Monitoring Ongoing Clinical Trials Under Fractional Brownian Motion With Drift

The standard Brownian motion (Bm) with a linear drift is a convenient statistical structure for monitoring ongoing clinical trials in practice for more than four decades (Lan and DeMets). Under this model, the most current one-point statistic is sufficient. However, in our experience, the sponsor and the data monitoring committee often would like to make decision or recommendation based on the "trend" observed from the history of data, not just a one-point snapshot. In this article, we introduce and advance the fractional Brownian motion (fBm) with drift model to formally accommodate this need. The possible dependence and/or the nonlinear trend (e.g., piecewise linear drift with change-point) of observations in clinical trials may come from uncontrollable factors such as patient entry processes may have seasonal patterns over time, patient survival time may depend on the practices of clinical centers, physicians or censoring time (Lai et al.). The violations of the standard Bm and the need for the fBm are discussed with illustrative examples. The common methods including conditional power and sample size re-estimation used for monitoring clinical trials are derived and implemented in the Dynamic Data Monitoring (DDM) system for practitioners under the fBm.