Impact of Endpoint Delay on the Efficiency of Multi Arm Multi Stage Trials

Multi-arm multi-stage (MAMS) is an efficient class of trial designs that helps to assess multiple treatment strategies at the same time using an adaptive design. These designs can substantially reduce the average number of samples required compared to an equivalent single stage multi-arm trial. However, if patient recruitment is continued while we await treatment outcomes, a long-term primary outcome leads to a number of 'pipeline' patients getting recruited in the trial, who do not benefit from the early termination of a futile arm. This study focuses on quantifying the efficiency loss a MAMS design undergoes, in terms of the expected sample size (ESS), because of outcome delay. We first estimate the number of 'pipeline' patients (recruited during the interim analysis (IA) while awaiting outcome data) analytically through different recruitment models, given the total recruitment time. We then compute the ESS accounting for delay and assess the Efficiency Loss (EL). The results indicate that more than 50% of the expected efficiency gain is typically lost due to delay when the delay is more than $1/3\text{rd}$ of the total recruitment length. Although the number of stages have little influence on the efficiency loss, the timing of the IA can impact the efficiency of MAMS designs with delayed outcomes; in particular, conducting the IAs earlier than an equally-spaced design can be harmful for the design. Finally, we conclude that, in order to gain maximum benefit of MAMS in terms of a reduced sample size in multi-arm trials, the outcome delay should be less than a third of the total recruitment length.