Small sample confidence intervals for the difference,ratio and odds ratio of two success probabilities

This paper discusses the problem of constructing small sample confidence intervals for the difference of success probabilities of two independent Bernoulli distributions. An algorithm is given based on an extension of Sterne's (1954) method for constructing small sample confidence intervals for a single success probability. These confidence intervals have several invariance and other desirable properties such as short lengths and monotonicity. A comparison is made with an algorithm due to Santner and Yamagami (1993) which is also based on an extension of Sterne's method. Our algorithm is found to yield shorter intervals for a majority of outcomes, and these outcomes are located in the central portion of the sample space. Santner and Yamagami's algorithm gives shorter intervals for outcomes in the northwest and southeast corners of the sample space (corresponding to large differences in the observed sample proportion of successes), and is computationally faster. Modifications of the algorithm for obtaining confidence intervals for the ratio and odds ratio are indicated.