Sequential tests and estimators after overrunning based on maximum-likelihood ordering

Often in sequential trials some additional data become available after a stopping boundary has been reached. A method for incorporating such information from overrunning is developed, based on a maximum-likelihood ordering of the sample space after overrunning. This yields a p-value for the primary test and a median-unbiased estimator and confidence intervals for the parameter under test. The context is that of observing a Brownian motion with drift, with either linear stopping boundaries in continuous time or discrete-time group-sequential boundaries. The methods apply to many clinical trials and are exemplified with data from a survival-analysis-based sequential clinical trial.