Information in adaptive optimal design with emphasis on the two stage case
Seminar Room 1, Newton Institute
In 1963, Box and Hunter, followed by many others, recommended selecting sequential treatments to maximize the increment of some information measure (e.g., the determinant of the Fisher information matrix). Under nonlinear regression models, because information is a function of unknown parameters, such increments must be estimated; and the information from different stages is not independent. To explore the accrual of information in adaptive designs, we study a basic one parameter nonlinear regression model with additive independent normal errors. The stage 1 treatment is taken to be fixed, the treatment allocation rule for stage 2 is taken to be a unique function of maximum likelihood estimates derived from stage 1 data. Although conditioning on the design is common in data analyses, we show in this scenario, that conditioning on the stage 2 treatment is equivalent to conditioning on the stage 1 data. This raises questions about the role of conditioning in the analysis o f adaptive designs. We also explore the efficiency conducting studies in stages and the effect of allocating different proportions of subjects to stage 1 versus stage 2.