A likelihood-based approach for computing the operating characteristics of the 3+3 phase I clinical trial design with extensions to other A plus B designs

Background In phase I clinical trials, the 3+3 algorithmic design has been unparalleled in its popularity. The statistical properties of the 3+3 design have been studied in the literature either in comparison with other methods or by deriving exact formulae of statistical quantities. However, there is still much to be known about its capabilities of describing and accounting for uncertainties in the observed data. Purpose The objective of this study is to provide a probabilistic support for analyzing the heuristic performance of the 3+3 design. The operating characteristics of the algorithm are computed under different hypotheses, levels of evidence, and true (or best guessed) toxicity rates. The dose-finding rules are further compared with those generated by the modified toxicity probability interval design and generalized for implementation in all A+B designs. Methods Our likelihood method is based on the evidential paradigm. Two hypotheses are chosen to correspond to two hypothesized dose-limiting toxicity rates, for example, H 1 unsafe versus H 2 acceptable. Given observed toxicities per dose, the likelihood-ratio is calculated and compared to a certain k threshold (level of evidence). Under various true toxicities, the probabilities of weak evidence, favoring H 1 and H 2 , were computed under four sets of hypotheses and several k thresholds. Results For scenarios where the midpoint of the two hypothesized dose-limiting toxicity rates is around 0.30, and for a threshold of k=2, the 3+3 design has a reduced probability (approximate to 0.50) of identifying unsafe doses, but high chances of identifying acceptable doses. For more extreme scenarios targeting a relatively high or relatively low dose-limiting toxicity rate, the 3+3 design has no probabilistic support, and therefore, it should not be used. In our comparisons, the likelihood method is in agreement with the modified toxicity probability interval design for the majority of the hypothesized scenarios. Even so, based on the evidential paradigm, a 3+3 design is often incapable of providing sufficient levels of evidence of acceptability for doses under reasonable scenarios. Limitations Given the small sample size per dose, the levels of evidence are limited in their ability to provide strong evidence favoring either of the hypotheses. Conclusion In many situations, the 3+3 design does not treat enough patients per dose to have confidence in correct dose selection and the safety of the selected/unselected doses. This likelihood method allows consistent inferences to be made at each dose level, and evidence to be quantified regardless of cohort size. The approach can be used in phase I studies for identifying acceptably safe doses, but also for defining stopping rules in other types of dose-finding designs.