Models for Count Data and Rates: Poisson Regression, Incidence Rate Ratios (IRRs), Overdispersion, and Offset Terms

This chapter introduces statistical methods for analyzing count outcomes and incidence rates in epidemiological research. Students learn to distinguish count data, which represent the number of events within a specified observational unit, from incidence rates, which account for the time individuals spend at risk. The chapter explains population at risk, follow-up time, and person-time, including person-years, person-months, and person-days. It emphasizes that case counts alone may be insufficient to compare disease occurrence between populations when observation periods or population sizes differ. Using examples involving respiratory infections, tuberculosis (TB), and disease incidence across geographical areas, students learn how to calculate incidence rates and why person-time is an essential denominator in cohort studies and epidemiological surveillance.

The second major focus is Poisson regression and the interpretation of Incidence Rate Ratios (IRRs). Students explore the Poisson distribution and its fundamental assumption that the expected value and variance of the count outcome are equal, known as equidispersion. The chapter introduces Poisson regression within the generalized linear model (GLM) framework using a logarithmic link function to examine relationships between disease counts and explanatory variables, such as age, environmental exposures, population density, and urban status. Particular attention is given to the offset term, \(\log(\text{person-time})\), which accounts for differences in observation time and allows the model to estimate incidence rates rather than merely compare absolute counts. By exponentiating regression coefficients, students obtain IRRs, which quantify relative differences in incidence rates between exposure groups or for a unit increase in a continuous predictor. The chapter also demonstrates how to implement these methods in R, including model estimation, confidence intervals, coefficient interpretation, and graphical presentation of epidemiological findings.

The final component addresses overdispersion, alternative count-data models, and epidemiological model diagnostics. Students learn that overdispersion occurs when the observed variance exceeds the mean, violating a central assumption of standard Poisson regression. The chapter explains potential causes, including unmeasured population heterogeneity, clustering, excess zeros, and model misspecification, and examines the consequences for standard errors, confidence intervals, and hypothesis testing. The chapter introduces two principal alternatives: quasi-Poisson regression, which adjusts the variance while retaining the same mean structure, and Negative Binomial regression, which explicitly accommodates additional variability. The material also distinguishes count models from rate models and emphasizes the importance of evaluating residual patterns, deviance-to-degrees-of-freedom ratios, influential observations, and functional relationships. Overall, this chapter equips Statistics students with the skills to select appropriate models for epidemiological count and rate data, interpret IRRs correctly, account for differences in person-time, assess model adequacy, and communicate reliable statistical evidence for public health research and decision-making.