Likelihood-weighted confidence intervals for the difference of two binomial proportions

Two new methods for computing confidence intervals for the difference delta = p(1) - p(2) between two binomial proportions (p(1), p(2)) are proposed. Both the Mid-P and Max-P likelihood weighted intervals are constructed by mapping the tail probabilities from the two-dimensional (p(1): p(2))-space into a one-dimensional function of delta based on the likelihood weights. This procedure may be regarded as a natural extension of the CLOPPER-PEARSON (1934) interval to the two-sample case where the weighted tail probability is alpha/2 at each end on the delta scale. The probability computation is based on the exact distribution rather than a large sample approximation. Extensive computation was carried out to evaluate the coverage probability and expected width of the likelihood-weighted intervals, and of several other methods. The likelihood-weighted intervals compare very favorably with the standard asymptotic interval and with intervals proposed by HAUCK and ANDERSON (1986), Cox and SNELL (1989), SANTNER and SNELL (1980), SANTNER and YAMAGAMI (1993), and PESKUN (1993). In particular, the Mid-P likelihood-weighted interval provides a good balance between accurate coverage probability and shea interval width in both small and large samples. The Mid-P interval is also comparable to COE and TAMHANE'S (1993) interval, which has the best performance in small samples.