On the Relationship between Logistic and Poisson Regression Models

Abstract

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.