A Semi-Markov Process for the K-in-a-Row Design

Abstract Estimating the value of a stimulus variable that has a prespecified percentage of successes is common in many fields, and known generally as “dose-finding”. In most practical applications, only a few values of the stimuli can be applied to the statistical units that participate in the experiment. K-in-a-Row Up-and-Down is a popular experimental procedure that sequentially allocates statistical units to the permissible values of a stimulus variable, using simple invariant rules. Despite having been in use for 60 years, K-in-a-Row’s statistical properties are still not broadly understood. We show that it is naturally modeled by a semi-Markov process. As far as we know, it is the first stochastic design in practical use that is represented by a semi-Markov process. The stationary distribution is characterized, assuming only that the success probability increases with the values of the stimuli. We prove the strong unimodality of the asymptotic distribution of the proportions of stimuli-specific allocations. Thus the mode of the stimuli-specific allocation serves as a summary measure of location for these designs, and we explicitly identify it.