A Comparison of Two Test Statistics for Poisson Overdispersion/Underdispersion

Affiliation(s)

Department of Biostatistics and Computational Biology, University of Rochester, 601 Elmwood Ave., Box 630, Rochester, NY 14642.

Department of Biostatistics & Bioinformatics, Rollins School of Public Health Winship Cancer Institute, Emory University 1365-B Clifton Rd. NE, Suite 4100, Room B4109, Atlanta, GA 30322.

Department of Biostatistics and Computational Biology, University of Rochester, 601 Elmwood Ave., Box 630, Rochester, NY 14642.

Department of Biostatistics & Bioinformatics, Rollins School of Public Health Winship Cancer Institute, Emory University 1365-B Clifton Rd. NE, Suite 4100, Room B4109, Atlanta, GA 30322.

ABSTRACT

Within the family of zero-inflated Poisson distributions, the data has Poisson distribution if any only if the mean equals the variance. In this paper we compare two closely related test statistics constructed based on this idea. Our results show that although these two tests are asymptotically equivalent under the null hypothesis and are equally efficient, one test is always more efficient than the other one for small and medium sample sizes.

Within the family of zero-inflated Poisson distributions, the data has Poisson distribution if any only if the mean equals the variance. In this paper we compare two closely related test statistics constructed based on this idea. Our results show that although these two tests are asymptotically equivalent under the null hypothesis and are equally efficient, one test is always more efficient than the other one for small and medium sample sizes.

Cite this paper

H. Wang, C. Feng, X. Tu and J. Kowalski, "A Comparison of Two Test Statistics for Poisson Overdispersion/Underdispersion,"*Applied Mathematics*, Vol. 3 No. 7, 2012, pp. 795-799. doi: 10.4236/am.2012.37118.

H. Wang, C. Feng, X. Tu and J. Kowalski, "A Comparison of Two Test Statistics for Poisson Overdispersion/Underdispersion,"

References

[1] L. D. Brown and L. H. Zhao, “A Test for the Poisson Distribution,” Sankhyā, Vol. 64, pp. 611-625

[2] D. Lambert, “Zero-In?ated Poisson Regression Models with an Application to Defects in Manufacturing,” Technometrics, Vol. 34, No. 1, 1992, pp. 1-14. doi:10.2307/1269547

[3] R. A. Fisher, “The Negative Binomial Distribution,” Annals of Human Genetics Vol. 11, No. 1, 1941, pp. 182187. doi:10.1111/j.1469-1809.1941.tb02284.x

[4] G. J. S. Ross and D. A. Preece, “The Negative Binomial Distribution,” The Statistician, Vol. 34, No. 3, 1985, pp. 323-336. doi:10.2307/2987659

[5] S. M. DeSantis and D. Bandyopadhyay, “Hidden Markov Models for Zero-In?ated Poisson Counts with an Application to Substance Use,” Statistics in Medicine, Vol. 30, 2011, pp. 1678-1694. doi:10.1002/sim.4207

[6] A. El-Shaarawi, “Some Goodness-of-Fit Methods for the Poisson Plus Added Zeros Distribution,” Applied and Environmental Microbiology, Vol. 49, 1985, pp. 1304-1306.

[7] Y. Xia, D. Morrison-Beedy, J. Ma, C. Feng, W. Cross and X. M. Tu, “Modeling Count Outcomes from HIV Risk Reduction Interventions: A Comparison of Competing Statistical Models for Count Responses,” AIDS Research and Treatment, Vol. 2012, 2012, Article ID 593569. doi:10.1155/2012/593569

[8] T. Loeys, B. Moerkerke, O. De Smet, et al., “The Analysis of Zero-In?ated Count Data: Beyond Zero-In?ated Poisson Regression,” British Journal of Mathematical and Statistical Psychology, Vol. 65, No. 1, 2012, pp. 163-180. doi:10.1111/j.2044-8317.2011.02031.x

[9] A. Khan and M. Western, “Does Attitude Matter in Computer Use in Australian General Practice? A Zero-In?ated Poisson Regression Analysis,” Health Information Management Journal, Vol. 40, 2011, pp. 23-29.

[10] B. T. Pahel, J. S. Preisser, S. C. Stearns, et al., “Multiple Imputation of Dental Caries Data Using a Zero-In?ated Poisson Regression Model,” Journal of Public Health Dentistry, Vol. 71, No. 1, 2011, pp. 71-78. doi:10.1111/j.1752-7325.2010.00197.x

[11] S. R. Hu, C. S. Li and C. K. Lee, “Assessing Casualty Risk of Railroad-Grade Crossing Crashes Using ZeroIn?ated Poisson Models,” Journal of Transportation Engineering-ASCE, Vol. 137, No. 8, 2011, pp. 527-536. doi:10.1061/(ASCE)TE.1943-5436.0000243

[12] D. B?hning, “A Note on a Test for Poisson Overdispersion,” Biometrika, Vol. 81, No. 2, 1994, pp. 418-419. doi:10.1093/biomet/81.2.418

[13] W. Cochran, “Some Methods of Strengthening χ2 Tests,” Biometrics, Vol. 10, No. 4, 1954, pp. 417-451. doi:10.2307/3001616

[14] C. Feng, H. Wang and X. M. Tu, “The Asymptotic Distribution of a Likelihood Ratio Test for the Poisson Distribution,” Sankhyā, 2012 (in press)

[15] C. S. Li, “A Lack-of-?t Test for Parametric Zero-In?ated Poisson Models,” Journal of Statistical Computation and Simulation, Vol. 81, No. 9, 2011, pp. 1081-1098. doi:10.1080/00949651003677410

[16] C. S. Li, “Testing the Lack-of-Fit of Zero-In?ated Poisson Regression Models,” Communication in StatisticsSimulation and Computation, Vol. 40, No. 4, 2011, pp. 497-510. doi:10.1080/03610918.2010.546541

[17] C. Rao and I. Chakravarti, “Some Small Sample Tests of Signi?cance for a Poisson Distribution,” Biometrics, Vol. 12, No. 3, 1956, pp. 264-282. doi:10.2307/3001466

[18] J. Van den Broeck, “A Score Test for Zero In?ation in a Poisson Distribution,” Biometrics, Vol. 51, 1995, pp. 738-743.

