Due to the staggered entry of subjects in phase I trials, some subjects will only be partially through the study when others are ready to be enrolled. Nonetheless, many phase I designs focus solely upon whether or not subjects experience toxicity, thereby determining the maximum tolerated dose (MTD) with a binomial likelihood using data from fully observed subjects. The time-to-event continual reassessment method (TITE-CRM) was the first attempt to incorporate information from partially observed subjects by using a weighted binomial likelihood in which the weights are based upon the actual toxicity time distribution. Unfortunately, it is difficult to accurately estimate the toxicity time distribution because only a small proportion of enrolled subjects will experience toxicity. Creators of the TITE-CRM propose the simple alternative of weighting subjects by the proportion of time observed, as well as two adaptive weights to adjust for late-onset toxicities. As a alternative to these approaches, we suggest assuming the toxicity times, as a proportion of the total time under observation, have a Beta distribution with parameters 1.0 and theta; we also allow theta to vary by dose. The value of theta allows us to reflect the occurrence of early- or late-onset toxicities without correctly specifying the actual distribution of toxicity times. Through this model, we do not necessarily expect to improve identification of the MTD, but rather hope to reduce the exposure of subjects to overly toxic doses. Through simulation, we examine how well our model identifies the MTD and allocates dose assignments in three scenarios investigated by previous publications.
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