On the bias of estimation of a Brownian motion drift following group sequential tests

Group sequential tests have been widely used to control the type I error ra te at a prespecified level in comparative clinical trials. It is well known that due to the optional sampling effect, conventional maximum likelihood estimates will exaggerate the treatment difference, and hence a bias is int roduced. We consider a group sequentially monitored Brownian motion process . An analytical expression of the bias of the maximum likelihood estimate f or the Brownian motion drift is derived based on the alpha spending method of Lan and DeMets (1983). Through this formula, the bias can be evaluated e xactly by numerical integration. We study how the Brownian motion drift and various alpha spending functions and interim analysis patterns affect the bias. A bias adjusted estimator is described and its properties are investi gated. The behavior of this estimator is studied for differing situations.