Confidence intervals for overall response rate difference in the sequential parallel comparison design

High placebo responses could significantly reduce the treatment effect in a parallel randomized trial. To combat that challenge, several approaches were developed, including the sequential parallel comparison design (SPCD) that was shown to increase the statistical power as compared to the traditional randomized trial. A linear combination of the response rate differences from two phases per the SPCD is commonly used to measure the overall treatment effect size. The traditional approach to calculate the confidence interval for the overall rate difference is based on the delta method using the variance-covariance matrix of all outcomes. As outcomes from a multinomial distribution are correlated, we suggest utilizing a constrained variance-covariance matrix in the delta method. In the observation of anti-conservative coverages from asymptotic intervals, we further propose using importance sampling to develop accurate intervals. Simulation studies show that accurate intervals have better coverage probabilities than others and the interval width of accurate intervals is similar to the interval width of others. Two real trials to treat major depressive disorder are used to illustrate the application of the proposed intervals.