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Confidence Intervals for the Weighted Sum of Two Independent Binomial Proportions

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  • Acknowledgements. The authors are grateful to Alan Agresti and Rob Cannon for helpful advice and constuctive review comments, and to Kathleeen Quan of DAFF Biosecurity for provision of the inspection data. This research was partially supported by ACERA and ARC grant DP110103159.

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Summary

Confidence intervals for the difference of two binomial proportions are well known, however, confidence intervals for the weighted sum of two binomial proportions are less studied. We develop and compare seven methods for constructing confidence intervals for the weighted sum of two independent binomial proportions. The interval estimates are constructed by inverting the Wald test, the score test and the Likelihood ratio test. The weights can be negative, so our results generalize those for the difference between two independent proportions. We provide a numerical study that shows that these confidence intervals based on large-sample approximations perform very well, even when a relatively small amount of data is available. The intervals based on the inversion of the score test showed the best performance. Finally, we show that as for the difference of two binomial proportions, adding four pseudo-outcomes to the Wald interval for the weighted sum of two binomial proportions improves its coverage significantly, and we provide a justification for this correction.

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