Cost-efficient fixed-width confidence intervals for the difference of two Bernoulli proportions

We study the properties of confidence intervals (CIs) for the difference of two Bernoulli distributions’ success parameters, ${p_x} - {p_y}$px−py, in the case where the goal is to obtain a CI of a given half-width while minimising sampling costs when the observation costs may be different between the two distributions. We propose three different methods for constructing fixed-width CIs: (i) a two-stage sampling procedure, (ii) a sequential method that carries out sampling in batches, and (iii) an $\ell $ℓ-stage “look-ahead” procedure. Under diverse scenarios, our proposed algorithms obtain significant cost savings versus their baseline counterparts. Furthermore, for the scenarios under study, our sequential-batches and $\ell $ℓ-stage “look-ahead” procedures approximately obtain the nominal coverage while meeting the desired width requirement. Our sequential-batching method is more efficient than the “look-ahead” method computationally, with average running times an order-of-magnitude faster over the scenarios tested. We illustrate our procedures on a case study comparing generic and brand-name drugs.