On the Relationship between Logistic and Poisson Regression Models
DOI:
https://doi.org/10.14738/aivp.1306.19688Keywords:
Logistic Regression, Poisson Regression, Generalized Linear Models, Regression Analysis, Log OddsAbstract
This study explores the relationship between Logistic and Poisson regression models, leveraging on the mathematical connection between the binomial and Poisson distributions, particularly when the probability of success (p) is small and the number of trials (n) is large. The research provides an algebraic derivation of the Logit and Log odds functions, grounded in probability theory, to highlight the theoretical parallels between the two models. Using the "Affairs" dataset in R Studio, both models were fitted to predict binary outcomes. A comparison of their performance, based on the Akaike Information Criterion (AIC), revealed that the Logistic regression model (AIC = 625.36) provided a superior fit to the data compared to the Poisson model (AIC = 684.71). Despite this difference in overall fit and divergent parameter estimates, the predicted probabilities from both models exhibited a strong correlation (95.2%), demonstrating their close alignment in practical applications. The findings suggest that while both models can be used for binary outcomes, Logistic regression is statistically preferred; however, their interchangeability under specific conditions offers valuable flexibility for practitioners in statistical modeling. This study contributes to pronounced understanding of Generalized Linear Models (GLMs) by quantifying the practical and performance trade-offs between these approaches.
References
Agresti, A. (2015). Foundations of linear and generalized linear models. New York: John Wiley & Sons.
Ardiles, P., Gutiérrez, L., & Palacios, J. (2018). Application of negative binomial regression in analyzing the relationship between hospitalization and air pollution. Journal of Environmental Health, *81*(5), 12–19
Collett, D. (2003). Modeling Binary Data, 2nd edition, London: Chapman & Hall/CRC.
Consul, P. C., & Famoye, F. (1992). Generalized Poisson regression model and its properties. Communications in Statistics - Theory and Methods, 21(1), 89-109. https://doi.org/10.1080/03610929208830766
Dunteman, G. H., & Ho, M. H. R. (2005). An introduction to generalized linear models (Vol. 145). Thousand Oaks, CA: Sage Publications.
Famoye, F. (2010). Bivariate negative binomial regression model. Journal of Applied Statistics, 37(6), 969-981. https://doi.org/10.1080/02664760902984618
Gagnon, R., Schemitsch, E., Zdero, R., & Bhandari, M. (2008). Application of Poisson regression in trauma research. Journal of Trauma and Acute Care Surgery, *64*(3), 703–710. https://doi.org/10.1097/TA.0b013e31816533f9
Ijomah, M. A., Biu, E. O., & Mgbeahurike, C (2018). Assessing Logistic and Poisson Regression Model in Analyzing Count Data. International Journal of Applied Science and Mathematical Theory ISSN 2489-009X Vol. 4 No. 1 2018
Ismail, N., & Jemain, A. A. (2007). Handling overdispersion with negative binomial and generalized Poisson regression models. Casualty Actuarial Society Forum, 2007(Winter), 103-158.
Land, K. C., McCall, P. L., & Nagin, D. S. (1996). A comparison of Poisson, negative binomial, and semiparametric mixed Poisson regression models. Sociological Methods & Research, *24*(4), 387–442. https://doi.org/10.1177/0049124196024004001
Long, J.S. (1997). Regression models for categorical and limited dependent variables. Thousand Oaks, CA: Sage.
Nelder, J. A. and Wedderburn, R. W. M. (1972). Generalized linear models. Journal of the Royal Statistical. Society Series A, 135, 370-384.
Omer, H., & Hussian, M. (2023). Generalized Poisson and negative binomial regression models for count time series data. Journal of Statistical Computation and Simulation, 93(5), 712-730. https://doi.org/10.1080/00949655.2022.2113345
Pearl, Raymond and Lowell J. Reed (1922) A further note on the mathematical theory of population growth. Proceedings of the National Academy of Sciences, 8,
Pearson, K. (1901). Mathematical contributions to the theory of evolution. X. Supplement to a memoir on skew variation. Philosophical
Sarvi, M., Droudgar, A., & Neyestani, Z. (2014). Socio-economic factors and tuberculosis incidence: A negative binomial and Poisson regression analysis. Epidemiology and Infection, *143*(1), 245–252. https://doi.org/10.1017/S0950268814000518
Takahashi, K., & Kurosawa, M. (2016). A regression correlation coefficient for Poisson regression models. Journal of Statistical Planning and Inference, 178, 1-13. https://doi.org/10.1016/j.jspi.2016.06.003
Verhulst, Pierre-Francois (1838) Notice sur la loi la population suit dans son accroissement. Correspondence Mathematique et Physique, Publiee par A. Quetelet, 10, 113 - 120.
Verhulst, Pierre Francois (1845) Recherdes mathematique sur la loi d'accroissement de population. Nouveaux Memories de L'Academia Royale des Sciences, des lettres et de Beaux-Arts de Belgique, 18, 1 - 38.
Yule, G Udney (1925) The growth of population and the factors which controls it. Journal of the Royal Statistical Society, 38, 1-59. affairs gender age yearsmarried children religiousness education
Zou, G., & Donner, A. (2013). Extension of modified Poisson regression models for correlated binary data. Statistics in Medicine, 32(16), 2856-2867. https://doi.org/10.1002/sim.5745
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Copyright (c) 2025 Oyowei, Esueze Augustine, Ilori, A. K., Awogbemi, Clement Adeyeye, Utalor, Ifeoma Kate, Egbuniwe Obiageli Nancy, Olowu Abiodun Rafiu , Alagbe, Samson Adekola , Arabi Tolu Kayode, Dariyem Naandi Kruslat
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