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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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Published in: | Environmental and ecological statistics 2012-06, Vol.19 (2), p.269-293 |
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description | 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. |
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G ; Bataineh, O. M</creator><creatorcontrib>Cadigan, N. G ; Bataineh, O. M</creatorcontrib><description>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. 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G</creatorcontrib><creatorcontrib>Bataineh, O. M</creatorcontrib><title>Inference about the ratio of means from Negative Binomial paired count data</title><title>Environmental and ecological statistics</title><addtitle>Environ Ecol Stat</addtitle><description>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.</description><subject>Bias</subject><subject>Binomial distribution</subject><subject>Biomedical and Life Sciences</subject><subject>Chemistry and Earth Sciences</subject><subject>Computer Science</subject><subject>confidence interval</subject><subject>Ecology</subject><subject>gears</subject><subject>Health Sciences</subject><subject>Hypothesis testing</subject><subject>Life Sciences</subject><subject>Math. Appl. in Environmental Science</subject><subject>Maximum likelihood method</subject><subject>Medicine</subject><subject>Physics</subject><subject>Random variables</subject><subject>statistical models</subject><subject>Statistics</subject><subject>Statistics for Engineering</subject><subject>Statistics for Life Sciences</subject><subject>Studies</subject><subject>Theoretical Ecology/Statistics</subject><subject>uncertainty</subject><subject>variance</subject><issn>1352-8505</issn><issn>1573-3009</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2012</creationdate><recordtype>article</recordtype><sourceid>M0C</sourceid><recordid>eNp1kEtPwzAQhC0EEuXxAzhhiXNgbcdxcoSKR0UFh8LZ2jp2SdXExU6Q-Pc4CgcuHEa7Ws3MSh8hFwyuGYC6iQwKyTJgo8oiKw_IjEklMgFQHaZdSJ6VEuQxOYlxCwA543JGnheds8F2xlJc-6Gn_YelAfvGU-9oa7GL1AXf0he7SdcvS--azrcN7ugem2BravzQ9bTGHs_IkcNdtOe_85SsHu7f5k_Z8vVxMb9dZkbk0GfoeLEWVojK1JXLDbM5KlWUSrFKOKcUSpODQ6xqBdIUskg5rrhEwaEUp-Rqat0H_znY2OutH0KXHmoGnPG8FIVILja5TPAxBuv0PjQthu9k0iMxPRHTiZgeiemxmU-ZmLzdxoa_zf-HLqeQQ69xE5qo31ccWA5JAioQP6F3ddI</recordid><startdate>20120601</startdate><enddate>20120601</enddate><creator>Cadigan, N. 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G</au><au>Bataineh, O. M</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Inference about the ratio of means from Negative Binomial paired count data</atitle><jtitle>Environmental and ecological statistics</jtitle><stitle>Environ Ecol Stat</stitle><date>2012-06-01</date><risdate>2012</risdate><volume>19</volume><issue>2</issue><spage>269</spage><epage>293</epage><pages>269-293</pages><issn>1352-8505</issn><eissn>1573-3009</eissn><notes>http://dx.doi.org/10.1007/s10651-011-0186-8</notes><abstract>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.</abstract><cop>Boston</cop><pub>Springer-Verlag</pub><doi>10.1007/s10651-011-0186-8</doi><tpages>25</tpages></addata></record> |
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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 |
title | Inference about the ratio of means from Negative Binomial paired count data |
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