Response-adaptive randomization procedures are appropriate for clinical trials in which two or more treatments are to be compared, patients arrive sequentially and the response of each patient is recorded before the next patient arrives. However, for those procedures that involve sequential estimation of model parameters, start-up designs are commonly required in order to provide initial estimates of the parameters. In this paper, a suite of such start-up designs for two treatments and binary patient responses are considered and compared in terms of the numbers of patients required in order to give meaningful parameters estimates, the number of patients allocated to the better treatment, and the bias in the parameter estimates. It is shown that permuted block designs with blocks of size 4 are to be preferred over a wide range of parameter values. For the case of two treatments, normal responses and selected start-up procedures, a design incorporating complete randomization followed appropriately by repeats of one of the treatments yields the minimum expected number of patients and is to be preferred.