Inference about the ratio of means from Negative Binomial paired count data

We derive some statistical properties of the distribution of two Negative Binomial random variables conditional on their total. This type of model can be appropriate for paired count data with Poisson over-dispersion such that the variance is a quadratic function of the mean. This statistical model...

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Bibliographic Details
Published in:Environmental and ecological statistics 2012-06, Vol.19 (2), p.269-293
Main Authors: Cadigan, N. G, Bataineh, O. M
Format: Article
Language:English
Subjects:
Bias
Binomial distribution
Biomedical and Life Sciences
Chemistry and Earth Sciences
Computer Science
confidence interval
Ecology
gears
Health Sciences
Hypothesis testing
Life Sciences
Math. Appl. in Environmental Science
Maximum likelihood method
Medicine
Physics
Random variables
statistical models
Statistics
Statistics for Engineering
Statistics for Life Sciences
Studies
Theoretical Ecology/Statistics
uncertainty
variance
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Summary:We derive some statistical properties of the distribution of two Negative Binomial random variables conditional on their total. This type of model can be appropriate for paired count data with Poisson over-dispersion such that the variance is a quadratic function of the mean. This statistical model is appropriate in many ecological applications including comparative fishing studies of two vessels and or gears. The parameter of interest is the ratio of pair means. We show that the conditional means and variances are different from the more commonly used Binomial model with variance adjusted for over-dispersion, or the Beta-Binomial model. The conditional Negative Binomial model is complicated because it does not eliminate nuisance parameters like in the Poisson case. Maximum likelihood estimation with the unconditional Negative Binomial model can result in biased estimates of the over-dispersion parameter and poor confidence intervals for the ratio of means when there are many nuisance parameters. We propose three approaches to deal with nuisance parameters in the conditional Negative Binomial model. We also study a random effects Binomial model for this type of data, and we develop an adjustment to the full-sample Negative Binomial profile likelihood to reduce the bias caused by nuisance parameters. We use simulations with these methods to examine bias, precision, and accuracy of estimators and confidence intervals. We conclude that the maximum likelihood method based on the full-sample Negative Binomial adjusted profile likelihood produces the best statistical inferences for the ratio of means when paired counts have Negative Binomial distributions. However, when there is uncertainty about the type of Poisson over-dispersion then a Binomial random effects model is a good choice.
ISSN:1352-8505
1573-3009
DOI:10.1007/s10651-011-0186-8