Small-Sample Confidence Intervals forp1–p2andp1/p2in 2 × 2 Contingency Tables

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Abstract
Consider two binomial populations II1 and II2 having “success” probabilities p 1 in (0, 1) and p 2 in (0, 1), respectively. This article studies the problem of constructing small-sample confidence intervals for the difference of the success probabilities, Δ = p 1 – p 2 and their ratio (the “relative risk”), ρ = p 1/p 2 based on independent random samples of sizes n 1 and n 2 from II1 and II2, respectively. These are nuisance parameter problems; hence the proposed intervals achieve coverage probabilities greater than or equal to their nominal (1 – α) levels. Three methods of constructing intervals are proposed. The first one is based on the well-known conditional intervals for the odds ratio ψ = p 1(1 – p 2)/p 2(1 – p 1). It yields easily computable Δ and ρ intervals. The second method directly generates unconditional intervals of the desired size. An algorithm is given for producing the intervals for arbitrary n 1 and n 2. The 2 × 2 case is given as an illustrative example. The third method constructs unconditional intervals based on a generalization of the classical (Fisher) tail method. Some comparisons are made.