Optimal allocations in sequential tests involving two populations with covariates

We consider the problem of testing which of two normally distributed treatments has the largest mean, when the tested populations incorporate a covariate. From the class of procedures using the invariant sequential probability ratio test we derive an optimal allocation that minimizes, in a continuous time setting, the expected sampling costs. Simulations show that this procedure reduces the number of observations from the costlier treatment and categories while maintaining an overall sample size closer to the “pairwise” procedure. A randomized trial example is given.