A convention in designing randomized clinical trials has been to choose sample sizes that yield specified statistical power when testing hypotheses about treatment response. Manski and Tetenov recently critiqued this convention and proposed enrollment of sufficiently many subjects to enable near-optimal treatment choices. This article develops a refined version of that analysis applicable to trials comparing aggressive treatment of patients with surveillance. The need for a refined analysis arises because the earlier work assumed that there is only a primary health outcome of interest, without secondary outcomes. An important aspect of choice between surveillance and aggressive treatment is that the latter may have side effects. One should then consider how the primary outcome and side effects jointly determine patient welfare. This requires new analysis of sample design. As a case study, we reconsider a trial comparing nodal observation and lymph node dissection when treating patients with cutaneous melanoma. Using a statistical power calculation, the investigators assigned 971 patients to dissection and 968 to observation. We conclude that assigning 244 patients to each option would yield findings that enable suitably near-optimal treatment choice. Thus, a much smaller sample size would have sufficed to inform clinical practice.