Estimation of hurst exponent for sequential monitoring of clinical trials with covariate adaptive randomization

Classical Brownian motion (BM) has been commonly used in monitoring clinical trials including those with covariate adaptive randomization (CAR). Independent increment property is commonly assumed in the sequential monitoring process of the clinical trials with CAR designs. However, in reality, correlation may exist in the error terms of the underlying model, resulting in dependent increment in the sequential monitoring process.

We conducted simulations for estimating the Hurst exponent to evaluate the stochastic property in the covariate adaptive randomized clinical trials under two scenarios: 1. CAR designs with independent and identically distributed error terms. 2. CAR designs with correlated error terms. The theoretical properties of covariate adaptive randomized clinical trials with correlated error structure were investigated. A test statistic including the covariance pattern of the error terms was proposed.

In our study, the sequential test statistics under CAR procedure is shown to be asymptotically Brownian motion when the error structure is correctly specified. Further, Brownian motion is a special case of fractional Brownian motion when Hurst exponent equals to 0.5. Our simulations are consistent with the theoretical asymptotic results.