Comparison of 'tail method' exact confidence limits for the difference of binomial probabilities

摘要

Suppose that Y (1) and Y (2) are independent and have Binomial(n (1),p (1)) and Binomial (n (2),p (2)) distributions respectively. Also suppose that theta=p (1)-p (2) is the parameter of interest. We consider the problem of finding an exact confidence limit (either upper or lower) for theta. The solution to this problem is very important for statistical practice in the health and life sciences. The 'tail method' provides a solution to this problem. This method finds the exact confidence limit by exact inversion of a hypothesis test based on a specified test statistic. Buehler (J. Am. Stat. Assoc. 52, 482-493, 1957) described, for the first time, a finite-sample optimality property of this confidence limit. Consequently, this confidence limit is sometimes called a Buehler confidence limit. An early tail method confidence limit for theta was described by Santner and Snell (J. Am. Stat. Assoc. 75, 386-394, 1980) who used the maximum likelihood estimator of theta as the test statistic. This confidence limit is known to be very inefficient (see e.g. Cytel Software, StatXact, version 6, vol. 2, 2004). The efficiency of the confidence limit resulting from the tail method depends greatly on the test statistic on which it is based. We use the results of Kabaila (Stat. Probab. Lett. 52, 145-154, 2001) and Kabaila and Lloyd (Aust. New Zealand J. Stat. 46, 463-469, 2004, J. Stat. Plan. Inference 136, 3145-3155, 2006) to provide a detailed explanation for the dependence of this efficiency on the test statistic. We consider test statistics that are estimators, Z-statistics and approximate upper confidence limits. This explanation is used to find the situations in which the tail method exact confidence limits based on test statistics that are estimators or Z-statistics are least efficient.