This article describes a new design, phase I LE /II dose searching, used in four prospective, randomized, multicenter clinical trials of escalating total doses of hyperfractionated radiation. This design combines an experimental protocol with a statistical application of ranking and selection theory. Its purpose was to identify (within a certain margin of error) a dose that achieved the highest rate of clinical response from a set of doses that were tolerable in terms of both acute (within 90 days) and late (more than 90 days) toxic effects (LE). We calculated the number of patients required to reliably test toxicity under various assumptions. To determine the maximum tolerated total dose for hyperfractionated radiation, we randomly assigned patients with tumors that responded to radiation theraphy in a dose-dependent manner from four body sites (lung, upper respiratory and digestive tract, bladder, and brain) to one of three regimens total doses (D 1 , D 2 , or D 3 ) differing by increments of 4.8 Gy. All patients received two fractions of 1.2 Gy each (separated by 4–6 hours) daily 5 days a week. The lowest total dose was set at the level considered tolerable with standard once-a-day radiation therapy. We tested tumor responses and late toxic effects of higher doses by assigning patients to these three regimens until acute effects and early estimates of late effects were found to be acceptable for the highest dose D 3 ; thereafter, regimen D 1 was closed, and additional patients were assigned to D 2 , D 3 and D 4 (an escalated total dose greater than D 3 by an increment of 4.8 Gy). The assignment of patients was performed in a weighted manner (1:1:2), so that greater numbers were assigned to the highest dose regimen (whether D 3 or D 4 ) to allow rapid evaluation of the feasibility of the highest dose. [J Natl Cancer Inst 83:1065–1071, 1991]