The problem of ordering outcomes in deriving exact confidence intervals
by
Graham Hepworth
The University of Melbourne

To derive an exact confidence interval for a proportion, the outcomes must be ordered in some way so that tail probabilities can be calculated. In most situations the ordering is unambiguous. However, when the event space is multidimensional, there may be different ways of ordering the outcomes, leading to different confidence intervals. One situation where this can occur is “group testing”, in which individual samples are pooled together and tested as a group for the presence of an attribute. If the groups have different sizes, the event space is multidimensional.

Three methods of ordering outcomes are considered in deriving exact confidence intervals for proportions estimated by group testing. Outcomes are ordered according to either (i) probability, (ii) MLE, or (iii) likelihood relative to the maximum. Method (i) turns out to be unacceptable because of its irregularities. Method (ii) is the smoothest and most flexible, but method (iii) has coverage closest to the nominal level, and that’s what matters most.

Date received: August 29, 2001