AŞIRI SIFIR DURUMUNDA BAZI SAYIM MODELLERİNİN KARŞILAŞTIRILMASI: BİR UYGULAMA

Abstract

Sayma verileri için literatürde farklı regresyon modelleri geliştirilmiştir. Bunlar arasında en bilinen regresyon modelleri Poisson ve negatif binomial regresyon modelleridir. Poisson ya da negatif binomial modeller eğer fazla sıfır değerli terimler yoksa uygun olur. Sayma verilerinde aşırı sıfır olduğunda eşit yayılım durumunda zero-inflated Poisson, aşırı yayılım durumunda zero-inflated negatif binom modelleri en çok tercih edilen modellerdir. Çok fazla sıfır olması durumunda kullanılan başka bir model de Poisson Hurdle ve negatif binomial Hurdle modelleridir. Bu çalışmada örnek bir veri seti için bu modeller karşılaştırılmıştır. Bu amaçla LL, AIC, BIC ve Vuong test istatistiği kullanılmıştır.

Keywords

• Hurdle model
• negatif binomial model
• poisson model
• sayma verisi
• sıfır şişirilmiş modeller

COMPARISON OF SOME COUNT MODELS IN CASE OF EXCESSIVE ZEROS: AN APPLICATION

Abstract

Different regression models have been developed in the literature for count data. Among these, the most well-known regression models are Poisson and negative binomial regression models. Poisson or negative binomial models are suitable if there are not many zero-valued terms. When there are excessive zeros in count data, zero-inflated Poisson models are the most preferred models in the case of equal dispersion, and zero-inflated negative binomial models are the most preferred models in case of overdispersion. Other models used in the case of too many zeros are the Poisson Hurdle and negative binomial Hurdle models. In this study, these models are compared for a sample data set. For this purpose, LL, AIC, BIC and Vuong test statistics were used.

Keywords

• Count data
• Hurdle model
• negative binomial model
• poisson model
• zero-ınflated models.

References

1. Akaike, H. (1973). Information theory and an extension of the Maximum Likelihood Principle. Second International Symposium on Information Theory. Budapest, Academiai Kiado, 267-281.
2. Allison, P. (2012). Do we really need zero-inflated models? Retrieved April 15, 2021 from https://statisticalhorizons.com/zero-inflated-models
3. Altun, E. (2018). A new zero-inflated regression model with application. Journal of Statisticians: Statistics and Actuarial Sciences. 11(2), 73-80.
4. Altun, E. (2019). A new model for over-dispersed count data: Poisson Quasi-Lindley regression model. Mathematical Sciences. 13(3). 241-247.
5. Baetschmann, G. & Winkelmann, R. (2017). “A dynamic Hurdle model for zero-inflated count data. Communications in Statistics-Theory and Methods. 46(14), 7174-7187.
6. Boucher, J.P. & Denuit, M. (2008). Credibility premiums for the zero‐inflated Poisson model and new hunger for bonus interpretation. Insurance: Mathematics and Economics. 42, 727‐735.
7. Cameron, A.C. & Trivedi, P.K. (2013). Regression analysis of count data, Econometric society monograph (2nd Edition). Cambridge University Press. England.
8. Cragg, J.G. (1971). Some statistical models for limited dependent variables with application to the demand for durable goods. Econometrica: Journal of the Econometric Society. 829-844.

