The use of weighted Z-tests in medical research

Traditionally the un-weighted Z-tests, which follow the one-patient-one-vote principle, are standard for comparisons of treatment effects. We discuss two types of weighted Z-tests in this manuscript to incorporate data collected in two (or more) stages or in two (or more) regions. We use the type A weighted Z-test to exemplify the variance spending approach in the first part of this manuscript. This approach has been applied to sample size re-estimation. In the second part of the manuscript, we introduce the type B weighted Z-tests and apply them to the design of bridging studies. The weights in the type A weighted Z-tests are pre-determined, independent of the prior observed data, and controls alpha at the desired level. To the contrary, the weights in the type B weighted Z-tests may depend on the prior observed data; and the type I error rate for the bridging study is usually inflated to a level higher than that of a full-scale study. The choice of the weights provides a simple statistical framework for communication between the regulatory agency and the sponsor. The negotiation process may involve practical constrains and some characteristics of prior studies.