Abstract

A large number of experiments in clinical trials, biology, biochemistry, etc. are, out of necessity, conducted in two stages. A first-stage experiment (a pilot study) is often used to gain information about feasibility of the experiment or to provide preliminary data for grant applications. We study the theoretical statistical implications of using a small sample of data (1) to design the second stage experiment and (2) in combination with the second-stage data for data analysis. To illuminate the issues, we consider an experiment under a non-linear regression model with normal errors. We show how the dependency between data in the different stages affects the distribution of parameter estimates when the first-stage sample size is fixed and finite; letting the second stage sample size go to infinity, maximum likelihood estimates are found to have a mixed normal distribution.