Analysis of covariance with pre‐treatment measurements in randomized trials under the cases that covariances and post‐treatment variances differ between groups

When primary endpoints of randomized trials are continuous variables, the analysis of covariance (ANCOVA) with pre‐treatment measurements as a covariate is often used to compare two treatment groups. In the ANCOVA, equal slopes (coefficients of pre‐treatment measurements) and equal residual variances are commonly assumed. However, random allocation guarantees only equal variances of pre‐treatment measurements. Unequal covariances and variances of post‐treatment measurements indicate unequal slopes and, usually, unequal residual variances. For non‐normal data with unequal covariances and variances of post‐treatment measurements, it is known that the ANCOVA with equal slopes and equal variances using an ordinary least‐squares method provides an asymptotically normal estimator for the treatment effect. However, the asymptotic variance of the estimator differs from the variance estimated from a standard formula, and its property is unclear. Furthermore, the asymptotic properties of the ANCOVA with equal slopes and unequal variances using a generalized least‐squares method are unclear. In this paper, we consider non‐normal data with unequal covariances and variances of post‐treatment measurements, and examine the asymptotic properties of the ANCOVA with equal slopes using the variance estimated from a standard formula. Analytically, we show that the actual type I error rate, thus the coverage, of the ANCOVA with equal variances is asymptotically at a nominal level under equal sample sizes. That of the ANCOVA with unequal variances using a generalized least‐squares method is asymptotically at a nominal level, even under unequal sample sizes. In conclusion, the ANCOVA with equal slopes can be asymptotically justified under random allocation.