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Background. The negative binomial distribution is used commonly throughout biology as a model for overdispersed count data, with attention focused on the negative binomial dispersion parameter, k. A substantial literature exists on the estimation of k, but most attention has focused on datasets that are not highly overdispersed (i.e., those with k$1), and the accuracy of confidence intervals estimated for k is typically not explored. Methodology. This article presents a simulation study exploring the bias, precision, and confidence interval coverage of maximum-likelihood estimates of k from highly overdispersed distributions. In addition to exploring small-sample bias on negative binomial estimates, the study addresses estimation from datasets influenced by two types of event under-counting, and from disease transmission data subject to selection bias for successful outbreaks. Conclusions. Results show that maximum likelihood estimates of k can be biased upward by small sample size or under-reporting of zero-class events, but are not biased downward by any of the factors considered. Confidence intervals estimated from the asymptotic sampling variance tend to exhibit coverage below the nominal level, with overestimates of k comprising the great majority of coverage errors. Estimation from outbreak datasets does not increase the bias of k estimates, but can add significant upward bias to estimates of the mean. Because k varies inversely with the degree of overdispersion, these findings show that overestimation of the degree of overdispersion is very rare for these datasets.

Description Calculates Fisher’s exact test, Blaker’s exact test, or the exact McNemar’s test with

Here is a quick example of the function rateratio.test. Suppose you have two rates that you assume are Poisson and you want to test that they are different. Suppose you observe 2 events with time at risk of n = 17877 in one group and 9 events with time at risk of m = 16660 in another group. Here is the test:> rateratio.test(c(2, 9), c(n, m)) Exact Rate Ratio Test, assuming Poisson counts data: c(2, 9) with time of c(n, m), null rate ratio 1 p-value = 0.05011 alternative hypothesis: true rate ratio is not equal to 1 95 percent confidence interval:

Here is a problem which motivates this package. Consider comparing the difference between two exact Poisson rates. Suppose the observed rates for the two groups are 2/17877 and 10/20000. Since the counts are low, an exact test would be appropriate. In the stats package (R version 2.11.0 (2010-04-22)), we could perform an exact test of the difference in Poisson rates by:

When we perform the two-sided Fisher’s exact test on this table we get> ft <- fisher.test(x)> ft Fisher's Exact Test for Count Data data: x p-value = 0.04371 alternative hypothesis: true odds ratio is not equal to 1 95 percent confidence interval: 0.03888003 1.05649145 sample estimates: odds ratio 0.2189021 The two-sided p-value is less than 0.05 but the 95 percent confidence interval on the odds ratio contains 1. What we want is a matching confidence interval that goes with the test, but what fisher.test outputs is a confidence interval that matches a different test. The confidence interval that fisher.test outputs matches with a two-sided Fisher’s exact test whose p-value is twice the minimum one-sided p-value. We call that test the central Fisher’s exact test, since the matching confidence interval is a central confidence interval (i.e., there is a maximum of α/2 probability that the true odds ratio is lower than the lower limit and analogously for the upper limit). The function exact2x2 gives the proper matching interval. Here are the two types of two-sided Fisher’s exact test. We use the option ”minlike ” for the usual two-sided Fisher’s exact test since it is based on the priciple of minimum likelihood.> exact2x2(x, tsmethod = "minlike") 1 Two-sided Fisher's Exact Test (usual method using minimum likelihood) data: x p-value = 0.04371 alternative hypothesis: true odds ratio is not equal to 1 95 percent confidence interval: 0.0435 0.9170 sample estimates: odds ratio

R topics documented: BlakerCI-package...................................... 1 binom.blaker.acc...................................... 3 binom.blaker.limits..................................... 5 BlakerCI-internal...................................... 7

Confidence intervals for a binomial parameter or for the ratio of Poisson means are commonly desired in high energy physics (HEP) applications such as measuring the detection efficiency of a device or algorithm, or measuring branching ratios. Due to the discreteness of the observed data, in both of these closely related problems the frequentist coverage probability unfortunately depends on the unknown parameter. Trade-offs among desiderata for sets of intervals have thus led to numerous sets of intervals in the statistics literature, while in the HEP literature one typically encounters only the classic intervals of Clopper-Pearson (central intervals with strictly no undercoverage but substantial over-coverage) or a few approximate methods which are known in the statistics literature to perform poorly. When strict coverage is relaxed, some sort of averaging is needed to compare intervals. In most of the literature, this averaging is over different values of the unknown parameter, which is conceptually problematic from the frequentist point of view in which the unknown parameter is typically fixed. In contrast, we perform an (unconditional) average over observed data in the ratio-of-Poisson-means problem. We are led to two recommendations (Clopper-Pearson intervals and intervals from inverting the likelihood ratio test) if strict coverage is desired (for central and non-central intervals, respectively), with Lancaster’s mid-P modification to either if only average coverage is desired. Key words: hypothesis test, confidence interval, binomial parameter, ratio of

Textbooks for introductory statistics courses use a sample size of 30 as a lower bound for large-sample inference about the mean of a quantitative variable. For binary data, there is no consensus bound, and rarely is the student told how to handle small samples. We suggest a guideline for interval estimation that ties in with the rule of 30 for quantitative variables and also gives direction for smaller samples: Form the usual confidence interval if at least 15 outcomes of each type occur, and otherwise use that interval after adding two successes and two failures.

Description Calculates exact tests and confidence intervals for one-sample binomial and one- or two-sample Poisson cases.