equivalent to conducting the conventional Z-tests commonly used to evaluate NI under FMM and SM.