Approximate confidence intervals for one proportion and difference of two proportions

Constructing a confidence interval for a binomial proportion or the difference of two proportions is a routine exercise in daily data analysis. The best-known method is the Wald interval based on the asymptotic normal approximation to the distribution of the observed sample proportion, though it is known to have a bad performance for small to medium sample sizes. Recently, Agresti and his co-workers proposed an Adding-4 method: 4 pseudo-observations are added with 2 successes and 2 failures and then the resulting (pseudo-)sample proportion is used. The method is simple and performs extremely well. Here we propose an approximate method based on a t-approximation that takes account of the uncertainty in estimating the variance of the observed (pseudo-)sample proportion. It follows the same line of using a t-test, rather than z-test, in testing the mean of a normal distribution with an unknown variance. For some circumstances our proposed method has a higher coverage probability than the Adding-4 method.