  Response-adaptive designs allow the randomization probabilities to change
during the course of a trial based on cumulated response data, so that a
greater proportion of patients can be allocated to the better performing
treatments. A major concern over the use of response-adaptive designs in
practice, particularly from a regulatory viewpoint, is controlling the type I
error rate. In particular, we show that the naive z-test can have an inflated
type I error rate even after applying a Bonferroni correction. Simulation
studies have often been used to demonstrate error control, but do not provide a
guarantee. In this paper, we present adaptive testing procedures for normally
distributed outcomes that ensure strong familywise error control, by
iteratively applying the conditional invariance principle. Our approach can be
used for fully sequential and block randomized trials, and for a large class of
adaptive randomization rules found in the literature. We show there is a high
price to pay in terms of power to guarantee familywise error control for
randomization schemes with extreme allocation probabilities. However, for
proposed Bayesian adaptive randomization schemes in the literature, our
adaptive tests maintain or increase the power of the trial compared to the
z-test. We illustrate our method using a three-armed trial in primary
hypercholesterolemia.
