Closed Sequential and Multistage Inference on Binary Responses With or Without Replacement

We consider closed sequential or multistage sampling, with or without replacement, from a lot of <italic>N</italic> items, where each item can be identified as defective (in error, tainted, etc.) or not. The goal is to make inference on the proportion, π, of defectives in the lot, or equivalently on the number of defectives in the lot <italic>D</italic> = <italic>N</italic>π. It is shown that exact inference on π using closed (bounded) sequential or multistage procedures with general prespecified elimination boundaries is completely tractable and not at all inconvenient using modern statistical software. We give relevant theory and demonstrate functions for this purpose written in R (R Development Core Team 2005, available as online supplementary material). Applicability of the methodology is illustrated in three examples: (1) sharpening of Wald's (1947) sequential probability ratio test used in industrial acceptance sampling, (2) two-stage sampling for auditing Medicare or Medicaid health care providers, and (3) risk-limited sequential procedures for election audits.