Exact Inference for Proportions From a Stratified Finite Population

Abstract

Abstract Auditors and others often encounter finite populations with a dichotomous characteristic from which they draw stratified samples. In auditing the dichotomy arises when a population item is classified as either in error or in compliance with some rule or regulation. Usually the proportion of errors is small. The auditing objective may require calculation of a p value for the sample outcome relative to a hypothesis, or a confidence bound for the proportion or total number of errors in the population. In sampling from L strata with hypotheses concerning the total number of errors in the population, the calculation of p values is not straightforward. The complication arises because the parameter of the null hypothesis does not completely specify the distribution of the test statistic. This distribution depends on an (L − 1)-dimensional nuisance parameter consisting of the number of errors in each stratum. Because confidence bounds can be obtained by inverting the hypothesis test, the same difficulty applies to calculating confidence bounds. This article tests H 1 using the maximum p value over the feasible set of nuisance parameters. It describes a fairly efficient search method for finding a global maximum p value. Confidence bounds are calculated by inverting the hypothesis test. The article also presents an heuristic expression for determining good starting values in the search for confidence bounds. The procedures are implemented on a standard statistical package and are available from StatLib. They seem to perform reasonably well with samples from a moderate number of strata with a small number of errors.