9. Cui, Y. & Yang, W. (2009). Zero-inflated generalized Poisson regression mixture model for mapping quantitative trait loci underlying count trait with many zeros. Journal of Theoretical Biology. 256, 276-285.
10. Dalrymple, M.L., Hudson, I.L. & Ford, R.P.K. (2003). Finite mixture, zero-inflated Poisson and Hurdle models with application to SIDS. Computational Statistics & Data Analysis, 41, 491-504.
11. Deniz, Ö. (2005). Poisson regresyon, İstanbul Commerce University Journal of Science. 4(7), 59-72.
12. Denuit, M., Maréchal, X., Pitrebois, S. & Walhin, J.F. (2007). Actuarial modelling of claim counts: risk classification, credibility and bonus-malus systems. John Wiley & Sons.
13. Erdemir, Ö.K. & Karadağ, Ö. (2020). On comparison of models for count data with excessive zeros in non-life insurance. Sigma Journal of Engineering and Natural Sciences, 38(3), 1543-1553.
14. Famoye, F. & Singh, K.P. (2006). Zero-inated generalized Poisson regression model with an application to domestic violence data”, Journal of Data Science, 4(1), 117-130.
15. Flynn, M. (2009). More flexible GLMs zero‐inflated models and hybrid models. Casualty Actuarial Society E‐Forum, 148‐224, Retrieved June 12, 2021 from https://www.casact.org/pubs/forum/09wforum/flynn_francis.pdf
16. Gerdtham, U.G. (1997). Equity in health care utilization: further tests based on Hurdle models and Swedish micro data. Health Economics, 6, 303-319.
17. Greene, W. (2005). Functional form and heterogeneity in models for count data. Foundations and Trends in Econometrics. 1(2), 113‐218.
18. Greene, W.H. (1994). Accounting for excess zeros and sample selection in Poisson and negative binomial regression models. NYU Working Paper. No. EC-94-10.
19. Hemmingsen, W., Jansen, P.A. & MacKenzie, K. (2005). Crabs, leeches and trypanosomes: an unholy trinity? Marine Pollution Bulletin, 50(3), 336-339.
20. Hofstetter, H., Dusseldorp, E., Zeileis, A. & Schuller, A.A. (2016). Modeling caries experience: advantages of the use of the Hurdle model. Caries Research, 50(6), 517–26.
21. Hurvich, C.M. & Tsai, C. (1989). Regression and time series model selection in small samples. Biometrika, 76, 297-307.
22. James, D. (2014). What is the difference between zero-inflated and Hurdle models? Retrieved April 16, 2021 from https://stats.stackexchange.com/questions/81457/what-is-the-difference-between-zero-inflated-and-hurdle-models
23. Karen, C.H.Y. & Kelvin, K.W.Y. (2005). On modeling claim frequency data in general insurance with extra zeros. Mathematics and Economics. 36, 153-163.
24. Khoshgoftaar, T.M., Gao, K. & Szabo, R.M. (2005). Comparing software fault predictions of pure and zero-inflated Poisson regression models. International Journal of Systems Science. 36(11), 707-715.
25. Kibar, F.T. (2008). Trafik kazaları ve Trabzon bölünmüş sahil yolu örneğinde kaza tahmin modelinin oluşturulması [Master Thesis]. Karadeniz Technical University Graduate Institute of Natural and Applied Sciences. Trabzon.
26. Lambert, D. (1992). Zero-inated Poisson regression, with an application to defects in manufacturing. Technometrics. 34(1), 1-14.
27. Lee, Y., Moudud, A., Noh, M., Rönnegård, L. & Skarin, A. (2016). Spatial modeling of data with excessive zeros applied to Reindeer Pellet-group counts. Ecology and Evolution. 6, 7047–7056
28. Mamun, A. (2014). Zero-inflated regression models for count data: an application to under-5 deaths [Master Thesis]. Ball State University Muncie. Indiana.
29. McCullagh, P. & Nelder, J.A. (1989). Generalized Linear Models (Second Edition). Chapman and Hall, New York, USA.
30. McQuarrie, A.D.R. & Tsai, C. (1998). Regression and time series model selection. World Scientific Publishing Company. Singapore.
31. Mouatassim, Y. & Ezzahid, E.H. (2012). Poisson regression and zero-inflated Poisson regression: Application to private health insurance data. European Actuarial Journal. 2(2), 187-204.
32. Mullahy, J. (1986). Specification and testing of some modified count data models. Journal of Econometrics, 33(3), 341-365.
