Volume 8 • Issue 2 • 1000338 J Biom Biostat, an open access journal ISSN: 2155-6180 Research Article McCracken and Looney, J Biom Biostat 2017, 8:2 DOI: 10.4172/2155-6180.1000338 Research Article Open Access Journal of Biometrics & Biostatistics J o u r n a l o f B i o m e t r i c s & B i o s t a t i s t i c s ISSN: 2155-6180 Keywords: p-Confidence; Exact methods; Interval length; One-sided interval; Sample size Introduction is study was motivated when one of the authors was approached by a clinical investigator in the dental school who was seeking advice on how to design a research study consisting of a sequence of independent trials, each with only two possible outcomes (“success” and “failure”). Based on a previously published study he had conducted [1], as well as his clinical judgment and experience, the investigator had strong reason to believe that the observed number of “failures” in the study he was planning would likely be zero. His question for us was, “Assuming that there will be no failures in my study, how many trials do I need to conduct so that I can be reasonably sure that the true probability of failure is no greater than .05”? Since the trials can be assumed to be independent, it is reasonable to assume that X=the number of “failures” in the clinical study follows a binomial distribution, with n=the number of independent trials and π=the probability of “failure” on any one trial. Many research studies in clinical areas and other applied fields meet the assumptions of the binomial distribution (Without loss of generality, we will refer to the two outcomes as “success” and “failure” and the outcome of primary interest as “success”). Furthermore, it is not uncommon in these studies for x, the observed number of successes in a sample of size n, to be zero. Examples can be found [2-6]. Only a few publications in the statistical literature have examined or compared methods for analyzing binomial data in which the observed number of successes is zero; for example, see [7-12]. Observing X=0 in a binomial sample lends itself to the Bayesian approach since one can condition on the observed data, and several authors have considered this approach [13-16]. In the present article, we approach the problem from a frequentist point of view; however, we also include a Bayesian approach for comparison purposes. We restrict our attention to the situation in which one is interested only in finding the one-sided (upper) confidence limit for the true value of the binomial proportion when the number of successes has already been observed to be zero. e use of one-sided confidence limits in this situation is controversial [8,10], but some authors have recommended their use [17,18]. Following our discussions with the clinical investigator, we decided that a one-sided upper confidence limit would be appropriate. Our review of the relevant literature indicated that there was no clear consensus on the best upper confidence limit to use when the observed number of successes is zero, especially if one wished to approach the problem from a frequentist perspective. Hence, in order to be able to provide a well-informed response to the dental researcher's question, we concluded that it would be worthwhile to systematically compare the most widely-used binomial confidence interval (C.I.) methods under the assumption that x=0. In Section 2, we describe each of the methods for finding confidence limits for a binomial proportion that we compare; in Section 3, we describe our methodology for comparing the performance of these methods; in Section 4, we provide a summary of the results of our comparisons; and in Section 5, we discuss our results and make recommendations concerning the best methods to use in practice. Methods In this section, we describe the eight methods we compared for finding π u , the upper 100 (1-α)% confidence limit for the true probability of success for a binomial random variable, under the assumption that the number of successes has already been observed to be 0. We selected these methods either because (1) they are commonly covered in introductory statistics or biostatistics textbooks, or (2) they have been recommended for general use when finding confidence limits for a binomial proportion. One method we did not include is the Wald interval, which continues to be one of the most widely used methods for finding confidence limits for a binomial proportion. e Wald interval is based on the normal approximation to the binomial, and the approximate *Corresponding author: Looney SW, Department of Biostatistics and Epidemiology, Augusta University, Medical College of Georgia, Augusta, GA 30912, USA, Tel: (706) 721-4846; E-mail: slooney@augusta.edu Received March 10, 2017; Accepted March 27, 2017; Published March 30, 2017 Citation: McCracken CE, Looney SW (2017) On Finding the Upper Confidence Limit for a Binomial Proportion when Zero Successes are Observed. J Biom Biostat 8: 338. doi:10.4172/2155-6180.1000338 Copyright: © 2017 McCracken CE, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. On Finding the Upper Confidence Limit for a Binomial Proportion when Zero Successes are Observed Courtney E McCracken 1 and Stephen W Looney 2 * 1 Department of Pediatrics, Emory University, Atlanta, USA 2 Department of Biostatistics and Epidemiology, Augusta University, Medical College of Georgia, USA Abstract We consider confidence interval estimation for a binomial proportion when the data have already been observed and x, the observed number of successes in a sample of size n, is zero. In this case, the main objective of the investigator is usually to obtain a reasonable upper bound for the true probability of success, i.e., the upper limit of a one-sided confidence interval. In this article, we use observed interval length and p-confidence to evaluate eight methods for finding the upper limit of a confidence interval for a binomial proportion when x is known to be zero. Long-run properties such as expected interval length and coverage probability are not applicable because the sample data have already been observed. We show that many popular approximate methods that are known to have good long-run properties in the general setting perform poorly when x=0 and recommend that the Clopper-Pearson exact method be used instead.