[19] O. Thas and J. C. W. Rayner, “Smooth Tests for the ZeroIn?ated Poisson Distribution,” Biometrics, Vol. 61, 2005, pp. 808-815. doi:10.1111/j.1541-0420.2005.00351.x

[1] L. D. Brown and L. H. Zhao, “A Test for the Poisson Distribution,” Sankhyā, Vol. 64, pp. 611-625

[2] D. Lambert, “Zero-In?ated Poisson Regression Models with an Application to Defects in Manufacturing,” Technometrics, Vol. 34, No. 1, 1992, pp. 1-14. doi:10.2307/1269547

[3] R. A. Fisher, “The Negative Binomial Distribution,” Annals of Human Genetics Vol. 11, No. 1, 1941, pp. 182187. doi:10.1111/j.1469-1809.1941.tb02284.x

[4] G. J. S. Ross and D. A. Preece, “The Negative Binomial Distribution,” The Statistician, Vol. 34, No. 3, 1985, pp. 323-336. doi:10.2307/2987659

[5] S. M. DeSantis and D. Bandyopadhyay, “Hidden Markov Models for Zero-In?ated Poisson Counts with an Application to Substance Use,” Statistics in Medicine, Vol. 30, 2011, pp. 1678-1694. doi:10.1002/sim.4207

[6] A. El-Shaarawi, “Some Goodness-of-Fit Methods for the Poisson Plus Added Zeros Distribution,” Applied and Environmental Microbiology, Vol. 49, 1985, pp. 1304-1306.

[7] Y. Xia, D. Morrison-Beedy, J. Ma, C. Feng, W. Cross and X. M. Tu, “Modeling Count Outcomes from HIV Risk Reduction Interventions: A Comparison of Competing Statistical Models for Count Responses,” AIDS Research and Treatment, Vol. 2012, 2012, Article ID 593569. doi:10.1155/2012/593569

[8] T. Loeys, B. Moerkerke, O. De Smet, et al., “The Analysis of Zero-In?ated Count Data: Beyond Zero-In?ated Poisson Regression,” British Journal of Mathematical and Statistical Psychology, Vol. 65, No. 1, 2012, pp. 163-180. doi:10.1111/j.2044-8317.2011.02031.x

[9] A. Khan and M. Western, “Does Attitude Matter in Computer Use in Australian General Practice? A Zero-In?ated Poisson Regression Analysis,” Health Information Management Journal, Vol. 40, 2011, pp. 23-29.

[10] B. T. Pahel, J. S. Preisser, S. C. Stearns, et al., “Multiple Imputation of Dental Caries Data Using a Zero-In?ated Poisson Regression Model,” Journal of Public Health Dentistry, Vol. 71, No. 1, 2011, pp. 71-78. doi:10.1111/j.1752-7325.2010.00197.x

[11] S. R. Hu, C. S. Li and C. K. Lee, “Assessing Casualty Risk of Railroad-Grade Crossing Crashes Using ZeroIn?ated Poisson Models,” Journal of Transportation Engineering-ASCE, Vol. 137, No. 8, 2011, pp. 527-536. doi:10.1061/(ASCE)TE.1943-5436.0000243

[12] D. B?hning, “A Note on a Test for Poisson Overdispersion,” Biometrika, Vol. 81, No. 2, 1994, pp. 418-419. doi:10.1093/biomet/81.2.418

[13] W. Cochran, “Some Methods of Strengthening χ2 Tests,” Biometrics, Vol. 10, No. 4, 1954, pp. 417-451. doi:10.2307/3001616

[14] C. Feng, H. Wang and X. M. Tu, “The Asymptotic Distribution of a Likelihood Ratio Test for the Poisson Distribution,” Sankhyā, 2012 (in press)

[15] C. S. Li, “A Lack-of-?t Test for Parametric Zero-In?ated Poisson Models,” Journal of Statistical Computation and Simulation, Vol. 81, No. 9, 2011, pp. 1081-1098. doi:10.1080/00949651003677410

[16] C. S. Li, “Testing the Lack-of-Fit of Zero-In?ated Poisson Regression Models,” Communication in StatisticsSimulation and Computation, Vol. 40, No. 4, 2011, pp. 497-510. doi:10.1080/03610918.2010.546541

[17] C. Rao and I. Chakravarti, “Some Small Sample Tests of Signi?cance for a Poisson Distribution,” Biometrics, Vol. 12, No. 3, 1956, pp. 264-282. doi:10.2307/3001466

[18] J. Van den Broeck, “A Score Test for Zero In?ation in a Poisson Distribution,” Biometrics, Vol. 51, 1995, pp. 738-743.

[19] O. Thas and J. C. W. Rayner, “Smooth Tests for the ZeroIn?ated Poisson Distribution,” Biometrics, Vol. 61, 2005, pp. 808-815. doi:10.1111/j.1541-0420.2005.00351.x