CONFIDENCE-LIMITS FOR SMALL PROPORTIONS IN COMPLEX SAMPLES

Exact upper confidence limits for small proportions in stratified samples are derived. An algorithm for their computation which employs a new normal approximation for the case of infinitely large strata and a finite number of defectives is proposed. Using selected examples it is shown that the usual confidence limits derived from the standard normal approximation can be highly misleading, and that the exact limits are not unacceptably conservative when compared to natural Empirical Bayes and appropriately defined pseudo-Bayes limits. The loss of efficiency of non-proportionate staratified designs, vis-a-vis simple random sampling or proportionate designs for setting confidence limits on small proportions is studied in a variety of examples. Exact upper confidence limits for small proportions are also derived for simple random samples of equal-size clusters, and a similar algorithm for their derivation is presented. These limits are compared to several Bayes credibility limits in selected examples. The loss in efficiency due to clustering is studied in a number of cases in terms of probability of coverage of false values.