In many fields such as acute toxicity & sensory studies, Phase I cancer trials, and psychometric testing, binomial regression models are used for analysis following sequential informative dose allocation. We assume the simplest general case in which a univariate binary response Y has a monotone positive response probability P(Y=1|x)=F(x) to a stimulus or treatment X; X values are sequentially selected, from a discrete set, to concentrate treatments in a region of interest under F(x). We call a positive response a toxicity and the stimulus a dose. From first principles, we describe dependencies that are introduced by sequentially choosing informative doses. We refute the prevailing notion that the number of dose-specific toxicities seen, conditional on the dose's allocation frequency, are binomial random variables; and characterize at finite sample sizes, bias of the observed toxicity rate at dose x for F(x).
This is important. Isotonic regression methods use dose-specific toxicity rates directly. Likelihood-based methods mask bias by providing first-order linear approximations to basic properties of estimators. We illustrate these findings in some common adaptive designs.