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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.

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In this article, some simple methods for testing and estimating the parameters of some discrete distributions are proposed. For hypothesis testing, a new test is obtained by combining the usual exact test and an alternative exact test. The exact properties of the usual exact test, the alternative exact test and the combined test are evaluated numerically for the binomial and Poisson distributions. Numerical studies show that the combined test is more powerful than the usual one while controlling the sizes satisfactorily. Furthermore, the combined procedure produces confidence intervals that are practically equivalent to the intervals based on some other complex methods. The methods are also illustrated for the hypergeometric and negative binomial distributions.

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

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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:

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

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

For many years, sampling theory has been based on Shannon and Nyquist who stated that band-limited signals can be exactly reconstructed using samples acquired at or higher than Nyquist rate (1949). Recently however, the focus has shifted to compressed sensing, where if the underlying signal is sparse, the signal can be represented by a small collection of linear projections. It is now understood that many