The effect of covariate mean differences on the standard error and confidence interval for the comparison of treatment means

The use of covariates is commonly believed to reduce the unexplained error variance and the standard error for the comparison of treatment means, but the reduction in the standard error is neither guaranteed nor uniform over different sample sizes. The covariate mean differences between the treatment conditions can inflate the standard error of the covariate-adjusted mean difference and can actually produce a larger standard error for the adjusted mean difference than that for the unadjusted mean difference. When the covariate observations are conceived of as randomly varying from one study to another, the covariate mean differences can be related to a Hotelling's T². Using this Hotelling's T² statistic, one can always find a minimum sample size to achieve a high probability of reducing the standard error and confidence interval width for the adjusted mean difference.