Derivation of Exact Distributions Following an Up-and-Down Design

Use of large sample theory in sequential designs, such as the up-and-down designs, to estimate parameters of interest may not be appropriate when the sample size is small, as is the case with many phase I clinical trials. In these situations, it may be desirable to derive the exact distribution of various statistics of interest. Having the exact distribution will enable us to calculate exact probabilities, moments or confidence intervals. We show how one can derive these exact distributions by enumerating all possible outcomes. We also derive a formula to obtain the total number of all possible treatment outcomes using the biased up-and-down design. These results will enable the experimenter to plan a phase I study with more confidence.