Analysis of covariance with pre‐treatment measurements in randomized trials: Comparison of equal and unequal slopes

Abstract

In randomized trials, an analysis of covariance (ANCOVA) is often used to analyze post-treatment measurements with pre-treatment measurements as a covariate to compare two treatment groups. Random allocation guarantees only equal variances of pre-treatment measurements. We hence consider data with unequal covariances and variances of post-treatment measurements without assuming normality. Recently, we showed that the actual type I error rate of the usual ANCOVA assuming equal slopes and equal residual variances is asymptotically at a nominal level under equal sample sizes, and that of the ANCOVA with unequal variances is asymptotically at a nominal level, even under unequal sample sizes. In this paper, we investigated the asymptotic properties of the ANCOVA with unequal slopes for such data. The estimators of the treatment effect at the observed mean are identical between equal and unequal variance assumptions, and these are asymptotically normal estimators for the treatment effect at the true mean. However, the variances of these estimators based on standard formulas are biased, and the actual type I error rates are not at a nominal level, irrespective of variance assumptions. In equal sample sizes, the efficiency of the usual ANCOVA assuming equal slopes and equal variances is asymptotically the same as those of the ANCOVA with unequal slopes and higher than that of the ANCOVA with equal slopes and unequal variances. Therefore, the use of the usual ANCOVA is appropriate in equal sample sizes.

Reply to: ‘Comment on: Effect of smoking on retrobulbar blood flow in thyroid eye disease’

“…Indeed, the estimator of the effect at the observed mean is not different between equal and unequal variance assumptions. 2 Also, we used type III sum of squares, which is more robust in obtaining the variance of treatment effect.…”

“…1 Because of the great genetic heterogeneity of RP, technologies based on the next-generation sequencing are increasingly used to screen mutations or identify novel causative genes in RP patients for investigative and diagnostic purposes. 2 Here, we report a Chinese family suffering from autosomal dominant RP (adRP). The genetic cause of the family was further investigated.…”

“…As a rule, a variable with a SD that is larger than one half of the mean value is accepted as nonnormally distributed. 2 Second, the sample sizes between groups (n = 16 in the smoker group and n = 35 in the nonsmoker group) were unequal; therefore, it is likely that the variances will be unequal. 3 The assumptions for ANCOVA include that the outcome variable is normally distributed and the variances are similar between groups (homogeneity of variance).…”

Longitudinal analysis of pre‐ and post‐treatment measurements with equal baseline assumptions in randomized trials

“…The ANCOVA has higher power to detect a treatment effect than the t ‐test under standard assumptions. In previous studies, we examined the asymptotic properties of four ANCOVA models under the cases that covariances and posttreatment variances differ between groups (Funatogawa & Funatogawa, ; Funatogawa, Funatogawa, & Shyr, ). The actual type I error rate of the usual ANCOVA with equal slopes (coefficients of pretreatment measurements) and equal residual variances is asymptotically at a nominal level only under equal sample sizes, but that of the ANCOVA with equal slopes and unequal residual variances is asymptotically at a nominal level even under unequal sample sizes (Funatogawa et al., ).…”

“…However, the efficiency of the latter model is relatively low. The ANCOVA with unequal slopes has higher efficiency but cannot keep a nominal type I error rate irrespective of variance assumptions (Funatogawa & Funatogawa, ). Yang and Tsiatis () showed the longitudinal model with equal baseline means and variances and unequal covariances and posttreatment variances has the same efficiency with the ANCOVA with unequal slopes.…”