The weighted mid-P confidence interval for the difference of independent binomial proportions

Abstract

The binomial distribution belongs to the class of standard distributions in Quantitative linguistics, see e.g. Kohler (1995), Altmann (1984), Altmann (1988), Altmann (1991), Uhliřova (1995a), Uhliřova (1995b), Schmidt (1996), Best (1997) and Wimmer & Altmann (1999). The proper estimation of a single binomial proportion and the estimation of the difference of two binomial proportions belong to the basic tasks in modelling dichotomic situations by using binomial models. The interval estimators for the difference of two independent binomial proportions, say δ = pt − pc, are required also in other areas of scientific research, e.g. in clinical trials, where the typical task is to compare a new treatment with a standard treatment (control). The recent statistical literature offers several exact and approximate methods to construct such interval estimators, see e.g. Newcombe (1998a), Newcombe (1998b) and Chan & Zhang (1999) for numerical comparison of several selected exact and approximate methods. In this paper we propose a new class of alternative interval estimators for the difference of two independent binomial proportions. In detail, we say that the interval estimator is exact (in a strong sense) if its minimum coverage probability (CP) is equal or greater to the prespecified nominal level 1−α, i.e. minCP(δ) ≥ 1−α, where δ = pt − pc, and such that −1 ≤ δ ≤ 1. We propose a class of weighted mid-P interval estimators parameterized by the parameter κ, 0 ≤ κ ≤ 1. If κ = 0, the mid-P interval estimator is an approximate estimator, however, exact in weak sense: i.e. such that the average coverage probability CP = ∫ CP(δ)d f (δ) ≥ 1−α for some smooth distribution f (δ) on the parameter space 〈−1,1〉. In this case, according to Newcombe (1998a), the CP should be close to 1−α, ideally a little over 1−α, with minCP(δ) a little under 1−α. If κ = 1, the minimum coverage