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Inference for a binomial proportion in the presence of ties

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  • Paul H. Garthwaite
  • John R. Crawford

Abstract

We suppose a case is to be compared with controls on the basis of a test that gives a single discrete score. The score of the case may tie with the scores of one or more controls. However, scores relate to an underlying quantity of interest that is continuous and so an observed score can be treated as the rounded value of an underlying continuous score. This makes it reasonable to break ties. This paper addresses the problem of forming a confidence interval for the proportion of controls that have a lower underlying score than the case. In the absence of ties, this is the standard task of making inferences about a binomial proportion and many methods for forming confidence intervals have been proposed. We give a general procedure to extend these methods to handle ties, under the assumption that ties may be broken at random. Properties of the procedure are given and an example examines its performance when it is used to extend several methods. A real example shows that an estimated confidence interval can be much too small if the uncertainty associated with ties is not taken into account. Software implementing the procedure is freely available.

Suggested Citation

  • Paul H. Garthwaite & John R. Crawford, 2011. "Inference for a binomial proportion in the presence of ties," Journal of Applied Statistics, Taylor & Francis Journals, vol. 38(9), pages 1915-1934, October.
    • Handle: RePEc:taf:japsta:v:38:y:2011:i:9:p:1915-1934
    DOI: 10.1080/02664763.2010.537649
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    References listed on IDEAS

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    1. Nicholas Longford, 2009. "Analysis of all-zero binomial outcomes with borderline and equilibrium priors," Journal of Applied Statistics, Taylor & Francis Journals, vol. 36(11), pages 1259-1265.
    2. Frank Tuyl & Richard Gerlach & Kerrie Mengersen, 2009. "The Rule of Three, its Variants and Extensions," International Statistical Review, International Statistical Institute, vol. 77(2), pages 266-275, August.
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