Statistical Inference for a Two‐Stage Adaptive Seamless Design Using Different Binary Endpoints

Adaptive seamless design, which integrates phases II and III into a single trial comprising two stages, is garnering increasing interest in the efficient drug development. The first stage involves selecting promising treatment group(s), followed by comparing the efficacy between the selected and control groups in the second stage. This study focused on a two-stage adaptive seamless design where treatment selection is based on a short-term binary endpoint, while the comparison is based on a long-term binary endpoint. Recently, exact and mid- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>p</mml:mi></mml:mrow> <mml:annotation>$$ p $$</mml:annotation></mml:semantics> </mml:math> tests were proposed in this setting. However, treatment effects at the second stage were estimated using the conventional maximum likelihood estimator (MLE), leading to upward bias owing to treatment selection. We propose the conditional mean-adjusted estimator (CMAE) and uniformly minimum variance conditional unbiased estimator (UMVCUE) to address the bias in this setting. Additionally, confidence intervals for exact and mid- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>p</mml:mi></mml:mrow> <mml:annotation>$$ p $$</mml:annotation></mml:semantics> </mml:math> tests were constructed using the Clopper-Pearson method. Simulation studies were performed to compare the six inference methods defined by combinations of the three estimators and two statistical tests. The simulation results showed that MLE of the treatment effect at the second stage exhibited a notable bias, while CMAE and UMVCUE substantially reduced the bias. The exact test was conservative in terms of the type-I error rate of the comparison at the second stage, while the mid- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>p</mml:mi></mml:mrow> <mml:annotation>$$ p $$</mml:annotation></mml:semantics> </mml:math> test yielded results close to the nominal level. In conclusion, we recommend statistical inferences based on the CMAE + mid- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>p</mml:mi></mml:mrow> <mml:annotation>$$ p $$</mml:annotation></mml:semantics> </mml:math> test or UMVCUE + mid- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>p</mml:mi></mml:mrow> <mml:annotation>$$ p $$</mml:annotation></mml:semantics> </mml:math> test in our setting.