Testing for a treatment-by-stratum interaction in a sequential clinical trial

We consider a two-arm sequential trial with two or more strata. The
trial is monitored and terminated under the assumption of a common
treatment effect (if any) in all strata. A secondary question at the end of
the trial is: Does the treatment effect differ across strata—that is, is there
a treatment X stratum interaction? We provide a test of the null hypothesis
of no interaction—a test that recognizes the sequential stopping rule and
allows for uneven accumulation of information in the various strata, a
common case. In the case of two strata, and either a group-sequential
design or a fully-sequential linear-boundary design, an optimal property
of the test is derived. A computational algorithm is provided, and two
examples summarized.