Generalized Linear Models (GLMs): Binary Outcomes and Count Data

This chapter introduces Generalized Linear Models (GLM) as a flexible extension of classical linear regression for analyzing health and environmental health data that do not satisfy the assumptions of ordinary linear regression. In Environmental Health Statistics, outcomes are often binary, such as disease versus no disease, or count data, such as the number of disease cases or hospital visits. GLM addresses these situations through three integrated components: the random component, which specifies the probability distribution of the outcome; the systematic component, which represents the linear predictor of explanatory variables; and the link function, which connects the expected outcome to the linear predictor. The chapter emphasizes that GLM provides a common framework encompassing linear, logistic, and Poisson regression.

HomeworkThe chapter then develops two important GLM applications in environmental health. For binary outcomes, students learn logistic regression, including probability, odds, odds ratio (OR), and the distinction between OR and risk ratio (RR). Particular attention is given to the rare disease assumption, because OR may substantially overestimate RR when the outcome is common. For count outcomes, the chapter introduces Poisson regression, the interpretation of coefficients as Incidence Rate Ratios (IRR), and the use of an offset to account for differences in population size or observation time. Environmental examples, particularly the relationship between PM2.5 exposure and respiratory disease, are used throughout the chapter, with R examples demonstrating how these models can be implemented and interpreted.

Finally, the chapter emphasizes that selecting an appropriate model requires careful attention to the data's distribution and characteristics, rather than automatically applying Poisson or logistic regression. For count data, students are introduced to overdispersion, Negative Binomial regression, and zero-inflated models when the Poisson assumptions are inadequate. The chapter concludes with goodness-of-fit and model diagnostics, including deviance, Pearson chi-square, AIC/BIC, residual analysis, leverage, Cook’s distance, and other diagnostic approaches. Thus, from a teaching perspective, the central message of this chapter is that students should learn to move systematically from the type of outcome → appropriate probability distribution → link function → regression model → effect measure → model diagnostics, so that statistical conclusions about environmental exposures and health outcomes are both appropriate and interpretable. The course material and homework click the link here (Material and Homework)