On the restricted maximum likelihood estimation in a two-arm randomized clinical trial for binary endpoints

Optimal design of superiority, noninferiority and equivalence two-arm clinical trials with binary endpoints remains a scientific topic of considerable theoretical and practical interest. In their seminal paper, Farrington and Manning (Stat. Med. 9 (1990) 1447–1454) demonstrate, upon systematic simulation, the superiority of the design based on the conditional maximum likelihood estimators (RMLE) to a number of viable alternatives. The approach based on the RMLE is the most popular option among practitioners with more than 700 citations up to date. They offer a closed-form formula for the maximum likelihood estimator, which is claimed to be a unique root to a certain cubic polynomial in a specified parameter-dependent interval. We identify a number of instances where this formula is not well-defined, and thus not universally valid. More specifically still, for infinitely many parameter combinations, the formula suggested in Farrington and Manning (Stat. Med. 9 (1990) 1447–1454) involves division by 0 which is an ill-defined concept. In this manuscript, we offer a full, mathematically rigorous theory of RMLE in this set-up, which has not been known to have been published previously. We thereby correct and extend the current results on the topic and provide a formula, which is indeed (universally) valid for all possible parameter combinations.