Sequential Analysis

Preliminaries of a Self-Designing Trial

Consider an equally randomized clinical trial to compare a treatment, T, and a control, C. Assume that the difference between the two arms is normally distributed with an unknown mean θ and known variance σ2. The methodology can be extended to the situation with unknown variance. Positive values of θ imply superiority of the treatment T. Suppose the block size at each stage to be {2Bi, i = 1, 2, …}, where Bi for each arm is pre-fixed. Let denote the mean difference for the ith block of data. i ∼ N(θ, σ2/Bi).

Let Si define the standardized statistic based on the ith block of data and Uj define the standardized statistic for the cumulated data up to the jth analysis; then

The one-sided null hypothesis, H0: θ ≤ 0, is tested against the alternative hypothesis H1: θ > 0.

The trial is reviewed after observing every 2Bi subjects at the ith interim analysis. If the cumulated information shows sufficient evidence that the new treatment is ineffective or inferior to the standard one, the trial should be terminated early. At each interim analysis, the futility boundary is constructed from the confidence limit of θ specified at significance level α0 and the expected mean difference δ. If Uj is below the futility boundary, the trial is terminated to accept H0 at the next step. Otherwise, a weight based on the conditional power is calculated using the cumulated data before the current step. The procedure is iterated until the total squared weight reaches one at step m, that is, and . Specifically, after observing the jth block of data,

The weight functions are constructed using conditional power except at the first step. Specifically, w1 = (B1/N1)1/2, where N1 = (zα + zβ)2σ2/δ2, and α and β are the specified Type I and II error rates, respectively. Note that α0 is the significance level used for futility monitoring that can be different from a. For j > 2,

is the naive estimator of θ using the cumulated data up to the jth block, and Nj is solved from an inequality aimed at achieving conditional power of 1–β. The final test statistic is constructed as a weighted average of the sufficient statistic of each

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