An improved double sampling procedure based on the variance.

Sample size calculations for a continuous outcome require specification of the anticipated variance; inaccurate specification can result in an underpowered or overpowered study. For this reason, adaptive methods whereby sample size is recalculated using the variance of a subsample have become increasingly popular. The first proposal of this type (Stein, 1945, Annals of Mathematical Statistics 16, 243-258) used all of the data to estimate the mean difference but only the first stage data to estimate the variance. Stein's procedure is not commonly used because many people perceive it as ignoring relevant data. This is especially problematic when the first stage sample size is small, as would be the case if the anticipated total sample size were small. A more naive approach uses in the denominator of the final test statistic the variance estimate based on all of the data. Applying the Helmert transformation, we show why this naive approach underestimates the true variance and how to construct an unbiased estimate that uses all of the data. We prove that the type I error rate of our procedure cannot exceed alpha.