33. Mwalili, S.M., Lesaffre, E. & Declerck, D. (2008). The zero-inflated negative binomial regression model with correction for misclassification: An example in caries research. Statistical Methods in Medical Research, 17(2), 123-139.
34. NNCS Statistical Software (2020). Negative binomial regression, Chapter 326. Retrieved April 15, 2021 from https://ncss-wpengine.netdna-ssl.com/wp-content/themes/ncss/pdf/Procedures/NCSS/Negative_Binomial_Regression.pdf
35. Peng, J. (2013). Count data models for injury data from the national health interview survey [Master Thesis]. The Ohio State University Graduate Program in Public Health, Columbus.
36. Ridout, M., Demetrio, C.G.B. & Hinde, J. (1998). Models for count data with many zeros. International Biometric Conference, Cape Town, Retrieved June 12, 2021 from https://www.kent.ac.uk/smsas/personal/msr/webfiles/zip/ibc_fin.pdf.
37. Ridout, M., Hinde, J. & Demetrio, C.G.B. (2001). A score test for a zero-inflated Poisson regression model against zero-inflated negative binomial alternatives. Biometrics, 57, 219-233.
38. Rose, C.E, Martin, S.W., Wannemuehler, K.A. & Plikaytis, B.D. (2006). On the of zero-inflated and Hurdle models for modeling vaccine adverse event count data. Journal of Biopharmaceutical Statistics, 16, 463-481.
39. Sakthivel, K.M. & Rajitha, C.S. (2017). A comparative study of zero-inflated, Hurdle models with artificial neural network in claim count modeling. International Journal of Statistics and Systems. 12(2), 265-276.
40. Sarul, L.S. & Şahin, S. (2015). An application of claim frequency data using zero inflated and Hurdle models in general insurance. Journal of Business, Economics and Finance, 4(4), 732-743.
41. Sellers, K.F. & Shmueli, G. (2010). A flexible regression model for count data. Annals Applied Statistics. 4(2), 943-961.
42. Shalabh, K. (2020). Poisson regression models, Chapter 15. Retrieved April 12, 2021 from http://home.iitk.ac.in/~shalab/regression/Chapter15-Regression-PoissonRegressionModels.pdf
43. Shmueli, G., Minka, T.P., Kadane, J.B., Borle, S. & Boatwright, P. (2005). A useful dis-tribution for fitting discrete data: revival of the conway-maxwell-Poisson distribution. Applied Statistics, 54, 127–142.
44. Sinharay, S. (2010). Discrete probability distributions, ETS. Elsevier. Princeton, NJ, USA.
45. Sugiura, N. (1978). Further analysts of the data by Akaike's Information criterion and the finite corrections. Communications in Statistics-Theory and Methods. 7(1),13-26.
46. Ver Hoef, J.M. & Frost, K.J. (2003). A bayesian hierarchical model for monitoring Harbor seal changes in Prince William Sound, Alaska. Environmental and Ecological Statistics, 10, 201–219.
47. Vuong, Q.H. (1989). Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica. 57, 307-333.
48. Wang, W. & Famoye, F. (1997). Modeling household fertility decisions with generalized Poisson regression, Journal of Population Economics.10, 273-283.
49. Workie, M.S. & Gedef, A.A. (2021). Bayesian zero‑inflated regression model with application to under‑five child mortality. Journal of Big Data. 8(4), 1-23.
50. Yang, J., Li, X. & Liu, G.F. (2012). Analysis of zero-inflated count data from clinical trials with potential dropouts”, Statistics in Biopharmaceutical Research. 4(3), 273-283.
51. Yau, K.K.W. & Lee, A.H. (2001). Zero-inflated Poisson regression with random effects to eval-uate an occupational injury prevention programme. Statistics in Medicine, 20, 2907-2920.
52. Yau, Z. (2002). Score tests for generalization and zore-inflation in count data modeling [Unpublished Ph.D. Thesis]. University of South Caroline. Columbia.
53. Yeşilova, A., Kaydan, M.B. & Kaya, Y. (2010). Modeling insect-egg data with excess zeros using zero-inflated regression models. Hacettepe Journal of Mathematics and Statistics, 39(2), 273-282.
54. Yip, K.C. & Yau, K.K. (2005). On modeling claim frequency data in general insurance with extra zeros. Insurance: Mathematics and Economics, 36(2), 153-163. Zwilling, M. L. (2013). Negative binomial regression. The Mathematica Journal, Wolfram Media. 15, 1-18.