Exact and Approximate Power and Sample Size Calculations for Analysis of Covariance in Randomized Clinical Trials With or Without Stratification

Analysis of covariance (ANCOVA) is commonly used in the analysis of randomized clinical trials to adjust for baseline covariates and improve the precision of the treatment effect estimate. We derive the exact power formulas for testing a homogeneous treatment effect in superiority, noninferiority, and equivalence trials under both unstratified and stratified randomizations, and for testing the overall treatment effect and treatment × stratum interaction in the presence of heterogeneous treatment effects when the covariates excluding the intercept, treatment, and prestratification factors are normally distributed. These formulas also work very well for nonnormal covariates. The sample size methods based on the normal approximation or the asymptotic variance generally underestimate the required size. We adapt the recently developed noniterative and two-step sample size procedures to the above tests. Both methods take into account the nonnormality of the <i>t</i> statistic, and the lower order variance term commonly ignored in the sample size estimation. Numerical examples demonstrate the excellent performance of the proposed methods particularly in small samples. We revisit the topic on the prestratification versus post-stratification by comparing their relative efficiency and power. Supplementary materials for this article are available online.

协方差分析（Analysis of Covariance，ANCOVA）常被应用于随机临床试验的数据分析中，用于校正基线协变量并提升治疗效应估计的精度。我们推导了非分层与分层随机化场景下，优效性、非劣效性及等效性试验中检验齐性治疗效应的精确功效公式；同时推导了存在异质性治疗效应时，检验总体治疗效应与治疗×层交互作用的精确功效公式，此时除截距项、治疗因素及预分层因素外的协变量均服从正态分布。该类公式对于非正态分布的协变量同样适用。基于正态近似或渐近方差的样本量计算方法，通常会低估所需的样本量。我们将近期提出的非迭代两步样本量计算方法适配至上述检验场景中。两种方法均考虑了t统计量的非正态性，以及样本量估计中通常被忽略的低阶方差项。数值算例结果表明，所提方法具有优异的表现，尤其在小样本场景下效果突出。我们通过对比预分层与后分层的相对效率与功效，重新探讨了这一研究议题。本文的补充材料可在线获取。