Asymptotic validity of Schoenfeld’s sample size formula for the Cox proportional hazards model via the Wald test approach

Se Yoon Lee

Abstract: We revisit the widely used sample size formula for the Cox proportional hazards model, originally proposed by Schoenfeld in 1983. The classical derivation, based on the score test, evaluates the Fisher information under the null hypothesis, overlooking key conditions required for its validity. Using a Wald test framework, we demonstrate that the derivation relies on the risk set proportionality property, wherein the ratio of at-risk counts in the treatment and control arms at observed times matches the randomization ratio. This property typically holds under the null hypothesis or when event rates are low, given a sufficiently large sample size. Our analysis clarifies the asymptotic validity of the formula and shows that violations of this assumption can lead to substantial loss of efficiency, particularly under alternative hypotheses. In contrast, simulation-based approaches remain robust. A retrospective analysis of the ADAURA trial illustrates how simulation-based power analysis could have shortened the study duration compared to the formula-based approach, while still maintaining the type I error rate at the nominal level and preserving the coverage properties of the confidence interval. This work highlights the limitations of the Schoenfeld formula in realistic trial settings and recommends simulation-based methods for planning survival trials, especially when a large treatment effect is expected.