Sample size for partially nested designs and other nested or crossed designs with a continuous outcome when adjusted for baseline

In a randomized controlled trial, outcomes of different subjects may be independent at baseline, but correlated at a follow-up measurement due to treatment. This treatment-related clustering at follow-up can arise for instance because the treatment is given in a group or because subjects are treated individually but by the same therapist (therapist effect). There is substantial literature on the design and analysis of such trials when estimation of the intervention effect is based on a follow-up measurement (eg, directly after treatment or at a later time point). However, often the baseline measurement of the outcome is highly correlated with the follow-up measurement, and this information can be used in the analysis. For a randomized design with a baseline and a follow-up measurement, we compare sample size requirements for analyses with and without adjustment for this baseline measure. We show that adjusting for baseline reduces required sample size. This reduction depends on the variance of the difference between arms at baseline, the variance of this difference at follow-up, and the correlation between the two. From this, we derive sample size formulas for partially or fully nested designs, and cluster randomized trials with treatment as a partially or fully cross-classified factor. Also, we discuss situations where clusters are already present at baseline or where treatment by cluster interaction is present. For the partially nested design, we work out practical design considerations (eg, use of content-matter input, design factors and optimal allocation ratio) and investigate small sample properties of the sample size formula.