Stochastically Curtailed Tests Under Fractional Brownian Motion

Abstract

Stochastic curtailment has been considered for the interim monitoring of group sequential trials (Davis and Hardy, 1994). Statistical boundaries in Davis and Hardy (1994) were derived using theory of Brownian motion. In some clinical trials, the conditions of forming a Brownian motion may not be satisfied. In this paper, we extend the computations of Brownian motion based boundaries, expected stopping times, and type I and type II error rates to fractional Brownian motion (FBM). FBM includes Brownian motion as a special case. Designs under FBM are compared to those under Brownian motion and to those of O'Brien-Fleming type tests. One-and two-sided boundaries for efficacy and futility monitoring are also discussed. Results show that boundary values decrease and error rates deviate from design levels when the Hurst parameter increases from 0.1 to 0.9, these changes should be considered when designing a study under FBM.

Statistical Monitoring of Clinical Trials Using Brownian Bridges

“…Of course, stopping a trial early due to efficacy or futility will also depend on other considerations, including whether the actual timing is too early or not, secondary objectives, planned subgroup analyses, safety, quality of life, translational research, etc. Additionally, the stopping boundary for the group sequential design using stochastic curtailment and Brownian bridge (for more than one interim analysis which will be explored in future research) would be less than those under the Brownian motion, again according to which the monitoring rule is very conservative especially during early analysis times (Zhang et al, 2015). This is due to the smaller variances of Brownian bridges which lead to the higher predicted probabilities of rejecting the null hypothesis.…”

“…For illustration purpose, we will calculate conditional power for one interim analysis as shown above for the rest of the paper. We can follow similar algorithms as in Zhang et al (2015) when there are multiple interim analyses in practice.…”

“…Also, the early treatment difference needed to reject the null hypothesis is more realistic than those using conditional power under the BM (e.g., Z values are 3.52 vs. 5.35 at = 0.20). We will discuss further regarding group sequential designs based on conditional power (Zhang et al, 2015) under the BB in the discussion. For BHAT, we can calculate the conditional probabilities of rejecting the null hypothesis given the similar positive trend of around ≅ 2 for the first three analyses in addition to the fourth one.…”

“…There are multiple methods developed for the interim monitoring of clinical trials, these include, for example, the well-known Pocock (1977) and O'Brien-Fleming (OBF, 1979) designs, alpha spending function (Lan & Demets, 1983), stochastic curtailment (conditional power, CP, Davis & Hardy, 1990;Zhang, Lai, & Davis, 2015) and repeated confidence intervals (RCI, Jennison & Turnbull, 2000, Zhang, 2011. Most of these methods are built upon Brownian motion (BM).…”