In Defence of Score Intervals for Proportions and their Differences

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Abstract
A recent article (Santner et al., 2007 Santner , T. J. , Pradhan , V. , Senchaudhuri , P. , Mehta , C. R. , Tamhane , A. ( 2007 ). Small-sample comparisons of confidence intervals for the difference of two independent binomial proportions . Computat. Statist. Data Anal. 51 : 5791 – 5799 . [Crossref], [Web of Science ®] [Google Scholar] ) asserted that a score interval for a difference of independent binomial proportions (Miettinen and Nurminen, 1985 Miettinen , O. , Nurminen , M. ( 1985 ). Comparative analysis of two rates . Statist. Med. 4 : 213 – 226 . [Crossref], [PubMed], [Web of Science ®] [Google Scholar] ) may have inadequate coverage. We re-visit the properties of score intervals for binomial proportions and their differences. Published data indicate these methods produce mean coverage slightly above the nominal confidence level 1 − α. We argue it is appropriate to align mean rather than minimum coverage with 1 − α, based on a moving average representation of the coverage probability. The poor coverage properties claimed by Santner et al. ( 2007 Santner , T. J. , Pradhan , V. , Senchaudhuri , P. , Mehta , C. R. , Tamhane , A. ( 2007 ). Small-sample comparisons of confidence intervals for the difference of two independent binomial proportions . Computat. Statist. Data Anal. 51 : 5791 – 5799 . [Crossref], [Web of Science ®] [Google Scholar] ) actually relate to an inferior version of the score interval (Mee, 1984 Mee , R. W. ( 1984 ). Confidence bounds for the difference between two probabilities . Biometrics 40 : 1175 – 1176 . [Web of Science ®] [Google Scholar] ).