the problem of constructing confidence intervals for the sum or difference of two binomial probabilities, based on samples of unequal size. Therefore, we address cases where the focus of attention is $\rho_{12}$ rather than $x_1-x_2$. Differences between lattice centres are permitted by our regularity conditions, but their role is not treated in detail.

Practical motivation {#sec1.3}
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The extra term in (\[eq1.2\]), relative to (\[eq1.1\]), is of significant interest to a practitioner, since it causes significant inaccuracy when the central limit theorem is used to approximate the distribution of $T$. The presence of this extra term motivates the continuity correction, and also the fiducial approach taken by Clopper and Pearson [@Clo34] and Sterne [@Ste54] to estimating a binomial proportion, as well as a large, more recent literature discussing methodology for solving problems such as constructing confidence intervals for the difference or sum of two binomial proportions. See, for example, Hall [@Hal82], Duffy and Santer [@Duf87], Lee et al. [@Lee97], Agresti and Caffo [@Agr00], Brown et al. [@Bro01; @Bro02], Zhou et al. [@Zho01], Price and Bonnett [@Pri04], Brown and Li [@Bro05], Borkowf [@Bor06], Roths and Tebbs [@Rot06], Wang [@Wan10] and Zieliński [@Zie10].

The practical motivation for the work described in this 