two-stage studies may be chosen optimally by minimising a single characteristic like the maximum sample size. However, given that an investigator will initially select a null treatment eﬀect and the clinically relevant diﬀerence, it is better to choose a design that also considers the expected sample size for each of these values. The maximum sample size and the two expected sample sizes are here combined to produce an expected loss function to ﬁnd designs that are admissible. Given the prior odds of success and the importance of the total sample size, minimising the expected loss gives the optimal design for this situation. A novel triangular graph to represent the admissible designs helps guide the decision-making process. The H 0-optimal, H 1-optimal, H 0-minimax and H 1-minimax designs are all particular cases of admissible designs. The commonly used H 0-optimal design is rarely good when allowing stopping for eﬃcacy. Additionally, the δ-minimax design, which minimises the maximum expected sample size, is sometimes admissible under the loss function. However, the results can be varied and each situation will require the evaluation of all the admissible designs. Software to do this is provided. Copyright © 2012 John Wiley & Sons, Ltd.