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Journal Article

# Small-Sample Confidence Intervals for p1 - p2 and p1/p2 in 2 × 2 Contingency Tables

Thomas J. Santner and Mark K. Snell
Journal of the American Statistical Association
Vol. 75, No. 370 (Jun., 1980), pp. 386-394
DOI: 10.2307/2287464
Stable URL: http://www.jstor.org/stable/2287464
Page Count: 9

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## Abstract

Consider two binomial populations Π1 and Π2 having "success" probabilities p1 in (0, 1) and p2 in (0, 1), respectively. This article studies the problem of constructing small-sample confidence intervals for the difference of the success probabilities, $\Delta \equiv p_1 - p_2$ and their ratio (the "relative risk"), $\rho \equiv p_1/p_2$ based on independent random samples of sizes n1 and n2 from Π1 and Π2, 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 $\psi \equiv p_1(1 - p_2)/p_2(1 - p_1)$. 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 n1 and n2. 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.

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