Dating back to Dixon and Mood (1948), an Up-and-Down procedure is a se- quential experiment used in binary response trials for identifying the stress level (treatment) corresponding to a pre-specified probability of positive response. In Phase I clinical trials U&D rules can bee seen as a development of the traditional dose-escalation procedure (Storer, 1998). Recently Baldi Antognini et al. (2008) have proposed a group version of U&D procedures whereby at each stage a group of m units is treated at the same level and the number of observed positive re- sponses determines how to randomize the level assignment of the next group. This design generalizes a vast class of U&Ds previously considered (Derman, 1957; Durham and Flournoy 1994; Giovagnoli and Pintacuda, 1998; Gezmu and Flournoy, 2006). The properties of the design change as the randomization method varies: appropriate randomization schemes guarantee desirable results in terms of the asymptotic behaviour of the experiment (see also Bortot and Giovagnoli, 2005). Results can be extended to continuous responses (Ivanova and Kim, 2009).
Other approaches for identifying a target dose, alternative to the nonparamet- ric U&D, are the parametric Continual Reassessment Method introduced by O'Quigley et al. (1990), and several recent modifications thereof. The debate on dose escalation procedures in the recent statistical literature continues to be very lively.