Power Calculation for Non-inferiority Test Based on Linear Combination of Two Correlated Binary Endpoints

In cancer research, multiple primary study endpoints, which may be correlated, are often considered. In practice, it is common to evaluate each clinical endpoint separately when performing sample size calculations. However, when the endpoints are correlated, treating them independently may lead to conservative sample size requirements. In this article, alternatively, we consider power calculation based on linear combination of two correlated clinical outcomes (e.g., binary responses) to address the correlation between them. For simplicity and for illustrational purpose, sample size formulae will be derived under hypothesis testing for non-inferiority. The results are evaluated both theoretically and via numerical studies. The results indicate that the required sample size is highly sensitive to the correlation between endpoints (ρ), event probabilities (p₁, p₂), and the non-inferiority margin (δ_Z). Stronger correlation increases informational redundancy and weakens the effective signal, raising the sample size requirement; when event rates are equal, variance is maximized, resulting in the highest sample size; conversely, a larger non-inferiority margin-implying greater tolerance for treatment differences-significantly reduces sample size. The weights (a, b) assigned to the composite endpoint are also crucial. With equal endpoint effects, balanced weights lower variance and reduce sample size, while unequal weights raise variance and increase sample size. Moreover, the assumed direction of treatment effects can substantially influence non-inferiority conclusions and the required sample size (e.g., [Formula: see text] vs. [Formula: see text]).