Optimal allocation of participants for the estimation of selection, preference and treatment effects in the two-stage randomised trial design

Outcomes in clinical trials may be affected by the choice of treatment that participants might make, if they were indeed allowed to choose (a so-called selection effect), and by whether they actually receive their preferred treatment (a preference effect). Selection and preference effects can be important, but they cannot be estimated in the conventional trial design. An alternative approach is the two-stage randomised trial, in which participants are first randomly divided into two subgroups. In one subgroup, participants are randomly assigned to treatments, while in the other, participants are allowed to choose their own treatment. This approach yields estimates of the direct treatment effect, and of the preference and selection effects. The latter two provide insight that goes considerably beyond what is possible in the standard randomised trial. In this paper, we determine the optimal proportion of participants who should be allocated to the choice subgroup. The precision of the estimated selection, preference and treatment effects are functions of: the total sample size; the proportion of participants allocated to choose their treatment; the variances of the outcome; the proportions of participants who select each treatment in the choice group; and the selection, preference and treatment effects themselves. We develop general expressions for the optimum proportion of participants in the choice group, depending on which effects are of primary interest. We illustrate the results with trial data comparing alternative clinical management strategies for women with abnormal results on cervical screening.