RESEARCH ARTICLE Efficient treatment allocation in 2 × 2 multicenter trials when costs and variances are heterogeneous Francesca Lemme 1 | Gerard J.P. van Breukelen 1,2 | Math J.J.M. Candel 1,2 1 Department of Methodology and Statistics, Maastricht University, Maastricht, The Netherlands 2 CAPHRI Care and Public Health Research Institute, Maastricht University, Maastricht, The Netherlands Correspondence Francesca Lemme, Department of Methodology and Statistics, Maastricht University, Maastricht, The Netherlands. Email: francesca. lemme@maastrichtuniversity.nl; francesca.lemme75@gmail.com Funding information Netherlands Organization for Scientific Research (NWO), Grant/Award Number: 400‐09‐396 At the design stage of a study, it is crucial to compute the sample size needed for treatment effect estimation with maximum precision and power. The optimal design depends on the costs, which may be known at the design stage, and on the outcome variances, which are unknown. A balanced design, optimal for homogeneous costs and variances, is typically used. An alternative to the balanced design is a design optimal for the known and possibly heterogeneous costs, and homogeneous variances, called costs considering design. Both designs suffer from loss of efficiency, compared with optimal designs for heterogeneous costs and variances. For 2 × 2 multicenter trials, we compute the relative efficiency of the balanced and the costs considering designs, relative to the optimal designs. We consider 2 heterogeneous costs and variance scenarios (in 1 scenario, 2 treatment conditions have small and 2 have large costs and variances; in the other scenario, 1 treatment condition has small, 2 have intermediate, and 1 has large costs and variances). Within these scenarios, we examine the relative efficiency of the balanced design and of the costs considering design as a function of the extents of heterogeneity of the costs and of the variances and of their congruence (congruent when the cheapest treatment has the smallest variance, incongruent when the cheapest treatment has the largest variance). We find that the costs considering design is generally more efficient than the balanced design, and we illustrate this theory on a 2 × 2 multicenter trial on lifestyle improvement of patients in general practices. KEYWORDS 2 × 2 factorial design, balanced design, heterogeneous costs, heterogeneous variance, multicenter trial 1 | INTRODUCTION Clinical trials are very important for the evaluation of medical and health interventions. In traditional randomized clin- ical trials conducted in a single center, a limited number of patients may meet the inclusion criteria or be willing to par- ticipate, leading to a small sample size and lack of power. Further, a single‐center trial does not allow testing treatment by center interaction, leaving the question of the generalizability of the treatment effect across centers unanswered. Therefore, clinical trials are increasingly conducted in multiple centers. At the design stage of a multicenter trial, 2 important statistical issues are (i) the optimal number of centers versus the optimal number of patients per center and (ii) within each center, the optimal treatment allocation ratio across treatment conditions. The first issue has been addressed, among others, by Fedorov, 1 Moerbeek et al, 2 and Raudenbush and Liu, 3 Received: 27 March 2017 Revised: 14 August 2017 Accepted: 23 August 2017 DOI: 10.1002/sim.7499 Statistics in Medicine. 2017;1–16. Copyright © 2017 John Wiley & Sons, Ltd. wileyonlinelibrary.com/journal/sim 1