Probability and Distributions for Environmental and Health Data

This chapter introduces linear regression as a fundamental statistical method for analyzing quantitative health data, particularly for examining relationships between environmental exposures and health outcomes. Students are introduced to simple and multiple linear regression. The chapter explains the intercept, regression coefficient, and error term, and how Ordinary Least Squares (OLS) estimates regression parameters. Particular attention is given to the assumptions underlying linear regression, including linearity, independence, homoscedasticity, normality of residuals, and the absence of serious multicollinearity. Examples involving PM2.5, respiratory disease, hospital admissions, and lung function help students connect the statistical concepts with health-data applications.

The chapter then focuses on interpreting regression coefficients and obtaining valid adjusted effect estimates. Students learn to interpret exposure effects per unit, per standard deviation (SD), and per interquartile range (IQR), allowing them to communicate regression results in ways meaningful for health research. The material also discusses covariate selection and controlling for confounding, emphasizing that covariate selection should be guided by theoretical and causal considerations, including the use of Directed Acyclic Graphs (DAGs), rather than relying solely on statistical significance. The chapter introduces crude and adjusted models and the change-in-estimate approach for assessing confounding. It also develops students' ability to evaluate regression assumptions through residual analysis, heteroscedasticity tests, leverage, and influential observations, including the Breusch–Pagan test and Cook's Distance.

Finally, the chapter extends linear regression to more complex analytical situations commonly encountered in health data. Students learn about multicollinearity, its consequences for standard errors and coefficient stability, and how to assess it using correlation and Variance Inflation Factor (VIF). The chapter also introduces non-linear exposure–response relationships, explaining why a straight-line model may be inadequate for patterns such as threshold, plateau, U-shaped, or J-shaped relationships. Polynomial regression and spline regression, including restricted cubic splines, are presented as more flexible approaches to modeling these patterns. The chapter also distinguishes prediction from statistical inference, including measures such as RMSE and MAE, training/testing data, and cross-validation. Overall, the chapter aims to develop students' ability not merely to run a regression model, but to construct, diagnose, interpret, and critically evaluate regression analyses in health-data research.  The course material and homework click the link here (Material)