Confidence Intervals for a Proportion Estimated From Pooled Samples Based on Firth's Corrected Score

Interval estimation of a proportion estimated from testing of pooled samples has been studied by a variety of researchers, who have examined both exact and asymptotic methods. Recent work in point estimation has seen the use of Firth's correction to maximum likelihood estimation to reduce bias effectively. We consider the Firth-corrected score statistic, using either expected or observed information, as the first derivative function of a penalized likelihood and develop score-based confidence intervals wholly within this framework. We evaluate their performance, comparing them by information type and to the existing, recommended asymptotic method of Gart. The methods are illustrated using data on West Nile virus infection in field-collected mosquitoes and on yellow fever virus infection in mosquitoes reported in Walter et al. The penalized likelihood approach using expected information is found to unify previously recommended point and interval estimates under one inferential framework. Detailed evaluation of coverage properties of the intervals over a range of pooling designs show improved performance for the intervals using observed information and skewness-correction to the score. We derive simple, interpretable expressions for the score and skewness-corrected score statistics, and give a quadratic formula for the skewness-corrected score interval for the case that all pools are the same size, which reduces to a skewness-correction for the standard Wilson interval in the standard binomial case.