How to calculate % information in longitudinal group sequential design

Question

I'm looking at using group sequential design for a study that will treat subjects with drug or placebo and then follow them for a year. The subjects get measured every 3 months for the year under study. They may become a "responder" at month 3, 6, 9, or 12, but they are not officially a "non-responder" until they have failed to respond for all 12 months.

My question is, group sequential designs dictate conducting interim analysis at certain milestones of % information gathered. In this scenario, I am trying to figure out how to calculate the amount of information gathered. Let's say I need 100 subjects for the power I want, and the design states IAs at 33% and 67% of information. I don't think I need to wait for 33 subjects to be studied for 12 months because some of them may be responders before 12 months. I am thinking a subject would count if they have either failed (not responded in 12 months) or responded (regardless of how long it takes them to respond).

Does that sound right? I am looking for a check on my logic since this design is new to me.

Answer 1

This is actually a very interested problem. I will formalize your question first.

Let $\pi_T$ and $\pi_P$ be the true responder rate for treatment and placebo, respectively. At the end of your study, you want to test $H_0:\pi_T \leq \pi_C$ or $H_0:\pi_T = \pi_C$, I assume.

In a group sequential design, you test $H_0$ before the end of the study and to do so appropriately, you need an unbiased or a consistent estimator for $\pi_T$ and $\pi_C$, respectively. As a side note, it probably would also be enough if you can just estimate either the ratio or the difference unbiased or consistently. However, since you test $H_0$ before the end of the study, many subjects are not non-responders or responders yet and they could still turn into either at a future time point. Therefore, unless you make any assumption about the distribution of the time when subjects respond and link this distribution to $\pi_T$ and $\pi_C$, I do not see how you could use subjects with an incomplete response to make inference about $H_0$.