In prospective clinical trials, safety and efficacy results should be monitored periodically. If early data provide convincing evidence of a superior therapeutic index for one of the treatments, then early trial termination would satisfy important ethical requirements and save valuable resources and time. The data obtained in these studies are often analyzed further to determine whether treatment effects differ in various subsets. In this paper we discuss the problems that can arise from frequently used inappropriate approaches to interim and subset analyses. The proper role of such analyses is then discussed, and valid and useful methods are described for deciding on early termination of negative as well as positive studies and for investigating subset effects.